Tactometria. Seu, Tetagmenometria. Or, The geometry of regulars practically proposed after a new, most artificial, exact and expeditious manner (together with the natural or vulgar, by way of mensurall comparison) and in the solids, not only in respect of magnitude or dimension, but also of gravity of ponderosity, according to any metal assigned. ... And withall, the like artificial practical geometry of regular-like solids (as I term them) in both the foresaid respects: and moreover, of a cylindricall body, for liquid or vessell-measure (commonly called by the name of gauging) as is for solid measure; ... And lastly, an A-tactometrie, or an appendix, for the most ready and exact discovering of the dimensionall quantity of any irregular kind of body, ... A work very usefull and delightfull for all such as are either ingenuously studious of, or necessarily exercised and employed in the practice of the art metricall. By J.W. Wybard, John. This text is an enriched version of the TCP digital transcription A67191 of text R201018 in the English Short Title Catalog (Wing W3736). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 634 KB of XML-encoded text transcribed from 190 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2018 A67191 Wing W3736 ESTC R201018 99825115 99825115 29488 2240885902

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 Early English books online. (EEBO-TCP ; phase 2, no. A67191) Transcribed from: (Early English Books Online ; image set 29488) Images scanned from microfilm: (Early English books, 1641-1700 ; 2143:11) Tactometria. Seu, Tetagmenometria. Or, The geometry of regulars practically proposed after a new, most artificial, exact and expeditious manner (together with the natural or vulgar, by way of mensurall comparison) and in the solids, not only in respect of magnitude or dimension, but also of gravity of ponderosity, according to any metal assigned. ... And withall, the like artificial practical geometry of regular-like solids (as I term them) in both the foresaid respects: and moreover, of a cylindricall body, for liquid or vessell-measure (commonly called by the name of gauging) as is for solid measure; ... And lastly, an A-tactometrie, or an appendix, for the most ready and exact discovering of the dimensionall quantity of any irregular kind of body, ... A work very usefull and delightfull for all such as are either ingenuously studious of, or necessarily exercised and employed in the practice of the art metricall. By J.W. Wybard, John. [28], 342 p. printed by Robert Leybourn, for Nathaniel Brooks, at the Angel in Cornhill, London : M DC L. [1650] "To the reader" is signed: J. Wybard, DM. Stained and torn with slight loss of print. Reproduction of the original in the Bodleian Library. eng Geometry -- Early works to 1800. eebo Tactometria. Seu, Tetagmenometria. Or, The geometry of regulars practically proposed; after a new, most artificial, exact and expeditious manner (toge Wybard, John 1650 336 97469 386 0 27 0 0 0 42.37 D The rate of 42.37 defects per 10,000 words puts this text in the D category of texts with between 35 and 100 defects per 10,000 words. 2012-05 Assigned for keying and markup 2012-05 Keyed and coded from ProQuest page images 2012-07 Sampled and proofread 2012-10 Rekeyed and resubmitted 2013-01 Sampled and proofread 2013-01 Text and markup reviewed and edited 2013-02 Batch review (QC) and XML conversion

TACTOMETRIA . SEU , TETAGMENOMETRIA . OR , The GEOMETRY of Regulars practically proposed ; After a new , most artificial , exact and expeditious manner ( together with the naturall or vulgar , by way of mensurall Comparison ) And in the Solids , not only in respect of Magnitude or Dimension , but also of Gravity or ponderosity , according to any Metal assigned . Together with severall usefull Observations and Experiments falling in by the way , concerning Measure and Weight . And withall , the like artificial practical Geometry of regular-like Solids ( as I term them ) in both the foresaid respects : And moreover , of a Cylindricall Body , for liquid or Vessell-Measure ( commonly called by the name of Gauging ) as is for solid measure ; it being therefore a most exact and expedite way of Gauging : With sundry new and exact Experiments , Observations , and Rules concerning the same . And lastly , an A-TACTOMETRIE , or an APPENDIX , for the most ready and exact discovering of the dimensionall quantity of any irregular kind of Body , whether solid or concave , which in it selfe will not admit of an ordinary or orderly way of measuring . And this from certain new and exact Experiments also , made for that purpose . A Work very usefull and delightfull for all such as are either ingenuously studious of , or necessarily exercised and employed in the practice of the Art METRICALL . By J. Wybard

LONDON , Printed by ROBERT LEYBOURN , for Nathaniel Brooks , at the Angel in Cornhill , MDCL .

〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 : Vel , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . PLATO . De quo PLUTARCHUS lib. 8 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , prob . 2. Omnia quaecunque à primaeva rerum natura constructa sunt , numerorum videntur ratione formata : Hoc enim fuit principale in animo Conditor is exemplar . BOETIUS Arithm. lib. 1. cap. 2. Veniet tempus quo ista quae nùnc latent , in lucem dies extrahat , & longioris aevi diligentia . SENECA , nat. quaest. lib. 7. cap. 25. Est natura hominum novitatis avida . PLINIUS .
TO All that are wel-willers to the Mathematiques , in generall : But more especially to the candid , impartiall , intelligent , and practicall Reader .

THere is now two Lustres , or one Decade of years elapsed , and some more time , ( gentle and courteous Reader , ) since I hapned first to hit upon this artificiall kind of Mensuration ( or more artificiall practicall Geometry for regular and regular-like Magnitudes , ) which I here deliver : and which was then but in two or three Particulars thereof , here first of all laid down in the three principall Propositions contained in the first Part , and the same demonstrated practically . And so having since by degrees , very much enlarged my Conceits & inventions herein ( and indeed as far I think as possibly may be , & that only by way of Mathematical exercitation & recreation from other Studies and employments ) I thought good at length , thus to put them together , ( with other things by the way , pertinent thereto , ) and so to exhibite and expose them to a publicke View , examination and triall in generall , and in speciall to thy candid and courteous censure : which I was the more emboldened and encouraged to do , considering that when at first I propounded those things which I had then conceived in this way , to severall able Artists , not only in severall parts of this Kingdome where I happened then to come , and with whom I had the opportunity to converse , but also in some parts beyond the Sea ( where not long after I fortuned to spend some time , for the prosecuting of other Studies , which I then cheifly aymed at and intended , and have since for the part followed and embraced ) most of which were Professours and Teachers of the Mathematicks , and so such as are usually soonest acquainted with all the new mathematicall inventions that are any way made known ) I found that it was to them a meer novelty , ( according as I conceived it would be ) and that moreover at first they somewhat doubted of the same , saying , that it were indeed an excellent way , if it would generally certain and true ; and so to some others of lesse judgement , it seemed to be a thing so very improbable ( they having made no triall thereof ) as that they would suddenly and unadvisedly conclude it to be impossible , only that it might hit right now and then by chance , but not constantly .

And then besides this , I having for my further satisfaction herein , taken the pains ( both at first , and also again of late ) to make a strict search and enquiry into all the chiefest Authours which have hitherto treated of practicall Geometry either in Latin or English , and more especially for instrumentall practice , in the way of manuall or mechanicall Mensuration ; could not find the way here proposed , so much as barely hinted by any of them in the least kind . And so have I now here at length ( by the favour and permission of God ) according to my earnest desire , brought that to a Generall , which at first I had conceived or apprehended only in a few Particulars , and so have compleated the Invention , as that I may think there is hardly any room left for additi on thereunto .

Here then first ( friendly and ingenious Reader ) shalt thou find the most artificiall and exquisite quadrature of a Circle , in a practicall , or organicall way , as to the immediate obtaining of it's Area in any measure appointed . For as Joseph Scaliger saith of the squaring of a Circle in a generall way , ( or of the generall quadrature of a Circle ) Elem. Cyclomet. 1. or Elem. Cycloperimet . Desin . 5. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Circulum quadrare , est Circuli areae aequale Rectilineum invenire . that is , To square a Circle , is to find a right-lined Plane equall to the Area of the Circle . Here shalt thou find that rectiline Plane to be the very Square of the Diameter , ( or of the Circumference ) according to Quantity discrete ; in which it is artificially diminisht by a Line of measure , so as to be made equall with the Circle it self , whereas naturally or geometrically it is greater then the Circle . And so I may say , that more properly , precisely and nearly to square a Circle , is to find an exact Square equall to the Circle given : And the like understand for the squaring of any other regular Figure ; and so thou shalt here find the like artificiall quadrature of all rectiline regular Planes as of a Circle , by their laterall , diametrall , and diagonall lines , for the immediate producing of their superficiall contents , which I have here performed ( by way of practicall demonstration ) in two of the first of them .

Then next shalt thou here find the like most artificiall and excellent cubing of a Globe or Sphear , for the immediate producing of its solid content in any measure assigned . And what was said before of the squaring of a Circle , both general and particular or special , the like I may say of the cubing of a Sphear ; That the same generally understood ; is to find a right-lined or plain Solid , equall to the solid Area of the Sphear : and so more particularly and properly , to cube a Sphear , is to find an exact Cube equall to the Spheare given ; which thou shalt here find to be made artificially ( in quantity arithmetical ) the very Cube of the Diameter ( or of the correspondent Circumference ) which naturally , or in quantity geometricall , exceeds the Sphear it self ; and this not only in respect of solid measure , but also of gravity or ponderosity , according to any Metall assigned : And the like understand for any other regular Solid : And so shalt thou here find the like most artificiall and admirable cubature of all the five famous plain ordinate Bodies in Geometry , or rectiline regular Solids , as of a Sphear , in both the foresaid respects ; and that not only as considered simply and absolutely in themselves alone , but also in relation to a Sphear , as being described either within or about the same . And what is here performed in all these regular Solids for their solid dimensions , by way of cubature ; the like is done in them for their superficiall dimensions , by way of quadrature ; ( and which therefore I conceive , may not altogether unaptly and improperly be termed the Quadrature of these Bodies , as to their superficiall or externall part . )

And then shalt thou here find after the like artificiall manner , the dimension both solid and superficiall of all such Solids , as are not exactly ordinate or regular , but somewhat like the same , and so which I call regular-like Solids , as namely right Cylinders and Cones ; and all Prisms & Pyramids constituted upon regular Bases ; and both which Dimensions aforesaid , may sometimes happen in these kind of Bodies , to be artificially of the same nature with those in exact regular Bodies : viz. cubatory and quadratary ; as I shall shew in their Dimensions ; though indeed in their solid Dimension , there is alwaies a Quadrature , in respect of the Base . And here likewise what is understood for solid measure , must be understood for gravity or weight , according to any Metall proposed .

And all these severall dimensions afore-named , are here performed by Lines of equall parts only , according to a decumane , decimane , or decimall division , in which therefore consisteth the excellencie of the performance : and the same demonstrated , not only in respect of the practicall use thereof , but also the theoricall grounds and reasons , according to certain Propositions laid down for that purpose . And which Lines are here generally set forth by Number , denoting or expressing their magnitudes from any Measure given or appointed .

And now , although that learned Mathematician , Mr. Edm. Gunter ( sometime Professour of Astronomy in Gresham-Colledge in London , and long since deceased ) a man excellent for Instrumentall inventions , hath ( among other Lines ) upon his Sector , certain Lines of quadrature , ( as he termeth them ) as for to make a Square equall to a Circle given , by the semidiameter thereof , & contrà : and so for the like quadrature of certain rectiline regular Planes , by their sides ; yet those are of a much different kind and nature from ours , ( as any one may plainly see ) being not severall Lines of measure divided into parts any way , whereby to give immediatly of themselves , the side of the equal Square arithmetically ( as I may so speak ) or in quantity discrete , as from any certain , set , denominate Measure , according to a common way of measuring ; as our Lines do , ( & consequently , the superficiall content of the Square , for the Area of the Figure ) but only one Line , drawn twice over upon the Sector ; or ( if you will ) two like , correspondent , or congruall Lines , drawn upon each leg or shank of the Sector , from the Center thereof , and undevided ; containing in them only certain points , in which the Sector must be opened , according to the semidiameter of the Circle given , or the sides of the other Figures to which they belong , being there expressed by the numbers of their Sides ; and so give the side of the equall Quadrat , only geometrically ( as I may so speake ) or by Line , crossing the Sector between the two points of quadrature at the ends thereof , and so parallel to the line for the semidiameter of the Circle , or side of other Figure , to which the Sector is opened : and which line being taken off from thence with Compasses , and so applied to any certain Line of measure divided , will then indeed give the side of the equall Square in the parts of measure , as our Lines of quadrature do : but yet Mr. Gunter doth not apply his Lines of quadrature to such a use , or any way mention the same , but only to the laying down of the exact side of the Square equall to a Circle given , &c. and so to reduce the Circle into a Square , geometrically , after a mechanicall manner : and which can hold but only in a small Circle , whose semidiameter ( or other ordinate Plane , whose side ) may not exceed a convenient extent or opening of the legs of the Sector , ( or of a pair of Compasses ) so as to take off the side of the equal Quadrat , in its due place upon the Sector : for indeed , the largest Sector or Compasses that are usually made , ( or can be made convenient for use ) will open or extend ( at the utmost ) but to a very short line , in comparison of those which frequently fall out to be measured ; and therefore his Lines ( or way ) of Quadrature , cannot extend to any large Dimensions , and so serve for a generall measuring as ours do , if they should be applied to the very same use that they are , namely , the superficiall dimension of Circles , and other ordinate Planes , ( in a quadratary way ) according to any measure assigned , and thereupon can serve but to very small purpose . Nor indeed had lacquainted my self with those his Lines of Quadrature , or any other upon his Sector , when I first apprehended and conceived in my mind this artificiall way of measuring which I here propose , ( though I had often seen that Instrument , and had much studied his Book in other things ) nor till I had in a manner perfected the same throughout all the Dimensions to be performed thereby , and so was come to the Close of this Book . Neither did any of those Artists , to whom I ever yet propounded the same ( as aforesaid ) mention to me thereupon in the least kind , Mr. Gunter's Lines of quadrature ( or any other upon his Sector ) in reference to any measuring ; although some of them I am sure ( if not all ) were well acquainted with Mr. Gunter's Sector , & all his other Instruments , & taught the use of them to others . But this only by the bye . Nor do his Lines of Superficies in generall , upon the Sector , being also two like , congruall Lines drawn on the two leg● thereof from the Center , and divided unequally into 100 parts , serve for the measuring of Superficiall Figures , as to the immediate producing of their contents simply by themselves , in any kind of measure given ; but only in a way of proportion , to find out the superficiall content of one Figure , by the superficiall content of another like ( or unlike ) Figure given or known ; together with severall other uses noted by him . And so his Lines of Solids in generall , being two like Lines , drawn upon the Sector in like manner , as the Lines of Superficies , and divided also into 100 , ( or rather 1000 ) parts unequally , serve only for the like uses in Solids , that the Lines of Superficies do in Superficies .

And then for his more particular Lines of Solids ( as I may term them ) upon the Sector , called the Lines of inscribed , and of equated bodies ; having reference in particular to the five foresaid plain regular Solids , and a Spheare : the first sort of them , in respect of the inscription of the said five bodies in a Spheare ; and the other sort , in respect of their equation to a Spheare , and also of one to another ; are of a different nature and kind from our artificiall Lines pertaining to these bodies , and a Spheare , for the immediate producing of their solid contents in any Measure appointed , according to an exact , absolute cubature , both simply in themselves , & also in respect of inscription & circumscription to a Sphear given , as aforesaid : but are the first of them , for the finding of the Sides of these five bodies , as being to be inscribed in a Sphear , by the semidiameter thereof given ; and the other , for the finding of their Sides , as being equall in magnitude to a Sphear , and this by the Diameter of the Sphear given ; & contrà : or as to be made equall one to another by their sides : and all this in a meer geometricall sence ; and thereupon these latter Lines of his , will give off from the Sector , the side of the Cube equall to a Sphear , by the Diameter thereof ; & to any of the other regular bodies , by the sides thereof given , in the same manner , that his Lines of quadrature give the side of the Square equall to a Circle , by the semidiameter thereof , and to the other regular Planes or Superficies , by their sides given ; these Lines of equated bodies ( as also those of inscribed bodies ) being of the like kind and nature with those of quadrature , ( which by the same reason , may aswell be called Lines of equated Superficies , for that they do not only equall a Square to a Circle , or other ordinate Plane given ; but also equall them all one to another ) and are drawn upon the Sector accordingly ; & so are of no further or better use then they are .

Then lastly , his Lines of Metals ( so called ) upon the Sector , ( being inserted with the lines of equated bodies , because there was spare room , and much of the like kind with them , and so are contrived together on the same two lines , on each leg of the Sector , being drawn from the Center ) are not like our lines of Metals ( as I may so term them ) which are for the immediate discovering of the weight of a Sphear , or other regular ( or regular-like ) body made of any Metall , in the very same ( cubicall ) manner , that their solid contents are obtained in any measure , by their respective artificiall Lines as aforesaid ; but those ( together with his Lines of Solids ) do serve only to find the proportion , as it were , between severall Metals ( as he saith ) in their magnitudes and weights , and that according to the experiments of Marinus Ghetaldus , in his book intituled Archimedes promotus ; that is ( as he saith ) In like Bodies of severall Metals , and equall magnitude , by having the weight of the one , to find the weight of the rest ; & contrà : together with two other uses noted by him .

These things ( courteous and judicious Reader ) I thought good here to insert by the way from Mr. Gunter , to shew the difference between his Lines and ours ; especially those of his , which for their use , may seem to come most nearly to ours , namely his Lines of Quadrature , and of equated Bodies : & which as they cannot be so generally usefull as ( or not considerable for their use , in respect of ) ours , according to what I shewed before ; so neither can they be altogether so exact in the performance of those small Dimensions which they can reach unto .

And what I have here performed Geometrically , or by Line ( or Scale ) in an artificiall way of measuring ; I have also set forth Arithmetically , or by Number , in a way of Proportion after the most exquisite manner that may be , as from the naturall Measure ; according to the same division or partition of the Unity , as is of the Lines of measure , both naturall and artificiall ; and by which therefore the artificiall measure may be readily deduced from the naturall , or the naturall Measure be reduced to the artificiall : together with a multitude of other metricall conclusions besides , in most of the geome●ricall Figures which I have here particularly handled , by way of practicall demonstration ; none of them having been done before by any man , that I do know of ; except those in the Circle , being set forth by Mr. Gunter , and from him by Mr. Wingate , and perhaps some others ; but yet not all of those Proportions in so ample termes , as I have here done them , which therefore I extracted again a-new .

And then moreover shalt thou here find , not only the most artificiall and expeditionall way of measuring thus all regular Bodies , and such as do come very near a regular forme ( which therefore I call regular-like ) but also of such as are of an irregular form ; and first , of concave Bodies , or Vessels for Wine and Beer , ( which commonly do somewhat imitate the form of a Cylinder , and may be called Cylindroidall , and so admit of a Cylinder-like dimension , being first reduced by art to a Cylinder : to which end I bestowed some considerable pains ( and a little cost too ) in the making of sundry experimiments , for the discovering of the true contents of the Standard-measures for Wine and Ale or Beer , pertaining to the Citie of London , ( which are kept at the Guild-hall ) as being commonly taken for the most generally received Measures for this purpose , throughout the Kingdome ; and by which I have seen the Measures which have been made for some eminent Towns farr remote from London , to be sized and sealed , ( though here I wil not cōtend about them ) and have here accordingly fitted a gauging-Line to each of them , according to our artificiall way of measuring , being also therefore Lines or Scales of equall parts in a decimall division , by which the liquid content of any Vessell will be obtained immediatly in Gallon-measure , after the same manner ( and with the same expedition in a manner , the irregularity of the Vessell being considered ) that the solid content of any exact Cylinder is had artificially , according to any Measure appointed .

And then last of all , do I here shew ( by way of Appendix ) the most easie and exact way for the discovering of the solid capacities of all other irregular kinds of Bodies whatsoever , both solid and concave , which of themselves are altogether unmeasurable in the usuall way of measuring ( or whose Dimensions can in no wise be taken by a Line of measure ) which therefore is a work of a contrary nature to all the former : and have here exemplarily illustrated the same from experiment , in a certaine regular solid body , easily and exactly measurable , for a confirmation thereof ; ( and which way also , hathnot been set forth by any man before , that I do know of ) Together with severall other new mathematicall experiments and observations , very usefull , and worth the noting . All which ( friendly Reader ) I commend to thy courteous consideration & acceptance ; hoping thou wilt receive no lesse delight ( and benefit also ) in the perusing and practising hereof , then I have taken delight and contentment in the study and exercise of the same , though surely with no small pains and industry , beside the expence of some time now and then , from my occasions of serious concernment , according as the same would reasonably permit : and which therefore I could do to no other end , then only to enlarge and advance ( so farr as here I might ) the practice and exercise of this so noble and admirable an Art , being drawn thereunto by that Genius , which hath heretofore much disposed and inclined me to mathematicall contemplation and exercitation in generall : so that , thy friendly acceptation hereof , is all I expect for my labour . And if any after me shall happen to raise any further Conclusions from what I have here laid down , in any particular thereof ; then surely will these my paines be yet thereupon so much the more to purpose . But now as I must expect this work of mine will meet with some Momaicall or Zoilan Spirit , so I shal not regard the same , or be terrified thereat , seeing that the best conceits and inventions of men that were ever yet published to the world , have been obnoxious to the obloquies and obtrectations of such malevolent and malignant spirits : and which hath been the complainr of the most learned men in all ages . And thus , courteous , ingenious , and ingenuous Reader , I friendlily bid thee fare-well , resting ,

London ; the first of May , 1650. Thine hearty wel-wisher , J. WYBARD , DM .
THE Generall Contents of this Work , consisting ( almost all ) in measure alone . PART I. SECT. I. OF the artificial Dimension of regular and regular-like Figures , in general Page 1. SECT. II. The Dimension of a Circle p. 9. The dimensional Proportions in a Circle , in the most exquisite terms , and all the variety that maybe . p. 19. The solid dimension of a Sphear . p. 21. The same another way ; as also another superficiall dimension of a Circle , artificially . p. 28. The superficiall Dimension of a Sphear in both those waies . ibid. and p. 29. The dimensional Proportions in a Sphear , both for solid and superficiall measure , like as in a Circle . p. 30. The solid Dimension of a Cylinder and Cone . p. 32. The same another way artificially . p. 39. SECT. III. The superficiall dimension of a Cylinder and Cone . p. 45. The dimensionall Proportions in the Cylinder and Cone both for solid and superficiall measure . p 54 SECT. IV. Of the differences between the naturall and artificiall Measure , in the dimension of Figures . p. 56. And of the grounds or reasons of the artificiall Mensuration . p. 59. PART II. SECT. I. OF the artificiall Dimension of right-lined regular Planes or Superficies in general ; and the same demonstrated particularly in two of the first of them . p. 71. As also the dimensionall Proportions of the same two Figures , expressed in all the variety that may be . p. 92 &c. The artificiall Dimension of Triangles in general . p. 78. And from thence , of any Rhombus , Rhomboides or Trapezium . p. 103. 105. SECT. II. Of the Dimension both solid and superficiall of regular-based Pyramids in generall , &c. p. 106. And the same demonstrated particularly in the three first kinds of them ; & that ( for solid dimension ) in three of the five plain ordinate Bodies , or rectiline regular Solids , viz. in the Tetrahedron , Octahedron , and Dodecahedron , with a brief description of all the said five Bodies . p. 108 , 116 , 119 , 127. The dimensionall Proportions in the said three first kind of Pyramids , both for solid and superficiall measure . between p. 136 and 137. Of the artificiall Dimension of regular-based Prisms . p. 137. SECT. III. Of the Dimension of the foresaid five famous ordinate Bodies ( commonly called the Platonicall Bodies ) both solidly and superficially , severall waies ; after the most exquisite manner that may be . p. 139. SECT. IV. The same Dimensions , with a multitude of other metricall Conclusions in the said five Bodies , expressed Arithmet●cally by way of Proportion , in the most exqu●site Terms that may be ; like those of the other Figures beforegoing , here particularly handled . p. 166 &c. SECT. V. Of the Dimension of all regular Solids aforesaid , in reference to grav●ty or weight , according to any Metall proposed ; in the same artificiall manner , as is for solid measure : And the same demonstrated particularly in a Sphear of a certain Metall proposed . p. 198. The P●oportions of gravity and magnitude together , pertaining to a spherical body of that metall , expressed in the like Terms with the other Dimensiōs beforegoing . p. 241. The like reason of Dimension , as before , for the gravities of all regular-like metalline Bodies . p. 246 and 258. W●●re , a demonstration , or illustration arithmeticall , of the artificiall Lines for the solid dimension of a Cylinder and Cone , in particular ; according to Theor. 3. and consequently thereby , for all the other kinds of Dimension in generall , according to Theor. 1 and 2. And withall the generall ground and reason of the same Dimensions fully declared . PART III. SECT. I. Of the measuring or gauging of Vessels , in generall . p. 262. SECT. II. Of the Quantities of the Wine and Ale-gallons , in reference to Gauging ; from sundry new and exact Experiments . p. 264. SECT. III. The practice of Gauging , from our artificiall way of measuring ; as also from the naturall ( by way of comparison ) for a confirmation of the other ; as in all the precedent Dimensions . p. 288. Appendix . FOr the most ready and exact discovering of the solid capacity or dimensionall quantity of any such irregular Body whether solid or concave , as cannot of it self be measured in a plain geometricall manner . p. 301. The Conclusion for the more speedy absolving or expediting of all the foregoing Dimensions . p. 338 The more particular Contents of this Work , as falling in by the way , in reference to the generall : consisting most of them in measure and weight together , and being experimentall . 1. THe Proportions or Comparisons of all the principall Metals , in gravity and magnitude , expressed in generall terms ; according to the experiments of Marinus Ghetaldus . p. 201 2. The same laid down particularly in our Troy-weight , in Sphears of equall magnitude . p. 237. 3. Of the true magnitude of the Roman Foot , and the same compared with our English Foot , according to the observations and experiments of our Countreyman Mr. John Greaves . p. 206 , &c. 4. The Roman Weight , and our Troy or Goldsmiths Weight compared together , from the said Mr. Greaves . And so from thence , a proportion set between them , in the least terms . p. 209. 5. Of the gravity of a certain serreall Sphear or Bulke , both in Avoirdupois and Troy-weight , from a new and exact experiment . p. 217 6. Severall Proportions set between the Troy and Avoirdupois-weights , compared together : And first in an Arithmeticall manner , according to the Doctrine of Proportions . p. 220 and 232. And then the same compared , and examined by the Ballance upon the weighing of severall Bodies . 223 & 276. 7. The proportion between forged or fine Iron , and cast or bullet-iron , deduced from new exact experiments p. 225. 8. Of the weight of water , in relation to it's solid measure ; or the ponderall and dimensionall quantity of water compared together , from severall new experiments , for discovering the solid measure of any irregular body thereby , whether solid or concave . p. 269 , 270 & 287. The same compared together severall wayes from the foresaid experiments , and that in measure both unciall and pedall , and in weight both Troy and Avoirdupois : And so the nearest weight of one Inch , and of one Foot cubique or solid of water : and conversly , the magnitude , or solid measure of one Ounce , and one Pound , &c. and so , some apt terms of proportion raised for a generall and ready use . p. 302 and 305. 9 How to find out the solid content of any irregular body by the weight of water , &c. p. 308. 10. How to find the exact quantity of water ( or other-like liquid body ) which is equall in magnitude or dimension to a solid body given . ibid. 11. How to find immediatly the gravity of the water ( or other like liquid body ) which is equall in magnitude to a solid body given , by the gravity of the solid body only : And so the manner of weighing a solid body in Water , whether it be heavier or lighter then the same . p. 310 and 311. 12. The three last Conclusions beforeing , exemplarily illustrated from exact experiments made upon a spherical Stone-Body , or Marble-Bullet ; and so confirmed accordingly . p. 313 , &c. 13. Of the difference of gravity in water ; or the different gravities of several ●●rts of water , in relation to our Atactometricall practice , or Dimension of irregular Bodies . 319. 14. The same most exactly experimented by the Ballance sundry times , upon eight several Waters ; and all compared together . p. 325.

WHosoever shall think good to make use of the Lines for Gauging , or for any other Dimensions described in this Book : or of any other Instruments for mathematicall practice in generall ; they may have the same very accurately made by Mr. Christopher Flower , dwelling in the Bulwark , neer the Tower of London : besides divers others about this City .

Courteous Reader ,

SEing that hardly ever any Book passeth the Presse , free from Typographicall errours , notwithstanding all the care and diligence that can be used ( and in which I was not here wanting ) especially Books of this nature : and then that they being commonly set at the end of the Book , are seldome taken notice of , till the Book be read throughout ; and so many times the mistakes of the Printer are by ignorant or malevolent Readers , put upon the Author as errours : Therefore the chief of those few faults which have here escaped , ( the most of them being but meerly literall ) I thought good to put at the beginning ; that so th●u mightest in the first place know them , and consequently amend them : assuring thee withall , that of that multitude of Numbers which are comained in this Book ( the most of them denoting Lines , and Proportions of measure ; and so upon which dependeth the speedy working and resolving of many excellent practicall Propositions : and are therefore specially to be regarded ; and in which a fault cannot be espied by bare inspection , ) there is not one of them defective in any one figure thereof ; such was my ( more then ordinary ) care therein .

ERRATA .

PAge 4. line 7. put ( before commonly . l. 8. dele of . l. 20. r. partior . p. 7. l. 28. for hath , r. both . p. 8. l. 22 , r. in respect of the Measure from which it is taken , as also of the Figure &c. l. 30. dele the. p. 32. l. 11. r. Sect. 4. p. 59. l. 14. for matter of , r. manuall or . p. 75. l. 2. r. whose . p. 91. l. 24. r. shall . p. 173. the beginning of the last line save one , r. And so . p. 214. l. 2. dele the comma at Measure , and put one after Diameter . p. 316. l. 14. dele the first as . Other faults thou maist meet with , which being only literall , and so not worth the noting here ; thou maist easily amend in the reading .

Tactometria , Or the most Exquisite practicall Dimension of all Regular , and Regular-like Figures in generall .
And first , of the Circle , Spheare , Cylinder , and Cone , in speciall . PART . I.
SECT. I. Of the nature and division of the Lines of Measure in generall , for the performing of all the aforesaid Dimensions .

FOras●uch as to the due measuring of any Magnitude or Quantity continuall ( in practicall Geometry ) there is required some certain Measure first to be given or appointed : Therfore first and more generally , by Lines , we do her understand any right Line assigned for a certain Measure ; such like as Euclid , Elem 10. Defin. 5. calleth 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , which is as much as certain , definite , determinate , speakable or expressible by voyce , or otherwise expressible by Number : And so the most Noble , illustrious , and learned , Franc. Flus . Candalla , a most diligent and industrious restorer of Euclid's Elements interprets it Certa , a line certain , as first put or proposed and made manifest , and divided into parts certain and known : And it is also called Famosa , a measure famous , that is , ( as P. Ramus , and Adr. Metius do note ) first Ram. Geom. lib. 1. El. 8. Et Schol. Mathemat . lib. 21 Adr. Met. Geom. pract. Par. poster . seu Gaeodaes . spoken or expressed , &c. But most of the Latine Geometers do call ! such a line Rationalis , a Rationall line , for that ( as Ramus saith in the places here cited ) it is rationall to it selfe , as are all magnitudes equall among themselves : and Clavius saith it is called Rationalis , because Clavius in Def. 5 , El. 10. it is alwayes put certain and known , whereas all other lines which are compared to this ( as lines infinite in multitude , may be according to Euclid in the place forenamed ) are not certain & known , though taken apart by themselves , they are , seeing that every one may be divided into any number of equall parts , and being compared to this for their Measure , they are all either symmetrall or asymmetrall , and so are said to be either 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or * * So put by Theon , whereas it should rather be 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ●s it is opposed privatlvely to 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , : Sec Flussate upon the place , and also in Proem . 1● . Elem. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , in Def. 6 , 7. explicable or inexplicable , rationall or irrationall ; ( but with this point we meddle not here , ) and so we here understand this Line to be most properly called Rationall , as comprehending or containing in it self the dimensionall reason of allother lines measurable thereby . And therefore here we ( for brevity ) will with most Latine Translatours and Commentators , and also our * * H. Billingsley , a Citizen of London , and Lord 〈◊〉 〈◊〉 . 1596 ▪ English Translatour of , and Annotatour on Euclid's Elements , understand any right line so first set , put or proposed , by the name of the Rationall Line ( and this may be applyed to any Measure whatsoever , and it is one of Euclid's Data in Lib. Dator . Defin. 1. ) or ( in respect of the ensuing work ) the prime , simple , true , or naturall Rationall line . And this we mean when any where we say simply the Rationall Line .

Secondly , and more especially by Lines we here understand any such line augmented or diminished by a certain convenient segment or portion of the same , for the more artificiall and speedy mensuration of the aforenamed Figures : And this Line we may ( not unaptly ) call the second supposed or artificiall Rationall Line , as being derived from the former , and so substituted in place thereof : like as in Numbers the Logarithmes are usually called Artificiall Numbers , as being substituted instead of the naturall numbers , from which they are deduced , and whose place they supply in a most excellent and admirable manner , by performing all arithmeticall operations with that facility , expedition , and compendiousnesse ( and exactnesse also in some cases , as I shall afterwards upon occasion shew ) which the naturall numbers themselves cannot , for that by these , the two most tedious and troublesome parts or species of Arithmetick ( to wit , Multiplication and Division ) are wholly avoyded and abolished , and that most difficult branch ( or operation ) thereof called the Extraction of Roots , is mightily abbreviated and facilitated : And so the Arithmetick performed thereby is usually called artificiall Arithmetick .

Now seeing that every continuall or continued Quantity falling under Measure ( in practicall Geometry ) is referred and reduced to the discrete , that thereby its dimension● may be made more manifest to us ; so that Geometry hath perpetually need of Arithmetick for the explicating and expressing of its magnitudes in their dimensions : Surely no kinde of Numeration can be so accommodate to this thing , as that which considereth the Intger of Measure ( as the Unit ) in a decumane , decimane , or decimall solution , for that by this the practicall , instrumentall , or mechanicall part of measuring , or of the art Metricall , commonly and improperly called by Ramus , Metius , and of some others Geodasia which properly signifies the division or partition of right lined Superficies , as Pediasimus , de mensuratione & partitione Terrae , well observes , saying Terrae mensuratio duas in partes dividitur , Geometriam , scil. & Geodaesiam : Areae namque secundum aertem mensuratio , & terrae mensuratio est et meritò Geometria vocatur ; Unius verò & ejusdem areae , seu loci divisio inter diversas personas , partitio quaedam est terrae , & jure optimo Geodaesia appellatur , ( and which from him Clavius noteth Geomet . pract . lib. 6. ) for that the greeke words , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , do ( poetically ) signifie the same that the Latine words divido and portior , and so of which , and the word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , Terra , comes 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , i. e. Ter●ae divisio seu partitio ) is made much more facile accurate and expeditious .

For indeed this Mathematicall solution of unity or continuity , is of all others the most absolute and certain , and the most perspicuous and rationall , and by how much the more numerous it is in the parts thereof , by so much the more exact it is , and consequently the work effe●●ed by it . And what utility it hath brought to the Mathe●●atiques in generall , may be sufficiently witnessed by that ●ost noble and usefull part of Geometry , called Trigonomeirie , and that in the Radius of a Circle ( which is the very Basis and Root of all Trigonometricall operations ) where ( to wit ) first , that greatly renowned Mathematitian Johannes Regiomontanus , having for a long time used the Sexagenary solution ( as Ptolemy and others before him ) did at last bethink himselfe of the Decimall , as being much better ( seeing that the Unit would perform a far greater Compendium then the Senarie ) and indeed the best of all ( and so put the Radius , to 10 millions of parts , and next after him , Rheticus in his great Trigonometrical Canon , to 10000 millions , and afterwards ( to make that his Canon most absolute and perfect ) he proceeded to 1000 milliots ( as I term them ) or millions of millions , whereby the art of Trigonometry & consequently other Mathematical arts , as Astronomy , Geography , &c. depending thereupon ) did become , in the practice , far more facile and expeditious then before : for where the first proportionall Term of the Trigonometricall proposition , is the Radius or totall Sine ( which very frequently happens , or may be so made for the most part , as the learned Pitiscus excellently sheweth amongst other his compends in working ) there the proposition or question is solved only by a * * This is to be understood in working by the Naturall Numb . conjunct or manifold Pitiscus , Trigo● . lib. 5. composition ( or mutuall implication or induction ) of the second and third terms , seeing that the Unit altereth nothing in a conjunct or manifold resolution , but the Number of composition immediately becomes the number of resolution , only distinguishing between the absolute or integrall , and the fractionall part thereof . And moreover the benefit of this solution of unity is excellently seen in that most excellent Arithmeticall operation , vulgarly called The Extraction of Roots , wherein ( to wit ) seeing the Roots of numbers not explicable or rationall ( which the Algebrists or Cossists commonly call Surd Numbers , and so their Roots surd Roots ) cannot be exactly had , then those numbers are reduced into some kinde of decimall parts ( or parts of a great denomination , as Ramus termeth them ) as C●ntesms , Millesmes , &c. and that figurate , as Quadrate , Cubique , &c. that thereby their Roots may be had more certain and neerer to the truth , Ramus lib. Geom. 12 de Quadrato & 24 de Cubo . then they can by the naturall or vulgar extraction , as Ramus sheweth in the aforesaid Books , where he ( the first ) shewed this See Wingate Arithm . 1 Book , ●1 Chap. 3 Sect. kinde of Extraction which to some seems to have been the very foundation of Decimall Arithmetick , although Ramus hath no where else in his Mathematicall Works made any other use of this kinde of numbring , or made any mention of the same : But indeed that of Regiomontanus in the Radius of a Circle seems to me to have given the first light thereof to the World ; so that the Trigonometricall Numbers which now we use , may be termed Decimall , as they are derived from that Radius : For all the Sines to a Quadrant , and the Tangents to an Octant or semi-quadrant , are decimall parts or fractions of the Radius , but indeed the least Secant is greater then the Radius : And so wee will here make use of this kinde of Numeration , as being the fittest for our purpose , as we said before : And indeed for that this present work of ours cannot conveniently be performed by any other .

Now the Geometricall Figures which we have here first named , and so for which we have first extracted these kinde of metricall lines ( or linear numbers ) are those foure which Archimedes himselfe more especially treated of , and which are as it were , the beginning and ground of all the rest , namely , the Circle , Spheare , Cylinder and Cone , and accordingly the like Lines may be extracted for all ordinate Planes and Solids whatsoever , as we afterwards shew ; for that to these foure may be aptly referred all other regular and regular-like Figures ; As to the Circle may be aptly referred all ordinate Planes , to the Sphear , all exactly ordinate Solids ; to the Cylinder , all ordinate-based Prismes ; and to the Cone , all ordinate-based Pyramids : And though a Cylinder and Cone , and the like , cannot properly be reckoned among Regular Figures , according to the strict acception of an ordinate or regular Figure in Geometry , yet in respect of the regularity of their Bases , and also the regularity and uniformity of their other , and more speciall superficiall part besides ( whether the same consist of one entire plane only , as in the Cylinder and Cone , or of severall planes equall and like , as in all right or erect regular-based Pyramids and Prismes ) the same may in a sort be termed regular ( especially the Cylinder and Cone ) and which therefore , for distinction sake , I call regular like solid Figures .

But now the quantities of the artificiall metricall Lines , first extracted for the foure Figures first before named , according to any prime Rationall Line , and that to a Decu-millenary division of the same , are numerally thus .

I. A Line for the most excellent superficiall mensuration ( or Diametrall Quadration ) of a Circle , is 1. 1284 ●eré . II. A Line for the most exquisite Solid dimension ( or Diametrall Cubation ) of a Globe or Spheare , is 1. 2407. III. A Line for the Square Solidation ( as I may term it ) or Rectangular Parallelepipedation of a Cylinder , by its Diameter and Axis or Altitude ( as to an exact quadrate Base ) is 1. 0838. IV. A Line for the like dimension of a Cone , is 1. 5632 ferè .

And so divers other the like Lines , for the dimension both of these , and also of other Figures hath Superficiall and Solid , I shall afterwards shew in their severall places .

Every such second supposed , or artificiall Rationall Line , must be divided decimally , as is the first , true , or naturall Rationall Line , from which it is taken , according as the length thereof can conveniently beare : But if the first Rationall Line cannot well admit of so many parts as are here set down , then they may be abbreviated or contracted ; as to Millesmes , Centesmes , or to prime or simple Decimals ( or Tenths ) only ; though indeed our least common Measure in use , ( viz. an Inch or Pollicar ) may be distinctly divided in●o 100 ( or 1000 ) parts , if it be rightly handled according to the more artificiall ( or Diagonall ) way , as hereafter we shall have occasion to shew . And so we have here set forth the Rationall line in parts of a large denomination , that so it might serve for more exactnesse in use , because the more parts , or the greater divisionall denomination the Integer of measure ( or the unite ) is of , the more exact will be the work performed by it , as I noted before , and as afterwards I shall make plainly to appear .

Furthermore , every such artificiall or supposed rationall line , may be said to be two fold , to wit , generall or universall , and speciall or particular : Generall , in respect of the number by which it is indicated and explicated , for that the same linear number , doth serve alike to all prime Rationalllines : And particular or speciall in respect of the Figure ( Superficiall or Solid ) to which it is appropriated and applyed , because that every such particular Figure as is mentioned in this Book , doth peculiarly and properly claim to it selfe ( and that severall wayes ) such a line for its more artificiall and expeditionall mensuration .

Now the use of the foure artificial Lines before-going , for the measuring of the foure first Figures aforenamed , we will deliver in the three practical or problematicall Propositions following in the next Section ; and withall the magnitudes and uses of all the other artificiall metricall Lines pertaining to the said foure Figures .

SECT. II.
PROP. 1.

If the Diameter of a Circle be taken by its proper Line of Measure ( according to any Rationall Line ) I say then , that the Quadrat of the Diameter shall be the Area of the Circle ( according to the same Rationall line ) which I prove thus demonstratively .

geometrical diagram

Now that there may be a comparing of our new or artificiall mensuration with the common or naturall , and thereby a confirmation of the same : I draw the right line C D , for the second , compound , supposed , artificiall ( or quadratarie ) Rationall Line , to the length of A B , the primary , simple , true , or naturall Rationall line , and moreover . 13 ferè of the same ( for so much is the additament or additionall segment in centesimall parts ) and then I divide C D likewise into 100 equall parts ( or first into 10 parts , and then only one of those parts into 10 parts , which will be sufficient , as in the Line A B ) which done ; I measure the line A B 2.00 ( for the Diameter of the Circle ) by the line C D ( the Line or Scale of quadrature ) and finde it to be thereof 1.77 ( for the line A B , simply upon the line C D , falleth about the middle between .88 and .89 , and so the double of A B in one entire line at length being measured by the Line C D will fall about the middle between 1.76 and 1.78 , which is about 1.77 ) which squared , gives 3.1329 for the Area of the Circle , agreeing with the former area exactly in the integrall part : But now the fraction-part of that area , viz. 1/7 being converted into decu-millesmes or square centesmes , is .1428 , which exceeds our measure by .0099 , viz. 99 square centesmes , which difference is not considerable in common practise , as I shall afterwards plainly shew : but yet our measure wanteth of the true area , found by the more new & true terms of Cyclometry , or Circular Tetragonisme ( which here we use in all Cyclometricall operations ) not so much by 12 square centesmes , viz. but 87 such centesmes as I shall streightway shew : For those of Archimedes , of the Diameter to the Circumference 7 to 22 subtripla-sesquiseptima , Archim . de dimē . Circ. prop. 2 , 3. and consequently of the Square of the Diameter to the Circle 14 to 11 supertriundecima , or supertripartiens-undecimas ( though sufficient enough for any ordinary mechanicall use , as hee only meant them , and that especially in smaller Circles , yet ) are in strictnesse of Art too large , and so give the area somwhat greater then indeed it is , and the more , the greater the Circle is , and hereupon hath arisen that difficult and curious question so much controverted among Artists about the Quadrature of a Circle ; and in which many learned men have bestowed great pains , as to the finding out of the neerest proportion between the Diameter and Circumference , &c. among which that excellent Artist * * Lib. de Circ. & Ad scrip . Here also is seen the excellent use and benefit of decimall Numeration in the Quadrature of a Circle . Snel . lib. Cyclomr prop. 31. See Lansberg . Cyclome● . lib 1. Porism● 3. to 29 places , agreeing with Ceulen . Ludolph van Ceulen ( aliàs Cullen & Collen ) hath hitherto generally carryed away the greatest commendations , having set forth the same in decimall terms to 36 places , which he willed to be engraven upon his Tombe-stone , as a Testimoniall and Memoriall of those his painfully sustained and finished Labours ( as W. Snellius noteth , who afterwards produced the very same number ) and which I have seen upon the same in the great Church in Leyden , called St. Peter's Church , there being drawn thereupon a large Circle , and upon the Diameter-line an unit with 35 cyphers for the number of the Diameter , and round about the Circumference , the number for the same to 36 places ; of which so many as are needfull here to deliver , as being sufficient for ordinary use , are 3.14159 , answering to the Diameter 1.00000 , or more briefly 3.1416 , the Diameter being 1.0000 ; which according to the foregoing Archimedean terms of 7 and 22 , will be 3.1428 , &c. But those of that famous Mathematician * Metius the elder , somtime Geometrician to the Estates of the confederated or United Belgick Provinces , of 113 and 355 do agree with the Ceulenian terms to the first seven places , viz. 3141592 , answering to the Diameter 1000000 , for that he demonstrateth the proportion of the Periphery to its Diameter , to be lesse then 3 17/120 ( that is 377/12 ) but greater then 3 15/106 ( that is 33/106 ) whose * * Metius in lib. advers. Quadratur . Circ. Simonis a Quercu . Et Adr. Met. in Lib. Geomet . pract . part . prior . Cap. 10. prop. 3. & part . post . Cap. 4. prop. 1. intermedian The mean Arithmeticall is in the least Terms ● 1801/12720 which is Decimally 3.1415880503 , &c. and the mean Geometricall ( or mean proportionall ) is decimally 3 . 1415880493 , &c. But his mean 3 16/1●3 is decimally 3.141592920 , &c. which exceedeth both the other , & that of Cullen is 3.14159265 , &c. proportion is ( saith he ) 3 16/113 ( or 355/113 ) and which is a little larger then that of Ludolph Van Cullen , yet so , as that the difference is lesse then 1/10●000 . And this proportion doth give the Area of the foregoing Circle 3 16/113 , which is by Decimall conversion of the fraction-part into square Centesms , 3.1416 , agreeing exactly with that which is produced by the Ceulenian proportion , and this is the true Area of the Circle : but the Area 3 1/7 produced by the vulgar or Archimedean proportions , being decimally to square Centesmes 3.1428 exceedeth this Area by 12 Square Centesmes ( and so neere the old and new Cyclometry do here agree ) & so the area of the circle found ▪ by our way 3.1329 . wanteth of the true area but 87 square centesmes , as I said before : and this defect ( not considerable in common practise , as I noted before ) happeneth , in regard that the double of the first line measured by the second , falleth not precisely on 1.77 , but is somwhat more , though indeed upon the said line it selfe , it is very hardly discernable , being but very little : Wherefore if both the Rational Lines were divided into more parts , as 1000 , 10000 , &c. then the work would prove still more and more exact ( but indeed here we could not draw a line actually capable of a greater decimall division , or denomination of parts then 100 , according to the plain , vulgar , simple , or naturall division of a Line , which here chiefely for plainnesse we have used : ) For first , if the prime rationall Line ( A B ) be divided into 1000 parts ; then the second ( C D ) will be of the same . 1.128 , and the first compared with the second ( being also divided into the like number of parts ) wil be .886 or neer thereabouts , which falleth about the middle between .88 and .89 of the said second line being 100 , for these two converted into Millesmes are but .880 & 890 , and so the double of the first Line measured by the second will be 1.772 , or neere thereabouts , which is somwhat more then 1.77 upon the second Line centesimally divided , for 1.77 converted into Millesmes , is but 1.770 , which shews that 1.77 was somwhat too little for to produce the true content of the Circle : Now 1.772 being squared , gives 3.139984 for the area of the Circle ( which is in square Centesmes , 3.1400 ferè ) and the area produced by the truest terms of Cyclometry , both Ceulenian and Metian , is to Milli-millesms , or square Millesms , 3.141593 ferè , ( whereas the vulgar area 3 1/7 is 3.142857 ) which our area is wanting of by .001609 ferè , viz. 1609 square Millesms , or but 16 square centesms , and agrees with the same in square tenths , viz. 3.14 ) which comes much neerer the matter then the centesimall operation , and indeed as neare as need be desired : but yet this inconsiderable difference ( by way of defect in ours ) happens also for the reason aforesaid , in that the double of the first line measured by the second is not just 1.772 , but somwhat more ( as the first line simply applyed to the second , is not precisely 0.886 but somwhat more ) and therefore if the first be divided into 10000 parts ; the second will be thereof 1.1282 ferè , according to the ancient or Archimedean Cyclometry , but 1.1284 ferè according to the later or Ceulenian and Metian Cyclometry : and so the first line compared with the second ( being also 10000 ) will be 0.8862 ( or somwhat more ) and thereupon the double of the first , being measured by the second , will be 1.7724 , which shews that 1.772 was somwhat too little to produce the true area , for this in decu-millesms is but 1.7720 , but our number is 1.7724 ( and indeed somwhat more ) which squared yields 3.14140176 for the C●rcular area , agreeing yet much neerer with the true area , which is now 3.14159265 ( to which comes very neere that of 3 16/113 being by decimall conversion of the fractionall part into square decu-millesms 3.14159292 , beginning but now to exceed the true area produced by the Ceulenian Cyclometricall terms ) our measure wanting thereof now but 19089 square decumillesms , which by contraction of the parts is hardly 2 square centesms .

And if we proceed one operation further , namely , to a centu-millenary solution of the rational Lines ( where A B being 100000 , CD will be thereof 1.12838 ferè ) our measure will be found to agree with the true naturall measure , exactly to square centesms , and to want thereof hardly 14 square millesms : for AB 2.00000 , being measured by CD , made 100000 , wil be found thereby 1.77245 , ( which shews that the artificiall or quadratary number in the precedent operation , was not exactly 1.7724 , but would fall about the middle between that and 1.7725 upon the artificiall line of measure , and so gave the area of the Circle short of what it should be ) which being squared affords 3.1415790025 , for the area of the Circle , which wanteth still of the true area ( being now correspondently 3.1415926536 ferè ) 136511 square centumillesms , which by contraction of the parts is but 1365 square decu-millesms , and about 14 square millesms , and not one square centesm .

And if we go on yet further to a milli-millenary solution of the Rationall Lines ( where AB being 1000000 , CD will be of the same 1.128379 ) the naturall and artificiall Measure will be found to agree exactly to square millesms : For AB 2.000000 entirely taken , being measured by CD ( put 1000000 ) wil be found 1.772454 ferè , whose Quadrat is 3.141593182116 ferè for the superficies of the Circle , which now exceedeth the true measure ( being here correspondently 3.141592653590 ferè ) by 528526 square milli-millesms , which by abbreviation or contraction , is 5285 square centumillesms , or but about 53 square decu-millesms , and not one square millesm .

By all which it is already sufficiently evident , that the more parts the Unit or the Integer of measure given , is divided into , the more exact will be the work performed by it , as I noted before , and as may be also further seen in the following dimensions .

But indeed if the Rationall Lines be divided but into 100 parts they will be sufficicient for any ordinary use : For the greatest difference arising here between the true naturall measure and ours , being that in the first operation ( by centesms ) where the superficiall content of the Circle found by our measure 3.1329 . falleth short of the true content ( found by the truest terms of Cyclometry ) 3.1416 , by 87 square centesm● , is in vulgar terms but ●/115 of the prime rationall Line A B ( as the Integer of the measure given ) squared , which is but as one part of a square unit divided into 115 parts ; and surely this difference is of no moment in common practise . And whereas in the other operations , the differences happening between the true content and outs , in parts of a greater denomination , may seem to such as do not well apprehend this matter , to be very great ; yet being reduced to vulgar Arithmeticall terms ( which are better understood by them ) will appear to be still lesser and lesser , and as nothing : As in the fifth or last operation of this first Example or Demonstration , being under a solution of the Rationall Lines to a million of parts , where the area of the Circle found by its proper Line of diametrall quadration , exceeds the true area by 528526 square milli-millesms , that is , 528526 parts of the prime Rationall Line being 1000000 parts squared , and so resolved into 1000000,000000 parts : which difference , though it may seem great by reason of the multitude of the decimall fraction-parts , yet considering the greatnesse of their denomination , and being reduced into common arithmeticall terms ( whose numbring part is 1 ) they will appear to be as nothing , being hardly 1/1●92●54 of the intire prime Rationall Line ( as the Intiger or Unity of Measure ) squared , which is but as one part of a square Unit containing 1892054 parts .

And if the Line of Measure first given , be so short as that it cannot be distinctly divided into 100 parts , according to its own selfe simply . then may 10 parts reasonably suffice : As suppose here a Rationall Line to be of the former line A B , .1 , and divided equally into 10 parts , viz. A b. ( which is A B 0. 10 ) then the second Line , or Line of quadration , viz. C d. will be thereof 1.1 ( which is as A B 0.11 ) answering analogically to A B 1.10 , and which wanteth of the true additionall parts .03 ferè , because the totall additionall centesimall segment of A B is . 13 ferè , as was noted before ) And let the Diameter of a Circle , be A b , 1.5 ( which is as A B 0.15 ) then the Peripherie wil be ( according to the same division ) A b 4.7 ( which is as A B 0.47 ) whose moiety A b 2.35 ( or A B 0.235 ) together with the semidiameter A b 0.75 . produceth the Circle it selfe in square Integers and parts of A b , 1.76 . Now the diameter A b 1.5 being measured by C d ( being also of a denarie or simple decimall division ) becomes but 1.3 , whose square is 1.69 for the superficies of the Circle ; which wanteth of the true content 7 square primes or tenths only , viz. 7 parts , of the prime Rationall Line , or Line of Measure first given , A b 10 parts ( as the Integer or Unit of measure ) squared , and so resolved into 100 parts , which in vulgar Arithmeticall esteem are hardly 1/14 of the same liue squared , the said line it selfe being then neere 4 parts . Or the diameter being taken in centesimal parts only of the Line A B , viz. 0.15 the content of the Circle wil be only in square centesms , .00176 , exceeding the Square of the quadratarie parts CD 0.13 , viz. 0.0169 by 7 square seconds , or centesms only , which in common accompt make but 1/1429 ferè of A B squared ( the line it selfe being then neer 38 parts ) which is much lesse then 1/14 for the difference between the two Measures : so that you may here see , how much 100 parts in the Line of Measure are better then 10 ; and therefore they that understand the more artificiall way of dividing lines ( commonly called the diagonall way ) may better divide the least Line of Measure , into 100 parts : For this latter assumed Line A b being but. 1 of the first Line A B , and but about 1/4 of an Inch or Pollicar may be distinctly divided into 100 parts , being taken according to the power thereof , so as that any number of centesms may be taken exactly therefrom .

geometrical diagram

And here wee now gather these profitable and delectable proportionall conclusions in the Circle , and that to a milli-millenary solution of the unite .

1 The proportion of the Diameter to the Circumference 3.141593 ferè Side of the Square ●quall to the Circle ( which is the most neare , precise , and proper squaring of a Circle . ) as 1.000000 to .886227 ferè Side of the inscribed square .707107 ferè , √ q 1/2

geometrical diagram

Contrarily. 2 The proportion of the Circumference to the Diameter , .318310 ferè Side of the Square equall to the Circle , ( which is the second most proper quadrature of a Circle . ) as 110 .282095 side of the inscribed Square . 2250●9

3 The proportion of the Quadrat . Diametrall or circumscribing to the Circle it self , as 1 to .785398 Circumferential .079577

Contrariwise. 4 The proportion of a Circle to the Quadrat diametrall or circumscribing as 1. to 1.273240 ferè . circumferentiall 12.566371 ferè .

And here you may observe the excellencie of these proportions , as also of those of the like kinde in the Sphear , and all other the Figures following ; in that the antecedent or first Term is always an Unit , accompanyed ( or supposed to be accompanyed ) with so many Cyphers , as are places in the second term , which may here signifie either a milli-millenary composition of the Unity , or the like resolution of the same , according as the Terms are taken ; by means whereof the solution of the question or proposition is mightily facilitated and expedited , according to what I said in the beginning : and therefore in the extraction both of these in the Circle , and also of those in all the other Figures following , I have used all possible industry and exactnesse .

PROP. 2. Of the solid dimension of a Spheare .

If the Axis or Diameter of a Spheare , be taken by its proper Line of solid Measure ; I say , that the Cube of the Diameter shall be the solidity of the Sphear , according to the prime Rationall Line : which I thus demonstrate practically .

ADmit the former Rationall Line A B ( divided as before ) and let the Diameter of a Spheare be of the same Line entirely taken 2.00 , so the solidity thereof ( according to the commonly received Archimedean proportion of the Cube of the Diameter to the Spheare , 21 to 11 , super-decupartient-undecimall ) will be 4 4/21 : which appears also by the Circumference of the greatest Circle of the Spheare , 6 2/7 ( in respect of the subtriple sesquiseptimall proportion of the Diameter to it ) for this being infolded with the diameter , gives the totall convex superficies of the Sphear , 1● 4/7 , whose 1/3 , viz. 4 4/21 being infolded in like manner with the Semidiameter ( A B 1.00 ) or the 1/6 viz. 2 2/11 ( answering to the square Plane or Base of the inscribed Cube ) infolded with the whole diameter , produceth the totall sphericall solidity 4 4/21 as before .

geometrical diagram

Again , suppose the Diameter of a Sphear to be of the lesser , or simple decimall Rationall Line A b , 1.5 ( which is in parts only of the greater or centesimall Line A B 0.15 ) so the greatest periphery wil be of the same , 4.7 ( in parts only of A B 0.47 ) as formerly in the second practical demonstration upon the Circle ) which two infolded together , do produce the superficies of the Spheare in square or superficiall Integers and prime decimall parts of A b , 7.05 ( in square centesimall parts only of AB 0.0705 ) whose 1/3 viz. 2.35 ( or 0.0235 ) being infolded with halfe the Diameter A b 0.75 ( or A B 0.075 ) or the 1/6 , viz. 1.175 ( or , 0.01175 ) with the whole Diameter , A b 1.5 ( or A B 0.15 ) do produce the totall Spheare in solid Integers and prime parts of A b , 1.762 ( in solid centesimall parts only of A B 0.001762 ) Now A b 1.5 for the Diameter of the Sphear , upon E f the proper correspondent Line of Cubation ( being A b , 1.2 ( viz. A B 0.12 ) which in Centesms make but 1.20 , wanting indeed of the true additionall parts .04 , for that the totall segment of A B in centesms , to be added , is .24 , as before hath been shewed ) divided also into 10 parts , is 1.2 , whose Cube is 1.728 , for the solid content of the Sphear wanting of the true content , only 34 cube-primes , or simple decimall parts , viz. 34 of 1000 , which in a common arithmetical accompt , are only about 1/29 of the Line A b ( as the Integer of measure ) cubed , and which difference will be in the centesimall operrtion ( by parts only of the Line A B ) but 34 cube-centesms , viz. 34 of 1000000 , or more vulgarly 1 of 29412 ferè : For that the Diameter of the Sphear being put A B 0.15 , the same will be upon the correspondent Line of Cubation E F 0.12 , whose Cube , 0.001728 , is for the solidity of the Spheare , which wanteth of the true solidity , 0.001762 , only as aforesaid .

And here we may observe by the way , how this Sphear and the second Circle before supposed , do agree , to wit , in that they both having the same diametral number , have also the very same dimensionall or are all number , the one superficiall , the other solid ; viz. in Integers and parts of the Line A b , 1.76 , &c. in parts only of the Line A B .0176 , in the Circle , and .00176 &c. in the Spheare : which will further appear , if their common Diameter be taken according to a more ample division ( or partiall denomination ) of the Rationall Line , and so consequently , the superficiall content of the one , and the solid content of the other , how far soever they be extended decimally , ( and then also indeed they will be found somwhat greater . ) And this may also plainly appear by the more usuall or common arithmeticall expression of their Diameter ( and so of their other parts of dimension ) viz. A b 1 1/2 , ( or A B 3/20 , in the least proportionall terms to A B 0.15 ) and so their periphery , in the reason of the common ( or Archimedean ) Cycloperimetricall terms ( 7 and 22 ) A b 4 5/● ; or rather of the Metian terms , 113 and 355 ( before noted ) 4 161/226 ( in proper parts answering to A B 3/2● is A B 1065/2260 ) whose 1/2 A b 2 161/452 ( or A B 1065/4520 ) together with the semidiameter A b 3/4 ( or A B 3/40 ) do produce the area of the Circle , 1 1387/18●8 ; which converted into square primes , or simple decimals , ( according to the division of the Line A b ) is 1.76 : or the area in parts only , 3195/180800 converted into square seconds or centesms ( according to the partition of the Line A B ) is 0.0176 , as before . Then the Diameter , and so the Circumference of the Spheare being the same with those of the Circle , these two conjunctly , produce the Sphericall superficies according to A b , 7 31/452 , ( in parts only according to A B , 3195/●4520 ) whose 1/6 viz. 1 483/2172 ( or 3195/27●2●● ) together with the diameter 1 ●/2 ( or 3/20 ) or the 1/3 viz. 2 483/1356 ( or 3195/135600 ) with the Semidiameter 3/4 ( or 3/40 ) do produce the Sphericall solidity , 1 4161/3414 , which in the decimall expression to millesms , or cubicall prime decimall parts ( according to the division of the Line A b ) is 1.767 ( and so much it is really ; exceeding the solidity formerly cast up , by 5 cube-primes : ) And the solidity in parts only , 9585/5424000 is in milli-millesms , or cube-centesms ( according to the division of the Line A B ) 0.001767 . So as that the superficiality of the Circle , and the solidity of the Sphear being put in vulgar fractional terms ( either proper in respect of parts only , or improper in respect of Integers and parts ) viz. the Circle 3195/1●●8 00 , and the Sphear 9585/5424 000 , they will be found to be proportionall among themselves , viz. as the Denominatours are each to other , so also are the Numerators ; for as 1808 -- 00 to 5424 -- 000 , so 3195 to 9585 , and so convertibly .

geometrical diagram

And here note , that what we have now done in the Diameter of the Circle and Another superficiall ( or quadrate ) dimension of a Circle , and solid ( or Cubick ) dimension of a Sphear : viz. By their Circumfe : rences : and the Artificiall Lines of Measure for the same . Sphear , to make the Square of the one , equal to its Circle , and the Cube of the other equall to its Spheare : The like may be done also for their Circumferences : so the Line for squaring the Circumference of a Circle , will be of the prime Rationall Line ( under a decu-millesimal par●ition ) 3.5449 . and for Cubing the greatest or true Circumference of a Sphear , will be 3.8978 : and which perhaps may be of more generall use then those for the diameters , because that commonly the Circumference of a Circle and Spheare ( in materiate things , where the Center is not apparent ) must first be had , before their diameter can exactly , ( especially in a Spheare ) for then that must be had by proportionall argumentation ; though indeed the diameter of a Spheare may be taken at first as well ( if not better ) by a paire of Callaper Compasses , where the same may be had ready upon occasion , and then the Line for the diameter may better be used .

And other Lines also of the like nature may be fitted to the Diameter The superficiall dimension of a Sphear , only by squaring its Diam . & Circumference . And the artificiall Lines of Measure for performing the same . and Circumference of a Sphear , as to Superficiall Measure : As namely to make their particular Quadrats , equall to the convex sphericall superficies : So the Line of measure for this purpose , will be for the diameter , of the Rationall Line ( deficiently ) * * ●q . Diam , ad periph . 1 viz. .1 -- . 68-169 . &c. 0.564 &c. ( viz. as A B 10000 , &c. — .4358 , &c. from the segment of diminution ) and for the ( greatest or true ) Circumference ( redundantly ) * * ●q . Periph . ad Diam . 1. as in page 19. and see page 30. & 31. Numb. 2. 1.772 &c. And hence we may here raise this practicall proposition . viz. If the Diameter or Circumference of a Spheare , be taken by their proper and peculiar artificiall Lines for superficiall measure ; That then their severall Quadrats , shall be equall to the Superficiall Area of the Spheare ( or to the Sphericall ) according to the prime or naturall Rationall Line .

ANd so supposing the former Spheare , whose diameter being first put ( of the prime Rationall Line ) exactly 21 , the sphericall convex superfice was found ( according to the true or greatest Periphery 65.97 &c. ) 1385.4424 . Now the said Diameter being taken by its proper Line of Quadrature ( in a centesimall partition ) will be found 37.22 , whose Square is , 1385.3284 , for the sphericall superfice ; agreeing almost exactly ( in the very parts of measure ) with the other .

And so again likewise the greatest Periphery of a Sphear , being first put ( of the Rationall Line ) exactly 66 , the true Diameter will be found ( of the same ) 21.008 &c. and thereby , the sphericall superficies , 1386.5579 ferè . Now the said Periphery being measured by its proper Line of Quadiature ( under a centesimall solution ) will be found , 37.24 ferè , which squared , gives 1386.8176 ferè , for the convex superfice , or the Surface of the Sphear ; exceeding the other , only about 1/4 of a square integer ( or unit ) as of the Measure first appointed .

And here we now gather these usefull and excellent proportionall Corollaries in the Spheare , viz.

1 The proportion of the Axis or Diameter to the Periphery of the greatest Circle , & contra : is the same with that of the Diam . of a Circle to its Circumference , & contra . Side of the Cube equall to the Sphear ( which is the neerest , precisest , & properest cubing of a Globe or Sphear . ) as 1. to .806040 Side of the Quadrat equal to the convex sphericall superfice , ( which may not altogether unaptly , be termed the squaring of a Sphear ; but most truly the squaring of the Sphericall : and that the neerest and properest also . ) ( ferè . 1.772454

2. The proportion of the greatest or true Circumference to the Side of the Cube equall to the Spheare ( which is the next most proper cubing of a Spheare . ) as 1. to .256556 Side of the Quadrat , equal to the convex superficies ; ( which may be termed he next most proper squa●ing of the Sphaericall . ) .564189

3. The proportion of the Cube of the Axis or Diameter to the Sphear it self as 1. to .523599 Greatest Periphery .016887 f.

Contrariwise .

4 The proportion of the total Sphear to the Cube of the Axis or Diameter . as 1. to 1.909859 Greatest Periphery . 59.217626

5 The proportion of the Quadrat of the Diameter . To the totall superficies as 1. to 3.14159 , &c. being the same with that of the Diam . to the Circumf . Periphery 0.318310 ferè , being the same with that of the Circumference to the Diam .

Conversly .

6 The proportion of the totall superficie of a Sphear to the quadrat of the Diameter Is the same with that of the Peripheriall Quadrat to the totall superficies , ( or the Circumference to the Diameter . ) Periphery Is the same with that of the Diametrall Quadrat to the totall superficies ( or the Diam . to the Circums . )

As for the proportion of the Diameter and greatest Periphery to the side of the inscribed Cube ; I shall deliver the same afterwards , among the proportions in the five plaine Regular bodies inscribed in a Spheare , Part 2. Sect. 3.

PROP. 3. Of the solid dimension of a Cylinder and Cone .

If the ( basiall ) Diameter , and the Axis of a right Cylinder , be taken by their proper Line of Measure , and the Quadrat of the Diameter be augmented by the Axis ; I say , that the resulting rectangle , regular-based Prisma or Parallelepipedon , shall be solidly equall to the Cylinder . And the like in a Cone . Both which I do here practically confirm .

geometrical diagram
geometrical diagram

But indeed ( which is the most absolute , compleat , and compendious way ) the very same artificial dimension which is used in a Cylinder ; wil hold good in a Cone , and so may be as properly and fitly applyed thereunto , ( according to the last Proposition ) whereby the solid content thereof shall be immediately produced , as that of a Cylinder ( even as if it were a Cylinder ) notwithstanding the continuall diminution of its body between the Base and the top or point . And the artificial Line of measure for this purpose ; I noted at the beginning , to be of the Rationall Line in generall , 1.5632 ferè , and which for the present purpose will here be ( according to the foregoing practicall demonstrations ) of the centesimall Line , A B 1.56 , which is here the Line I K , ( divided as the former Lines ) By which the basiall Diameter of the fore-supposed Cone , ( naturally A B 0.25 ) being taken , the same I finde to be thereof , 0.16 , whose Quadrat , 0.0256 , is for the Base of the Cone : And then the Axis ( naturally A B 0.75 ) being also measured by the same Line , becomes thereof , 0.48 ferè , which wholly infolded with the whole artificiall Base , produceth the solidity of this accute-angled Cone , 0.012288 ferè , which exceedeth the true solidity before-noted , viz. 0.012175 , by 113 solid centesms only , and which in vulgar terms , make but 1/8850 ferè , viz. as one part of a solid Integer or unit , divided into 8850 parts .

geometrical diagram

And so a Cone being of equall Altitude and Diameter ( in the Base , and so of equall base ) with the foresaid Cylinder , the sub-triple thereof shall be the totall Oxygoniall Cone , viz. 153.938040 ( or 153 106/113 ) which by the first base 38 1/2 , is 154 exactly ; and which agreeth exactly with the Circle last handled whose Diameter was put 14 ) as appeareth by the most usuall dimension of a Cone , by infolding the Base with a trient of the Altitude , which here being ( A B ) 4.00 , produceth the Conicall solidity as before , ( or a trient of the base with the Axis produceth the same . )

geometrical diagram

And here observe , that as in the Circle and Sphear , we shewed , how that the superficiality of the one , and both the superficiality and solidity of the other , were obtained artificially , not only by the bare Quadration and Cubation of their Diameters , but also of their Circumferences , by the like artificiall Lines of measure accommodated to them : So likewise in the Cylinder and Cone , may the solidity be had , not only by the induction , implication , or involution of the Quadrat of their Basiall Diameter into their Axis ( as we have already shewed in both of them ) but also of the Quadrat of their Basiall circumference into their Axis , these being ( both of them , ) The solid dimension of a Cylinder and Cone by their Basiall Peripheries , and their Axes together . And the artificial Lines of measure for the same . taken by one and the same Line of measure : So the Line for squaring the circumference ( of the Base ) of a Cylinder , ( for its Base ) will be of the Rationall Line , 2.32489 : and for squaring the Basiall circumference of a Cone , 3.3531 ; which two Lines being divided as the former , and the Basial circumferences ( as also the Axes ) of these two Bodies taken thereby , may somtimes prove to be of more satisfaction then those for the basiall Diameters , to wit , where the basiall centers of these Bodies are not apparent , ( especially if the Bases of the Cylinder or Cone to be measured be very large ) according as I noted before in the Circle and Sphear . But indeed , where the Diameters may be first exactly taken , either in these two Figures , or in the other two before going , it will be much easier and readier in practise , then by the Circumferences , in regard both , that the artificiall Lines of measure serving thereunto , are much shorter , and also that the Diametrall numbers being much lesser or smaller then the circumferentiall , the Arithmetical operations following thereupon , in casting up the superficiall and solid contents , wil be sooner expedited , unlesse the same be performed Geometrically ( as I may term it ) or Instrumentally , viz. by Scale and Compasses , or the like ( as I shall in the very close of this Book by way of conclusion declare ) for then the latter may be performed thereby , even as soon as the former .

And now you may observe here by the way , how that although the base of a Cylinder and Cone be a Circle , and its Area be here had also , by the only quadration of the Diameter or Circumference , as before was done in the Circle , whereby the same artificiall Lines of measure that are there used , might also perhaps seem to serve here , ( and which indeed in some sort may , as I shall straight way shew ) yet for the obtaining of the solidities of these Bodies , by the implication or induction of the Quadrat of their Basial Diameter or Circumference into their Axis ( either of these two basiall Lines , with the Axis , in each of these two Bodies , being taken by one and the same proper , distinct , artificiall Line of measure , as before hath been sufficiently shewed ) the Lines there appropriated to a Circle cannot here hold for their Bases ; for the Diameters or Circumferences here of being taken by those Lines , wil be either greater or lesser ( in the number or parts of measure ; according to quantity discrete ) then by their own proper Lines ; ( and so their Quadrats for the Bases , and consequently the solid contents wil be greater or lesser then they ought to be ; what artificiall Line of measure soever , the Axis should be measured by . ) For so in a Cylinder and Cone , the Diameter of the Base being taken by the artificiall Line belonging to the Diameter of a Circle , for its quadrate dimension thereby ( being simply and absolutely considered in it selfe alone as a Circle ) will be found in the Cylinder to be somwhat lesse , and in the Cone much greater , ( in the number or parts of measure , as in a Quantity discrete ) then being taken by its proper artificiall Line for the quadrate dimension of the Base , as in reference to the solid dimension of the Cylinder and Cone , performed wholly by the said Line of Measure , from the Basiall Diameter and the Axis together .

And then , both in a Cylinder and Cone , the circumference of the Base being taken by the artificiall Line , belonging to the circumference of a Circle simply , for its quadrate dimension thereby ; will be found lesse ( as falling under the notion or nature of a Quantity discrete , as aforesaid ) then by its proper , respective , artificiall Line , for the squaring of the base thereby , as in relation to the solid dimension of the Cylinder and Cone , wholly performed by the said Line , from the Basiall Circumference , and the Axis together .

As in the Cylinder and Cone last handled ; where the Diameter of the Base being put naturally ( A B ) 7.00 , the same was there found to be artificially ( G H ) 6.46 ferè , and ( I K ) 4.48 ferè , and which will be found by the Line ( of Quadrature ) pertaining to the Diameter of a Circle , simply ( C D ) to be 6.20 , viz. somwhat lesse then that of G H , and much greater then that of I K.

And so there again , the Circumference of the Base being naturally ( A B ) 22 ferè ; the same will be found to be artificially , by the Line for squaring the ( Basiall ) Circumference of a Cylinder ( as in reference to its solid dimension by the Circumference and Axis together ) 9.46 ; and by the Line for squaring the basiall Circumference of a Cone ( as in relation to its solid dimension in the like manner ) 6.56 , and which by the Line for the Circumferentiall quadration of a Circle , simply , will be found 6.20 ( as the Diameter before , by its proper Line of quadrature ( C D ) which is lesse then either of the other .

Therefore , if the basiall Diameter or Circumference of a Cylinder and Cone , be taken by the Lines of ( Diametrall and Circumferentiall ) quadration , properly , peculiarly , and simply pertaining to a Circle , and so its Quadrat be made the Base of the Cylinder or Cone ; then must the Axis be taken by the Prime Rationall Line : ( And so the dimension will be mixt . ) For that here the Area of the Base , will fall immediately , in the true , naturall , measure , ( as under the dimensionall reason of the prime , true , or naturall Rational Line ) according as hath been demonstrated before in the dimension of a Circle .

As here the Basiall diameter of the Cylinder and Cone , naturally ( A B ) 7.00 taken by the Line C D is 6.20 ( or 6.2 , as I shewed even now ) whose Quadrat . 38.4400 ( or 38.44 only ) for the Base , ( which was found before , to be most truly and naturally , 38.4845 ) being drawn into the Axis , put before , naturally , ( A B ) 12.00 , will give the Cylindricall solidity ( in the dimensionall reason of A B ) 461.280000 , ( or 461.28 only ) and so the Conicall solidity , 153.76 ; which differ ( by way of defect ) from the true , naturall solidities ( produced wholly by the Line A B ) viz. of the Cylinder , 461.814120 , about 1/2 of a cubique Integer or Unit , as of A B cubed : and so of the Cone , viz. 153.938040 , about 1/6 of a cube-integer only , as of the same Line Cubed ; And the solidity of the Cylinder , produced artificially by its proper Line of measure ( G H ) viz. 461.968812 ferè , differeth therefrom ( by way of excesse ) only about 1/6 of a cube-integer ; and the solidity of the Cone produced in like manner , by its proper artificiall Line of measure ( I K ) viz. 154.140672 ferè , only about 1/5 of a cubique or solid Integer of the naturall measure ( as of A B considered cubically ) as before hath been shewed ,

All which severall solid dimensions ( both naturall and artificiall , and mixt of both ) doe so neerly agree one with another , as that their differences are altogether inconsiderable . And the same will happen in the solid dimensions of these two Bodies , by the other Line of Quadrature pertaining simply to a Circle , as to the quadration of their Basiall Peripheries , for their basiall Area's , and withall by the prime or naturall Rationall Line , as to the dimension of their Axes or Altitudes .

And now , as for the Axis ( or Altitude ) of a right or erect Cone , you may here observe , that though the same cannot immediately be taken Instrumentally ; or by a Line of Measure ( as it is within the body of the Cone ) as that of a Cylinder ( being parallel to , and so agreeable with the side , ) by reason of the inequality of its Body betweene the Base and the Cuspe , or verticall point ; but being obtained purely Geometrically , must be had by mediation of the Side ( as being the side of a rectiline rectangle Triangle , subtending the right angle , and therefore potentially equall , or equally potent to the two containing , comprehending , or including sides . by E. 1. p. 47. ) and of the basiall Ray , ( being one of the sides about the right angle ( and most commonly the lesser , ) and so the side of the Cone , and the radial line of its base , being first taken Instrumentally or Mechanically , the Axis will be had Trigonometrically , according to the reason of the fore-cited prop. of Euclid . ) Yet may the same be obtained most readily ( out of the Cone ) by a Line of measure , if from a Plane constituted in the top , or verticall point of the Cone , parallel to the Base , you let down a Perpendicular-line to the Plane on which the Base is , ( and which may fall precisely upon the Basiall periphery ) for that ( being measured ) shall be equall to the true Axis of the Cone . And thus also may the altitude of any oblique , scalene , or inclined Cone be taken ; and also of an oblique , scalene , or inclined Cylinder ; if from the top of the Cone to the Plane in which the Base is set , or from the superiour base of the Cylinder , to the Plane of the inferiour base , be let fall a perpendicular-line ; for that ( being measured ) shall be the altitude of the oblique , or inclined Cylinder or Cone ; and so being infolded with the whole base , if a Cylinder ( whether measured by the naturall or the artificiall Line of measure ) shall make the solidity of the same : or being infolded with a trient of the base , if a Cone , ( or the base with a trient of that ) and measured by the naturall Line ; or else the whole perpendicular of altitude with the whole base , if measured by the artificiall Line proper to a Cone , shall produce the solidity of the oblique Cone ; seeing that such a Cylinder and Cone , is equall to a right Cylinder and Cone , having the same base and altitude See also E. 11. p. 31 , & E. 1. p. 35 , 36 , 37 , 38. with it , ( according to the reason of E. 12. p. 11 , & p. 14. )

And after the same manner will the true altitude of any Pyramid , whether right or oblique , be had , as of a right or oblique Cone , and the altitude of an oblique or inclined Prisme , as of an oblique or inclined Cylinder ; the dimension of which bodies , especially the Pyramidal ( both solidly and superficially ) according to our new artificiall way ( together with the naturall or vulgar , by way of dimensionall comparison ) I shall shew next after the dimension of right-lined ordinate Planes or Superficies , seeing that upon any of them may be erected or constituted a Pyramid or Prisme ; as a Cone or Cylinder upon a Circle .

SECT. III. Of the Superficiall Dimension of a Cylinder and Cone .

ANd as we have here shewed the most artificiall and expeditious dimension of the Cylinder and Cone , in respect of their Stereometry , or solid measure ; so we shall likewise demonstrate their dimensions in respect of Planometry , or superficiall Measure . And the artificiall Lines for the performing hereof , I finde , for the Cylinder , ( in respect of its Diameter & Side together ) to be of the prime Rational Line , the same with that which was formerly found to be for the obtaining of the Superficies of a Sphear , by the only quadration of the Diameter , viz. 0.564 , &c. And for the Cone ( in relation to its basiall diameter , & its side conjunctly ) 0.79788 ( which according to the parts of diminution , is as A B 1.00000 — .20212 ) and in relation to its basiall periphery and side together , it will be 1.4142 , &c , ( viz. √ q 2 ) which severall Lines being exactly set off from the prime Rationall Line , and then divided as the same , and so the basiall diameter , and the side of a Cylinder , and of a Cone ; and also the basiall Circumference , and the side of a Cone together , be taken by their peculiar , respective , distinct artificiall Lines of Measure , and so multiplyed together , their severall products shall be the superficiall contents of the Cylinder and Cone , according to the prime Rationall Line : where , by the superficies of a Cylinder and Cone , must be understood without their Bases ; for so the Superficies of these two Solids are generally taken by Artists , and are called by the Latin Geometers , especialy Ramus , ( in one word ) Cylindraceum , and Conicum , that is as much as to say in English , the Cylindraceall or Cylindricall , and the Conicall ; and so the Superficies of a Spheare is called by them Sphaericum , viz. the Sphericall . But now for a briefe dilucidation of these superficiall dimensions in the Cylinder and Cone , I shall lay down an example in each of these three artificiall Metricall Lines ( though here I do not draw them , but only expresse them by number ) whereby the verity of these our dimensions also , may plainly appeare , to those that shall please to make tryall thereof . And first we may hereupon raise this practicall Proposition , viz.

If the ( basiall ) Diameter , and the Side of a right Cylinder ; And the basiall Diameter or Periphery , and the Side of a right Cone , be taken by their proper and peculiar , distinct artificiall Lines of Measure , and the same be severally infolded together : That the resulting rectangle Parallelogram , shall be equall to the Cylindricall and Conicall Superficies , ( according to the reason of the prime Rationall Line . )

THerefore , suppose here first the Cylinder last handled , whose Diameter was put ( of the Prime Rational Line ) exactly 7 , and so the Circumference ( according to the most common Cycloperimetricall terms ) was exactly 22 , and the altitude ( which is the same with the side ) was put exactly 12 , and these two infolded together , do produce the Superficies of the Cylinder ( without the two Bases ) exactly , 264 : whereby this Cylinder becomes absolute in all its dimensions ; But by the other Cycloperimetricall terms , the Circumference was but 21.99 &c. ( or 21 112/113 ) whereby , the Superficies becomes now ( to square centesms of the Rational Line ) but 263.8938 , ( or more vulgarly 263 101/113 ) Now if the Diameter of the Cylinder naturally 7 , be measured by its proper artificial Line of superficiall measure ( being as A B 0.56 , made 100 ) the same will be found , 12.41 ferè , and the Side of the Cylinder naturally 12 , being measured by the same Line , will be found , 21.27 ferè , which two multiplyed together , produce 263.9607 ferè , for the superficies of the Cylinder , agreeing with the true superficies , exactly in Integers ( or Units ) of measure , and differing therefrom in the fraction-parts ( by way of excesse ) but as 1/14 of a square integer or unit . And if the first or naturall Line of measure be made 1000 , and so the second or artificiall Line ( being thereof 0.564 ) be also made 1000 , then the said Diameter taken thereby , wil be 12.407 , and the Side wil be 21.269 , which two multiplyed together , will produce the Cylindrical Superfice , 263.884483 , differing from the true one , viz. 263.893783 ( now by way of defect ) only as 1/108 ferè , of a square integer or unit of the appointed measure . Which excellent compend in the superficiarie dimension of a Cylinder by its Diameter and Side only , will further and fullier appear , if it be compared with the most naturall or vulgar way , in respect of the Side and Diameter only given : where , by the Diameter , the Circumference being proportionally obtained , must then be multiplyed into the side , to make the Superficies ( for that the Superficies of a right Cylinder is most naturally , a Plane made of the Circumference and the side thereof ) and the obtaining of this by the Diameter and Side ; Mr. Oughtred in his Book entituled The Circles of Proportion , Part 1. Chap. 7. Sect. 10. delivers in this proportionall manner .

As 7 to 22 , Or 1 to 3.1416 .

So the Diameter and side multiplyed together ,

To the Superficies ( viz. without the two bases . ) And so in this our Cylinder , the Diameter being put 7 , and the side 12 , I say ,

As 7 to 22 , or rather , 1 to 3.14159 &c. ( or 113 to 355 ) So 7 into 12 , viz. 84 ,

To 264 , the Superficies ,

Or rather 263.89378 &c ( or 263 101/113 ) as first by the Circumference and side together : which two severall operations you see do consist of two severall Multiplications , and one Division , whereas ours consisteth of one Multiplication only , to wit , of the Diameter and Side together .

And from this Analogicall operation used by Master Oughtred , for the discovering of the superficies of a right Cylinder , by its side and ( basiall ) Diameter , you may here note , how that it is demonstrated by Archimedes , Lib. 1. de Sph. & Cylind . prop. 13. that the Superficies of a right Cylinder ( without the Bases ) is equall to a Circle , whose Semidiameter is a mean proportional Line between the side of the Cylinder , and the Diameter of its Base .

And so in this our Cylinder , the side being 12. and the Diameter 7 , the mean Proportionall between them , wil be 9 3/19 , or rather by an immediate decimal extraction of the parts , 9.16515 &c ( whereas the other is decimally by conversion of the parts , but 9.15789 &c. ) which being put as the Semidiameter of a Circle , and so the Square thereof 84 , the Area of the Circle wil be found 263.89378 &c. agreeing exactly with the Superficies of the Cylinder . Or the Area is by the Metian Cyclometry , 263 101/113 , for the Superficies of the Cylinder , as before . ) For seeing here , that the Semidiameter of the Circle equall to the Superficies of a right Cylinder is the mean proportionall betweene the Side and ( basiall ) Dia meter of the Cylinder ; and that the Analogie holds the same from the Quadrat of a Circles Semidiameter to the Circle it selfe , as from the Semidiameter to the Semicircumference , or the * * Quae est ratio totius ad totum , ●adem est ratio dimidii ad dimidium : Et sie alic●rus partis ad aliquam partim consimilim sin iogneminem . Diameter to the Circumference , ( only the one Analogie is Lineall , and the other Superficiall . ) Therefore the Analogie of the side and ( basial ) Diameter of a right Cylinder infolded together , to the Superficies thereof ( without the two Bases ) wil hold the same , as of the Diameter of a Circle to its Circumference .

Again , supposing the basiall Diameter and the side of a right , erect , or Isoskelan Cone , to be the same with those of the foregoing Cylinder , viz. 7 and 12 ; the superficies will be halfe the superficies of the Cylinder : for that the superficies of a right or Isoskelan Cone is a rectangle Plane made of the basiall semi-periphery and the side ( or the basiall periphery and the semi-side ) and so the true Conicall superfice or surface , will be 131.9469 , ( or 131 107/113 , agreeing decimally with the former , or by the most common accompt , 132 exactly ) And the finding of thetrue , genuine superficies of a right Cone by the side and basiall Diameter , the foresaid Mr. Oughtred also sheweth in the 8 Sect. of th● same Chapter before named , by this Analogie , viz.

As 7 to 22 , Or 1 to 3.1416 :

So the Semidiameter of the Base multiplyed with the side ,

To the Superficies ( viz. without the Base ) And so the side of the Cone being here put 12 , and the basiall Diameter 7 , I say ,

As 7 to 22 ( or 113 to 355 ) or rather 1 to 3.14159 &c.

So 3 1/2. ( or 3.5 ) with 12 , viz. 42. to 132 exactly :

Or more truly , 131 109/113 , or 131.9469 , just as before , for the Conical . So that in both these wayes also , are two severall Multiplications and one Division , besides the bipartion or mediation of the basial diameter : or the same is performed by halfe the side with the whole Diameter , ( for both come to one passe . )

And from hence you may observe , that Archimedes , lib. 1. de Sph. & Cyliud . prop. 14. ( from whence the foresaid Analogie is deduced ) demonstrateth the superficies of a right or Isoskelan Cone ( without the Base ) to be equall to that Circle , whose semidiameter is the mean proportional line between the side of the Cone , and the Semidiameter of its Base . And so in this our Cone , the Side being 12 , and the basiall Semidiameter 3 1/2 ( or 3.5 ) the mean proportionall between them is 6 6/13 , or rather by immediate decimall production of the parts , 6.48074 &c. ( whereas the other is by decimal reduction , but 6.461538 &c. ) which being put for the semidiameter of a Circle , and so the Quadrat thereof 42 , the Area wil be found 131.9469 &c. agreeing exactly with the foregoing Superficies of the Cone : ( or the Circular Area , is from the Metiau Cyclometry , 131 107/113 , which is decimally , 131.9469 &c. agreeing exactly with the true Conicall Superficies , found out decimally at first .

And here you may by the way take notice Archi●● . lib. 1. de Sph. & Cyl . pro. 15 of the next Proposition of Archimedes concerning the superficiary dimension of a Cone , laid down Analogically thus , viz. That the superficies of every Isoskelan Cone , hath that rationality to the Base , as the side of the Cone hath to the Semidiameter of the Base . For so here ,

As 12 ( the side )

To 3 1/2 , or 3.5 ( the Radius of the Base )

So 131 107/113 , or 131.9469 ( the true superficies )

To 38 219/45●● or 38.4845 the Base ; ( And so was the same found to be before , in the solid dimension of a Cone , and of a Cylinder , ) And contrariwise will the reason of the Base be to the superficies of the Cone , as of the Semidiameter of the Base to the Side of the Cone : For so ,

As 3 1/2 or 3.5 , ro 12 ;

So 38 219/452 or 38.4845 , to 131 107/113 , or 131.9469 . Now the Base being added to the foregoing Superficies , there will arise the totall Superficies of the Cone , 170 195/452 , or 170.4314 &c. which by the most common accompt , will be 170 1/2 , the Conical being just 132 , and the Base 38 1/2.

And the very same Analogie with the former , holdeth in the right , erect , or rectangle Cylinder , from the Superficies to the double of the Base , ( or both the Bases joyntly ) and contrar●ly ; because that the Superficies of the Cylinder is double to the Superficies of the Cone , having the same Side , and Diameter in the Base , as I noted before : And so here ,

As 12 ( the Side )

To 3 1/2 or 3.5 ( the basiall ray )

So 263 101/113 or 263.89378 &c. ( the Superficies )

To 76 219/226 , or 76.9690 &c. ( the double of the Base , or the aggregate of the two Bases . ) And so contrarily , will the Analogie hold from the double of the Base , or the two bases conjunctly ( 76 219/226 , or 76.9690 &c. ) to the Cylindrical Superficies ( 263 101/113 or 263.89378 &c. ) as from the Radius of the Base ( 3 1/2 or 3.5 ) to the side of the Cylinder ( 12. ) And lastly , if to the foresaid Cylindricall Superficies 263 101/113 , or 263.8938 ferè , you adde the aggregate of the two Bases , 76 219/226 , or 76.9690 , you will have the total externall superficies of the Cylinder , 340 195/116 , or 340.8628 ferè , which by the most vulgar accompt , would be 341 exactly ; the Cylindraceal ( or Cylindrical Superfice ) being just 264 , and the two Bases together . 77

But now for a more speedy dimension of the Superficies of the foregoing Cone , and first by the Side and basial diameter only , together : I say , that if the side of the Cone ( naturally 12 ) be measured by its proper Line of Measure for this purpose ( which is in centesimal parts only of the prime Rational Line , 0.80 ferè ) under a centesimal solution ; the same wil be found thereby , 15.04 ; and the basial Diameter ( naturally 7 ) measured by the same Line , will be found 8.77 , which two being multiplyed together , do produce 131.9008 , for the Conical Superfice , which wants of the true content , only 1/22 ferè of a square or superficial Integer or Unit of the Measure first assigned . And so again , if the basial Periphery of this Cone , naturally , 22 ferè : be taken by its proper , respective , artificial Line of Measure for this purpose , ( being of the prime Rational Line , in a centesimal solution , 1.41 ) centesimally divided ; the same wil be found thereby , 15.56 ferè : and the Side of the Cone being also taken by the same Line , will be found 8.48 ; which two multiplyed together , do yeild 131.9488 ferè , for the Superficies of the Cone , which agreeth somwhat more neerly with the true Conical , viz. 131.9469 , then that which was produced in this kinde by the basiall Diameter and the Side together , viz. 131.9008 . for this latter ( by the basial periphery and the Side ) differeth from the true one ( by way of excesse or redundancie ) but 19 square or superficial centesms ferè , viz. 19 of 10000 , which in a vulgar Arithmetical expression , are hardly 1/526 of the prime Rational Line ( as the Integer or Unit of measure ) squared : so that both these wayes for the finding of the superficies of a Cone , consist also but of one Multiplication only , as that for the Cylinder .

But for finding the superficies of a Cylinder in this manner , by an artificiall Line of measure , peculiarly fitted to the Circumference and Side together , the same is not to be done , seeing it is performed immediately by the naturall Line of measure , or the Prime Rationall Line it selfe , with the same expedition as that in the Cone by an Artificiall Line ; for that the Plane made of the Circumference and Side of the Cylinder , taken by the prime or natural Line of measure first appointed ( according as I said at first ) doth constitute the true Cylindraceall , or the Cylindricall superfice . And so from the premisses it appears , that as a Cylinder in regard of solid dimension , is the triple of a Cone having the same Base and Axis , or Altitude : So in respect of superficiary dimension , it is the double of a Cone having the same Base and Side or Longitude .

And as for the Bases of a Cylinder and Cone , if they be required in their superficiary dimensions , the same may most readily be obtained by either of the Lines of Quadrature pertaining to a Circle , according as the Diameter or Circumference thereof shall happen to be taken : For so the basiall Diameter or Circumference of the foregoing Cylinder and Cone ( naturally 7.00 , and 22 ferè , viz. 21.99 ) being measured by the said Lines of diametrall and circumferentiall quadration ( under a centesimall solution ) will be found each of them severally ; to be 6.20 ( as I noted before in their solid dimensions ) which squared , yeilds 38.4400 for the Base ; and the true base was formerly found , 38.4845 ; from which ours differs , ( by way of defect ) not so much as will make in vulgar terms , 1/22 of a square Integer or Unit.

And thus having sufficiently declared and demonstrated the dimension both Solid and Superficiall of a Cylinder and Cone ( both theorically , and ) practically , according to the Instrumentall part of Geometry ; and that as well naturally ( by way of metricall comparison ) as artificially , for the confirmation and verification of our artificial way of measuring : I shall next lay down the same Dimensions numerally , in Terms analogicall ( by way of comparison ) from the naturall Measure to the artificiall ; and that under an ample or numerous solution of the unity ( according as I did in the Circle and Spheare ) by means whereof , the artificiall Measure may readily be deduced from the naturall ; or the naturall measure be reduced to the artificiall : And first for the Cylinder ,

The true or natural Measure is to the artificial , in respect of the Axis , or Altitude , conjunctly with the ( basial ) Diameter as 1. to .92264 ferè Solid measure . Circumfer . .430127 . Side ( which in a right , erect , or rectangle Cylinder , is equall to the Axis or Altitude ) with the ( basial ) Diameter , as 1. to 1.772454 ferè , ( being the same with that which was noted before in the Spheare , for the proportion of the Diameter to the side of the Quadrat equall to the Spherical . ) Superficiail Measure .

Then for the cone ;

The natural Measure is to the artificial , in respect of the Axis ( or Altitude , ) conjunctly with the basiall Diameter as 1. to .639839 Solid dimen . Periphery .298377 . Side ( or Longitud ) Diameter 1.253314 Superficiaty Dimension Periphery .707107 [ ferè √ q 1/2.

The last of which proportions , is the same with that of the Diameter of a Circle to the side of its inscribed Quadrat , noted before in the dimension of a Circle .

SECT. IV. Shewing briefly , the theoricall reason of the differences happening between the naturall and artificiall Measure , in the superficiall and solid contents of Figures . And moreover , some observations concerning the grounds and reasons of the Artificiall Mensuration in generall .

ANd now again , as for the differences happening between the superficiall and solid Contents of Figures , found by the naturall or vulgar way of measuring , and our artificial way ; we have formerly shewed , how small and inconsiderable they generally are ▪ and also the practical , instrumental ( or Geometricall ) reason thereof ; viz. that the severall Lines of dimension in the severall Figures ( either naturally belonging to them , or artificially and commonly abscribed to them ) as namely , the Diameters and Circumferences of the Circle and Sphear ; and so the Diameters . Circumferences and Axes , or Altitudes , and Sides , of the Cylinder and Cone , before going , ( and so of all the other Figures following respectively ) taken by their proper , respective artificiall Lines of Measure , are seldome or never exact and precise indeed in the parts of measure , but either deficient or redundant in the same , and so give the are all contents of those figures either a little lesser or greater , then indeed they are ( though for the most partlesse , especially in the two first decimal partitions of the Lines of measure , viz. into Centesms , or only into prime or simple Decimal parts , or Tenths ) as appeared formerly ( and will also afterwards ) by a continuall decimall augmentation ( or subdecuplation ) of the parts of those Lines , whereby the Contents were had still neerer to the truth ; Which reason ariseth from ( and so dependeth upon ) the more true , natural , theoricall ( or the Arithmetical ) reason of these differences , lying in the extraction of the Square and Cube-Roots : For that the Roots of numbers not exactly Square and Cubical , cannot be exactly had , but are alwayes defective , so as that they being inverted , or drawn again into themselves , do not render the numbers precisely , out of which they were extracted ; but the further that the extraction is extended or continued decimally , by the adjection or apposition of more Figures , ( or of Cyphers where there is need ) the neerer still to the truth will the Root be had ; as I noted at the * * Where I might also have particularly expressed one thing more ( to the young practitioner ) amongst other particulars concerning the decimal solution of Unity , ( and which I may here , not altogether un-opportunely do , though it be there included and understood in the Generall , ) viz. That thereby , all the tedious and troublesome operations of Arithmetick in Fractions , by the vulgar way , are wholly avoyded : for that here all fractional numbers , whether comming alone by themselves , or together with integrall numbers , are wholly and universally wrought as integrals , without any manner of preparatory operation , ( as Reduction or transmutation of terms , one way or other , ) which in the working of vulgar Fractions is necessarily required . beginning , where I took occasion to speak of the excellencie of Decimall Arithmetick , as in reference to the work in hand : So that what errour ( though inconsiderable ) may at any time arise in our work , doth proceed primarily and principally from the extraction of the Square and Cube-Roots ; our artificiall Lines of measure giving immediately the Square and Cube-Roots of the Figures to which they are appropriated and applyed , ( or the sides of their equall Squares and Cubes , as nearly , almost as may be ) according as their dimension is superficial or solid , as being naturally ( as it were ) procreated or derived from them ; except it be in the Cylinder and Cone , and the other regular-like Solids following ; as regular-based Pyramids and Prisms : but else in all truly ordinate Superficies and Solids ; as the Circle and Spheare before-going , and so the equiterminall and equiangular Superficies and Solids following , it holdeth so : And so likewise in the Cylinder and Cone , and other regular-like Solids , both for solid and superficiarie dimension , where there is a congruity between their Lines of dimension , by which their solid and superficial Conetnts are obtained , as the Diameters , Peripheries , Sides , and other dimensionall Lines of their Bases with their Axes or Altitudes for solid dimension , and with the sides of their Bodies for superficiall dimension . And therefore if there were no defection in the aforesaid Radicall extractions , there would be none in our work ; for that all our artificiall Lines of measure ( or Lines of artificiall measure ) being thus Radically produced , are themselves in the nature of Roots quadrate and cubique . And so also , all the Numbers or Terms of Quadrature and Cubature , &c , by which the sides of the equall Squares and Cubes of Figures , ( in the naturall Mensuration ) and also some other particular lines , or numbers of dimension , in the artificiall Measure , deduced from the naturall ) are proportionally obtained severall wayes ( according to unity ) as from their severall lines of dimension before named ( and which also are immediately given by our artificiall Lines of measure ) are produced by Radical extraction , from the respective figures , to whose dimension ( superficiall or solid ) they serve , as being their Roots , or the sides of their equall Squares and Cubes , according to their foresaid severall dimensionall Lines , put unitly . But indeed , the greatest errour that can commonly arise here , ( in respect of the difference of our measure from the true measure , whether superficiall or solid , ) will be of no moment , as I have shewed before ( and shall also shew after ) in severall examples . And if our measure agree with the true measure , but exactly to integers or units ( as it almost always doth , and much neerer also ; as even to small parts of an Integer or unit , ( superficiall or solid ) of the appointed Measure ) it will be sufficient in any matter of mechanicall Mensurations ; for which , this our artificiall way of Measuring , was chiefely devised and intended .

But now further , as to the ground and reason , briefly , of this kinde of dimension ( or of these artificial metricall Lines , ) the same may be understood Concerning the grounds and reasons of the Artificiall measure in generall . to be two-fold , to wit , generall or universall ; and speciall or particular ; that consisting in Unity alone ; this in the solution of Unity : For the generall reason is by it selfe absolute , simple , and certain , without relation or limitation to any the proper , compounding , denominate ( or other ) parts of the Rationall Line , ( as being the Integer or Unit of measure ) but considereth the same generally ( and absolutely in it selfe ) as some one entire or whole thing , ( of which I shall ( God willing ) speake more fully afterwards , in the close of the second part of this Book , as being a place convenient . ) But the particular reason ( which I shall chiefely insist and proceed upon ) is limited and confined to the certain , set , or commonly known parts of every particular rationall Line ( they being considered discretely or Arithmetically ( taken as a common , known Measure , and that , according to several places & customs , ( or such like other parts , as the Geometer or Artificer shall in his minde think fit privately to impose on the same for his use ) and so commeth by them to that of the Generall . ( Every particular , figurate Quantity or Magnitude , measurable in this artificiall manner , being here considered in those parts , according to their powers Quadrate or Cubick ( or the Parts considered in their said powers Arithmetically , in every particular Figurate Magnitude , according as the dimension is superficiall or solid . ) For every common or customary Line of measure is usually divided into certain denominate parts , of which it doth primarily and properly consist : As our Foot is vulgarly said to be divided into 12 parts immediately , called Inches , which do compose or constitute the same : Or ( as in divers Countries beyond the Sea ) a foot may be understood with us to be composed first ( and that most nearly ) of 4 Palms , or Hand-breadths ; a Palm ( being composed ) of 4 Digits , or Finger-bredths ; and a Digit of 4 Grains or corns of Barley ; according to the Latin Distich ,

Quatuor ex granis , Digitus componitur unus : Est quatèr in Palmo digitus : quatèr in Pede palmus .

And this according to the description of Vitru● . Lib. 3. Architect , cap. 1. the ancient Roman Foot by Vitruvius and others : So that thus the Foot contains 16 Digits , answering to 12 inches with us , for that a * * See Circles of Propo●tion , Part 1 Chap. 9. Sect. 4. Palme ( namely Palmus minor ) is said to be 3 of our Inches , and so 4 Palms 12 Inches : But we in England usually taking no notice of the Palme and Digit in measuring , but only of the Foot and Inch ( besides the Yard and Ell &c. ) divide the Foot immediately into 12 parts , called by the Latines unciae & Pollices or Pollicaria , and so also was the Roman Foot anciently divided ( and still is ) and so is the Foot in many other places . So that every such greater Measure is commonly composed of the next lesser , being some certain times reiterated or repeated : As our greatest common Geometricall Line of measure , viz. a Perch or Pole . ( for Land-measure ) is commonly composed of * * Statute measure A● . 33. Edw. 1. other ( customary ) measures there be , as the Pole or Perch of 18 feet , usuall for Wood-land measure , &c. 16 1/2 Feet , a Foot being composed of 12 Inches ( as I said before ) and an Inch of three barley-corns in length , picked out of the middle of the Eare : but a Barley-corn being the least of all Measures ( or rather no measure at all , being but the very beginning of measure , as an Unit is usually counted no number it selfe , but only the beginning of Number ) cannot be composed of any other .

And although Arithmetick in generall , naturally taketh no notice of these lesser Measures as the proper composing parts of the greater Measure given ; but immediately considers every particular Measure ( as an Unit ) according to a simple or naturall Arithmeticall division into parts ( or a division into parts simply or arithmetically denominate ) as halves , quarters , and the like ; or more especially ( and more exquisitely ) as Decimall Arithmetick ; into Tenths , Centesms , &c. and so taketh the Contents of Superficies and Solids , to such parts thereof , Square and Cubick ( or otherwise Superficiall and Solid ) in generall ; as Square and Cubick parts of a Foot , and of an Inch ; square parts of a Perch or Pole ; and so also , square parts of an Acre , &c. Yet this being understood only by Artists , and so not sufficient to satisfie the vulgar : these simple or naturall Arithmeticall parts ( or parts meerly divisionall ) must at last be reduced to the proper compounding , denominate , or commonly known parts of the Measure given or appointed ; ( or the Geometricall or mensurall parts of the said Measure ( as I may term them ) they being by themselves alone , put as measures certain , and intire , and so compounded of , ( or dividuall into ) other the like kinde of parts , or inferiour Measures , ( according as I noted even now ) and so are to be considered as Quantities continuate : ) As the parts or fractions of a Foot into Inches ; of a Perch or Pole into Feet , ( if they be required ) and so likewise of an Acre into Perches , &c.

And now again , the speciall , particular , or partiall reason aforesaid , of this artificiall dimension ( or of the Artificiall Lines of Measure ) or the Reason of the Parts , as I may term it , from what I have declared before : ) may be considered in a twofold respect : viz. either more generally ; as relating to the denominate , compounding or Geometricall parts of every Measure first given or appointed ( as the prime or naturall Rationall Line ) only for the discovering or producing of the artificiall Lines of measure , as considered in the generall Reason , ( or the Reason of the Whole ; , as I may call it , from what I have said before concerning the same ) such as are all the artificiall Lines beforegoing , and also the other following , being expressed by Quantity discrete , or number , which shew their Magnitudes ( or quantities in measure ) from the intire , prime , or naturall Line of Measure in generall , from which they are to be taken ; And which therefore I may call ( not unaptly , I conceive ) their Indicant or Exponent Numbers . Or more particularly , precisely and properly ; as relating meerly to the foresaid parts of every particular Measure given ( considered discretely or Arithmetically ) for the constituting of the said artificiall Lines of Measure so , as to give the superficiall and solid contents of Here note another the like artificiall dimension of Figures ( as the former ) not mentioned before ; which is only according to the compounding , denominate ( or Geometrical ) parts of Measures given or appointed . Figures , quadrately and cubickly , &c. ( or the sides of their equall Squares and Cubes , &c. ) from their severall lines of dimension belonging to them , and by which severally they may be thus artificially measured ; as the forementioned artificiall Lines do ) only , ( or for the most part ) according to those parts , ( being considered meerly in the nature of such parts of the prime or naturall Line of measure appointed ; which otherwise taken apart by themselves alone , may be put as measures entire ( or Integers of measure ) as I noted even now ; and then are considerable in our generall Reason of Measure , or Reason of the Whole , before named . ) And which Lines therefore being most properly considered and laid down from the foresaid parts of the natural Measures appointed , ( as before , the Lines are from the whole intire Measures themselves ) will alter continually in every particular Figure , ( in respect of quantity discrete , according to their Arithmeticall Indices or Exponents , which expresse their magnitudes in the forenamed parts of Measure ) not only in regard of the different dimensionall lines thereof , by which it may be propounded to be artificially measured , as aforesaid , and so to which they are respectively fitted : ( as the artificiall Lines before-mentioned do , according to their like Indices or Exponents before declared ; especially for the most part ; though it doth happen otherwise somtimes , as that one and the same artificiall Line , is found either wholly throughout , or sufficiently in part , ( that is , in respect of the fractionall or decimall part of the naturall Line of measure , from which it is taken , shewed by the Arithmeticall Index or Exponent , how far soever the same be continued or extended decimally ) to serve unto severall Dimensions ; as I shew afterwards , in Part 2. Sect. 3. ) but also , in respect of each one and the same particular dimensionall line thereof , by it selfe alone , according as the said parts of the Measure proposed , do ( arithmetically ) alter : whereas the arithmeticall Exponents of the first mentioned Lines , relating generally to any whole Measure appointed ( as I noted at the beginning ) do continue the same ( without alteration ) in every particular , distinct Dimension of one and the same Figure , according to its severall lines of Dimension aforesaid . And so these two severally mentioned ( or supposed ) sorts of artificiall Lines , will hereupon differ in every severall dimension of one and the same Figure , in respect of quantity discrete or Arithmeticall , ( according to their Indices or Exponents of measure ) though they doe not , in respect of quantity continuate or Geometricall ; or of Measure it selfe in generall , as I shall shew by and by : And then also they differ herein ; That as the former artificiall Lines being immediately of the whole intire naturall Lines ( either redundantly or deficiently , according as their Arithmeticall Exponents do shew ) considered without respect of parts , compounding or Geometricall , ( but only Arithmeticall , or meerly divisionall , as is alwayes necessarily required , for the exactnesse of measure ) and so giving the Areall contents of Figures accordingly ; are themselves to be considered ( in the artificiall Dimension ) as whole intire Lines of measure in like manner : These latter mentioned ( or supposed ) Lines must ( for the contrary respects aforesaid ) be considered ( in the like Dimension ) as Lines of measure containing ( or compounded , as it were , of ) certain parts , answering ( Arithmetically ) to the primary or compounding parts of the naturall Lines , from which they are ( most properly ) derived , and so which they do artificially represent , and consequently , according to which ( chiefly ) they give the superficiall and solid contents of Figures ( as aforesaid . ) And which Lines ( for the reasons before alleadged , as also for distinctiorsake ) we may well call , the particular or second artificiall Lines ( or the Lines of parts ) as the other may be called the prime or integrall artificiall Lines ( or Lines of the Whole ) And so the Measure arising there from may accordingly be called , the one , the prime or integrall Measure ( or Measure of the Whole ) as having its denomination simply and absolutely from the whole intire naturall Line of Measure appointed : And the other the particular or partiall Measure ( or Measure of the parts ) as being denominated chiefly from the parts of the said Line of Measure . ) And therefore , as those first Lines do artificially represent the respective naturall Lines from which they are taken , considered simply and intirely in themselves , as the Integer ( or Unity ) of Measure severally : So these second Lines , do accordingly represent the said naturall Lines , as they are composed of certain denominate , or mensurall parts ; ( viz. of some inferiour or lesser Measure , considered as a part of some greater Measure , and so some certain times iterated or ennumerated , for the making up of the same , according as I lately shewed , ) And so these second Lines are really none other then the first , divided into the like number of parts , as are the composing , constituting , ( or Geometricall ) parts of the respective naturall Lines from which they are deduced ; and which parts of these ( supposed ) second artificiall Lines , we may conveniently call ( by way of distinction from other parts ) their prime or Geometricall parts : and these being then divided severally into some certain number of parts , as is requisite for the exactnesse of Measure , as aforesaid ( according as the correspondent parts of the naturall Lines are ) especially decimall ; we may call the same , their second or Arithmeticall parts , ( or the particles of measure , as being indeed only the parts of the other Parts . ) So that these second Lines are never exactly decimall , as the prime Lines perpetually are , unlesse it happen , that the prime or compounding parts of the naturall Measure appointed be in a decimall number , for then as the naturall Line , so likewise the correspondent particular , or secondary artificiall Line will be exactly decimall ; their said prime parts being divided decimally ; but yet however , they do generally perform the Dimensions wrought by them , after a decimal manner ( though secondarily , viz. after a reduction of the Measure taken by them , into their prime parts , &c. As first ( for example ) in the naturall Mensuration ; the most common , mechanicall way of measuring by the foot with us , is , as the same is vulgarly divided into 12 Inches , which are , as its composing , denominate , or Geometricall parts ( for so every measure is commonly said to be divided into the parts , of which it is prop●rly composed ) and each Inch divided into some certain parts ( as is necessary for exectnesse in measuring ) which may best be decimall : and so the lines measured hereby , do fall out most frequently , in Feet , Inches , and parts of Inches together : Now if the sides or other dimensionall lines of any Superficies or Solid propounded to be measured , be thus found ( in this mixt measure ( as they will for the most part ) then must the same be first of all reduced into Inches , &c. before the content of the Figure , superficiall or solid , can be conveniently cast up , seeing that the Measure thus taken is mixt ( as it were ) of severall parts or kindes of measure , and so is of different denominations ; and therefore must be reduced into one kinde of measure ( or one mensurall denomination : ) And then if the parts of the Inch be decimall , the work will be afterwards performed in a decimal manner , in Inch-measure only : But if the said lines thus measured , be found wholly in Feet ( without Inches , &c. ) then will the contents be had immediately in whole Foot-measure , which otherwise must be had in the like measure by Reduction from Inch-measure . And so seeing that the second artificiall Lines doe represent the naturall Lines , only as being composed of certain denominate , or mensurall parts , and so are to be considered themselves accordingly , as aforesaid : Therefore , if the side or other dimensionall Line of any Figure measured thereby , for the obtaining of its content superficiall or solid , by way of Quadrature , Cubature , &c. and so artificially representing the Side of the equall Square or Cube , &c. be found mixtly , in Integers and prime parts , &c. of the same Lines ; ( as for the most part they will ) then must the Measure so taken , be first reduced wholly into their said prime parts , &c. for the casting up of the content as aforesaid , ( according to what was shewed before in the naturall Mensuration , after the most plain or vulgar way by a common naturall Line of parts ) And then those artificiall prime , or mensurall parts , being divided Decimally ; the dimension of the Figure proposed , will be performed in a Decimall manner , according to those parts only : But if the side , or other line of the Figure to be measured , doe fall out in Integers only of the said artificiall Lines , ( viz. in whole Lines without any parts ; ) then will the Areall content be obtained immediately , and exactly , in Integers of the naturall Measure appointed ( according to the reason of the prime artificiall Lines , or Lines of the whole Measure ) which otherwise can be had in the integrall measure ( or the measure denominated only from the whole naturall Line ) only , by way of reduction from the primary , compounding , denominate , o●metricall parts of the same , in which it is first found , as before was shewed .

And now therefore it appears from hence , that the artificiall Dimension performed by these second or particular artificiall Lines , or Lines of parts , so called ; or the artificiall Lines , as they are considered meerly in the particular or speciall reason of the artificiall Dimension ( or Reason of the Parts ) taken in the latter respect ; is not considerable in comparison of the artificiall Dimension performed by the prime or integrall ( or more generall ) artificiall Lines , or Lines of the Whole ; or the artificiall Lines , as they are considered in the generall reason of the said artificiall Dimension ( or Reason of the Whole ) they giving the sides of the equall Squares , and Cubes , &c. of Figures ( and so their superficiall and solid contents accordingly ) immediately in In●egers ( and decimall parts ) of the prime Rationall Line , of the naturall Measure appointed ; which the particular Lines do give ( for the most part ) mixtly in Integers and compounding parts and particles together ( or in Integers and parts primary , and secondary ) and so we must come at last to the Areall content in the former Measure , by way of reduction from those Parts , as aforesaid . But yet however ; for the variety of operation and Art , in this kinde of Mensuration , I thought it would not be amisse to manifest thus much concerning this latter way ; that so the ingenious Reader that shall please to exercise himselfe ( practically or experimentally ) in this artificiall way of measuring , may ( by comparing the effects or results of the generall and particular , or speciall reason thereof together , both in the extraction or production of the artificiall Lines themselves , & also in the said two severall wayes of working by them ) receive a more full satisfaction , and the thing it selfe be accordingly confirmed .

And now seeing that the aforesaid parts of Measures given , are various ( as in quantity Geometricall , so usually in quantity Arithmeticall ) according as the Measures themselves are in magnitude various : therefore I shall first and principally prosecute our said speciall or partiall Reason of the artificiall dimension , and that chiefly in relation to the first , or more generall acception or consideration thereof , ( namely , for the producing of the artificiall Lines , as they are considered in the generall Reason , by the parts of the Measure given ( or the prime artificiall Lines by the second , as we have differenced or distinguished them first of all , in Quantity discrete , or in the Number of their measure from the naturall Line , according as the same is understood , either without or with the foresaid kinde of parts . )

And this I shall accordingly lay down in three speciall Theorematicall Propositions ( contained in the second part of this Work ) answering to the three principall Problematicall or practicall Propositions beforegoing ( laid down in the Figures particularly handled in this first Part , namely the CIRCLE , SPHEARE , and CYLINDER , together with the CONE ) and which will generally serve for all other ordinate or regular , and regular-like Figures whatsoever , for the like occasion ( according to what I noted at the beginning ) And the last of these only I shall demonstrate or illustrate by Number , as being sufficient for all , though indeed there needeth no manner of demonstration or illustration of any of them , they being all so very plain and perspicuous .

printer's device of Robert Leybourn, featuring an old man standing by a olive-tree with falling branches (McKerrow 310) NOLI ALTUM SAPERE

PART II. Containing the most artificial and expeditious practicall Dimension , of all right-lined ordinate or regular , and regular-like Figures in generall .
SECT. I. Proposing the foresaid Dimension in all right-lined regular Planes or Superficies in generall : And demonstrating the same particularly in two of the first of them .
THEOREME I. Exhibiting particularly , the fore mentioned Lines , for the Quadrature of a Circle ; from our particular or speciall grouna and reason before declared . And consequently , the Lines for the like dimension of all recti●ine or angular ordinate Planes in generall .

IF the Diameter of a Circle equal to the Quadrat from the Parts aforesaid , of the Rationall Line , be found ; The same shall be the respective Line of Quardature , according to the Parts : And the reason between that , and the correspondent or congruall Tetragonismall Line , according to the whole Measure will be such as the reason between the Parts , and the Whole ; which is as the reason of their Squares . And the like for the Peripherie .

geometrical diagram
geometrical diagram

As a , d , b , or a , d , c , 90º — 10,0000000 Rad. to a b , or a , c , 12 — 1,0791812 .

So a b d , or a c d , 60º — 9,9375306 . S , A. to a d , 10.3923 &c. — 1,0167118 .

geometrical diagram

Or again , by the same reason ;

As b , a , d , or c a d , 30° — 9,6989700 . S , A to b , d , or c d , 6 — 0,7781512

So a b d , or a c d 60° — 9,9375306 . S , A to a d , 10.3923 — 1,0167118 . as before . And other wales also may the same be found out ; either by Sines alone , or by Sines and Tangents together , according to Axiom . 1 Planor Pitisc . and Consect. 1 and 2. And Prop. 1. and 2. Planor . in Trigon. Brit. and more particularly cap. 4. before cited , probl. 1. 2. 3. But this way of working is the most plain and vulgar .

Now the Diameter or Perpendicular of the Trigon , a d , 10.3923 being drawn into the semi-base ( or semi-side ) b d , or c d , 6 ; there resulteth the Area of the Trigon ( according to the reason of E 1. p. 41. ) 62.3538 . Or the same Trigonal Area will also be produced , by comparing each side of the Trigon severally , with the semi-aggregate of all the sides , and then infolding the said semi-aggregate and the difference of each side thereto , continually together ; for so the Root quadrate of the total Result , shall be the Area of the Trigon : As ;

See upon this kind of Triangular dimension ( or Geodesie , as Ramus terms it ) Ram. lib. 12 Geomet . theor . 9. but more espcially the reason thereof demonstrated by him , in fine lib. ult-Schol . mathemat .

geometrical diagram
The artificiall Dimension of Triangles in generall .

And here our artificial Mensuration , may be applied to any other kind of Triangle , either regular-like , or altogether irregular , ( as I may term them ) viz. Isoskelan or Skalene , for the finding of their areal contents ; but it will be in a way somwhat different from that of the regular or Isopleural and Isogonial Triangle ; ( being indeed according to the vulgar or natural way of measuring , ) viz. in that , as there the Area is had immediatly by the only squaring of some one dimensional line of the Triangle , as the Side or Perpendicular , &c. and so the same is the Quadrat thereof in Quantity arithmetical or measure numeral : here it will be the Rectangle made of the whole Base and Perpendicular together ( as it is naturally and really the half of that Rectangle , by E. 1. p. 41. before cited ) which two lines , if they happen at any time to be equal , then will the Area be produced under the form of a perfect Quadrat , from either of them , as that of an Isopleuron , or regular Trigon ; seeing that the Rectangle made of them both , is no other then the exact Quadrat of either of them singly : And so therefore , after this latter way may an Isopleuron also be artificially measured , the way being general . And this resembles the artificial dimension both solid and superficial of regular ▪ like Solids , as the Cone , and all regular-based Pyramids , &c. and so differeth from that particular and peculiar dimension of the ordinate or regular Triangle ( and of all other regular Planes ) as the dimension of regular-like Solids doth from that of Solids exactly regular : And so , as the solid content of any oblique , Scalene , or inclined Cone or Pyramid , &c. is obtained both naturally and artificially , as that of a right , upright , or Isoskelan Cone or Pyramid &c. ( as was shewed before in the dimension of a Cone , and shall be afterwards in the dimension of Pyramids ) So is here the superficial content of any oblique Scalenal or irregular Triangle obtained ( as well artificially as naturally ) like that of an upright or Isoskelan Triangle by any side thereof , put as the base , and the perpendicular lot fall from the opposite angle thereto , whether that side need to be continued out or produced , or not ; seeing that every Skalene Triangle is equal to an Isoskelan , having the same base and perpendicular of altitude , ( or they being constituted upon the same base , or equal bases , and in the same Parallels , as Euclid , and his interpreters do speak ) by E 1. p. 37. and 38. as every oblique or Skalene Cone and Pyramid , &c. is equal to a right , or Isoskelan Cone and Pyramid , upon the same ( or equal ) base , and of the same altitude , by the reason of E. 12 , p. 11. and 14. beforecited in the Cone , &c. and p. 5. and 6. afterwards in the Pyramids , &c. And now the Line of measure , for the performing of this latter or general artificial dimension of Triangles , wil be the same with that , which was noted formerly for the superficial dimension of a Cone , by its side and basial Periphery together , viz. of the Rational Line in general , 1.4142 , &c. √ q 2. And here the proportion of the natural Measure to the artificial , in the two forenamed lines of a Triangle together , will be the same with that in the foresaid Conical dimension ; viz. 1. to .7071 &c. √ q 1/2 But indeed seeing that the Area of any Triangle may be obtained as readily in a manner by the natural Line of measure , or common way of measuring , as by this latter kinde of artificiall Line , or general artificial way of Triangular dimension : ( according to what I noted even now about the same . ) Therefore I shal presse this point no further , then what is only for variety of Art and operation in this kind .

geometrical diagram

SUppose next the side of the ordinate Pentagon , A B C D E , to be the same with that of the ordinate Trigon before going , ( as from the Rational Line A B , ) 12.00 ; then the true Area thereof , will be found , 247.7487 &c. and not 240 , as Ramus makes it , shewing the Geodesie ( as he terms it ) of ordinate Polygons in general , and particularly of an ordinate Pentagon , whose side is 12 ; where he makes the Ray of the inscribed Circle , ( which in Ram. Geom. lib. 19. el. 1. p. Ryff in Epitome Rami. And Oront . Fin. lib. Geom. pract. cap. 24. this pentagonal Figure , is F G ) to be exactly 8 , ( and consequently the Ray of the circumscribing Circle F C or F D , just 10 ) and thereby the Area , but 240 ; and which from him P. Ryff also hath in his Epitomy of Ramus : And so Orontius Fineus also measures this Pentagon ; and thus they take it one from another ( as Ramus from Fineus , and Ryff from Ramus ) without any further examination ; And hence they give the faid Pentagon , absolute in all its dimensional Numbers . But indeed , the side of an ordinate Pentagon , being 12 , the Ray of the inscribed Circle , wil be truly , 8.26 ferè ; and the Ray of the circumscribing Circle ( which here we need not ) wil be 10.21 ferè , and so the Pentagonal area will be exactly ( to square centesms ) 247.7487 . All which we shall here demonstrate Trigonometrically . For seeing that an ordinate Pentagon is a Triangulate , consisting of , ( or resolvable into ) five equal and like Isoskelan or equicrural Trigons , meeting vertically in the center of the Pentagon , ( or of its circumscribing Circle ) and so whose Bases are the sides of the Pentagon , and whose shanks are Raies of the said Circle ; such as is the Triangle C F D ; the vertical ( or Centricall ) angle , F , will answere to a Fifth of the said circumscribing Circle's circumference , as being measured thereby ; which the side of the Pentagon C D ( as the base of the Isoskelan Trigon ) subtending , the same is consequently the Pempta-chord ( as it were ) or Hypodia-pempte of the foresaid Circle , as the Hypotenuse , Inscript , or Chord Pentagonall thereof ; the said circumscribing Circle being Pentachordal . Which Isoskeles being biparted or bisected into two Orthogonials , viz. C G F , and D G F , by its perpendicular F G ( which is a Ray of the inscribed Circle , as aforesaid ) whose Hypotenusals ( specially so called ) F C , and F D , as the two equal sides of the Isoskeles , are two Raies of the Pentagon's circumscribing Circle , as aforesaid ; the semi-base of the Isoskeles , C G or D G , as the lesser containing side of the right angle in the two foresaid orthogonial Trigons ; will be 6 ( as being the semi-side of the Pentagon ) and he Angle C F G or D F G , will answer to a Tenth of the foresaid ambient Circle's circumference , as being halfe the vertical angle of the Isoskeles C F D ; and so consequently , the basial or greater acuteangle F C G , or F D G , will be the complement of the other two angles to a Semi-circle ( according to E. 1. p. 32 ) or the Complement of that other acute angle to a Quadrant : and hereby the perpendicular of the Isoskeles F G , as the perpendicular let fall from the center of the Pentagon to its side , ( or the Ray of the inscribed Circle ) being the thing next to be inquired , will be found . And therefore in the rectangle Triangle , C G F , or D G F , seeing all the angies are given , together with the lesser containing side C G or D G ; the greater and common containing side F G required , will be had these two several ways following : and first according to Axiom . 2 Planor . Pitisc . and Prop. 3. Trigon. Brit. &c. before-cited , most easily and readily by Logarithmo-trigonometrical supputation , or artificial Trigonometry , thus ;

As C F G , or D F G , 36º — 9,7692187 . S A to C G , or D G , 6. — 0,7781513 .

So F C G , or F D G , 54º — 9,9079576 . S A to F G , 8.26 ferè , viz. 8.25829 &c. — 0,9168902 .

geometrical diagram

Or secondly ; seeing that in a recti-line rectangle Triangle , any side may be put for the Radius of a Circle , by Axiom . 1 Planor . Pitisc . and Prop. 1 Planor . Trigonom . Brit. Therefore in the foresaid Rectangle C G F , or D G F , the lesser containing side C G or D G given , being put as Radius ; the greater ( and common ) containing side , F G sought for , will ( by the foresaid Axiom , and Prop. 2. Planor . Trigon. Brit. ) be the Tangent of its opposite angle F C G , or F D G ; Whereupon it followeth accordingly , in this Trigonometrical Reason ;

As C G , or D G , — 10,0000000 . R. to F G — 10,1387390 , T A , 54ᵒ.

So C G , or D G , 6 — 0,7781513 to F G , 8.258 &c , — 0,9168903 as before .

Which being infolded with the semi-perimeter of the Pentagon , A B C G , 30 , there will result the Area of the Pentagon , 247.7487 , as I said at first . Or the said F G , as the perpendicular of the foresaid Isoskeles C F D , being infolded with the semi-side of the Pentagon , C G or D G , as the semi-base of the Isoskeles , produceth the Area of the Isoskeles , C F D , 49.54974 , &c. which augmented by the number of the composing Isoskelan Trigons , produceth the total Pentagon , 247.7487 &c. as before ; and which is most readily and accuratly obtained by Logarithmical numeration , only by a simple composition of Numbers , thus ;

F G , 8.258 &c. — 0,9168903 A

C G , or D G , 6 — 0,7781513 A

Isoskeles C F D , 49.5497 &c. 1,6950416 aggreg . A

5 A 0,6989700 aggreg . A

Pentagon , 247.748745 &c. — 2,3940116 . aggreg .

geometrical diagram

Or , according to the first operation , more briefly thus ;

F G , 8.258 &c. — 0,9168903 A

A B C G , 30 — 1,4771213 A

Pentagon , 247.7487 &c. — 2,3940116 , as before .

geometrical diagram

Now as to the Radius of the circumscribing Circle , which Ramus , & the others beforenamed , do make exactly 10 , ( though here we make no use thereof , yet for the truth's sake ) we will demonstrate the same to be 10.21 ferè , thus : And first in the rectangle Triangle C , G , F , or D G Faforesaid ; the lesser acute angle C F G or D F G , and the lesser side about the right angle , viz. C G or D G ( as the semi-side of the Pentagon ) subtending the said acute angle , as the right Sine thereof , being first of all given : the Hypotenuse F C , or F D , being the Ray of the ambient Circle , sought , will be found thus ,

As C F G , or D F G , 36º — 9,7692187 . S A. to C G , or D G , 6 — 0,7781513 .

So C G F , or D G F , 90º — 10,0000000 . R. to F C , or F D , 10.21 ferè , viz. 10.2078 &c. — 1,0089326 .

Or again secondly in the same Triangles , the greater acute angle F C G , or F D G ( being half the Pentagonal angle ) and the greater and common side about the right angies , viz. F G , subrending the two said equal acute angles , ●eing only first given ; the Hypotenusal side , F C , or F D , inquired , will be had thus ;

As F C G , or F D G , 54º — 9,9079576 S A to F G , 8.25829 &c — 0,9168902

So F G C , or F G D , 90º — 10,0000000 R. to F C , or F D , 10.2078 &c. 1,0089326 , as before .

Or again , thirdly , in either of the said rectangle Triangles , where the foresaid greater acute angle being only known and the lesser side about the right angle , viz. C G , or D G , and the same put as Radius ; the side subtending the right angle , viz. F C , or F D inquired , wil be the Secant of the said greater acute angle , F C G , or F D G , ( as the greater and common side about the right angle , and the common Subtense of the said two acute angles , viz. F G , was said before in the like case , to be the common Tangent of the same angles ) whereupon it followeth in this Trigonometrical Analogie ;

As C G , or D G — 10,0000000 , R. to C F or D F — 10,2307813 , Sec. 54ᵒ.

So C G or D G , 6 — 0,7781513 . to C F or D F , 10.2078 &c. 1,0089326 , as before .

Or fourthly , in either of the said Rectangles , the lesser acute angle C F G , or D F G , being only given , and the greater containing side of the right angle , viz. F G , and the same put for Radius ; the Hypotenusal F C , or F D , will be the Secant of the said lesser acute angle , ( as the lesser containing side C G or D G will consequently be the Tangent of the same acute Angle . ) Therefore it followeth ;

As F G — 10,0000000 to F C or F D — 10,0920424 . Sec. 36ᵒ.

So F G , 8.258 ●c . — 0,9168902 to F C or F D 10.2078 &c. — 1,0089326 , as before ,

Or fifthly and lastly , in the Isoskeles C F D , where all the angles and the Base C D ( as the side of the Pentagon ) being known ; the side of the Isoskeles F C or F D sought for , will be obtained by the most common Trigonometrical operation , thus ;

As C F D , 72º — 9,9782063 . S A. to C D , 12 — 1,0791812 .

So C D F , or D C F , 54º — 9,9079576 , S A to C F or D F , 10.2078 &c. — 1,0089325 , as before .

Now the Radius of the circumscribing Circle , viz. F C or F D ( to which A F is equall ) being these five severall ways found out , to be 10.21 ferè ; the same with the Radius of the inscribed Circle , viz. F G found before , 8.26 ferè , will make up the totall Diameter or Perpendicular of the Pentagon , viz. A G ( which is the altitude thereof ) to be 18.47 ferè ; which according to the accompt of Ramus , and the other Auth●rs before named , would be exactly 18. For that here you may observe by the way , how that in all such ordinate Planes , as have their angles and sides in an uneven or unequall number ( and so the angles and the sides severally , are not exactly opposite one to another , that is , angle to angle , and side to side , but contrarily , ( as the Trigon , Pentagon , Heptagon , Hennea-gon , and the like ; the Diameter is composed of the semidiameters of the Circle circumscribing and inscribed : And in all such ordinate Planes , as have their angles and Sides in an even or equal number , ( and so the angles and sides severally , are directly opposite one to another , that is , angle to angle , and side to side ) as the Tetragon , Hexa-gon , Octa-gon , Deca-gon , and the like ; the Diameter is no other then that of the circumscribing Circle . And so by these Diameters may the areal contents of these Figures be obtained artificially as by their sides , according as I noted at first : For so the artificial Line of measure for squaring of an ordinare Pentagon by its said Diameter or Perpendicular , will be found ( by the reason of the foregoing Theoreme , &c. ) to be of the prime Rational Line in general , 1.1732 ferè . And from The artificiall Line in general for the quadrate dimension of an ordinate Pentagon by it's Diameter or Perpendicular of altitude . the premisses it appeares , that if the Diameter of a Circle be measured by any of the artificial Lines pertaining to the Diameters of the latter kind of ordinate Planes here mentioned , the Quadrat thereof shall be equal to the respective inscribed Figure ; where , by the Diameters , are meant their angular Diameters , or longer Diagonals passing between two extreamly opposite angles , through the Center of the Figure ; where as there is to be understood another Diameter , passing beween two opposite sides , through the Center , ( and to which is equall in the Hexagon the line passing between the ends of the two opposite sides , and so subtending the Angle of the Figure , and which is more peculiarly and specially called the Diagonall-line , or Diagonie of the same Figure ) which is no other , then the Diameter of the inscribed Circle ; And so by this other Diametrall line , may the said Figures be also artificially measured as before , by Lines of Measure appropriated thereunto ; And therefore , if the Diameter of a Circle be taken thereby , the Quadrat thereof shal be equal to the respective circumscribed Figure : And so also may these ordinate Polygonal or Polypleural Planes in general , be artificially measured by their shorter Diagonal-lines , which subtend their angles singly , ( and are equal , neither to the diameters of their Circles circumscribing , nor inscribed , nor composed of both ) & are more peculiarly called their Diagonials or diagonies ; and which is in the Hexagon , the Side of its inscribed Trigon ; in the Octagon , the side of the inscribed Tetragon ; & in the Decagon , the side of the inscribed Pentagon , &c. And which sort of line is drawn and handled ( amongst others ) in the next Pentagonal figure , set forth for the dimensional proportions in a Pentagon . And so the artificial Line for the quadrate dimension of an ordinate Pentagon by its said diagonie , or angular Subtense , will be of the prime Rational Line , ( in the general Reason of Measure ) 1.2336 ferè .

geometrical diagram

And thus having demonstrated the dimensions of the three first ordinate , Planes , geometrically , in every respect : we she next of all deliver the same purely arithmetically , by way of proportion , as we have extracted them in all the variety thereof that may be ; some of which may be of singular use , not only for the more easie and speedy superficial dimension of these Figures simply , according to the natural Measure ; but also for the solid dimension of the Pyramids raised or constituted upon them ( and so consequently of the soresaid five plain , or rectiline regular Solids , after the natural and vulgar way of measuring them , which is the most difficult ) as to the speedy discovering of the semidiameters of the Circles circumscribed to their Bases ; and so thereby , and by the side of the Pyramid together , the Axis or Altitude of the same ; as I shall shew in the next Section ; like as I did before , for finding the Axis of a Cone . And first therefore for Linear Proportions ( as I may term them ) or Proportions of Linear dimension , in the ordinate or regular Trigon.

1 The Side ( a , e or a , ● ) is to the Raie of the Circle , circumscribing ( a. o. ) as 1. to .57735 √ q 1/3. inscribed ( o. u. ) .288675 . And so to the diameter or Perpendicular ( a. u. ) as 1. to .8660254 √ q 3/4 or . 75. And vice-versâ , the Perpendicular ( a , u ) is to the Side ( a , e , or a , i ) 1.1547 .

geometrical diagram

2 The Radius of the circumscribing Circle ( a , o ) is to the Side ( a , e , or a i , ) as 1. to 1.73205 √ q 3. Raie of the inscribed Circle ( o , u , ) 0.5 dupla . And consequently to the Diameter or Perpendicular , ( a , u ) as 1. to 1.5 subs●squi ●●tera .

3 The Radius of the inscribed Circle ( o , u ) is to the Side ( a , e , or a , i ) as 1. to 3.4641 Ray of the circumscribing Circle , ( a , o , ) 2. subdupla . And so to the Perpendicular or diameter ( a , u ) 3. sub●ripla .

Whereby it appeares , that the Radius of the circumscribing or containing Circle , is 2/3 of the Trigonal Perpendicular or diameter , and the Radius of the inscribed or contained Circle is 1/3 of the same , and which is farther manifest by E. 14 p. 18.

Secondly , for superficiall Proportions in the ordinate Trigon ; or Proportions of superficiall dimension :

4 The Quadrat of the Side is to the Trigon it selfe , as 1. to .4330127 Diameter .57735 . √ q 1/3.

Which latter , is the same with that of the side to the Raie of the circumscribing Circle .

And contrarily ;

5 The totall Trigon is to the Quadrat of its Side as 1. to 2.309401 Diameter , 1.73205 . √ q. 3

And so consequently .

6 The Side , is to the side of the Quadrat equall to the Trigon , as 1. to .658037 . Diameter , .759836 ferè .

Next for the like proportions in the ordinate Pentagon ; and first in respect of Linear dimension ;

1. The side of the Pentagon ( a. b. ) is to the Ray of the Circle circumscribing ( a. g ) as 1. to .85065 . inscribed ( g. h. ) .68819 . And so to the Diameter or Perpendicular of the Pentagon ( a. h. ) as 1. to . 1.53884 . Diagonall , or the Subtense of the pentagonall angle ( b. e. ) * * Flussat . El. 16. p. 2. cutting the Diameter by extream and mean proportion ) as 1. to 1.618034 ferè .

4. The Radius of the ambient Circle ( a. g. ) is to the Side of the Pentagon , ( a. b. ) as 1. to 1.17557 Ray of the inscribed Circle , ( g. h. ) .809017 ( And so to the Pentagonall Diameter ( a. h. ) as 1. to 1.809017 . ) Pentagonall Diagonie , or angular Hypotenuse ( b. e. ) as 1. to 1.902113 .

geometrical diagram

3. The Radius of the inscribed Circle ( g. h. ) is to the Side of the Pentagon , ( a. b. ) as 1. to 1.453085 Ray of the ambient Circle , ( a. g. ) 1.236068 ( ferè . ( And so to the Diameter or Perpendicular ( a. h. ) as 1. to 2.236068 ) Diagonal or Hypothenusal ( b. e. ) as 1. to 2.351141

4. The Diameter of the Pentagon ( a. b. ) is to the Side ( a. b. ) as 1. to . 649839. Ray of the Circle ( of which two circular Raies it is composed . ) circums . ( a. g. ) as 1. to .552786 inscrib . ( g. h. ) .44721 Pentagonall Diagony ( b. e. ) as 1 to 1.051462 .

And to its greater portion or segment ( f. h. ) ( being parted by the said Diagonall or Hypothenusall , according to extream and mean reason , as aforesaid ) as 1. to .618034 , and to its lesser portion● ( a. f. ) as 1. to .381966 : which two segments are Algebraically or Cossically , the irrationall Apotomies , √ 1 1/4 ( or 1.25 ) — 1/2 ( or . 5 ) for the greater segment ; and 1 1/2 ( or 1.5 ) — √ 1 1/4 ( or 1.25 ) for the lesser segment ; according to a Cossicall invention of the said two segments , agreeing exactly with the Trigonometricall .

geometrical diagram

5. The Diagonall or Hypotenusall ( b , e ) is to the Side , ( a , b ) as 1. to . 618034. Ray of the Circle circumscrib . ( a. g. ) as 1. to .525731 inscribed , ( g. h. ) .425325 And so to the Diameter , ( a , h ) as 1. to .951056 . And to the greater segment thereof ( f , h ) ( as being divided by the Diagonall , according to extream and mean reason , as aforesaid ) as 1. to .587785 And to the lesser segment of that division or partition , ( a , f ) as 1. to .363271 .

Which two Diametrall segments , are Cossically the irrationall Apotomies √ 1.130635 &c. — 0.475528 , for the greater segment ; and 1.426584 &c. — √ 1.130635 &c. for the lesser segment , according to a Cosficall computation of these two segments , which we find to agree exactly with the Trigonometricall , as the former .

Which Pentagonall proportions before-going , ( those of the Diagony being secluded ) would be according to the Dimension of the foregoing ordinate Pentagon , by Ramus &c. exactly rationall in the least terms , thus ;

1. The Side of the Pentagon to the Ray of the Circle circumscribing , as 6 to 5 , sesquiquinta , ( which is decimally irrationall , 1. to .833333 infinitely , & defective . ) inscribed , as 3 to 2 sesquialtera , ( which is decimally irrational , 1. to .66666 &c. infinitly , and defective . ) And so to the Diameter , as 2 to 8 , subsesquialtera ( which is decimally irrationall 1. to 1.5 , ( or integrally , rationall , 10 to 15 , defective . )

2. The Radius of the circumscribing Circle to the Radius of the inscribed Circle , as 5 to 4 , sesquiquarta , ( which is decimally irrationall . 1. to .8 ( or integrally , rationall , 10 to 8. deficient . And consequently to the Pentagonall Diameter , as 5 to 9 , sub-superquadriquinta , or subsuperquadrapartiens-quintas , ( which is decimally irrationall , 1. to 1.8 ( or integrally , rationall , 10 to 18 deficient . )

And so the Radius of the inscribed Circle , to the said Diameter , as 4 to 9 , subdupla-sesquiquarta , ( which is decimally irrationall , 1. to 2.25 , ( or integrally rationall , 100 to 225 ) and excessive or redundant . )

By which the reciprocall proportions may be had by inversion of the Termes : And here the proportion of the Pentagonall Diameter to the side , wil be the same with that of the side to the Raie of the inscribed Circle , before noted , viz. 3 to 2 sesquialtera .

Then secondly in relation to superficiarie Dimension , the Pentagonall proportions will be exactly , as followeth .

1. The Quadrat Laterall is to the Pentagon it selfe , as 1. to 1.720477 Diametral 0.726543 ferè Diagonial 0.657164 ferè

And contrariwise

2. The Pentagon is to the Quadrat Laterall as 1. to .581234 Diametral 1.37638 Diagonial 1.52169

And so ,

3. The Side is to the Side of the Pentagonall Quadrat , as 1.10 1.31167 . Diameter .85237 . Diagonial .810656 .

Which three last Proportions are the most precise and proper Tetragonismal termes of an ordinate Pentagon ; but chiefly the first of them .

And which superficiall Proportions ( secluding those sor the Diagony ) would be , according to the former Linear Proportions , deduced from the foregoing Pentagonall dimension of Ramus , &c. in these Terms ;

1. The Quadrat Laterall to the Pentagō it selfe , as 1. to 1.666666 infinitely , & defective ; being in vulgar terms , as 1. to 1 ●/● , or 5/1 sub-super-bitertia , or sub-super-bipartienstertias . Diametrall . .740740 , infinitely , & excessive ; being more vulgarly , 1 to 20/27 , superseptu-partiens-vigesimas ; which is very neer sesqui-tertia in the least terms , it being superpartient 7/20.

Contrarily ;

2. The Pentagō to the Quadrat Laterall as 1. to .6 ( or integrally , 10 to 6 ) excessive ; being more vulgarly , 1 to 3/5 , super-bitertia , viz. superpartient 2/3. Diametrall 1.35 , ( or integrally , 100 to 135 , ) defective ; being in vulgar terms , as 1 to 17/2● , sub-super-septupartiens-vigesimas .

And so ,

3. The Pentagonall . Side to the Side of the Pentagonall Quadrat , as 1. to 1.29099 , desicient . Diam . .86066 , redundant .

But the former or Linear Proportions are erronious ; Therefore , the latter or superficiall .

Then as for the Proportions of dimension in the Tetragon ; there be onely these two considerable ; viz.

1. The Side is to the Diagony ( or circumscribing Circle's Diameter ) as 1. to 1.4142 , &c. √ . 2. ( which is the generall Linear or Scalar Number , for Diagonial Quadrature , noted before )

geometrical diagram

And Vice-versâ .

2. The Diagony is to the Side , as the diameter of a Circle to the side of it's inscribed Quadrat , noted formerly in the dimension of a Circle ; viz. 1. to .7071 , &c. √ 1/2 :

And so is the side to the semi-diagonie , ( or the circumscribing Circle's semi-diameter . )

And here , as the side of the Tetragon , is the Tetrachord , or the Chord tetragonall of the circumscribing , comprehending , or containing Circle , ( as subtending a Quadrant of it's Peripherie ) So is the Diagony , the like Chord of the Circle described out of some angular point of the Tetragon , according to the side thereof , as the Radius ; viz. the Circle described about the diagoniall Quadrat ; the said Diagony being then the Side of the Tetragon inscribed ; And so those Circles are Tetrachordall .

And here we may take in by the way , another kind of tetragonall Plane , very variable in respect of it's angles , but regular-like , ( as I may term it ) being equilaterall , though notequi-angular , called Rhombus , being the onely obliquangle equilaterall Concerning the dimension of a Rhombus . Parallelogram , and which therefore is a Quadrat , as it were , dislocated in it's Terms , or compressed in the angles , as Ramus speaketh lib. Geomet . 14. el. 7 , 8 , and so is a Triangulate , consisting as it were , of two equall and like isoskelan Triangles , meeting upon one common base ; And if the Triangles happen to be equilaterall , equiangular , or exactly regular ; then will the Rhombus be artificially in the nature of an exact Quadrat , as that kind of Triangle is , and so all other regular Planes artificially are : and so will admit of the like artificiall , or quadrate dimension , by it's side , ( or by either of the Diagonall-lines which may be drawn in it , and so by which it may be resolved into 4 equall and like rectangle Scalenons ; the lesser or obtuse angle-diagonall , which is the common base of the said two Isopleurons , being equall to the side of the Rhombus , as being a side of it's composing ordinate Triangle ) according to an exact Quadrature . And the artificiall Line for this particular Rhomball Dimension , will be ( according to the reason of the foregoing Theoreme , &c. ) of the rationall Line in generall . 1.0745 , &c.

geometrical diagram

And by the artificiall Line for the dimension of Triangles in general , may the Area of any Rhombus be obtained , it's two diagonal-lines being measured thereby ; for so the rectangle Parallelogram resulting therefrom , shall be equall to the Rhombus ; as the Rectangle of those two lines , is naturally and really double to the Rhomb ; or the Rhomb is half the Rectangle made of those two lines , being measured by the naturall Line of measure ; viz. the Rectangle from one whole Diagonall infolded with half the other . And thus also may be artificially obtained the Area of any Rhomboides or Trapezium .

SECT. II. Setting forth the Dimension , both solid and superficiall , of regular-based Pyramids in generall , and their Compounds : And demonstrating the same particularly in the three first kinds of them .

ANd now having sufficiently shewed our artificiall Dimension in the three first rectiline or angular ordinate Planes in particular ; namely the Trigon , Tetragon and Pentagon , simply in themselvs , ( but chiefly the Trigon and Pentagon , as being in them , only requisite ) and so consequently the like Dimension of all ordinate polygonall or polypleurall Planes whatsoever , by the same metricall reason : We shall next proceed to the like kind of dimension in them , as in order and relation to all Pyramidall Bodies , both prime or simple , and compound , ( as I may so speak ) or Pyramids , and Pyramidates , as being their Bases : or the dimension of these Solids , being founded , ( as it were ) constituted , or erected upon such Planes ; and so denominated from them accordingly , as I have said before . And this I shall particularly shew in the three first sorts of Pyramids , constituted or raised upon the three first Planes before named ; and more especially for that , as those three Planes do concurre superficially , to the composition , ( or to the superficiall composition ) of the five famous ordinate Bodies , or rectiline regular Solids , as I said before ; namely , the Trigon , to the Tetrahedron , Octahedron , and Eicosahedron ; the Tetragon to the Hexahedron , and the Pentagon to the Dodecahedron : so the three kinds of Pyramids erected or constituted upon them , do concurre in like manner solidly , to the composition , ( or to the solid composition ) of the said five Bodies , as I shall shew particularly in each of them .

And seeing that of all the kinds of Pyramids ( which may be as infinite in number , as the Figures , for their Bases , upon which they are raised and constituted , and so from which they take their special denomination , as whether the same be trigonall , tetragonall , pentagonall , or howsoever polygonall , and so the respective Pyramids be denominated accordingly ) there is but one kind exactly ordinate or regular , and so is specially and peculiarly , ( for the excellency thereof ) called Tetraedron , and by Euclid , simply by the name of Pyramid , in E 13. p. 13 ; it consisting of four equall ordinate Trigons compact together by solid angles ( by E 11. d. 26 ) which therefore are in number subtriple the plain , superficial , or Trigonal angles constituting the same , ( so that to the constitution of one solid angle , do here concurre three superficiall angles ; and therefore this solid angle is contained under two plain right angles precisely , and so is 2/3 of a solid right angle , as it's composing or basiall angle is 2/3 of a plain right angle ) and so the angular lines , or sharp edges , called the sides of the Pyramid , ( made by the connexion of the sides of the ordinate trigonall Planes ) are in number , subduple the trigonall sides constituting the same ) And so any one of the said Trigonall Planes may be put for the Base of this Pyramid , and thereupon the solid angle opposite thereunto , made by the inclination , connexion , or concursion of the other three like Planes , ( according to their verticall angles ) shal be the top or verticall point of the same ; between which , and the Center of the Base , shall be adjudged the perpendicular altitude , ( or the Axis ) thereof : We shall first therfore shew our artificial dimension ( both solidly & superficially ) of the first kind of Pyramid , in the first of the foresaid five regular Bodies , namely the Tetrahedron ; and that by the Side of the Base , ( which is the generall side of this Body ) in answer to the foregoing quadrate dimension of an ordinate Trigon by it's side ) and the Axis together ; which we shall compare with the naturall or vulgar dimension , according as we have done in all the Figures beforegoing ; whereby the same may withall be understood by such as are yet to learn .

geometrical diagram

And though a Cone do somewhat resemble a Pyramid , and a Cylinder , a Prisme ; yet a Cone cannot properly be called a Pyramid ( as some do call it ) nor a Cylinder a Prisme ; they being not plain Solids , rising from a rectiline or angular Base ; but various , or variable gibbous Solids , rising from a curviline , obliqueline , or circular Base ; but yet a Cone may be understood to comprehend in it's selfe , any kind of Pyramid ; and so a Cylinder to comprehend any kind of Prisme ; their Bases being the Circles circumscribed to the Base of the Pyramid or Prisme , and so their whole Bodies circumscribed to the whole Pyramid or Prisme , viz. their concave superficies just touching the angular lines , or the sides of the respective Pyramid or Prisme inscribed . And contrarily , may any Pyramid be understood to comprehend or include a Cone , & any Prisme a Cylinder ; the rectiline Planes of the Pyramid and Prism , just touching the convex Superficies of the inscribed Cone and Cylinder , ( and so the angular or rectiline base of the Pyramid and Prism , compleatly circumscribed to the circular base of the Cone and Cylinder , ) according to the nature of Mathematical Inscription and Circumscription , as you may see it fully set forth in E. 4. d. 1 , 2 , 3 , 4 , 5 , 6 , for Superficies ; And E , 11. d. 31 and 32 , for Solids .

But now to the main thing in hand , to wit , the solid dimension of the foregoing Pyramid , according to our artificial and compendious way ( as before in the Cone ) and here , by the side of the Base , & the Axis , or line of altitude ; the artificial Line of Measure The artificiall Line for the solid dimension of a Trigonall Pyramid by the side of it's base and its Axis or Altitude together . for which purpose , I find ( according to the reason of the 3d. Theoreme , &c. following ) to be of the prime or natural Rational Line in general , 1.9064 , ferè ; which being duly set off therefrom ; the side of the aforesaid Tetrahedrum's Base ( put naturally 12.00 ) will be found thereby , 6.29 , ( in a centesimall partition ) whose Quadrat is 39.5641 , for the artificiall base : And the Axis or Altitude of the Tetrahedrum , found geometrically before , 9.7979 , &c. ( which according to a centenarie solution of the unit , or of the Rationall Line , is 9.80 ferè . ) will be found by the foresaid Line , 5.14 , for the artificiall Axis ; which being wholly infolded with the whole Base , will produce the rectangle regular-based Prisme , or Parallelipipedum , 203.359474 , sor the artificial solidity of the Tetrahedrum ; which differeth from the true , natural , solidity , viz. 203.646753 , ( by way of defect ) not so much as 1/3 of the prime Rationall Line cubed , or 1/3 of a cube-unit . And by a further solution of the two Lines of Measure , ( viz. the naturall and artificial Rationall Line ) the difference will be found much lesse , according to what I have said , and also plainly demonstrated in all the precedent Dimensions .

Now for the superficiarie dimension of this kind of Pyramid ; if it be exactly Concerning the superficiall dimension of atrigonal Pyramid . ordinate , as the Tetrahedron , then one of the Planes being had , the whole Superficies is easily had , by the quadruplication of that Plane : As the Plane of the foregoing Tetrahedrum , being found by the natural Dimension , 62.3538 ; the whole superficies wil be 249.4152 : And which may be artificially obtained by either of the Lines for the quadrate dimension of an ordinate Trigon ; but most readily and properly , by that for the side . As the side of this Tetrahedrum being 12 , wil be found by the Line of Lateral quadration of a Trigon , 7.90 ferè ( as before the side of the Trigon simply ) and so the Square thereof , 62.4100 ferè , for the Plane of the Tetrahedrum ; and the quadruple of this , is 249.6400 ferè , for the total superficies thereof ; which exceedeth the former , or true superficies , not so much as 1/4 of a square-unit , or Integer .

geometrical diagram

Now for a triall of this dimension artificially , and that by the foresaid perpendicular-line , and the Side of the base of the Pyramid together , whereby the rectangle Plane made of them , shall agree with the rectangle Plane made naturally of the semi-perimeter of the Base , and the said hedrall perpendicular , for the true Pyramidall superfice , viz. without the proper base : The artificiall Line of Measure specially serving hereunto , I find ( according to the reason of the 3d. Theoreme , &c. ) to be of the Rationall Line in generall , deficiently , 0.8165 ferè , ( which is Apotomally in Number , The artificial Line for the superficial Dimension of a trigonall Pyramid , by the Side of its Base & the perpendicular-line of its laterall trigonall Plane , ( or the altitude of the same ) together . according to our generall reason of Measure , 1 — .1835 ferè ) which being set off from the said Line , and the side of the foresaid Pyramid's base measured thereby ( in a centesimall partition ) will be found , 14.70 ferè ; and the Perpendicular being also measured by the same Line , will be found , 12.73 . ferè ; which two being multiplied together , there will result the rectangle Parallelogram , 187.1310 ferè , for the Pyramidall superfice , before-mentioned ( viz. without the Base ) which exceedeth the like superficies formerly found most truly , 187.0614 , not fully so much , as wil make in vulgar terms , 1/14 of a square-unit .

And thus much for the Dimension both solid and superficiall of the first kind of Pyramid in generall , whether ordinate or inordinate , so as the Base be ordinate : And so in speciall , of the first of the five ordinate plain Bodies , namely , the Tetrahedron ; according to a Pyramidall dimension only .

As for the first kind of Pyramidate , called in generall Prisma , which I have mentioned before in the first Part ; I shall afterwards speak a little of the like dimension thereof , as is of a Pyramid it selfe .

geometrical diagram

But now the whole difficulty of this Dimension of the Icosahedrum , ( and ●o of the other two soresaid Bodies ) consists in the investigation of the Axis , or line of altitude , of the compounding Pyramid ( as it did before in the Tetrahedron ) if the same be inquired geometrically ( as Geometricians speake ) to wit , by first having the side of the ordinate Solid only , and so coming at length , after many tedious and troublesome operations , both arithmetical and geometrical , to the said perpendicular-line of altitude , ( which is no other then the Radius of the Spheare inscribed within the plain Solid ) which way therefore I shal here pretermit , as being needless for me to demonstrate ; the same being shewed by divers practicall Authours , especially Ramus and Clavius in Latine ; and more abundantly in English , by our Countrey-man , Mr. Digges long since , in his learned Discourse of geometrical Solids , annexed to his Pantometria , as a part thereof ; to which Authours I therefore refer the diligent practizer for a full satisfaction in this poynt ; my intention in this place , being only to bring in our new or artificial way of Pyramidal Dimension , in the aforesaid ordinate Bodies , for the more easie and speedy obtaining of their solidities in a Pyramidal way , the Hexahedron being excepted . And therefore I shal here only shew , how , instrumentally or mechanically to get the said Pyramidall Axis or altitude , by getting first the altitude of the whole body , ( which 〈◊〉 no other then the total Dimetient of the inscribed Spheare ) being according to what I shewed before , for getting the altitudes of Cones and Pyramids in general . Therefore , if from the superiour Plane of the Icosahedrum , being produced or extended , that is , from the inferiour or interiour superfice of any Plane placed upon the uppermost Plane of the Icosahedrum , a perpendicular be let down to the inferiour or opposite Plane or Base thereof , in like manner produced , that is , to the superiour or exteriour superfice of a Plane placed under the Icosahedrum ( and so , upon which the same lyeth ) the said perpendicular-line ( being accurately measured ) shall give the altitude of the whole Icosahedrum , whose half shall be the Axis or Altitude of the composing Pyramid sought for .

So that the great difficulty in the dimension of the Icosahedrum , arising by the foresaid geometrical investigation of this Pyramidall altitude , is by this means quite taken away , and the thing made very easie . Ane therefore whensoever you would measure an Icosahedron ( or a Dodecahedron , for the same reason holds in it for the altitude ) it is best to use this way ; for so the solidity of the same wil then be obtained with little labour , especially according to our foregoing artificial pyramidal Mensuration ; For the said Axiscr altitude of the Icosahedron's cōposing Pyramid , being taken by the foresaid artificial Line for the solid dimension of a Trigonal Pyramid , and the same be tiplied into the Quadrat of the side , taken by the same Line ; the product shal be the solid content of the said compounding Pyramid ; whose vigecuple wil be the solidity of the total Icosahedrum . But how , readily to obtain the Axis of the compounding Pyramid , both of this , and also the other ordinate bodies ( where need is ) I shall afterwards shew among the dimensional Proportions in these Bodies ; by first having the side of the Body , and which is very easily taken , without any trouble at all .

And so the totall superficies of this Solid , wil readily be had by the Line for the Lateral quadrature of an ordinate Trigon ( according as I shewed before in the superficies of a Tetrahedron ) for the side thereof being measured thereby , its Quadrat wil be the area of one of the bases or Planes , whose vigecuple wil be the total Icosahedral superficies .

geometrical diagram
geometrical diagram

And if the solidity of an Octahedrum be immediatly requiree ; then the total Axis or Diagony thereof ( which is double to the Axis of the foresaid Pyramid , and so no other then the Diagony of it's base , or the Diameter of the Circle circumscribing the same , and also the Axis or Diameter of the Sphear circumscribing the Octahedron ) being infolded with a trient of the compounding Pyramids base , ( which is no other then the Octahed●on's Lateral Quadrat , as was shewed before ) or the whole base wi●h a trient of the said Axis or Diagonial ; there wil immediatly result the solidity of the Octahedrum : For so the true Axis or Diagonial of the foresaid Octahedrum , 16.97 , &c. sufficiently produced by Radical extraction , or otherwise , viz 16.97056274 , &c. being infolded with a trient of the aforesaid pyramidal base , viz. 48. ( or the whole base 144 with a trient of the axis , viz. 5.65685424 , &c. ) there , wilimmediatly result the true total Octahedral solidity , 814.587012 as before . And so the total axis or Diagony of the Octahedrum , found by the former artificial Line , ( in a centesimal solution ) 11.77 ferè , being infolded with the foresaid total common base of the two compounding Pyramids , produced by the same Line , 69.2224 ; there wil also immediatly result the solidity of the Octahedrum , 814.747648 ferè , which comes much neerer the true solidity , then the former dimension ; this differing there from ( now by way of excesse ) not so much as 1/6 of a cube-unit , or integer , of the appointed measure ; which Octahedral solidity is exactly quadruple to the foregoing Tetrahedral solidity , these two Bodyes having here one and the same side in measure . And here therefore it appears in brief , that if the total Axis or Diagonial , and the Side , of an Octahedrum , be taken by the foresaid artificial Line of measure for a Tetragonal Pyramid , and the Quadrat of the Side be augmented by the Axis ; the resulting rectangle regular-based oblong Prism , or Parallelepipedum , shal compleatly contain the solidity of the Octahedrum .

As for the Diagonial , Axis , or Altitude of the Octahedron , the same may be also obtained instrumentally or mechanically , according to what I shewed in the Tetrahedron , and Pyramids in general ; and also for the altitude of the Icosahedron , and Dodecahedron ; the altitude of this Body being considered according to a perpendicular-line comprehended between its two opposite angles , for as much as it is composed of two equal and like quadrangular Pyramids joyned together in their bases , as I said before .

As for the superficiarie dimension of this kind of Pyramid which now we have in hand , the The superficiall dimension of a tetragonal Pyramid ; And the artificiall Line for performing the same by the Side of the base , and the perpendicular line of the trigonall Plane together . same may be most readily performed in the same artificial manner as that of a trigonal Pyramid , to wit , by the side of the Base , and the perpendicular-line of it's triangular Plane together . And the artificial Line of Measure for this purpose , I find ( according to the reason of the same Theoreme ) to be of the Rational Line in general , 0.7071 , &c. √ 1/2 , which is Apotomally in Number , as 1 — .2929 ferè ) which Line being set off there from , and divided in a due manner , and then the two forenamed Lines of the Pyramid commensurable by the same , be accordingly measured thereby , the product arising by their mutual multiplication , shal be the superficies of the Pyramid , ( to wit , without the base ) agreeing with that which is produced naturally , by the semiperimeter of the base , and the foresaid trigonal perpendicular multiplied together , according as I fully demonstrated before in the superficiary dimension of the trigonal Pyramid . As in the foregoing tetragonal Pyramid , the side of the base being 12 , the perimeter thereof wil be 48 , and so the semiperimeter 24 : & the side of the Pyramid , or of its trigonal Plane , being the same with the side of the base , the perpendicular of the said Plane , wil be 10.39 , &c. as was shewed in the superficiarie dimension of the Tetrahedron , and before that , in the dimension of the Trigon alone ; which two multiplied together , ( the said perpendicular being further produced , as formerly ) there wil arise 249.4153 , for the superficies of this Pyramid , without the base ; agreeing with the total superficies of the foregoing Tetrahedrum , the Planes of that & this , being all one . Now the side of the base of this Pyramid ( naturally 12.00 ) being measured by it's proper Line , for superficial measure , wil be found artificially , the same that the Diagony or Diameter of its base is naturally , viz. 16.97 , &c. and the foresaid perpendicular wil be found by the same Line , ( in a centesimal partition ) 14.70 ferè ; which two multiplied together , do produce 249.4590 ferè , for the Pyramidal superfice aforesaid ; exceeding the true superfice , only so much as 1/23 of a square-unit or integer , it being decimally , .0437 , or 437 of 10000.

And thus may the superficies of an Octahedrum be obtained , it being only double the superficies of its compounding Pyramid : For so , the superficies of the foregoing Octahedrum , wil be by this latter or artificial Measure , 498.9180 ferè . And therefore if the side of an Octahedrum , and its hedral perpendicular , be taken by this artificial Line , and one of them be doubled ; the Rectangle Plane made thereof , wil be the total superficies of the Octahedrum : And so the hedral perpendicular of the foresaid Octahedrum being taken by this Line , and doubled , wil be 29.39 . which multiplyed by the side of the Octahedrum , found by the same Line , 16.97 ; the Plane produced therefrom , wil be , 498.7483 , for the superficies of the Octahedrum ; which comes a little nearer the true superficies , then the former ; that exceeding the same only , .0874 ferè , which in vulgar terms , is not fully 1/11 ; and this wanting thereof , but .0823 , which in vulgar account , is hardly 1/12 ; the true superficies being 498.8306 , ( double to the superficies of the foregoing Tetrahedrum : ) so that the side and hedral perpendicular of an Octahedron being taken by the prime Rational Line , and doubled ; the Plane arising from their mutual multiplication , shal be the true superficies of the Octahedrum ; being no other then that of the basial semiperimeter , and the trigonal perpendicular of its tetragonal compounding Pyramid , doubled ; As before , the double product of 24 and 10.39 , &c. ( or 12 , and 20.78 , &c. ) being now the single product of 24 , and 20.78 , &c. Or the side being quadrupled , and the said perpendicular taken single , shal together produce the same ; being no other then that of the whole basial perimeter of the foresaid Pyramid , and the perpendicular of its triangular Plane together , for the double superficies of the Pyramid ; which is the supersicies of the Octahedrum : As heer the product of 48 , and 10.39 , &c. Or the superficies of this Solid , may be had againe artificially by the Lines of quadrature pertaining to the Trigon , as I shewed before for the superficies of a Tetrahedrum ; for so , one of the Planes or bases being had ; the Octuple thereof shal be the total Octahedral superficies .

geometrical diagram
geometrical diagram

Or again , seing that the rectangle Solid ( Prisma or Parallelepipedum ) contained under the Perpendicular from the Center of any regular plain Body , to any of it's bases or Planes , and a trient of the total superficies , comprehends the solidity of the whole body ( according to what I shewed formerly in the dimension of a Sphear , for the producing of its solid content by the semidiameter , and a trient of the Superficies , the plain Solid arising therefrom , being equal to the spherical Solid : or which is all one , the plain Solid made of the whole Diameter , and a sextant of the spherical superficies , ) in as much as that which is contained under the said Perpendicular , ( which is here no other then the Axis of the compounding Pyramid , and the semi-axis of the inscribed Sphear ) and a trient of one of the bases or Planes ( which is the base of the compounding Pyramid , as aforesaid ) comprehends the solidity of one of the compounding Pyramids ; and so consequently that which is contained under the said Perpendicular , and a trient of all the bases or Planes together ( as being the bases of all the compounding Pyramids together ) must needs comprehend the solidity of all the Pyramids together , and so of the whole ordinate Body ; ( as the rectangle Plane contained under the perpendicular from the Center of any rectiline regular Plane , or superficial Figure , to any of it's sides , and the semi-perimeter of the same , comprehends the Area of the whole Figure ( as I shewed formerly in the Pentagon , and which answereth to that of the dimension of a Circle , for the producing of it's Area by the semidiameter , and semi-periphery , the rectangle Plane or Parallelogram resulting therefrom ( by their mutual implication ) being equal to the Circular Plane : or which is all as one , the rectangle Plane made of the whole diameter & a quadrant of the Periphery , or of the whole Periphery & a quadrant of the diameter ) in as much , as that which is contained under the said perpendicular , ( which is no other then the perpendicular of altitude of the Figure's compounding Trigon , and the Radius of the inscribed Circle ) and half the side , ( as being , half the base of the said Trigon ) comprehends the supercies , or area , of one of the compounding Trigons ; and so , that which is contained under the said perpendicular and half the perimeter of the Figure ( as being half of the bases of all the compounding Trigons together ) must needs comprehend the superficial Content of all the said Trigons together , and consequently of the whole regular Figure it self . ) Therefore the Base of the Dodecahedrum being 61.9371 , &c. the total superficies thereof , wil be 743.2462 &c. whose subtriple , 247.7487 , &c. being augmented by the foresaid perpendicular ( or axis of the compounding Pyramid ) 6.68 , &c. there wil result the total Dodecahedral solidity , 1655.2339 , &c. exactly as before : which may be plainly seen by the subsequent Logarithmical operations , whereby these Pyramidal and Dodecahedral dimensions , are most readily and accurately performed .

Therefore first ,

Side of the Dodecahedron , or of its compounding Pyramid's Base , 6.

Again , 2ly .

Or again , 3ly .

Which our Measure found by the foresaid artificial Line of Pyramidal consolidation , falleth short of indeed , about 3 integers , or units , and an half : But then if the said Pyramidal Line of Measure be increased in its parts , by a subdecuple solution of the former ; the side of the aforesaid Pyramids base ( naturally 6.000 ) wil be found thereby , 4.985 , ferè , whose Quadrat is 24.850225 ferè , for the artificial base : and the Axis of the said Pyramid ( naturally 6.681 ) wil be found thereby , 5.551 ferè ; which two by a conjunct composition , wil produce 137.943598975 ferè , for the solidity of the Pyramid ( which now exceeds the true solidity , being 137.936158730 , hardly so much as 1/134 of a solid integer or unit ) whose duodecuple , 1655.323187700 , is for the solidity of the Dodecahedron ; which now differeth from the true solidity , being correspondently 1655.233904760 ( by way of excesse ) hardly 1/1● of a solid integer or unit .

As for the superficies of a Do●ecahedron , the same may be readily obtained by any of the three Lines of quadrature pertaining to an ordinate Pentagon , but that for the side is the most fit and proper ( though all of them wil produce the same thing ) for so the side of the Dodecahedron being taken thereby , the Square thereof sh●l be the basial or hedral area , whose duodecuple wil be the total Dodecahedral superficies . But a farre better way for the dimension both solid and superficial of this , and the other plain regular Bodies , I shal shew in the next Section , which wil be wholly taken up about the said five Bodies .

As for the superficial Dimension of a pentagonal Pyramid seing it is but the Concerning the superficial dimension of a Pentagon all Pyramid ; And withall , the artificiall Line for performing the same , by the side of the base , and the perpendicular of the trigonal Plane together . same with that , which I have fully shewed in the two preceding Pyramids , both naturally and artificially , in the most ready manner that may be ; to wit , artificially , by the side of the Base , and the perpendicular-line of the triangular or laterall Plane together : Therefore I shal not need to insist upon the same in this last Pyramid here particularly handled , by way of exemplary illustration ; But shal only give the artificial Line of measure for the performance thereof , as I find it to be from the natural Line of measure , or prime Rational Line in general , deficiently , 0.63245 , &c. ( that is Apotomally in number , from our general reason of Measure , as 1 — .36754 , &c. the said artificial Line being √ 2/5 of the natural Line , in it's power quadratick .

And thus may the Dimension both solid and superficiall of this and all other Pyramids , be performed artificially by the other dimensional lines of their Bases ( specified before in the dimension of those basial Figures simply , and by which we said , they might be also artificially , or quadratically measured ; ) as , the Diameters or Perpendiculars , and Diagonals , together with their Axes or Altitudes , for solid measure ; and with the perpendicular-lines , or altitudes of their triangular Planes , for superficial measure : And which , though it be needles , that by the side of the Base , ( with the Axis and trigonal Perpendicular ) being most ready , and also most proper , according to what I noted in the beginning of this Section , in the basial Figures : Yet having for variety of Art in this kind , not spared the pains of extracting or eradicating the artificiall Lines serving thereunto ; I thought it might not be amiss , to set them down here also ; as they are from the natural Line of measure , or prime Rational Line in generall , in a decu-millenary solution ; those for solid dimension , ( as also the other beforegoing ) being all of them , thereof redundantly ; and those for superficial dimension , all of them deficiently .

The artificiall Line of Measure , is for the Pyramid , Trigonall , in respect of the basial Diameter , conjunctly with the Axis , or Altitude of the Body , 1.7320 √ q 3 Solid measure . Tetragonal , 1.8172 √ c 6 Pentagonal , 1.6043 . Trigonall , Perpendic line , or altitude of the trigonal or lateral Plane . 0.7598 √ qq 1/3 Superfie measure Tetragonal , 0.8409 √ qq 1/2 Pentagonal , 0.7846 ferè .

And for the Pentagonall Pyramid , in regard of the Basial Diagony with the Axis , or Altitude , 1.6589 ferè . Solid measure . Trigonall perpendic . 0.8045 ferè . Superficiall measure .

All which Pyramidal Dimensions beforegoing , in a Linear , instrumental , or geometrical way , I shal next briefly expresse in an Arithmetical way , in the proportional terms following , from an ample solution of the unit , ( according as I did before in the Cone and Cylinder , ) by which , the artificial Measure may be readily produced from the natural ; or the naturall Measure be reduced to the artificiall . Therefore ,

The naturall Measure is to the artificiall● in the Pyramid Trigonal , or Tetrahedral in respect of the Axis , or Altitude of the Body , conjunctly with the Side of the Base , as 1. to .524587 Solid dimension . Diam . .577424 Tetragonal , or Pētahedral Side .693373 √ c 1/2 Diam . .550321 Pētagonal , or Hexahedral , Side .830824 Diam . .623323 Diagō . .602815 Trigonal , Altitude , or perpendicular of the trigonal or lateral Plane conjunctly with the Side 1.224745 √ q 1 1/2 Superficiall dimension . Diam . 1.316074 √ bq3 Tetragonal , Side 1.414213 √ q2 Diam . 1.189207 √ qq2 Pentagonal , Side 1.581139 √ q 2 1/2 Diam . 1.274597 Diagō . 1.243014
Place this between Page 136 and 137.

And what hath been here delivered concerning the artificial Dimension of Pyramids upon regular Bases ; the like is to be understood for the dimension ( both solid and superficiall ) of Prisms upon the like bases , by artificiall Lines of measure peculiarly appropriated and applied to them for that purpose , in regard either of any of the basiallines aforenamed , with the Axis or Altitude for solid measure , or with the same Concerning the dimension of Prisms . for superficial measure , ( seeing the Axis or Altitude and the side , is all one in right or upright Prisms , as in right Cylinders , which I shewed formerly . ) But these I shal here passe by , leaving them to the industry of the ingenious Practitioner , that shal please to exercise himself therein ; and the rather for that the artificial Lines of measure pertaining to the Pyramids , wil also serve ( if need be ) for the dimension of the corresponding Prisms , ( as I shewed formerly between the Cone and Cylinder seeing that a Prism is only triple its correspondent Pyramid , according to E , 12. p. 7. as a Cylinder is triple its correspondent Cone , according to E , 12. p. 10 , as I noted formerly ; And as the proportion of the Prism to the Pyramid is triple , for solidity ; so it is double for superficiety ; the similitude , and so the reason between a Pyramid and Prism , being the same ( in both Dimensions ) with that between a Cone and Cylinder : And so the Lines of artificial Dimension , pertaining to the trigonal or tetrahedral Pyramid , wil serve for the trigonal or pentahedral Prism ; And the Lines for the tetragonal or pentahedral Pyramid , will serve for the tetragonal , or hexahedral Prism : And the Lines for the Pentagonal , or hexahedral Pyramid , wil serve for the pentagonal , or heptahedral Prism ; and so forward : For as the Pyramid begins à Quaternario , So the Prism , à Quinario .

And by the same artificial manner of measuring ( as by the natural ) may be obtained the solid content of any oblique or inclined Pyramid or Prism upon a regular base , as readily as of a right or upright one , according Concerning the dimension of oblique or inclined Pyramids & Prisms . as I shewed formerly for oblique Cones and Cylinders ; the true altitudes of these Bodies ( not their Axes ) being considered ; seeing that every such oblique kind of Body , is equal in solidity to the right or erect body , having the same ( or being of equal ) base and altitude ; by the reason of E , 12. p. 5 and 6 , and p. 11 and 14 , and also E. 11 , p. 30 and 31 , as I noted formerly .

SECT. III. Exhibiting a more speciall and peculiar artificiall way of measuring both solidly and superficially , the four plain ordinate Bodies , or rectiline regular Solids before handled , then was before ; & this , after the most exquisite manner that may be . Together with the like artificiall dimension of the other like regular Body , severall waies .

HAving now shewed our artificial Dimension several waies , in Pyramids , and their Compounds , or Pyramidates ; and consequently of the five fore-named ordinate plain Bodies , or rectiline regular Solids , in a Pyramidal way : ( the Hexahedron excepted . ) We shal next come to shew the dimension of the said Bodies in a farre better way ; and indeed , in the most excellent artificial , and compendious manner that can possibly be found out : and that for superficial measure , with the same speed , ease and exactnes , and so in the very same manner , as that of the Circle , Trigon , God 〈…〉 , &c. And for solid measure , as that of a Cube , by the natural or vulgar way of measuring the same ( which weshal also here shew by the like artificial way with the rest ) And this only by squaring and cubing any one of their dimensional lines ( as in the Sphear ) by artificial Lines of measure convenient for the purpose : Which may indeed , I almost despaired of , in regard of the great difficulty which I found to be in the solid dimension of these Bodies , by the usual or natural way , according to their Pyramidal compositions , ( the Hexahedron excepted ) especially the the two last and greatest of them , namely the Icosahedron and Dodecahedron . But yet considering the excellency of them in their compositions , constitutions , and structures above any other solid Figures ; from whence they are called by Pappus , and other of the Greeks , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , i , e. ordinata benè ordinata ; and commonly the Pythagorean , and Platonicall Bodies , as being first invented ( as is generally supposed ) by Pythagoras , and afterwards set forth briefly by * * Plato in Timaeo , de Anima Mundi , seu Natura . Plato in the composition and fabrick of the World , as in the Heaven , and the four Elements ; and so are also called therefrom , the Cosmical or mundane Bodies ; and so that in the knowledge and understanding of the natures , properties and affections thereof , lieth as great a difficulty , nicety , and curiosity of Geometry , as may be ; in so much as that Proclus makes the singular and admiable end of the Mathematiques , to be in the knowledge of these five Bodies , in respect of their constitution , adscription , and comparation , collation , or application among themselves : And so thereupon our Countreyman Billingsley , in his learned Annotations upon Euclids Elements in English , after the 25th . defin . of the 11th . book , speaking of the dignity and excellency of the said 5 Bodies ; saith , that they are as it were , the end , and perfection of all Geometry , and for whose sakes was written , whatsoever was written in Geometry : Therefore for the avoiding and removing of all difficulties in their Dimensions , and so the facilitating of the same , as to the obtaining of their solid & superficial Capacities ( especially solid ) as easily and exactly , as of any other Figure whatsoever , and that several wayes , both naturally and artificially ; I resolved ( by Gods assistance ) to prosecute my metrical conceits and invention herein , as far as in any other Figure .

And seing that the sides of these Bodies may more readily and accurately be taken by a Line of measure , then any other their lines of dimension adscribed to them , ( and indeed very accurately , without any trouble ) as being their only natural lines of dimension , and so only apparent of of themselves in them ; ( according to what I formerly said for rectiline Planes or Superficies in general : Therefore I shal here briefly demonstrate their dimensions both solid and superficiall ( by way of example ) artificially by their sides only ; and withall , shal by the way , give artificial Lines for the like dimension of them , by some other of their dimensional lines adscribed to them . And first for solid dimension , the Lines for the Sides of these Bodies , ( or for the cubick dimension of them by their Sides ) I find , by the reason of the 2d. Theoreme , &c. following , to be of the prime Rational Line in general , ( in a decu-millenary solution , ) as followeth .

The artificial Line for solid Measure , or Line of Cubature is for the side of the Tetrahedron , 2.0396 redundant Octahedron , 1.2849 Icosahedron , 0.7710 deficient Dodecahedr . 0.5072

Which Lines being duly set off and divided , as the former , and so the Sides of these bodies measured thereby ; the Cubes thereof , shal be the solid contents of the same , according to the dimensional reason of the prime Rational Line : which I shal briefly illustrate in three of these Bodies , by the foregoing Exammples laid down in their Pyramidal dimensions , viz. the Tetrahedron , Octahedron and Dodecahedron ; to which , I shal here add the like for the Icosahedron .

geometrical diagram
geometrical diagram
geometrical diagram

IV. The side of an Icosahedron being put naturally , 12.00 , the solidity thereof wil be found by the natural Pyramidal Dimension , upon the very point of 3770 : For the Axis of the compounding Pyramid , wil be found ( by the proportion of the side of an Icosahedrum to the Axis of its inscribed Sphear , or altitude of its own body , hereafter declared ) to be 9.069 , &c. which with a trient of the base ( being the same with that of the foregoing Tetrahedrum and Octahedrum , viz. 62.3538 , &c. ) viz. 20.7846 , &c. wil produce the solidity of the compounding Pyramid , 188.498398 . ( which by the proper artificial Pyramidal Line , wil be 188.325116 , ) whose vigecuple , 3769.967960 , is the solidity of the Icosahedrum : All which you may see most accuratly produced by the several Logarithmetical operations or artificial Numerations following , according to those formerly in the Dodecahedron . And therefore first .

geometrical diagram

Again , 2ly .

Or again , 3ly .

Or 4ly . ( to come towards our present way of measuring here proposed ) the same wil be most readily produced in a cubical manner , without any trouble of Calculation , or arithmetical and geometrical operation ; and that first by comparing the Cube of the Icosahedrum's side with the Icosahedrum it self , according to the most exquisite terms of proportion , noted in the next Section ; and this only by one simple composition of Numbers , from the artificial Numeration , thus ;

Or 5ly . and lastly ; the same from thence , by finding the content of the Cube equal to the Icosahedrum according to the most exquisite Proportion of the side of the Icosahedrum to the side of that Cube or Hexahedrum , noted immediatly after the former Proportion ; ( to which doth answer exactly , our present artificial Mensuration , o● organical , or mechanical Cubation ) As ,

Which seueral Logarithmical operations , do agree wel with that Algebraical or Cossical computation of Mr. Diggs in his forementioned discourse of geometrical Solids . Probl. 14. where having Cossically cast up the solid content of this very Icosahedrum , he saith at last , that the same being reduced into rational numbers , wil fal between 3769 and 3770.

And with these several artificial Numerations , wil be found to agree very nearly , our present artificial Mensuration : For the side of this Icosahedrum being measured by its proper Cubatorie Line , ( taken in a centenary solution ) wil be found thereby , 15.56 ( as before , the side of its equal Cube , by the artificial Numeration ) which cubed , yields , 3767.287616 , for the solidity of the Icosahedrum , which indeed wanteth of the true solidity , between two and three integers of the appointed measure ; but one example is not to be regarded : But however therefore , proceeding here in a more ample or numerous solution of the Line● of measure both natural and artificial , equally ; the solidity of this Icosahedrum wil be thus artificially produced very near the true content . As if the prime or natural Line be made 1000 parts , and so also the second , artificial , or Cubatorie Line ; then the side of this Icosahedrum measured thereby , wil be found 15.564 ferè , whose Cube is 3770.193726 , &c. for the solidity of the Icosahedrum , which exceedeth the true Content , not so much as 1/4 of the prime or natural Line , cubed ; or 1/4 of a cube-integer , or unit .

Now for the like superficiary Dimension Concerning the superf●cial dimension of the 4 regular Solids before-going , by way of exact Quadrature . of these Bodies , the artificial Lines of quadrature for this purpose , I find ( according to the reason of the first and 2d. Theoremes &c. ) to be of the prime Rational Line in general ( under the former solution ) as followeth .

The artificiall Line for superficial measure , or Line of Quadrature , is for the Side of the Tetrahedron , 0.7598 √ qq. 2/● Octahedron , 0.5373 ferè Icosahedron , 0.3398 Deficiēt Dodecahedr . 0.2201 ferè

By which Lines duly set off and divided , the sides of these Bodies being measured ; their Quadrats shal● be the total superficies ; which I shal also briefly illustrate by help of the examples beforegoing in the Dimension of the Trigon and Pentagon , which I made use of formerly in the superficial Dimension of the Tetrahedron and Octahedron in a Pyramidal way .

Therefore first ; the Side of the foregoing Tetrahedrum being naturally 12.00 , the true totall Supersicies thereof was formerly found , 249.4153 , ( which by the Line of quadrature pertaining to the side of an ordinate Trigon , was found , 249.6400 ferè . ) Now the Line of quadrature peculiarly appropriated to the side of a Tetrahedrum ( not having respect to the base or Plane simply , as being an ordinate Trigon ) being centesimally divided ; the side of this Tetrahedrum measured thereby , wil be found , 15.79 , whose Quadrat is 249.3241 , for the total superficies of the Tetrahedrum , which wanteth of the true superficies in vulgarterms , only 1/11 ferè , of a square integer or unit , of the measure appointed .

II. The side of the foregoing Octahedrum being the same with that of the Tetrahedrum , viz. 12.00 ; the superficial Content thereof wil be double to that of the Tetrahedrum , ( as I shewed before ) viz. 498 8306. Now the side of this Octahedrum being measured by its proper Line of quadration ( in a centesimal solution ) wil be 22.33 , whose Square is 498.6289 , for the Octahedro●'s total Superficies , wanting of the true content , only 1/5 of a square-integer . Or the said Line being made 1000 , it wil give the Octahedron's side , 22.335 ferè , which squared , giues the Octahedron's superficies , 498.852225 ferè , which exceeds the true Content , viz. 498.830633 ferè , hardly 1/46 ( in vulgar terms ) of a square-unit .

III. The side of the former Icosahedrum being the same with the side of the Tetrahedrum and Octahedrum ; the superficies thereof wil be quintuple the superficies of the Tetrahedrum , and so double-sesquialter the superficies of the Octahedrum , ( according to what I have formerly spoken in the superficial Composition of these Bodies ) viz. 1247.0766 ferè : Now the side of this Icosahedrum , being taken by its proper tetragonismal Line , under a centesimal partition only , wil be found 35.31 , whose Quadrat is 1246.7961 , for the superficies of the Icosahedrum ; which wants of the true superficies , scarc●ly 1/3 of a square-integer . But yet the side of the Icosahedrum being measured by its said Line under a millesimal solution , wil be found upon the point of 35.314 , which quadrately , is 1247.078596 ferè , for the Icosahedrum's superficies , which now exceeds the true superficies , ( being 1247.076581 ) in vulgar terms , hardly 1/4●6 of a square integer or unit .

IV. The Side of the foregoing Dodecahedrum , being , 6 , the true totall superficies thereof , was formerly found 743.2462 , &c. Now the Line peculiarly appertaining to the side of a Dodecahedrum , for the quadrate dimension of its Superficies ( not as relating to the Base or Plane thereof simply , as being an ordinate Pentagon ) being first laid down under a centesimal partition , and the side of this Dodecahedrum measured thereby , wil be found 27.26 , whose Quadrat is 743.1076 , for the total Superficies of the Dodecahedrum , which wanteth of the true superficies , only about 1/7 of a square-integer or unit .

geometrical diagram

II. Then for the superficial dimension of this Body by its said Axis or Diagoniall ; I find the artificial Line for that purpose , to be the very same with that which was formerly found for the superficial dimension of a tetragonal Pyramid , by the side of its The superficiall dimension of the Hexahedron artificially by its Axis or Diagony , according to quadration . base , and the perpendicular of its trigonall plane together , viz. 0.7071 , √ 1/2 : By which the said Axis or Diagony being taken ; the Square thereof shal be the total Superficies of the Hexahedrum : And therefore if the Diameter of a Sphear be taken by this Line , the Quadrat thereof shal be the inscribed Hexahedron's superficies .

III. As for the solid dimension of this Body artificially , by the Diagony of the base ; the Line of measure serving thereunto , I find to be the very same with that which was formerly noted for the dimension of a The solid dimension of the Hexahedron by the Diagony or Diameter of its Base , according to Cubature , artificially . Tetragon by its Diagony , and also for the superficial dimension of a Cone by its whose basial periphery and side together , viz. 1.4142 , √ 2 : By which the basial Diagony being taken , it wil be found to agree with the side taken by the prime Rational Line ; and so being cubed , must needs produce the same Hexahedral solidity .

IV. Then for the superficial dimension of this Solid , artificially , by its said basial Diagony or The superficial dimensiō of the Hexhaedron by its basial or hedral Diagony , according to quadrature artificially . Diameter ; the Line of quadrate dimension for this purpose , I find ( according to the reason of the 1 & 2d. Theoremes , &c. ) to be of the Rational Line in general , ( defectively ) 0.57735 , &c. which is Apot●mally in Number , 1 — .42265 , &c. the said artificial Line being √ 1/3 of the natural Line , taken in its quadratique power or capacity . By which Line ( first duly divided ) the basiall or hedral Diagony being taken , its Quadrat shal be the total superficies of the Hexahedrum .

V. And so may the superficies of this Body , be wholly obtained in the like manner , by its Side ( which may be The superficial dimension of the Hexahedron artificially by its Side , according to one exact quadrature . termed the most precise and proper squaring of a Cube , as to it's superficiety , or superficial part ; as also the two former waies by the Diagonial of the Base & of the total Hexahedron it self ) and the Line of quadrature convenient for this purpose , I find ( by the reason of the foresaid Theoremes ) to be of the prime rational Line in general ( deficiently ) 0 4082 , &c. the Apotomal segment , or parts of diminution , being .5917 , &c. the Line it self being √ 1/● of the foresaid Rational Line , in its power or capacity tetragonical . By which ( first duly divided ) the Side of the Hexahedrum being measured ; its Quadrat shal be the total Hexahedral superficies ; Which way , as also that by the Diagony of the base , as they are more artificial and excellent in themselves , so also more ready , for getting the superficies of an Hexahedron , then the vulgar or natural way , by squaring the side taken by the natural Line of measure , and then sextuplating that square , as being the Base . And here it appears , that if the Diameaer of a Sphear be taken by this last artificial Line , the Quadrat thereof shal be the total convexe superficies of the circumscribed Hexahedron .

And this ordinate Body here last handled , may also be conceived or considered together with the other 4 before-going , under a Pyramidal cōposition & resolution ( as al plain Solids generally are ; even Pyramids themselves ; as I shew'd before in the first and most simple kind of all , to wit , the trigonal , and that in the Tetrahedrum it self ) according to its Bases or Planes , which are to be understood , as the bases of so many Pyramids equal and alike , by which they are only externally eminent or apparent , their whole bodies beside , being internally latent , and so do meet in their vertical angles , in the center of the hexahedral Body ; ( or of its ambient Sphear , ) This Body being composed superficially or externally of , or contained internally under , six equal Tetragons , ( according to E , 11. d. 25 ) compact together by solid ( right ) angles , which are in number , subtriple the plain , superficial , or hedral ( right ) angles constituting or including the same , as in the Tetrahedrum and Dodecahedrum . And hereupon this regular Solid wil also admit of a Pyramidal dimension , for the obtaining of its solid Area , like as the other ; which wil be most easily perform'd , seing that the Axis or Altitude of its compounding tetragonal Pyramid , is equal to half the side ( or altitude ) of its body ; and which conjunctly with a trient of the Base ( or the whole Base with a trient of that ) wil produce the Content of one of the composing Pyramids ; whose sextuple wil be the Content of all the Pyramids together , and so of the whole Hexahedrum : Or ( more briefly ) the said Pyramidal Axis conjunctly with the whole Base , wil produce the Content of three of the said Pyramids , for half the Hexahedrum : And therefore hence it is , that the side of this Body , ( which is its altitude , and so double to the altitude of its composing Pyramid ) being infolded with the Base ( that is , cubed ) produceth immediatly the Content of all the 6 supposed Pyramids together , for the ●o●al Hexahedrum .

And as the Hexahedrum ; so also may the other four ordinate plain Bodies be artificially measured , both solidly and superficially , ( or cubically and quadrately ) by their other several lines of dimension , beside their Sides , ( as I noted before ) as either by their Axes , Diameters , or Diagonials , which in the Octahedron , Icosahedron , and Dodecahedron , are no other then the Axis or Diameter of their circumscribing , comprehending or containing Sphear , as in the Hexahedron , ( as I noted occasionally in them before ) or by their Altitudes , ( which in the Dodecahedron and Icosahedron , are the same with the Diameter of their inscribed Sphear ; as also in the Hexahedron , it being there the same with the side , and in the Octahedron , is the same with its Diagoniall , or Axis , being also the circumscribing Sphear's Axis , as I shewed formerly . ) Or by their Basiall or hedrall Diameters or perpendiculars ; and in the Dodecahedron , by its basiall or hedral Diagony also , according as I shewed in the Dimension of a pentagonal Pyramid , and also of a Pentagon it self simply . The artificial Lines , ( or Lines of quadrature and Cubature ) for some of which Dimensions , that are most material , ( though indeed none of them are absolutely needful , the Dimensions of these Bodies being most easily and accurately performed by their Sides , as I have suffciently shewed before ) I shal here deliver , as I find them to be in relation to the prime Rational Line in general , under a decumillenary solution , ( according to the reason of the 2d. Theoreme , &c. ) as followeth ; beginning with solid dimension , as I have done in all the former Solids , as being the most considerable in these and all other solid Figures : the first of the two numbers belonging to each Body , noted with the letter c. expressing the quantity of the Line for cubical or solid dimension , and the second number , noted with q , the quantity of the Line for quadrate , or superficiary dimension .

The artificiall Lines of Measure , or Lines of Cubature , and Quadrature , are for the Axis of the Tetrahedron 1.6654 ferè . c. 0.6204 . q. Octahedron 1.8172 ferè . c. √ c 6. 0.7598 . q. √ qq 1/3. Icosahedron 1.4666 ferè . c. 0.6464 ferè q. Dodecahedrō 1.4215 . c. 0.6168 , ferè , q. Altitude of the Icosahedron 1.1654 . c. 0.5136 . q. Dodecahedrō 1.1296 . c. 0.4901 . q.

By the first Section of which Lines ( being duly set off and divided ) the Axes of these Bodies being taken , the Cubes and Squares thereof , shal be their several solidities and superficialities , according to the nature of the Line by which they are measured : whereby it appears , that if the Axis of a Sphear be taken by these Lines ( except those which are for the Tetrahedrum ) its Cube and Quadrat shal be the solidity and superficiality of the inscribed Octahedrum , Icosahedrum , and Dodecahedrum , according to the respective Lines of measure , by which it is taken . And so the like may be done for the solidity and superficiality of the inscribed Tetrahedron , by the Diameter of its ambient Sphear : the Lines of measure for which purpose , I find to be of the Rational Line in general , 2.4980 , for cubique dimension , and 0.9306 , &c. for quadrate dimension .

By the second Section of these Lines , the Altitude of an Icosahedrum and Dodecahedrum being taken ; their Cubes , and Quadrats , shal be the solid and superficial Contents of their proper Bodies , according to the nature of the Line by which they are measured : And therefore , if the Diameter of a Sphear be taken by these Lines , the Cube and Quadrat thereof , shal be the solidity and superficies of the circumscribed Icosahedrum and Dodecahedrum , according to the respective Line , by which it is measured .

And the like may be done in the Tetrahedrum and Octahedrum circumscribed ; and in all the five Bodies both inscribed and circumscribed , may the same be done also , by the Circumference of the greatest or central Circle of the Sphear ( circumscribing and inscribed ) for finding their sold and superficial Contents : But the Lines of measure for these last-named Dimensions , as also for the cubique and quadrate dimension , ( or the cubing and squaring , as I may so term them ) of the four Bodies here last handled , by their hedral Diameters or Perpendiculars , and hedrall Diagony also in the Dodecahedrum , ( as I have already shewed in the Hexahedrum ) I shal here omit , as needlesse and superfluous ; and shal shew the chief of these Dimensions , as also all the other beforegoing in these Bodies , which I have practically demonstrated by way of example , in an arithmetical manner , from their several artificiall Lines of measure , ( or Lines of Cubature and Quadrature ) expressed only by Number ; together first with the Linear dimensions of these Bodies , ( in regard of their foresaid several lines of Dimension , except those here last of all named ) leading thereunto ; in a way of Proportion , after the most exquisite manner that may be , as from the natural Measure , according as I have done in all the precedent dimensions . But before I go to these , I think it very fit and expedient ( for a conclusion of this Section ) to give the practicall Reader to observe by the way , for the more ease and conveniency in this kind of Mensuration , or metricall practice , ( or more artificial kind of practical Geometry ) which here we handle ; & so to avoid multiplicity of artificiall metrical Lines or Scales , arising by the manifold particular dimensions here considered and declared ; what Lines of measure here already delivered and expressed by Number , do agree , either in the whole throughout , or in part only , in their measure or magnitude from the prime Rational Line , so farre as is generally needful for ordinary use , which is to centesimal parts only of the Rational Line from which they are taken , and which division in that , for setting off the artificial Lines therefrom , and so also in the artificial Lines themselvs for measuring thereby , is generally sufficient for ordinary use , as I have both said , and exemplarily shewed in most of the precedent Dimensions ; it having considerably failed but in three of all the foregoing practical demonstrative Examples , which was first , in the solid Content of the Sphear last handled , produced by the Line of Cubature pertaining to the Diameter : and secondly , in the solid content of the Dodecahedrum in the Pyramidal dimension , by the Line pertaining to a regular-based pentagonal Pyramid in general ; where yet it did agree sufficiently with the true , natural dimension , in the solidity of the compounding Pyramid it self ; and therefore not so considerable in the Dodecahedrum : and then in the solid content of the Icosahedrum produced by the Line of Cubature belonging to its side : But two or three particular Examples are not to be regarded , in respect of the general ; seing that the like artificial dimensions of the same kind of Figures , may in other Examples hi● right enough ; the reason of these defections having been shewed sufficiently at first , especially in the last Section of the first Part. And therefore observe ,

I. That the artificial Line , for the quadrate dimension or squaring of a Circle , by its Diameter , ( expressed first of all by Number , from a decumillesimal solution of the prime Rational Line , viz. 1.1284 ferè ) and the Line for the cub●que dimension , ( or cubing ) of a Dodecahedron by its perpendicular-line of altitude ( or inscribed Sphear's dimetient ) under the same solution of the Rational Line , viz , 1.1296 ) wil agree in part , viz. in a centesimal solution of the Rational Line , being thereby 1.13 ferè , which I shewed in the dimension of Circle , to be the Line C D , according to the primary Line A B.

II. That the Line for the quadrate dimension of an ordinate Pentagon by its Diameter or Perpendicular , ( expressed numerally , 1.1732 ferè ) and the Line for the cubick dimension of an Icosahedron by its perpendicular of altitude ( or inscribed Spheare's Axis ) noted Numerally , 1.1654 ) wil agree sufficiently in part , viz. in centesimal parts of the Rational Line , being thereby , 1.17 , the first compleat , the second incompleat ; which difference is not considerable .

III. The Line for the dimension of a Tetragon by its Diagony , ( or circumscribing Circle's Diameter ) and for the artificial dimension of Triangles in general ; and the Line for the solid dimension of an Hexahedron by its basial or hedral Diagony ; and also the Line for the superficiary Dimension of a Cone by its Side and total basial periphery together , do agree in the whole throughout , viz. 1.4142 , &c. infinitely , being √ 2 of the intire prime Rational Line ( as the unit or integer of Measure ) in it's power quadraticall : with which very nearly agrees the Line for the cubing of a Dodecahedrum by its Axis or Diagoniall , ( or circumscribing Sphear's Dimetient ) in centesimal parts of the Rational Line , viz. 1.42 compleatly .

IV. The Line for the Quadrate superficiall dimension of a Sphear , ( or the quadration of the Sphaerical ) by its Diameter or Axis ; and the Line for the square-like superficiary dimension of a Cylinder ( or the rectangular parallelogrammation of its Superficies ) by its Side and Diameter together , do meet in the whole throughout , viz. 0.56418 &c , infinitely .

V. The Line for the quadrate superficial dimension of a Tetrahedron ( or quadration of its superficies ) by its Side ; and for the like dimension of an Octahedron by its Axis or Diagoniall ( or comprehending Spheare's Diameter ) and for the rectangular superficial dimension of a trigonal Pyramid , by the perpendicular of its base , and of its other or lateral triangular Plane together , do happen to be all one infinitly , in parts of the Rational Line , viz. 0.7598 , &c. being √ 1/3 of the intire prime Rational Line , in its biquadratique potentiality , or capacity : with which agreeth in part , viz. to centesimal parts , the Line of quadrature for the Side of a Pentagon , ( noted formerly by Number , 0.7624 ferè ) viz. 0.76 compleat , which in the Dimension of a Pentagon , was demonstrated by the Line P Q. in parts of the Line A B. So that here one and the same Line wil serve thus farre , for these 4 several dimensions . And very nearly with this Line , agrees the Line for the cubing of an Icosahedrum by its side ( noted formerly in parts of the Rational Line , 0.7710 , &c. ) taken centesimally , viz. 0.77 .

VI. The Line for the square-like solidation ( as I may term it ) or rectangle-parallelepipedall dimension of a trigonall Pyramid by its Axis or altitude , and basial Diameter or perpendicular together ; and for the solid dimension of an Hexahedron by its Axis or Diagoniall , ( or ambient Sphear's dimetient ) do agree in the whole throughout , viz. 1.73205 , &c , infinitely , being √ 3 , of the integrall Rational Line , taken in its power zenzicall or tetragonicall .

VII . The Line for the rectangle-parallelipipedal Solidation of a tetragonall Pyramid by it's Axis or Altitude , and basiall diagony or diameter tagether ; and fo● the cubique dimension ( or Cubation ) of an Octahedron by its Axis or Diagoniall , ( or containing Spheare's axis or diameter ) are both one throughout , viz. 1.817 , &c. infinitely being √ 6 of the prime Rational Line considered in its cubical capacity , or comprehensibility .

VIII . The Line for the square-like , or rectangle-superficiary dimension of a tetragonall Pyramid by the side of its Base , and the perpendicular of its trigonal Plane together ; and for the exact quadrate-superficiary dimension of an Hexahedron , ( or the quadration of its superficies ) by its Axis or Diagoniall ( or circumscribing Spheare's Diameter ) do agree in the whole , viz. 0.7071 , &c. being √ 1/2 infinitely , of the prime Rational Line , in its power or capacity tetragonical .

IX . The Line for the rectangle-parallelepipedation , or parallelepipedal consolidation of a pentagonall Pyramid ( as to an exact quadrate base ) by the diagonal line , or angular subtense of its Base , together with the Pyramids Axis or altitude ( noted formerly by number , 1.6589 ferè ) and the Line for the exact Cubation of a Tetrahedrum by its Axis ( noted in like manner , 1.6654 ferè ) wil sufficiently agree in part ; as to centesmes of the Rational Line , viz. 1.66 ; this latter compleat , the former incompleat : but with so smal a difference , as that one and the same Line may thus farre serve indifferently for both these dimensions viz. 1.66 .

X. The Line for the rectangle-superficiary dimension of a pentagonall Pyramid by the foresaid diagoniall of its Base , and the perpendicular of its trigonal Plane together , ( viz. 0.8045 ferè ) and the Line for the like dimension of a Cone , by the diameter of its base , and its side together , ( 0.7979 ferè ) do agree in part , as to centesmes of the Rational Line , viz. 0.80 . the first compleat , the second incompleat .

XI . The Line for the exact quadrate supersicial dimension of a Tetrahedrum , by its Axis , ( 0.6204 ) and the Line for the like dimension of a Dodecahedrum by its Axis or Diagoniall ( or ambient Sphear's dimetient ) viz. 0.6168 ferè , wilsussiciently accord , as to centesimal parts , viz. 0.62 ; the first compleat , the second incompleat .

XII . Lastly , the Line for cubing of a Dodecahedron by its side , ( 0.5072 ) and the Line for the squaring ( as I may so term it ) of an Icosahedron , as to its superficial part , by its perpendicular of altitude ( or inscribed Sphears dimetient ) viz. 0.5136 ) wil sufficiently concurre in part , as to centesimal parts of the Rational Line , viz. 0.51 ; the latter compleat , the former incompleat .

So that here you may see , how that of all the several geometrical dimensions before particularly expressed , being in number 62 , as requiring so many several lines of Measure , or Lines of quadrature , cubature , &c. the Lines for 28 of them , are contracted into 13 , in respect of a centesimal solution of the Rational Line from which they are taken , but no further , ( according to these 12 Notes or observations ) And the Lines for 15 of the same dimensions , are contracted into 6 , in regard of an infinite resolution of the Rational Line , according to the 3. 4. 5. 6. 7. 8 Notes . And so with the other 34 dimensions not here named , having their particular Lines of measure differing in the whole , ( in respect of the fraction-part of the Rational Line , though not of the integral part ) the artificial Lines for all the 62 dimensions aforesaid , wil be contrived into 46 , according to a centesimal partition of the prime or natural Rational Line . And the like agreement of Lines as is here demonstrated , may fall out between these and other the like Lines , for other Dimensions not here particularly expressed ; and also between other Lines , which are none of them heresetdown : which I referre to the ingenious practiser to consider , according as he may have occasion offered .

SECT. IV. Expressing the manifold Dimensions in the five plain ordinate , or regular Bodies , Arithmetically , by way of Proportion , in the most exquisite manner that may be .

ANd so having in the Section immediatly beforeing , shewed the Dimension both solid and superficial of the 5 plain ordinate ( Pythagorean or Platonick ) Bodies , according to our artificiall way of measuring , in the most exquisite manner that may be : ( or instrumentall Cubature and Quadrature , ) We shal in this Section , lay down the same Dimensions , with variety of others , in these Bodies , by Number , in the most exquisite Terms of Proportion that may be ; such as have not yet been done ( no more then the former artificial way of measuring the same ) by any that I could ever meet with , or hear of : and which must needs very much exceed those , tedious , obscure , confused , Cossical Terms which Mr. Diggs in his forementioned discourse of these Bodies , hath Theorematically delivered ; where yet , he hath left out the most material and usefull ones , for the ready and speedy discovering of their solid and superficiall Contents , as being inscribed in , and circumscribed to , a Sphear , in relation both to the Axis or Diameter , and the greatest Periphery of the Sphear , circumscribing and inscribed .

And here I shal first begin with the Linear dimensions of these Bodies , in all the variety thereof , according to the fore-named several Lines of dimension belonging to them , as usually adscribed to them for their Dimensions , ( as I did in the other ordinate Figures before-going , namely , the Circle , Sphear , Trigon , and Pentagon ) and this , in relation to the first Dimension in Geometry , called in general , from the Greeks , Euthymetrie , or Mecometrie , and fromthe Latines Longimetrie ; And which ( in respect of the different kinds of Lines ) I may call more generally from the Greeks , Grammemetrie ; and from the Latines , Linemetrie .

Therefore ,

1 The Tetrahedron's Side , is to its Axis , as 1. to .816497 ferè . √ 1/3 alike . ambient Spheares Axis , as 1. to 1.224745 ferè . alike .

2 The Tetrahedron's Axis is to its Side , as 1 to 1.224745 ferè . alike . ambient Spheares Axis , as 1. to 1.5 , sub-sesquialtera .

3 The Axis of a Sphear , is to the inscribed Tetrahedron's Side , as 1 to .816497 ferè . √ 1/3 alike . Axis , as 1 to .666667 ferè . viz. .666666 infinitely , ses-quialtera .

4. The Side of the Octahedron , is to its Axis or Diagonal , or circumscribing Spheare's Dimetient , as 1 , to 1.414214 ferè √ 2. Hexahedron , 1.73205 . √ 3. Icosahedron , 1.902113 . Dodecahedron . 2.802517 .

Contrarily ,

5. The Axis , Diagoniall , or angular Diameter ( or the ambient Sphear's Dimetient ) of the Octahedron , is to the Side , as 1 to .707107 ferè . √ 1/2. Hexahedron , .57735 . √ 1/3. Icosahedron , .525731 . Dodecahedron . .356822 .

6 The Diameter of a Sphear is to the Side of the circumscribed . Tetrahedron , as 1. to 2.44949 ferè . Octahedron , 1.224745 supdupla . Hexahedron , 1. aequalis , seu una . Icosahedron , 0.66158 . Dodecahedron . 0.449028 ferè .

Contrariwise .

7. The Side of the Tetrahedron , is to the inscribed Spheare's Dimetient , as 1. to .408248 . Octahedron , .816497 . dupla . Hexahedron , 1. aequalis , as before . Icosahedron , 1.511522 . Dodecahedron . 2.227033 . ferè .

And so the two last of these Proportions , are consequently of the Sides of those two Bodies to their Altitudes .

And by the 6th . Section of Proportions , as also by the 5th . Sect. of the like proportions , in relation to the correspondent Circumference of a Sphear , you may observe , how that a Tetrahedrum and an Octahedrum being circumscribed to one Sphear , the side of the Tetrahedrum wil be exactly double to the side of the Octahedrum : And so by the 7th Sect. beforegoing , you may observe contrarily , how that these two Bodies hauing one and the same side , the Diameter of the Sphear inscribed in the Octahedrum , wil be exactly double to the Diameter of the Sphear inscribed in the Tetrahedrum . And the like with these , you may also observe afterwards in the 5 and 6 Sections pertaining to the correspondent Circumference of a Sphear . And the same proportion wil the Tetrahedrum here hold to the Octahedrum , both for solid and superficiall dimension , as it doth for lateral dimension , as I shal shew afterwards .

8. Tetrahedron's Axis is to it s inscribed Sphear's Axis , as 1. to .5 dupla . And therefore contrarily .

9. The Axis of a Sphear is to the Axis of its circumscribing Tetrahedron , as 1. to 2. subdupla .

10. The Axis of Tetrahedron's ambient or externall Sphear , is to the Axis of its inscribed or internal Sphear , as 1 , to .3333 , &c. infinitely , viz. 3. to 1 , tripla . And therefore conversly ,

11. The Axis of Tetrahedron's inscribed Sphear , is to the Axis of it's circumscribing Sphear , as 1. to 3 , subtripla .

12. The Axis , Diagoniall , or angular Diameter of the Hexahedron , Or their circumscribing Sphear's Dimetient , is to their inscribed Sphear's Dimetient , as 1. to .57735 . √ ●/3. Octahedron , Dodecahedron , Icosahedron . .794654 .

And so consequently , the latter of these two Proportions , is to be understood of the Axes of those two Bodies , to their Altitudes .

Conversly .

13. The Inscribed Spheare's Dimetient , is to the circumscribing Spheare's Dimetient , ( or the Axis , or Diagoniall ) of the Hexahedron , as 1. to 1.73205 , √ 3. Octahedron , Dodecahedron , Icosahedron . 1.258401 ferè

And so the latter of these two Proportions , is of the Altitudes of those two Bodies to their Axes .

By which two Sections of proportions in these 4 Bodies , and by the two last proportions in the Tetrahedrum next before-going , viz. Sect. 10 and 11. it appeareth , that these five regular Bodies , in respect of their Sphericall inscriptibility and circumscriptibility , do require three several distinct Sphears , circumscribing or containing , and inscribed or contained : That is , they being all severally inscribed within one Sphear , cannot then also be exactly circumscribed about ( or cannot compleatly comprehend with in them ) one Sphear , but three several Sphears ; whereof that which is inscribed in the Tetrahedrum wil be the least , and that inscribed in the Hexahedrum and Octahedrum , wil be equal , and that which is inscribed in the Dodecahedrum and Icosahedrum wil be also as one , and the biggest of all .

And so again contrarily , these five Bodies being severally circumscribed about ( or comprehending in them ) one Spheare , cannot then again be exactly comprehended or contained of one Spheare , but must have three severall comprehending , containing or including Sphears ; of which , that for the Tetrahedrum , wil be the greatest ; that for the Hexahedrum and Octahedrum wil be alike ; and that which is for the Dodecahedrum and Icosahedrum wil also be of equal magnitude , and indeed the least of all : ( See M. Diggs his Discourse upon these Bodies , Probl. 17. ) And the like to these may be observed in these 5 bodies , as being inscribed in , or circumscribed about , one Sphear , in respect of the Circles circumscribing their Bases , the Diameters of these Circles and of the Bodie's circumscribing and inscribed Sphear , being compared together : And therefore .

14. The Diameter of the Sphear circircumscribing the Tetrahedron , is to the Diameter of the Circle circumscribing their Base , as 1. to , .942809 . Hexahedron , — Octahedron , .816497 ferè . Dodecahedron , — Icosahedron . .607062 .

And so the second and third of these Proportions , are also of the Axes or Diagonies of these 4 Bodies , to the diameters of their Base's ambient Circles : which in the Hexahedron , is of the corporal Diagony , to the basial or superficiall Diagony .

15. The Diameter of the Spheare inscribed in the Tetrahedron , is to the Diameter of the Circle circumscribing their base , as 1. to 2.828427 . Hexahedron , — Octahedron , 1.414214 . ferè . Dodecahedron , — Icosahedron . 0.763932 .

And to the second of these Proportions , is consequently of the Hexahedron's Side to its Basiall Diagony .

The Converse of these are ,

16. The Diameter of the Circle circumscribing the Base of the Tetrahedron , is to the Diameter of their circumscribing Sphear , as 1. to 1.06066 . Hexahedron , — Octahedron , 1.224745 . Dodecahedron , — Icosahedron . 1.647278 .

And so the second and third of these Proportions , are also of the diameters of the basiall ambient Circles of these four bodies , to their Axes or Diagonies : which in the Hexahedron , is of the basiall or hedrall diagony , to the totall corporall diagony .

17. The Diameter of the basial or hedral ambient Circle of the Tetrahedron , is to their inscribed Sphears Diameter , as 1. to .353553 . Hexhaedron , — Octahedron , .707107 ferè Dodecahedron , — Icosahedron . 1.309016 .

And so the second of these Proportions , is of the Hexahedron's basial Diagony or Diameter , to its Side .

And the last of these Proportions , is of the basiall ambient Circle's Diameter , to the Altitudes of those two Bodies .

So that you may here see by these 4 Sections of proportions , how that these 5 bodies being described either within or about one Sphear , have only three several circumscribing or containing Circles , for their Bases ; whereof , that which is for the Tetrahedrum is the largest ; that for the Hexahedrum and Octahedrum are both one , and the next to it ; and that which is for the Dodecahedrum and Icosahedrum are also equal , and the least of all ; which Euclid E. 14. p. 5 , and 21 , speaks of , only in respect of these bodie's inscriptibility in one Sphear . As for the basial or hedral inscribed Circles of these 5 Bodies , whether inscribed in , or circumscribed about one Sphear ; there is no such parity or agreement amongst them , but they are all different one from another .

And as for the proportions of the sides of these Bodies to the Diameters of their hedral Circle's whether circumscribing or inscribed ; and to their hedral Diameters or perpendiculars , &c. the same are to be had in the three ordinate Planes before handled , viz. the Trigon , Tetragon & Pentagon : those for the Tetrahedron , Octahedron and Icosahedron , out of the Trigon ; those for the Hexahedron , out of the Tetragon ; and those for the Dodecahedron , out of the Pentagon ; But indeed , that of the Hexahedron's side to its hedral ambient Circle's Dimetient ( or its hedral Diameter ) & é contra ; is also noted in the 15. and 17. Sections .

And thus much for the linear proportions , in respect of the severall lines of Dimension in these Bodies , being considered both simply or absolutely in themselves , and also as being inscribed and circumscribed , and that in relation to the Diameter of the Sphear circumscribing and inscribed . And from these we shal proceed to the like proportions , in respect of the Circumference answering to the said Sphear's Diameter , which I have not yet found touched upon by any man , in any kind whatsoever . And therefore .

1. The Circumference of a Spheare's greatest , or Centrall Circle , is to the Side of the inscribed Tetrahedron , as 1. to .259899 ferè . Octahedron , .225079 . Hexahedron , .183776 . Icosahedron , .167345 . Dodecahedron . .113580 . exactly .

Vice-versâ .

2. The Side of the Tetrahedron , is to the circumscribing Sphear's greatest circumference , as 1. to 3.847649 . Octahedron , 4.442883 . Hexahedron , 5.441398 . Icosahedron , 5.975664 . Dodecahedron 8.804369 .

3. The Periphery of a Sphear's greatest Circle , is to in inscribed Tetrahedrum's Axis , as 1. to .212206 . And so contrarily .

4. The Axis of a Tetrahedrum , is to its ambient Sphear's greatest or true Periphery , as 1. to 4.712389 .

5. The Periphery of a Spheares greatest Circle , is to the side of the ambient . Tetrahedron , as 1. to .779697 ferè . subdupla . Octahedron , .389848 . subdupla . Hexahedron , the same , as the Circumference to the Diameter , the Side of this body being equall with the Spheares Diameter . Icosahedron , .210589 . ferè . I. Dodecahedron . .142930 . D.

Contrariwise ,

6. The side of the Tetrahedron , is to the inscribed Spheares greatest Periphery , as 1. to 1.28255 . dupla . Octahedron , 2.56510 . dupla . Hexahedron , the same as the diam , to the Circum● . for the reason aforesaid . H Icosahedron , 4.748589 . Dodecahedron . 6.99643 .

7. The Periphery of a Sphear's largest Circle , is to the Axis of its ambient Tetrahedrum , as 1. to .63662 :

And again conversly ,

8. The Axis of a Tetrahedrum is to it s inscribed Sphear's greatest Periphery , as 1. to 1.570796 .

As for the proportions of the circumscribing Spheare's greatest or Diametral Circumference of these Bodies , to their inscribed Spheare's like Circumference , & contrà : And of the greatest Circumference of the Sphear both circumscribing and inscribed , to the Circumference of the basial or hedral circumscribing Circles of these Bodies , & contrà : they wil be the same with those which are already expressed between the Diameters , viz. first in respect of the Sphear circumscribing and inscribed , between themselves ; and then of both these Sphears severally with the Circles circumscribing the bases of these Bodies , as being described either within or without one Sphear ( or several Sphears of one magnitude . )

Having thus expressed the Linear proportions in these Bodies , or the proportions of linear dimension , as many as ( and indeed many more then ) are here absolutely needful for the measuring of them : I shal now come to shew the proportions both superficial or quadrate , and solid or cubique deduced from thence , and both these conjunctly ( for brevity sake ) in reference to the other two Dimensions in Geometrie , called from the Grecians , Embadimetrie ( & Plat●metrie ) & Stereometrie ; and from the Latins , Plan●metrie & Solido-metrie : beginning with the latter of these as being most worthy and most considerable in solid Figures , as I have said before , the superficial dimension in them being not so useful or material , and also much more easily obtained ; especially in these five Solids ; whether the same be done naturally or artificially ; And therefore first in respect of these Bodies considered simply and absolutely by themselves , without any inscriptibility and circumscriptibility , either spherical or mutual ; the cubical proportions noted by the letter c , and the quadrate proportions by the letter q. wil be as followeth ; the bodies being placed in order according to their magnitudes increasing , both solidly and superficially .

1. The Laterall Cube and Quadrat of the Tetrahedron , is to the Bodies solidity and superficies , as 1. to .11785113 . c. T. 1.7320508 . q. T. Octahedron , .47140452 . c. O. 3.464102 ferè q. O. Hexahedron . 1.000000 . c. H. aequalis . sub-sextupla . 6.000000 . q. H. aequalis . sub-sextupla . Icosahedron , 2.181695 . c. I. 8.660254 . q. I. Dodocahedron . 7.663119 . c. D. 20.645729 . q. D.

Hence ,

2. The Side of the Tetrahedron , is to the Side of the Cube and Quadrat equall to the Bodies solidity and and superficies , as 1. to .49028 . c. T. 1.316074 . q. T. Octahedron , .778346 ferè . c. O. 1.86121 . q. O. Hex●●●dron , 2.44949 . q. H. Icosahedron , 1.29697 . c. I. 2.94283 . q. I. Dodecahedron , 1.971523 . c D. 4.543757 . q. D.

And hereby it appears , that these 5 Bodies having all the same side : the Octahedrum is quadruple to the Tetrahedrum in solidity , and double in superficiality : ( & which I have shewed before in their dimensions ) The Hexahedrum is bigger then both those together , both in solidity and superficiality : the Icosahedrum is greater then those 3 together , in solidity , but lesser in the superficies : and is herein quintuple to the Tetrahedrum , and so double-sesquialter to the Octahedrum ( as I have also shewed before ) And the Dodecahedrum is larger then all the other together , both in solid and superficial dimension .

3. The Cube of Tetrahedron's Axis is to its Solidity as 1.10 21650635. Quadrat Superficies 2.59807621

Hence ,

4. The Axis of the Tetrahedron is to the side of the Cube equall to its Solidity as 1. to 600468. Quadrat Superficies ●611855 ferè

Then for the Tetrahedrum in reference to a Sphear , by way of inscriptibility therein , as to its Dimetient , it followeth .

5. The Diametral Cube of a Sphear is to the inscribed Tetrahedron's Solidity as 1. to .0641500299 . Quadrat Superficies 1.1547005

Hence .

6. The diameter of a sphear is to the side of the Cube equal to the inscribed Tetrahedron's Solidity as 1. to .4003123 . Quadrat Superficies 1.0745699 .

Next for the other 4 Bodies , both simply in themselves , and also in relation to their ambient Sphear together , in respect of their common Axis or Dimetient , it followeth .

7. The Cube and Quadrat of the Axis , or Diagoniall of the Octahedron , or of the circumscribing Spheares Dimetient , is to the Bodies solidity and superficiality , as 1. to .166667 ferè c. O. 1.7320508 q. O. Hexahedron , .19245009 . c. H. 2.00000 . q. subdupla . H. Icosahedron . .3170189 . c. I. 2.393635 . q I. Dodecahedron , .348145 . c. D. 2.628656 ferè . q. D.

Hence ,

8. The Axis , Diagoniall , or angular Diameter of the Octahedron , or the circumscribing Spheares Axis or Diameter , is to the side of the Cube and Quadrat equall to the Bodies solidity & superficiality as 1. to .550321 . c. O. 1.316014 . q. O. Hexahedron , .57735 . c. the same as of the Spheares Diam . to the inseribed Hexabedrons side , and so of its own Axis or corporal Diagony , to its Side , noted before . H. 1.41421 . q. H. Icosahedron . .68186 . c. I. 1.54714 q. I. Dodecahedron . .703483 ferè c. D. 1.621313 . q. D.

By which proportions , ( as also by the 1. 2. and 5 Sections following of the like proportions , in relation to the greatest Periphery of a Sphear ) it appears , that these five Bodies being all described within one & the same Sphear , do retain the same order and rank among themselves , in respect of their magnitudes or dimensions both solid and superficiall , as they do , being of one common , lateral dimension , but not in that proportion and difference of dimension either solid or superficiall ; for that here the lateral Dimension being different in them all , and that by way of diminution from the least body to the greatest , the superficiall and solid dimension become thereby different in them all in like order , by way of augmentation ; but nothing so much , as being all under the same lateral dimension . And hereby it also plainly appears ,

I. That the Hexahedrum is triple the Tetrahedrum inscribed in the same Sphear , and so E 14. p. 32. And ,

II. That the Hexahedrum is to the Octahedrum within the same Sphear , solidly , as it is superficially , according to E 14. p. 27 ; and also as the Side of the Hexahedrum is to the Radius of the Sphear by the same Prop. And.

III. That the Octahedrum is superficially sesquialter the Tetrahedrum , according to E 14. p. 14. and so ( by the same Prop. ) the Tetrahedrum is Basially or Hedrally , sesquitertian the Octahedrum . And ,

IV. By the last named Sections of solid and superficial Proportions , together with the 3. and 5. Sections of Linear Proportions beforegoing , between the Diameter of a Sphear , and the sides of the 5 Bodies inscribed thererein ; and also the first Section of Proportions , between the greatest Periphery of a Sphear , and the sides of the Bodies inscribed ; it appeareh , how that the Octahedrum is to the Triple of the Tetrahedrum inscribed in the same Sphear , as its side is to the side of the Tetrahedrum ; and so E , 14. p. 22. And ,

V. That the Dodecahedrum is to the Icosahedrum in the same Sphear , both solidly and superficially , as the Hexahedrum is to the Icosahedrum laterally , according to E , 14. p. 9 and 11.

Then for these Bodies in relation to a Sphear , in respect of their circumscriptibility about the same , as to its Dimetient ; and so for the Hexahedrum , Dodecahedrum , and Icosahedrum , simply and absolutely in themselves also , in respect of their Dimetients of altitude , being all one with their inscribed Sphear's Dimetient : their proportions both cubatory and quadratory , wil be as followeth : the Bodies being here placed in order according to their magnitudes both solid and superficial decreasing , which is according to their due hedral order , in respect of their denominations from the numbers of their bases or hedral Planes . And therefore ,

9. The Diametrall Cube and Quadrat of a Sphear , is to the solidity & superficies of its circumscribing Tetrahedron , as 1. to 1.7320508 . c. T 10.3923048 . q. T Hexahedron . the same as its Lateral Cube & Quadrat . H Octahedron , 0.8660254 . c. O 5.1961524 . q. O Dodecahedron . 0.693786 . c. D 4.162719 . q. D Icosahedron , 0.631757 c. I 3.79054 q. I

And so the 4 last of these Proportions , are consequently , of the Cubes and Quadrats of the Altitudes of those two Bodies , to the solid and superficiall capacities of the Bodies themselves .

10. The Dimetient of a Sphear , is to the side of the Cube and Quadrat equall to the solidity and superficies of the ambient . Tetrahedron , as 1. to 1.200937 . c. T 3.223710 . q. T Hexahedron , the same as in Sect. 2. the side of this Body agreeing with the inscribed Sphears dimetient . H Octahedron , 0.953193 . c. O 2.279507 . q. O Dodecahedron , 0.885269 . c. D 2.040274 . q. D Icosahedron , 0.858058 . c. I 1.946932 . q. I

And so the 4 last of these proportions , are of the Altitudes of those two Bodies , to the sides of the Cubes and Quadrats , equal to their solidities and superficieties .

Whereby , as also by the 3 , and 7 Sections of Proportions following , it plainly appears , how that these 5 Bodies being all described about one Sphear , the Tetrahedrum is the biggest of all , both in solid and superficiall dimension , and is exactly double to the Octahedrum in both these dimensions ( as it was shewed before to be in its lateral dimension ) and very near as bigg as the Hexahedrum and Octahedrum both together , in respect of both dimensions : And the Icosahedrum is the least of all in both these dimensions ; which is almost contrary to the former course holden in these Bodies , being described within one Sphear .

N●w for the like Proportions , in relation to the Circumference of a Sphear's greatest Circle ; and first , in respect of these Bodies spherall inscriptibility ; it followeth ,

1. The Cube and Quadrat of the greatest or centrall Circumferēce of a Sphear , is to the solid & superficiall capacity of the inscribed Tetrahedron . as 1. to .0020689369 . c. T .1169956 . q. T Octahedron , .0053752557 . c. O .1754934 . q. O Hexahedron , .00620681069 . c. H .202642367 . q. H Icosahedron . .01022434 . c. I .2425259 . q. I Dodecahedron , .01122822 . c. D .266338 . q. D

Therefore .

2. The circumference of a Sphear's greatest Circle , is to the side of the Cube and Quadrat equall to the solid and superficiall content of the inscribed Tetrahedron . as 1. to .127423 . c. T .342046 . q. T Octahedron , .175173 . ferè . c. O .418919 . q. O Hexahedron , .183776 . c. before for the side of the Hexahedron inscribed . .450158 . q. before for the side of the Hexahedron , inscribed . Icosahedron . .2170427 . c. I .492469 . q. I Dodecahedron . .223925 . c. D .51608 ferè q. D

Then for the proportions cubatorie and quadratarie , in relation to a Sphear's greatest , Diametral , or true Peririphery , in respect of these Bodies Spherall circumscriptibility , it followeth ,

3. The greatest Peripheriall Cube and Quadrat of a Sphear , is to the solidity & superficiality of its comprepending , containing , or including Tetrahedron , as 1. to .055861296 . c. T 1.0529606 . q. T Hexahedron , .032251534 . c. H .6079271 . q. H Octahedron , .027930648 . c. O .526480314 . q. O Dodecahedron . .2237567 . c. D .4217715 . q. D Icosahedron , .02037514 . c. I .3840622 . q. I

Whereupon .

4. The greatest , or Diametrall Pe●iphery of a Spheare , is to the Side of the Cube and Quadrat equall to the solidity & superficiality of the circumscribing or cōprehending Tetrahedron , as 1. to .3822701 . c. T 1.0261387 . q. T Hexahedron .318309 . c. the same as of the circumf . to the Diam . the side of this Body agreeing with the Sphears diameter . H .7796968 . q. — H Octahedron . .30340798 . c. O .7255896 . q. O Dodecahedrō .2817898 . c. D .649439 . q , D Icosahedron .2731284 . c. I .619727 . q. I

Lastly , for these Bodies , and a Globe or Spheare compared wholly together , both solidly and superficially ; and that according , both to their inscription and circumscription , the Proportions wil be as followeth , ( the first or uppermost of the two Numbers belonging to each Body in the two first Sections , being for solid comparison , and the second for superficiall )

5. A Globe or Sphear , is to the inscribed Tetrahedron . Solidly and superficially together , as 1. to .12251753 T .3675526 ferè . T Octahedren .318309886 . agreeing with divers of the former proportions . .551328895 . Hexahedron .3675526 ferè . agreeing with the superficiall comparison in the Tetrahedron . . 6366●977 . Icosahedron .6054614 . I .76191789 . I Docecahedrō .6649087 . D. .8367272 . D

Contrariwise .

6. The Tetrahedron . Is to the circumscribing Globe or Sphear , solidly and superficially together , as 1 , to 8.162097 T 2.720699 T Octahedron . 3.14159265 , agreeing with many of the foregoing proportions . O 1.81379936 — O Hexahedron 2.720699 . agreeing with the superficiall comparison in the Tetrahedron H 1.5707963 — H Icosahedron . 1.6516327 ferè . I 1.312477 I Dodecahedrō 1.503966 . D 1.195133 . D

7. A Globe or Sphear , is to its ambient Tetrahedron both solidly and superficiarily together ( in one and the same reason ) as 1. to 3.30797337 . Hexahedron 1.9098593 . the same as of a Sphear to the Cube of its Axis , noted formerly ; the side of this Body agreeing with the Axis of the Sphear . H Octahedron 1.65398668 . subdu●plum Tetrahedri . Dodecahedrō 1.325034 . Icosahedron 1.206567 .

Conversly, 8. The Tetrahedron is to the inscribed Globe or Sphear , both solidly and superficially together in one , as 1. to .30229989 Hexahedron .52359877 , the same as of the Cube of a Sphear's Axis , to the Sphear it selfe ; the Cube of the Axis being here the ambient Hexahedrum . H Octahedron .604599788 . double to the Sphear inscribed in the Tetrahedrum . Dodecahedrō .7546973 . Icosahedron .8287974 .

By the two last of which Sections of solid and superficiall Proportions or comparisons between the 5 ordinate Bodies and a Globe or Sphear , it appeareth ; that any of the said Bodies being circumscribed about a Sphear , the solid and superficiall capacity thereof , wil be one and the same numerally , ( or in the number of a given Measure ) viz. where the said two severall Dimensions of the inscribed Sphear are alike in number : That is , two like Bodies exactly encompassing or environing two severall , distinct Sphear's , whereof the solidity of the one , & the superficiety of the other , are numerally alike ; there wil the solid capacity of one of the like ambient Bodies , and the superficiall capacity of the other , be also alike in the number of measure . And so likewise , two distinct Sphears being inscribed in two several like Bodies , whereof the solid measure of the one , and the superficial measure of the other , are numerally the same ; there wil the solidity of one of the inscribed Sphears , and the superficiety of the other , be also numerally the same : and which I have not found to be observed by any before .

Many more Proportions might here have been raised , if they were needfull ; as namely of the Bodies among themselves in respect of their mutuall inscription and circumscription ; and those also which are the converse of many of the former , to wit , the Proportions of these Bodies solidly and superficially , to the Cubes and Quadrats of their Sides , and of their Axes and Altitudes ; and so of the Diameter and greatest Circumference of their circumscribing and inscribed Sphears , whereby the sides , Axes and Altitudes of these Bodies , and so the Axes or Dimetients , and greatest Peripheries , of their circumscribing and inscribed Sphears , might be obtained , by having the solidities and superficialities of the Bodies only ; and that after one Radical extraction , quadrate or cub●que according as I delivered in the Circle and Sphear , for the obtaining of their Diameters and Peripheries , by their superficial and solid Contents ; and so in the ordinate Trigon for the finding of its side , and Diametral or perpendicular line ; and in the ordinate Pentagon , for its side and Diametrall or perpendicular , and Diagonal-line , by their are all or superficial Contents : But that these kind of proporti 〈…〉 not so usefull , being indeed more of curiosity then 〈◊〉 sith the superficial and solid Contents of Figures are more usually inquired out by their sides , Diameters , and other ▪ the Lines of Dimension , then these lines are by their superficiall and solid Contents ; for that the thing chiefly 〈…〉 in the dimension of all Figures , is their superficiall 〈…〉 Contents ( and in solid Figures , chiefly the so●●● Conte●● 〈◊〉 I said before ) which must be obtained by 〈◊〉 of Dimension .

〈…〉 ●●ving handled the five famous ordinate ( Py 〈…〉 tonick ) Bodies , or the angular , or recti 〈…〉 both geometrically , in an instrumen 〈…〉 our artificial way of measuring ) 〈…〉 the most exquisite proportional 〈…〉 that may be , and that chiefly in reference to their solid and and superficiall dimensions : I shal next come to the second theorematicall Proposition before-mentioned , in which , our more particular or special reason of our artificial instrumentary dimensions ( or mechanicall Cubature and Quadrature ) of these Bodies , like as first of a Sphear , ( as to the obtaining of the severall artificiall Lines of measure for performing the same ) is contained .

THEOR. II. Explicating particularly the foregoing artificiall Lines for the cubick dimension , ( or Cubing ) of a Sphear ; from our particular or speciall Reason of Dimension : And consequently , the Lines for the like dimension of all the other regular Solids .

IF the Axis or Dimetient of a Sphear , equal to the Cube of the Parts of the Rational Line , be had ; The same shal be the correspondent Line of Cubature , according to the said Parts . And the reason of that to the congrual Cubatorie Line , in reference to the whole intire Measure , wil be as the reason of the Parts to the Whole ; which is as the reason of their Cubes . And the like for the correspondent Periphery , or the Circumference of the greatest Circle .

ANd the like in both these , for the superficiary ( or quadrate ) dimension of a Sphear ( or quadration of the Sphericall ) only respect being here had to the quadrate parts of the Rational Line , as there to the cubique : and so indeed this may properly enough be referred to the first Theoreme , the reason here , being the same with that .

And the same reason holdeth in all the five fore-named ordinate plain or angular Bodies , both for solid or cubique , and for superficiaty or quadrate dimension , whether simply in themselves , or in relation to a Sphear , by way of inscription and circumscription , and that by any of their dimensional lines formerly named , as their sides , and Axes or Diagonials , or angular Diameters and Dimetients of altitude : And so by their circumscribing and inscribed Sphear's Dimetient , and also greatest Circumference , and other lines of dimension , according to the several Dimensions and dimensional Proportions beforegoing : And also in respect of their relations one to another , by way of mutual inscriptibility and circumscriptibility .

SECT. V. Shewing the Dimension of all exactly ordinate , or regular solid Bodies , artificially , for Gravity or Weight , as is for solid Measure : And demonstrating the same particularly , in the first ordinate Solid here handled , namely a Sphear .

ANd the like reason of Dimension to that before-going , wil hold in a Sphear , and the five plain regular ( Platonick ) Bodies , for gravity , or Quantity ponderall ( according to any Metal & Weight assigned ) as for solid magnitude , or Quantity dimensional ; there being generally the same mathematicall reason of these two Quantities , in so much , as that they are usually by Mathematicians compared together in several kinds of Bodies : Or divers kinds of bodies are compared together among themselves in this twofold reason of Quantity ; to wit , Magnitude or dimension , and Gravity , or Ponderosity , as you may see in Archimedes , Ghetaldus , and others : And so these two do proportionally answer each other , in so much , as that one may be deduced from the other ; as Gravity from Magnitude , and Magnitude from Gravity ; or Weight from solid Measure , and contrà : And therefore the gravity or ponderosity of each one of the foresaid regular solid bodies taken in some certain magnitude or bignesse , being first known according to some certain Metall , Weight , and Measure appointed ; there may be artificiall Lines of measure extracted for every severall kind of Body , according to the said particular Metall , Weight and Measure , ( and that according to the foresaid severall Lines of Dimension in those Bodies , by which they have been shewed to be artificially measured ) so as that any one of the said dimensional lines of each Body in any magnitude whatsoever , being measured by its proper artificial Line or Scale for this purpose , & cubed , the same shal be the weight of the metalline Body proposed : And which wil therereupon hold in a Sphear , and all the 5 plain regular Bodies , not only as considered simply and absolutely in themselves alone , but also as in relation one to another , by way of inscription and circumscription ; and so in the said 5 bodies , not only in the said relations to a Sphear , as being inscribed in , or circumscribed to the same , but also mutually one to another , as was said before for solid ( and superficiall ) measure : And so the Diameter or Circumference of a Sphear , being taken by their proper , respective artificial Lines of Measure for this purpose ( according to any certain Body , Metall , Weight , and Measore assigned ) the several Cubes thereof , shal be the weight of the respective Body inscribed , or circumscribed , according to which is proposed ; And the first of these , is the same ( in respect of a Sphear's Diameter ) with the artificiall dimension of the four greatest of the said 5 regular plain bodies , by their Axes , Diagonies or angular Dimetients , ( these being all one with their circumscribing Sphear's Dimetient , as I have shewed before ) And the latter agrees with the like dimension of the three last of those bodies , by their Altitudes , ( being the same with their inscribed Sphear's Dimetient ; and is also in the Hexahedron , the same with its Side , as I have likewise shewed before ) And so the reason of the artificial Lines for this cubical dimension of the foresaid Bodies , for weight , as for solid measure , may be partly referred to the foregoing 2d . Theoreme ; the difference being , that respect must be here had to the ( compounding , denominate ) parts simply , of the Weight proposed , in such manner , as is there to the cubical parts of the Measure proposed ; So that these artificial Lines may be easily produced therefrom ; And therefore I shall not need here to raise a particular Theoreme upon the same . All which might be performed also Arithmetically by way of Proportion , from the natural Measure appointed , according as all the former Dimensions : And both these waies I shal here particularly demonstrate in the first regular Solid beforegoing , to wit , a Sphear , or Globe , simply in it self , as comming most in use ; and that in the most usual and usefull Metall for this purpose . But first I wil give the Proportions or comparisons of all the usuall or principall ( or commonly received ) kinds of Metals , according to the experiments and observations of Marinus Ghetaldus , in his Archimedes promotus ( who is generally supposed to have come the nearest to the truth herein , of any man that hath ever yet written hereof ) according as they are there delivered by him in the second Table of that Tractate , next after after Theor. 9 or prop. 17 : which Mr. Gunter in the 5th . chap. of the 3d. Book of his Sector , hath expressed in the same terms , but more decimally , and that in whole numbers , by changing the natural or vulgar fractions of those numbers into decimal , and so expressing his natural mixt or heterogeneall numbers , in whole numbers , after a decimall manner , putting the first number 10000 , whereas Ghetaldus puts it but 100 ; which I have here collected orderly into this Table following , in proportion direct and reciprocall , in respect of the equal Magnitudes and gravities of like Bodies of different Metals .

In like Bodies of severall metals and equall magnitude , having the weight of the one , to find the weight of the rest . Gold . 10000 3895 ☉ In like Bodies of severall metals and equall weight , having the magnitude of the one , to find the magnitud of the rest . The converse of the former . Quick-silver . 7143 5453 ☿ Lead . 6053 6435 ♄ Silver . 5439 7161 ☾ Bresse . 4737 8222 ♀ Iron . 4210 9250 ♂ Tinne . 3895 10000 ♃

Which meralline Proportions or comparisons . Mr. Oughtred in his foresaid Book of the Circles of Proportion , Part. 1. chap. 10. hath expressed in the Terms following , being deduced from Ghetaldus his first Table of Comparisons ( if I much mistake not ) which is from Prob. 5 , and 6 , or Prop. 12 , and 13 , of that Treatise .

Gold 3990 Quicksilver 2850 Lead 2415 Silver 2030 Brasse 1890 Iron 1680 Tinne 1554

geometrical diagram

As suppose here again the former bullet of 9li. avoirdupois , whose diameter being put 4.00 inches , the Circumference wil be 12.57 inches ferè which by its proper Line of Cubature , wil be found 2.08 , as the Diameter before by its proper Line , and therefore by Cubation , must needs produce the same ponderosity as before .

But here note , that an Iron-Sphear of 4 inches the diametral magnitude in Ghetaldus his measure , wil weigh with him , 12li. 2. ounc. 1 scrup. 14 gr. and 2/37 of a grain ; for so I find it to be by the former proportional numbers , by comparing this Sphear with a Sphear of Tinne of the same diametrall magnitude , according as Ghetaldus sheweth in his foresaid book , and from him Mr. Oughtred in his fore-named book : and which is also expresly noted by Ghetaldus in a Table , wherein he hath set down the weights of a Sphear , in all the foresaid Metals , from 1/4 of an inch the diametral magnitude , to 12 inches , or the whole Foot , proceeding all along by quarters of inches . But now 9li. avoirdupois is in our Troy-weight , by Assize or Goldsmiths weight ( according to the commonly received proportion of the Pound-avoirdupois to the Pound-troy , 60 to 73 ) but 10 57/60 pounds , or 10.95li. exactly ; which is 10 pounds , 11 2/5 or 11.4 ounces ; or according to the common division of the ounce-troy by penny-weights , is 10li. 11 oun. and 8 penny-weights exactly .

Now Ghetaldus divides a Pound into 12 ounces ; an ounce into 24 scruples ; and a scruple into 24 grains : so that his ounce weigheth 576 grains , and his pound , 6912 grains : And we divide our Pound-troy , into 12 ounces ( as he doth his pound ) but the ounce-troy we usually divide into 20 penny-weights , and a penny-weight into 24 grains , so that our ounce-troy weigheth but 480 grains , and consequently our Pound-troy 5760 gr. All which sheweth , that either his Weight , or Measure , or rather both of them , do differ from ours , as I shal further shew : As for the Measure which hee useth , he saith it to be the ancient Romane Foot , divided into 12 unciae , or inches as ours is , and which by his description and delineation thereof , in his forementioned book , seemeth to differ but very little from our English Foot , ( if any thing at all ) and that deficiently , and which Mr Oughtred in his forenamed book also observeth : But Mr. John Greaves sometimes Professour of Geometry in Gresham Colledg London , and now Professour of Astronomy Concerning the magnitude of the Romane Foot . in the University of Oxford , hath in his Discourse of the Roman Foot , &c. ( published by him in English , Anno 1647 ; and deduced , not only from divers Authours , but also from his own observations and experiments , which in that his learned discourse he seemeth to have made with greatpains and industry , in his travails in forraign parts , but especially in Italy , and there at Rome ) more clearly expressed the Dimension or magnitude of the true Romane Foot ; having done the same not onely linearly ( by the half thereof , as Ghetaldus hath done ) but also Numerally , in comparing it with the standard measures of England , and divers other Nations : For the draughts or delineations thereof in Books , cannot give us the true length , in respect of the divers accidents happening to the paper whereon it is imprinted , and especially the contraction or shrinking of it after the impression , ( as both these authours do give us to observe ) which while it was moist in the Presse , received the true Measure , but afterwards being dried , loseth somewhat thereof ; and so Ghetaldus in the last page of his foresaid book , giveth us to note particularly in his draught there of half the Roman Foot , and how much is there to be added to it , to make up the true measure . For although Mr. Greaves doth conclude upon the same Measure of the Roman Foot , which ( he saith ) Ghetaldus doth , ( for the truest Measure ) yet if we compare their draughts or delineations of the half Foot , together , ( in their foresaid books ) we shal find that of Mr. Greaves , to be shorter then that of Ghetaldus , by almost 1/8 of Ghetaldus his Inch , as it is there set out ; or 1/10 of our English inch exactly . And such like disagreement is to be found among other Authours in their delineations or draughts of the same Roman Foot , for the reason aforesaid .

Now the Foot which the said Mr. Greaves ( among such a diversity of opinions concerning the true Roman Foote , as are to be found , and so many Feet as are taken to be Romane ) pitcheth upon for the most genuine and true Roman Foot , ( being led not only by the anthorities of Ghetaldus , and divers other learned and judicious men , ( as he saith ) but also by his own observations and experience ) is that which is commonly called by writers , Pes Colotianus , from the place where it is ( or sometime was ) to be found , namely , in hortis Colotianis in Rome , upon the Monument of Cossutius ( which now he saith to be remov'd thence ) The Roman and English Foot compared together . which Foot he comparing with our English Foot , which he took from the iron-Yard , or Standard of 3 Feet , at the Guild-hall in London ( for there is no single Foot-standard ) findeth to be 967 such parts , as the English Foot contains 1000 ; and so the English Foot to contain 1034.13 , such as the said Roman Foot contains 1000 : whereby this Romane Foot should be 11.6 ( or 11.604 exactly ) such parts as our English Foot is 12 ; viz. 11.6 ( or 11.604 ) inches , and so wanting of the English Foot , only 0.4 inch , or 0.396 inch exactly . But there are two other Romane Feet reckoned by him , which ( by his account ) do come nearer to our English Foot ; the first whereof is that on the Monument of Statilius , in hort is Vaticanis in Rome , which he observed to be 972 such parts as the English Foot is 1000 , ( and to be 1005.17 of the Pes Colotianus , being 1000 ) whereby this Foot wil be 11.7 ferè , of the English Foot , being 12 , viz. 11.7 inches ferè , it being 11.66 ; which wanteth of the English Foot , but , 0.34 inch . And the other Foot is that of Villalpandus , deduced from the Congius of Vespasian in Rome , which he saith to be 986 parts of ou● English Foot containining 1000 , ( and 1019.65 , of Pe● Colotianus , being 1000 ) and so is 11.8 of the English Foot being 12 , viz. 11.8 inches , which wants of the whole English Foot , only 0.2 inch . But the ancient Greek Foot doth by his observation more nearly agree with our English Foot , then this last Roman Foot , being ( by his collation ) 1007.29 , of the English Foot 1000 , which is hardly .09 of an inch above a Foot English , it being 12.087 inches english .

As for the Weights used by Ghetaldus in his forenamed Treatise , which he saith to be the Weights used in his time ( which is not very long since ) they are surely the Roman weights ( for he lived at Rome as I take it , when he wrote that book ) which have continued the same for many ages without alteration , as some writers do attest : And which Mr. Greaves in his Discourse of the Denarius ( annexed to that of the Roman Foot ) which he puts as an undeniable principle and foundation from whence the weights of the Ancients may be deduced , as the Roman Foot for the Principle of their Measures ) The Roman Weight , and our Troy or Gold-smiths Weight , compared together . having collated with the Troy-weights from our English Standard for Gold and Silver , by grains thereof , saith , that the Roman Pound both ancient and moderne containeth 5256 such grains , and so the Roman Ounce both ancient and modern , 438 of the same grains , the Troy-pound containing 5760 grains ( as I noted before ) and so the Troy-ounce , 480 : whereupon the * * Thomasius in the end of his Dictionary , reducing the weights and measures , &c. of the Auncients , to those in use with the English nation ; saith , that the Roman Pound is 10 oun. and an half Troy : and so the Roman ounce is 3 quarters and an half of an ounce-troy . By which accompt , the Roman pound wil contain but 5040 english graines , or such as our troypound contains , 5760 , which comes short of Mr. Greaves his accompt by 216 graines And so the Romane Ounce will containe but 420 such graines , which falleth short of Master Greaves his observation , by 18 grains . But herein wee rather give credit to the later and exacter observations and experiments of Mr. Greaves . But yet how our Troy-weight may have been altered since Thomasius his time , I know not . Romane Pound and Ounce should be ( in the least terms ) but 73/80 of our English pound and Ounce Troy-weight , and so the proportion of the Roman pound & ounce to our Troy-pound & ounce ( for the conversion or commuration of the Roman weight to our Troy-weight ) as 80 to 73 ( which is the commonly received proportion of the Ounce - avoirdupois to the Ounce - troy , for the like conversion , as I shal shew afterwards , and which is decimally , as 10000 to 9125 exactly . ) As for the division of the Romane ounce immediately by scruples , in number XXIV which Ghetaldus useth , Mr Greaves speaketh not of it , but only by drams , in number VIII , as the Ounce is commonly divided by the Physitians of all Countreys , and the physicall or medicinal weights which we use in England , are the same with the Troy-weights for Gold and Silver ; only the Ounce-troy is commonly by our Goldsmiths divided into 20 penny-weights ( as I shewed before ) a penny-weight consisting of 24 grains : and the Ounce by Physitians is universally divided into 8 drachmes ; a Drachme into 3 scruples ; and a scruple ( by an evil custome received in shops ) into 20 graines , which ought to have ( according to the ancient custome ) Morellus in cap. 1. Prolegom . ad composit . medicament . 24 grains , and so be equal with a penny-weight ; so that the Number of grains in the medicinall Pound and Ounce , is the same with that in the Troy : For the scruple being 20 grains , the Dram wil be 60 , the Ounce 480 , and so the Pound 5760 , as before-noted : whereas else the Scruple being made 24 grains , ( as anciently it was ) the dram would be with us , 72 gr. the Ounce 576 , and so the Pound 6912 gr. as Ghetaldus hath it ; but the number of grains contained in the Roman Pound or ounce , with the Romanes themselves , Mr. Greaves sheweth not , whereby we might find what the difference is between their grain and ours : But collating the number of grains contained in the Pound or Ounce used by Ghetaldus ( which we take to be the same which Mr. Greaves noteth for the Roman-pound and Ounce , both ancient and modern ) with the number of grains from the English-Standard for Gold and Silver , contained in the said Roman-pound and Ounce , ( which we shewed even now from Mr. Greaves ) we shal find the Grain ( such as the Roman-pound contains 6912 , and the Ounce 576 , according to Ghetaldus , as we lately shewed ) to be ( in the least terms ) but 73/96 of the English grain , ( such as the Romane pound contains 5256 , and the Romane Ounce 438 , according to the observations and experiments of Mr. Greaves ) So that the Romanegrain should be to the Troy grain , from the English-Standard ( for the conversion of Romane grains to our Troygrains ) as 96 to 73 ; & so consequently the Roman-weights in generall ( as pounds and ounces ) reduced into the proper grains , wil be to our Troy-weight in graines , accordingly .

And according to these collations and proportions of these two Weights the one to the other ; the former iron-Spheare weighing with Ghetaldus 12 li. 2. oun. 1 scrup. 14 2/37 gr. or 84134 2/37 graines Roman ( according to his measure of the Spheare's Diameter by inches of the Roman Foot ) will be in Troy-weight from the English Standard , 11 li. 1 oun. 2 drams , 16 1●4/111 gr. ( or according to the vulgar Division of the ounce-troy by penny-weights , 11 li. 1 oun. 5 penny-weights , and 16 1●4/111 gr. ) or 63976 104/111 gr. which is very neer 63977 gr. For first , I say ,

As 80 to 73 , So 12 li. 2 oun. 1 scrup. 14 gr. Roman , ( viz. 12 li. 2 oun. 38 gr. ) or 146 38/576 oun. Roman , to 133 6●●1/23●40 ounces-troy ; which is 133 oun. and 137 gr. ferè : and these reduced into libral weight , are 11 li. 1 oun. 2 dr. or 5 p. w. and 17 gr. ferè , as before .

Then secondly ; As 96 to 73 , So are 84134 2/37 gr. Roman , to 63976 104/111 gr. Troy , as before .

Which exceeds the former troy-weight of this Spheare ( deduced from its avoirdupois-weight , according to the common proportion , 60 to 73 ) viz. 10 li. 11 oun. 3 drams , and 12 gr. ( or 10 li. 11 oun. and 8 penny-weight , as before ) or 63072 gr. by 1 oun. 7 dr. and 5 gr. or 1 oun. 17 p. w. and 17 gr. or 905 gr. which difference in the Troy-weight here , may arise not only from the difference between Ghetaldus his Weight and Measure , and ours , but also from some difference in the Metal it self , which I shal speak of afterwards . Now if we shall reduce his measure of the Spheare's dimetient , being 4/12 , or 4 inches of the Roman Foot , ( which we shewed before , to be the Pes Colotianus , as being most approved of by him , for the true ancient Roman Foot ) to inches of our English Foot ( according to Mr. Greaves his foresaid collation of these two Feet together ) the said Spheare's Dimetient will be lesse then 4 inches of the English Foot , viz. but 3.868 inches-english ( the whole English Foot , or 12 inches . English , being 12 〈…〉 inches Romane , according to Mr. Greaves his pro 〈…〉 Foot to the English Foot , 1000 to 〈…〉 before noted ) and so the former 〈…〉 the Sphear , 12li. 2 oun. 1 scrup. 14 2/37 gr. with Gheta●dus , or 11 li. 1 oun. 2 dr. 17 gr. ferè , with us in Troy-weight , ( which according to the common proportion of the Troy librall weight , to the avoirdupois librall weight , 73 to 60 , is 9 li. 2 oun. and 1.057 dr. avoirdupois ) answering to the Spheare's diametrali magnitude of 4 inches upon the Roman Foot , will answer to 3.868 inches upon the english Foot . Or againe ; if we shall reduce our english measure of the said Spheare's dimetient , 4 inches ( and so commonly holden to weigh 9 li. avoirdupois , which is 10.95 li. troy , as we shewed before ) to Roman measure in inches , we shall find the same ( according to the former Pedal Collations ) to be more then 4 Roman inches , viz. 4.136 inches , and the gravity of an iron Sphear of this Diameter , will be in Ghetaldus his weight , 93044.86 graines , or 13 li. 5 oun. 4 dr. 20.86 gr. ( or according to Ghetaldus his Division of the ounce ) 13 li. 5 oun. 12 scrup . and 21 gr. ferè ) which is in our Troy-weight ( according to the former Collations of these two weights ) 70752.86 gr. or 12 li , 3 oun. 3 dr. 12.86 gr. ( or by the common division of the ounce-troy by penny-weights , 12 li. 3 oun. 8. p. w. and 0.86 gr. ) * * Archimed . promot . theor . 9. prop. 17. For seing that Spheares of the same kind , are among themselves in gravity , as the Cubes of their Dimetients are in magnitude : Therefore the weight of the iron-Spheare , whose Diameter is 4 inches Romane-measure , or 3.8 68 inches english-measure , being found as before ; the weight of the other Spheare of the same kind , whose diameter is 4 inches english-measure , or 4.136 inches Roman-measure , will be found also as before : Or more readily , by having the weight of such a Spheare , whose Diameter is one inch , which by Ghetaldus his Measure and weight , is 1314 22/77 graines . or 2 oun. 6 scrup. 18 22/●7 gr . and so in our troy-weight , 999 71/111 grains , or 2 oun. 1 p. w. and 16 gr. which weight therefore will answer to an iron-Spheare whose Diameter is 0.967 inch , english-measure , for this answers to one inch-Roman ; and so the weight of such a Spheare who●e Diameter is one inch-english ( which is 1.034 inch-romane ) should be by this accompt , 1105.51 graines-English , or 2 oun. 2 dr. 25.5 gr. or 2 oun. 6 p. w. and 1.5 gr. troy ( which are in Roman weight , 1453.83 gr. or 2 oun. 4 dr. 13.8 gr. or by Ghetaldus his division of the ounce ; 2 oun. 12 scrup . and 13.8 gr. )

Or ( here briefly to shew the use of the former proportional Numbers for Metals among themselves ) in respect of two Spheares of different kinds of Metals , and like magnitude ) the same weight of the foregoing Spheare of Iron , of 4 inches english-measure , or 4.136 inches Roman-measure , the diameter maybe produced as before , by the weight of a Sphear of any other Metal first had , being of the same magnitude : As for example , a Sphear of Tinne , ( for so Ghetaldus commonly deduceth the weights of other metalline Sphears from a stanneal Sphear , or a Sphear of Tinne ) whose diametrall magnitude is 4 inches-english , or 4.136 inches-Romane , I find to weigh by Ghetaldus , 86066.5 grains ( or 11li. 5 oun. 10 scrup . and 2.5 gr. ) which is with us in Troy-weight , 65446.4 gr. ( or 11li. 4 oun. 2 dr. 46.4 gr. or 11li. 4 oun. 6p . w. 22.4 gr. ) Now therefore , according to the foresaid propo●●onall Numbers for Tinne and Iron , I say ; As 3895 to 4210 ( which is with Ghetaldus in his fore-mentioned second Table of the Comparison of divers kinds of Bodies in gravity and magnitude , as 38 18/19 to 42 1/19 ) or more accurately ( the former termes being incompleat and unabsolute ) As 1554 to 1680 ( which is with Ghetaldus in his foresaid first Table of the like Comparisons , as 1 to 1 3/37 , that is , as 37 to 40 , and which is decimally , as 1. to 1.081,081 , infinitly ; or integrally , 1000 to 1081 , or 1000,000 to 1081 , 081 compleatly ) So is 86066.5 gr. ( or 12li. 5 oun. 10 scr. 2.5 gr. ) the Sphear of Tinne in Ghetaldus his weight , or 65446.4 gr. ( viz. 11li. 4 oun. 2 dr. 46.4 gr. ) the same Sphear in our Troy-weight , to 93044.86 gr. ( or 13li. 5 oun. 12. scr. 21 gr. ) the Sphear of Iron in Ghetaldus his weight ; or to 707.2.86 gr. ( viz. 12li. 3 oun. 3 dr. 12.86 gr. ) the same Sphear in our troy-weight , as formerly : which being converted into Avoirdupois-weight , ( according to the former terms of proportion between these two Weights ) wil be 10.0961 . li. viz. 10li. 1 oun. 8.576 dr. according to the common division of the Pound-avoirdupois into 16 ounces , and so of the ounce into 16 drams , and under which division or parts of weight , they go not ; this kind of weight serving with us for al kind of coarser or grosser Commodities , it being the common Mercatory weight , ( and called by a French name , Avoirdupois ) and so for the weighing of all Metals , but Gold and Silver , to which only the Troy-weight is assigned : Neither indeed is there any need of such exactnesse in the weight of a bullet or any other Body of Iron , or other like base Metall , as to graines , but only to satisfie Art it self in the curiosity thereof .

But now whether the common Tenent of an iron-bullet of 4 inches the diameter , to weigh just 9 li auoirdupois , were deduced from any certain and exact experiment or no , is a question here to be made ; Nor was it my purpose to trie the same , having not conveniences and accomodations thereunto : but only having the true weight of a Sphear or Bullet of Iron of any magnitude , in a certain measure given ( as Inches ) thereby to shew a way for the exact and most speedy obtaining of the weight of a Sphear or bullet of any magnitu●e whatsoever , which I have partly declared & demonstrated already , and shal more by & by : And looking in severall books of Gunnery , wherein are set down the weights of iron-bullets ( or round shot , as they call it ) fitted to all the usual pieces of Ordnance with us in England , according to the diameters of their bore , mouths , or concavities ( abating usually 1/4 of an inch of the Gun's said diameter , for the diameter of the bullet ) I found in one of them , the weight assigned to an iron-bullet of 4 inches diameter or crassitude , to be just 9li. and in another book this weight assigned to a bullet of 4 1/2 inches the diameter and so in other books , other weights assigned to a bullet of the same magnitude : And therefore finding such a discrepancie among our Masters of Gunnery in this thing , so that I could not discover from them any certainty in the weight of an iron-Sphear or bullet , at a certain magnitude or crassitude : I repaired one day in August 1648 , unto Mr. John Reynolds , one of the Clarks of the Mint in the Tower of London , ( and sometime Assay-master at Goldsmiths-hall ) a man much noted by artists for his industry and ingenu●ty in the Mathematiques ; ( and so indeed I find him to be ) who very courteously entertaining me at his house in the Tower , with discourse about many exexperiments made by him in the Mathematicks , and shewing the same to me ( as he had once done in some of them , long before ) among which were those concerning the weights of Metals , and their proportions one to another : I desired of him to be resolved in this point of art concerning the weight of a Sphear or bullet of Iron , according to some certain magnitude or crassitude , knowing him to have all accomodations fit for the finding out of the same ; and he thereupon produced me an experiment made by him not long before , upon a large bullet , which he then shewed me ; but I observed the same to be very unfit to ground an experiment upon , being not only much rusty , but also having severall holes and cavities therein , ( which seemed to be chiefly from the antiquity of it ) which might wel hinder the finding of the true weight , according to its diametrall magnitude , and which himselfe then doubted much of : whereupon I importuned him for another the like experiment , which might be exact ; And so both for his own satisfaction , and mine also , he soon after , got another bullet , which was very sound & solid , and clear enough from rust , and also as round every way , as could well be imagined ; whose Diameter we thereupon took by a pair of Callaper Compasses ( conceiving it to be a surer way to find the same , then by the Circumference ) as precisely as possibly we could , and measuring the same upon a line of inches , found it to be 4.8 inches ; and then for my further satisfaction , I took the Circumference of the bullet with a thred , by means of the small crease or Circle which encompassed the bullet exactly in the middle , being made by the Mould in which it was cast , and measuring the thred upon the said line of inches , we found it to be 15.25 inches , which by Cycloperimetricall proportionality , gives the diameter as before ; so that we might wel conclude , the magnitude of this bullet to be rightly found by us . Then for the weight thereof , we tried the same with all possible precisenesse , both by Avoirdupois-weights , and Troy-weights , and found it to weigh , 15li. 12 1/4 oun. avoirdupoiz ; and 19li. 1 25/32 oun. Troy , viz. 19li. 1 oun. p. w. 15 gr. According to which experiment , an iron-bullet ( made of cast iron , such as bullets are usually made of , and which weigheth much lighter then forged iron ) of 4 inches the diameter , wil weigh upon the point of 9li. and 2 oun. avoirdepois , which differs not much from the common Tenent .

And so according to this our experiment ( to which we wil adhere for the finding out of the weight of any Sphear or bullet made of cast-iron ) the artificial Line of measure , or cubatory Line of gravity , for the speedy discovering of the weight of any Sphear or bullet whatsoever , made of cast-iron , by the Diameter or Circumference thereof , as was formerly shewed , wil be 1.91 inch , for the Diameter , and 6.01 inches for the Circumference ; and these in respect of librall weight , as the two former Lines , viz. 1.92 inch , and 6.04 inches , deduced from the former common Tenent of the weight of an iron bullet of 4 inches diameter , which differ but very little one from another , as they are here set , from a centesimal division of the Inch. Now these two latter Lines being divided as the former , and the Diameter and Circumference of the foresaid Sphear or bullet measured thereby , the same wil be found severally , to be 2.09 ferè ( which by the other Lines were 2.08 ) which cubed , yields 9.129329 ferè . for the weight of the bullet , which is 9 li. 2 oun. 1 dr. averdepoiz , differing from the true weight ( by way of excesse ) only 1 dr. which is not considerable .

But now what sort of Iron , Marinus Ghetaldus in his experiments upon the same , for the weight thereof , and so its proportion to other Metals , meaneth , whereby a Sphear of 4 inches diameter , english-measure , ( or 4.136 inches Roman-measure ) should weigh according to him , 10.096 li. avoirdupois ( as being deduced from 12.283 li. troy and that from 13.461 li. Roman ) we know not , if Mr. Greaves his foresaid collations and comparisons of the Romane weight and Measure ( used by Ghetaldus ) with the English , be true , ( as we are willing to beleeve they are ) and also the foresaid proportion of the Troy-pound-weight to thè Avoirdupois-pound-weight , as it is commonly holden , viz. 73 to 60 ; although the aforenamed Mr. Reynolds will have it to be , as 17 to 14 ; deducing the same from a generall The Troy and Avoirdupois Pound compared together . Maxime , and undeniable Principle ( as he saith ) that 136 pounds-Troy , and 112 pounds Avoirdupois are equilibral , or equivalent in weight , which are in the least terms of proportionality , 17 and 14 ; and which indeed I find by experience to be the truer , as I shall afterwards shew : And this is to be understood properly ( as the Terms here stand , from the greater to the lesse ) for the reduction of Troy-pound-weight ( or Troy-pounds ) to Avoirdupois-pound-weight ( or Avoirdupois-pounds ) in asmuch as 73li. Troy , make but 60li. Avoirdupois ; or rather 17li. troy make but 14li. avoirdupois : and so the Avoirdupois Pound , is ( in the least terms ) 73/60 , or rather 17/14 the Troy-Pound ; whereas else , if we will precisely compare the Pound-troy simply , as the lesse , with the Pound-avoirdupois , as the greater ; then must the Terms be rather taken the contrary way : And thereupon the Pound-troy will be to the Pound-avoirdupois , as 60 to 73 , or rather , 14 to 17 : And so the Pound-troy will be ( in the least terms ) 60/73 , or rather 14/17 of the Pound-avoirdupois . And which several Terms of proportion , though they seem , to differ but very litle in the Reason thereof it selfe ; yet may the difference of the weights of things , produced severally thereby ( by way of conversion or reduction of one kind of weight to the other ) be many times considerable ; and the more , the greater that the quantity of the weight so reduced , is ; as I shall plainly shew afterwards : And so I will here first compare these two severall Proportions together ( and that according to the common acception of the Terms , from the Pound-troy to the Pound-avoirdupois , viz. the greater Term as antecedent , to the lesse , as consequent , and so the reasons of the Terms will be of the greater inequality ; and the rationality of the first , or common Terms , will be ( by prolation , from the Parabole of the said Terms , or Quantity of the Reason ) super-tredecupartiens-sex agesimas , viz. superpartient 13/6● : and of the other Terms , supertripartiens quartas-decimas , viz. superpartient 3/14 , and so the difference of the Reasons , or the differentiall Reason , but that of 1. to 420 , viz. 1/420 , according to a plain or simple subduction , or differencing of Reasons , by reduction of them to one common or conjunct Consequent , which imitates the subduction Rom. Arithm. l. 2. c. 2 & 3. à Laz. Schon . Et Clav. de Proport . composit . in fine 9 Elem. Eucl. of Fractionall Numbers , by reduction to one common or conjunct Denominatour ; and which is the most genuine , proper , and rational subduction , or discrimination of Reasons ( both as Clavius , Ramus and divers other of the best Authours do teach ) thus ;

Or more plainly thus , after the manner of Fractions .

And which may also be seen by the like operation in the superpartient termes only , thus :

And not according to that operation of Reasons or Proportions , which some do falsly and improperly call the subduction or subtraction of Reasons ( as Clavius also saith in the place fore-cited ) being indeed the true division or Resolution of Reasons , and which is as the division of Fractions ; whereby they make the quotall Reason to be the differentiall or residuall Reason ; and which would then be here by prolation , of the greater inequality , viz. 1 1/510 , according to the termes 511/510 ( whereas the other or true differentiall Reason , is of the lesse inequality , by very much ) but approaching very nigh unto a Reason of equality , or unity , or a Reason singular and individual ; and this latter is also unexpressible or indenominate as the former , and is had by a conjunct composition only of the alternate or heterologall terms , thus .

Anteced . 73. 17 1022 511 Quotient . Conseq . 60 14 1020 510 Quotient .

Or more plainly thus ,

Or yet more plainly and readily , by permutation of the termes of the dividing or resolving Reason , in respect of their places ; for so the work of Resolution wil be changed into a Composition of the termes , according to the ordinary Multiplication of Fractions , thus ,

And the latter of these Proportions between the Troy and Avoirdupois-pound , makes the Pound Troy to be more of the Pound Avoirdupois , or to come nearer the same in quantity , then the other or common Proportion doth ; as may be more plainly discerned , by reducing the said severall Proportions into decimal Termes , for so , the Proportion of the Pound-troy , to the Pound-avoirdupois from the Termes of 73 to 60 , wil be , as 1.0000 . to .8219 , and from the terms of 17 to 14 , as 1.0000 to .8235 . Or by taking the Terms the contrary way , ( as for the conversion or reduction of Avoirdupois-pounds to Troy-pounds ) the Pound-avoirdupois wil be to the Pound-troy from the termes of 60 to 73 , decimally , as 1.00000 , &c. to 1.21666 , &c. infinitely ; and from the Terms of 14 to 17 , as 1.00000 , &c. to 1.21428 , &c. and so here the Pound Avoirdupois is by these latter Terms , lesse of the Pound Troy , ( and so comes nearer the same ) then by the other Terms . And the Reason of these latter Terms ( as they here stand , from the lesse to the greater , ) wil be greater then the Reason of the first or common Terms ( according to the exact comparing of Proportions or Reasons together , as you may see in the fore-cited places of Ramus and Clavius ) in regard that the two severall Reasons being reduced to one common Consequent ( as aforesaid ) the new , compound or correspondent Antecedent of the latter Proportion , wil be greater then the like Term of the first ; or the foresaid Antecedent of the latter Proportion , wil be more of the common Consequent , then the like Antecedent of the first Proportion ( as in the former operations , the contrary happened , where the Terms of Proportion were put the contrary way , viz. from the greater Term as antecedent , to the lesse , as Consequent ) as you may here plainly see by these severall subsequent operations .

Or more plainly thus , in a Fractionall manner .

Or again , the same wil appear by a contrary operation , which is by reducing the two severall Proportions ( as the terms be here put ) to one common Antecedent ; for so the new , compound , or correspondent Consequent of the latter Proportion or Reason , wil be lesse then the like Te●● of the first or common Proportion , ( and thereby the Reason of these latter Terms , wil accordingly be greater then that of the other Terms ( according to the fore-cited Authours ) as you may plainly see by this next operation following ; the compound Antecedents of the operations next beforegoing , being here changed into the like Consequents .

And here the difference of the Reasons ( or the differentiall Reason ) wil be that of 2 to 1241 , viz. 2/1241 , as you may see in the first of these two operations before-going .

And now according to these latter termer termes of Proportion of the Po●nd-troy to the Pound-avoi●dupois , viz. 17 to 14 ; the foresaid Sphear of Iron wil be in avoirdupois weight , 10.1158 li , which ( by conversion of the fraction-part into the proper parts of this weight ) is 10 li. 1 oun. 13.6 dr. whereas by the common terms , it was 10 li. 1 oun. 8.6 dr. and so the difference of weight , 5 dr. exactly . But I conceive ( as in all probability and reason I should ) the Iron used by Ghetaldus in his experiments , to be the finer sort of iron , or forged Iron , which weigheth heavier then cast , coarse , or drossie iron , the proportion of weight between them , being ( as I have deduced it from the experiments of Mr. Reynolds ; and partly of my self also , upon this Metal ) in general , such as is between 1000000 and 951832 arguing from forged Iron to cast iron , as from the more to the lesse : and so contrarily from cast-iron to forged iron , as from The Proportion between forged iron and cast-iron . the lesser to the greater , the proportion of weight , wil be such as of 1000000 to 1050605 : For the said Mr. Reynolds found the weight of a cube-inch of fine Iron , which had been kept in the Treasury of the Tower of London , ever since King Henry the 7th . his time or longer ( and that very neatly in a velvet-Case ) to be 4.169 ounces Troy , viz. 4 oun. 3 p. w. & 9.12 gr. troy . which is 0.3474li. troy : And from the former Bullet of cast-iron of 4.8 inches the Diameter , which we found to weigh 19 pound , 1 oun. 15 penny-weight , and 15. grains troy , or 229. ounces , 15 p. w. and 15 gr. ( which is 229 25/32 ounces troy ) we gather the weight of one cube-inch of cast-iron to be 3.968 oun. troy ; which is 0.33068li. troy . And so according to this experiment , a Sphear or bullet of one inch Diameter , made of fine , or forged Iron , wil weigh a .18288 oun. troy , which is 0.1819li. troy : and so a Sphear or bullet of 4 inches Diameter , made of the same metal , wil weigh 11.642 poundstroy ( or 11li. 7 oun. 5 dr. and 38.17 gr. or 11li. 7 oun. 14 p. w. and 2.17 gr. ) which being compared with the weight of an iron-Sphear of 4 inches Diameter ( ●nglish-measure ) decuced f●om Ghetaldus before into our Troy-weight , viz. 12.283li. &c. troy , or 12li. 3 oun. 3 dr. 12.86 gr. or 12li. 3 oun. 8 p. w. and 0.86 gr. ) it will be found to want thereof , 7 oun. 5 dr. 34.11 gr. or 7 oun. 13 p. w. and 22.11 gr. ( according to the difference between 12.283483 li and 11.642044 li. being 0.641339 li. ) And so the weight of a Sphear of fine Iron of 1 inch diameter , here found 2.18288 oun. troy , viz. 2 oun. 1 dr. 27.78 gr. or ● oun . 3 p. w. 15.78 gr. being compared with the weight of a ferreall or iron-Sphear of the same Diameter , deduced formerly from Ghetaldus into our troy-weight , viz. 2.30315 oun. or 2 oun. 2 dr. 25.5 gr. or 2 oun. 6. p. w. and 1.5 gr. it wil be found to want thereof 57.7 gr. viz. 2 p. w. 9.7 g. which operations agreeing so nearly one with another , it is thereby manifest , that the Iron meant by Ghetaldus , is the finer sort of Iron , or purely forged iron ; But that we cannot make his experiments and ours exactly to agree in a Sphear of this Metall ; may happen not only in respect of the difference between his weight and measure and our● , and the uncertainty of the proportions between them , whereby the one might be exactly reduced to the other ; but also in respect of the difference between the Metal used by him and us ; for that all Iron ( or other Metal ) of the like sort , is not always of the same gravity or ponderosity precisely because all is not of a like finesse or coarsnesse , and so that which is finest , wil still be heaviest .

Now Ghetaldus beginning to find out the weights of metalline Spheares , according to the usuall known Weights with him , found that no diligence or industry of man could make a Spheare so exact as it ought to be , and therefore he procured a Cylinder to be made , and that of Tinne , equall in height to the Diameter of its Base ( which were of a certaine magnitude in inches of the Roman Foot ) for that this might be turned in a Lathe much exacter , and more easily then a Spheare : and hereby found , that a Cylinder made of Tinne , being of one inch or 1/12 of the Roman Foot in its crassitude and altitude , did weigh 3 oun. 4 scrup . which reduced into graines of his weight , is 1824 gr. whose 2/3 being 1216 gr. is the weight or gravity of a Spheare of the same Metall , whose diameter is equal with the diameter or height of the Cylinder ; according to the demonstrations of Archimedes , lib. 1. de Sph. and Cyl . prop. 32. It being there shewed by him , that a Cylinder , whose Base is equall to the greatest Circle in a Spheare , and its altitude equal to the diameter of the Spheare ( or of the said Circle ) is sesquialter the said Spheare . And so Ghetaldus having found the weight of one Spheare of Tinne ( at a certaine magnitude ) from thence he deduceth the weight of any other Spheare , and that not only of the same Metall , but also of any other metall , by the proportions of Tinne to the other Metals ( in like Bodies of equal magnitude ) found out by him at first .

Now the foresaid weight of 1216 gr. ( or 2 oun. 3 scr. 16 gr. ) for a Spheare of Tinne of 1 inch the diameter with Ghetaldus , will answer to the like kind of Spheare whose diameter is 0.967 inch from the English Foot , ( according Mr. Greaves his foresaid comparison of the Roman Foot with the English ) which being converted into our Troy-weight ( according to Mr. Greaves his foresaid comparison of the Romane weight with our Troy-weight ) is 924 2/3 gr . ( viz. 1 oun. 7 dr. 24 2/3 gr . or 1 oun. 18 p. w. 12 2/3 gr . ) and hence we gather the weight of a stanneall Spheare , whose diameter is 1 inch , or 1/12 of the English Foot , to be 1022.6 gr. english , ( viz. 2 oun. 1 dr. 2.6 gr. or 2 oun. 2 p. w. 14.6 gr. ) whereby we may easily obtaine the weight of any other Spheare of the same Metall , and also of any Sphear of any the other metals , by means of the foregoing proportionall Numbers between Tinne and those other Metals : For so a Sphear or Bullet of Gold ( supposed fine ) of one inch diameter English-measure , will be found , ( according to the proportion of 3895 to 10000 , which is with Ghetaldus in his second Table of the comparison of several sorts of Bodies in gravity and magnitude , 38 18/19 to 100 ) or rather ( the former termes being uncompleat ) of 1554 to 3990 ( which is with Ghetaldus in his first Table of the comparison of the same Bodies in gravity and magnitude , 1 to 2 21/37 , and that 's as 37 to 95 in the least rationall and absolute termes ) to weigh 2625.6 gr. english , ( which is 5.47 oun. ●roy , viz. 5 1/● ferè , being 5 ou. 3dr . 45.6 gr. or 5 oun. 9 p. w. 9.6 gr. ) And so the like Body of fine Silver , of the same magnitude , will be found , ( according to the proportion of 3895 to 5439 , which is with Ghetaldus in his foresaid second Table of the Comparison of severall Bodies in gravity and magnitude , 38 18/19 to 54 22/57 ) or more accurately ( the former termes being not absolute or comp●eat ) of 1554 to 2170 , ( which is with Ghetaldus is his forementioned first Table of the like Comparisons , 1 to 1 44/11● , and that is as 111 to 155 , in the least term●s rationall and absolute , and which are by decimall numeration , as 1. to 1.396 , 396 infinitly ; or integrally , 1000 to 1396 , or 1000 , 000 to 1396 , 396 exactly ) to weigh 1427.95 gr. english ( viz. 2 oun. 7 dr. 47.95 gr. troy , or 2 oun. 19 p. w. 11.95 gr. ) In like manner will be deduced the weight of Iron and the other baser sort of Metals , from Tin●e ( in Bodies of like form and magnitude ) But because the other Me●als besides Gold and Silver , as usually weighed with us , by the common Mercatory weight aforesaid , called Avoirdupois . Therefore for the more immediate and speedy obtaining of their weights from Tinne ( in sphericall bodies ) it is best to have the weight of the foregoing Stanneall Spheare of one inch diameter , viz. 1022 ●/5 gr . english , or 2.1304 oun , troy , ( viz. 2 oun. 2 p. w. 14.6 gr. as before ) reduced into avoirdupois unciall weight , and that decimally , whereby the weight of the like kind of Spheare of any other magnitude may be readily obtained ; and so consequently the weight of any other metalline Spheare therefrom , in its proper weight of Avoirdupois . Which said weight therfore of 2.1304 oun. troy , will be inavo●rdupois unciall weight ( according to the commonly received proportion of the Troy ounce-weight to the avoirdupois ounce-weight , 73 to 80 ) 2.3347 oun. ( viz. 2 oun. 5.355 dr. ) But according to Mr. Reynolds his proportion of these two weights the one to the other , which is , as 51 to 56 ( being deduced from his proportion of the Troy-pound-weight to the Avoirdupois-pound-weight noted before , and which proportions of weight are the nearest aod truest that may be , as I shall straightway shew ) the same spheare will be 2.33928 oun. avoirdupois ( viz. 2 oun. 5.428 dr. ) which exceedeth the former weight only .073 of a dram , which in this kind of weight is altogether inconsiderable .

Or for the ready obtaining of a Sphear of Tinne , of any magnitude , ( and so of any other Metal therefrom ) in librall-weight , it is convenient to have the foresaid Sphear of Tinne of 1 inch diameter , in libral-weight , both Troy and Avoirdupois , and that decimally ; which wil here be in Troy-weight , 0 . 17753li . and in Avoirdupois-weight , ( which is here chiefly to be regarded ) according to the commonly received proportion of the troy-weight to the avoirdupois weight ) 0 . 1459189li . and according the other ( and better ) proportion , 0 . 146205li . the difference of which from the other ( by way of excesse ) being hardly , 0 . 0003li . viz. 3 parts of a pound-avoirdupois , divided into 10000 equal parts ; Which difference of weight , although it be here of little or no value ( as also that in the operation immediatly preceding ) in regard of the smalnesse of the Body here handled , whereby the difference between the common Proportions of the Troy-weight to the Avoirdupois-weight , and the other proportions , may seem to be very small , and inconsiderable : Yet if we go to to reduce a metalline ( or other ) Body of a greater magnitude or dimenon , out of one of these Weights into the other ( according to the said severall proportions beforegoing ) we shal find the difference of weight ( in one and the same kind ) to be stil greater , and the greater or weightier that the Body is , the greater wil be the difference of the weight produced by the said different termes of proportion , in respect both of librall and unciall weight , insomuch as that the difference of weight will many times be considerable : As in the foregoing Sphear of iron , of 4 inches the diameter , ( english-measure ) whose weight being found from Ghetaldus , to be 13 . 461li . Romane , and from that , to be in Troy weight ( according to Mr. Greaves his collations of these two kinds of weight the one with the other ) 12 . 283li . and then the same converted into Avoirdupois weight , according to the two severall proportions between these two weights ; the difference of weight was there found to be 5 drams auoirdepois , which is almost 1/3 of an ounce . And greater differences of weight ( in this respect ) then this , I shall shew by and by , and also more afterwards .

Now therefore , for the proportion between the Troy-ounce , and the Avoirdupois-ounce The Troy and Avoirdupois Ounce compared together ; as also the severall Proportions assigned between them . according to the Terms 51 and 56 ( deduced from the foresaid Maxime or Principle of librall weight , that 136li . Troy , and 112li . Avoirdupois are eqvilibral , or equiponderant , and so accordingly from thence , 1632 oun. Troy and 1792 oun. Avoirdupois , which are in the least terms of Proportion , 51 and 56 ) the same being compared with the foresaid common termes , 73 and 80 , wil seem indeed to differ but little therefrom in the Reason it self , and that deficiently , as the terms here stand , from the lesse to the greater , like as the correspondent Terms before for librall weight , did from the common terms , being taken from the greater term to the lesse , viz 7● to 60 , and

Thomasius in the fore-cited place of his Dicti●n●●y , saith , that the Goldsa●ith , ( or Troy ) Ounce hath to the Avoirdupois Ounce , proportion ●●squiund●cimal , So that ( to wit ) 11 ounce troy are exactly equall to 12 ounces avoirdupois : and which ( he saith ) he proved by a m●st just and exact Balla●c● . But this Prop●rtion is som●what greater then either o● the other two here noted , & makes the ounce Averd. to be more of the ounce Troy then either of those tw● .

Mr. wingate in his A●ithm . l. 1 c. 1. Sect. 34. saith , that the avoirdupois . P●und is composed of 14 ●un . troy , and 12 p. w. viz. 14.6 oun. troy ; which is according to the commonly r●ce●ved Prop●●tiors between the Avoirpois and Troy-weight : But according to the other proportions , it will be most truly but 14 oun. 11 p. w. and 10 2/7 gr . troy .

17 to 14 : the Reason of the common terms , as they here stand from the lesse as the Antecedent , to the greater , as the Consequent , being by prolation , of the less in●quality , or inequality of the lesse , sub-super-septupartiens septuagesimas-tertias , viz. sub-super-partient ●/73 ; and of the other Terms , ( standing correspōdently ) sub-super-quintupartiens quinquage simasprimas viz. sub-superpartient 5/51 ; and so the difference of the Reasons , or the differentiall or residuall Reason ( according to the plain , simple and proper subduction of Reasons before shewed ) that of 1 to 560 , vix . 1/56● ; And this , for the conversion or reduction of Troy-unciall weight to the like Avoirdupois-weight ; and so the Ounce-troy wil be ( in the least terms ) 80/73 or more truly , 56/51 the Ounce-avoirdupois : Or else , the Terms , in this position , from the lesse to the greater , might seem rather to be taken contrarily ( according to an exact comparison ) for the proportion of the Avoirdupois-ounce , ( as the lesse ) to the troy-onnce ( as the greater ) for so the Avoirdupois-ounce , wil be 73/8●1 or rather 51/56 of the Troy-ounce . And these latter Terms of proportion , do make the Ounce Troy more to exceed the Ounce-avoirdupois , then the first or common Terms do , and so make the Ounce-avoirdupois to come short of the Ounce-troy ( or be lesse of the same ) accordingly ; as may more easily be seen , by reducing of the foresaid Terms of proportion into decimall termes ; for so , the proportion of 73 to 80 , will be as 1.00000 to 1.09589 , and of 51 to 56 , as 1.00000 to 1.09804 ferè : And contrarily ; by permutation of the Terms , the proportion of 80 to 73 , will be decimally , as 1.0000 to .9125 exactly ( as I shewed formerly , upon another the like occasion , ) and of 56 to 51 , as 1.0000 &c. to .9107 &c.

And therefore , now to shew ( by the way ) the difference between the foresaid The two foregoing several Proportions between the Troy and Avoirdupois Weights compared , and examined experimentally , by the Ballance , in the foregoing Iron Bullet . severall proportions of the Troy and Avoirdupoiz weights both librall and unciall , the one to the other , by comparing them with the weight of some Body , taken both by Troy and Avoirdupois-weights , and that especially in one and the same Ballance , and so converting it out of the one kind of weight into the other , by the said two severall kinds of proportions between them , whereby may be known which of these are the neerer and truer , as most agreeing with the Ballance it selfe : We will here take the foregoing Cannon-bullet of cast-iron , of 4.8 inches the diameter , whose weight we found ( as I have noted before ) in the Tower of London , by an exact Ballance , with Weights both Troy and Avoirdupois , to be 19li . 1 oun. 15 p. w. 15 gr. Troy ( which is 19li . and 1 25/32 oun. or wholly in librall-weight , 19 29/12● or 19 . 1484375li . exactly ) and 15li . 12 1/4 oun. Avoirdupois ( which is wholly in librall-weight , 15 49/64 or 15 . 765625li . compleatly ) Now if we shall convert the said Bullet from its Troy-weight to Avoirdupois-weight , according to the common proportion of the Troylibrall-weight to the Avoirdupois-librall-weight , viz. 73 to 60 , we shall find the same to be ( in the least termes ) 15 1725/2336 or 15 . 73844li . Avoirdupoiz , which wanteth of the true weight from the Ballance , 0 . 02718li . whcih is by reduction or conversion into the proper denominate , compounding parts of this weight 0. oun. 6.958 drams avoirdupois , which is upon the point of 7 dr. and that 's almost halfe an ounce avoirdupois . But according to the other proportion of the Troy librall-weight to the Avoirdupois librall weight , viz. 17 to 14 , the said bullet will be found to weigh ( in the least terms ) 15 837/1088 or 15 . 7693li . Avoirdupoiz , which exceedeth the weight , not fnlly one dram , being but 0.94 dr. as will appeare by reduction of the fractionall termes into the proper denominate parts of this weight , and which is not considerable .

Againe ; if we will reduce the said Bullet taken wholly in Troy-unciall-weight , being 229 25/32 or 229.78125 oun. exactly , into Avoirdupois unciall-weight , according to the said severall proportions of the one to the other , we shall find the very same differences necessarily to happen : for according to the common proportion of the Troy-unciall-weight to the Avoirdupois , viz. 73 to 80 , it wil be found ( in the least termes ) 251 119/146 or 251.815 oun. avoirdupoiz , which wanteth of the true or ballance-weight , ( being 252.25 oun. ) 0.435 oun. which by reduction , gives 6.96 drams as before : But according to the other proportion of the Troy unciall-weight to the Avoirdupoiz , viz. 51 to 56 , it will be found 252 21/68 oun. or 252.3088 oun. avoirdupoiz , which differeth from the ballance-weight ( by way of excesse ) only 0.0588 oun. which is by reduction , 0.94 dr. as before .

And so again contrarily , if we work from the Avoirdupoiz-weight to the Troy-weight , we shal find the like proportional differences of weight accordingly . For first if we convert the foresaid bullet , out of its true avoirdupois-weight from the Ballance , 15 40/64li . or 15.7656 , &c. into Troy-weight , according to the vulgarly received proportion of 60 to 73 , for librall-weight ( being the converse of the former ) we shal find the same to be ( in the least termes ) 19 697/384● , or 1815104li. Troy , which differeth from the true weight of the Ballance ( 19 19/12● or 19.1484375 li ) by way of excesse , 0.0330729li. which is by conversion into the proper , compounding denominate parts of this weight 0. oun. 3 dr. 10.5 gr. troy , or 0. oun. 7 p. w. ●2 . 5 gr. But according to the other proportion of 14 to 17 , it wil be found ( in the least termes ) 19 129/896 , or 19 . 1439732li . Troy , which differeth from the true weight ( by way of defect ) only 0 . 0044643li . which is by a continuall reduction into the least parts of weight , but 25.7 gr. or 1 p. w. and 1.7 gr. which is of little or no value in this thing .

And so likewise if we reduce the said Bullet out of its true avoirdupois weight taken wholly in ounces ( according to the Ballance ) being 252 1/4 or 252.25 oun. into Troy weight by ounces ; first , according to the common proportion of 80 to 73 , for unciall-weight ( being the converse of the former ) we shall find the same to be ( in the least terms 230 57/320 oun. troy , or 230.178125 compleatly , which exceedeth the true weight , 229 25/32 oun. troy , or 229.78125 exactly , by 0.396875 oun. troy exactly , which is by reduction into the least parts of weight , 3 dr. 10.5 gr. or 7 p. w. 22.5 gr. as before from librall weight : But according to the proportion of 56 to 51 , it wil be found to be ( in the least terms ) 229 163/224 oun. troy . or 229.727678 , which is deficient from the true weight , only 0.053572 oun. which by the like reduction , is but 1. p. w. and 1.7 gr. as before from librall weight .

By which operations it is sufficiently evident , that these latter Proportions between the Troy and Avoirdupoiz weights , are the truer , ( and indeed the nearest and truest that may be found ) and which I shall ( upon the like occasion ) further confirm afterwards by a double experiment from the weight of a liquid body , in the measuring of Vessels .

And thus much by the way concerning Weight and Measure in generall , in reference to the work here in hand , ( being the like artificial Dimension of metalline regular Bodies , for the speedy discovering of their gravities or weights , and more particularly of a ferreall or iron Sphear , as was formerly of a Sphear in generall for solid measure ) being induced thereunto by Ghetaldus in his foresaid work of Metals , in which he differeth from us in both these , as we have abundantly shewed .

And so these operations beforegoing in the particular metalline Sphears aforenamed , for the weight thereof , are from the experiments of M. Ghetaldus , & reduced from his weight and measure to ours , according to the observations and experiments of our Countreyman Mr. Greaves upon the same , and his collations of them together , as aforesaid ; And also in one of them , from my own experience , according to our English weights and measure ; with which we must rest contented , til some other experiments be produced , both in these , and also in the other Metals , from our English weight and measure , and which we may expect from Mr. Reynolds aforesaid , who hath taken great care and pains , and used much industry therin .

As for the weights of Metals compared in Sphears of one and the same magnitude , set down in the latter part of Mr. Ponds Almanack , in Troy-weight ( where also are noted the foresaid common proportions between Troy and Avoirdupois weights ) they are arcording to the experiments of Ghetaldus , being deduced from his proportional or comparative numbers , into the parts of Troy-weight , though not very precisely ; which therefore I have here put most correct and exact thus ; supposing ( with him ) first a Sphear of Gold , to weigh just one pound-troy , and from thence , the other metalline Sphears of the same magnitude , to weigh accordingly , as followeth .

oun . p. w. gr. mi. gr. 1. G. 12. 00. 00. 00. 5760. ☉ 1 2. Q-S . 8. 11. 10. 6. ferè 4114 2/7 ☿ 2 3. I. 7. 5. 6. 6. 3486 6/19 ♄ 3 4. S. 6. 10. 12. 12. 3132 12/19 ☾ 4 5. B. 5. 13. 16. 8. 2728 8/19 ♀ 5 6. I. 5. 1. 1. 5. 2425 5/19 ♂ 6 7. T. 4. 13. 11. 7. 2243 7/19 Jupit ; 7

But now to return a little to our foregoing work of the artificiall dimension , or diametral and Circumferentiall cubation of a Sphear of cast-iron , for the weight thereof , which as we shewed before by the Integer of weight it self ( or by librall weight ) So we will next shew how to perform the same by the composing , denominate parts therof immediatly , viz. ounces avoirdupoiz ( or uncial weight ) And the Line or Scale of equal parts for this purpose , in respect both of the foresaid common Tenent of the weight of an iron-Sphear or bullet , and also our own experiment , I finde ( according to the reason of the precedent second Theoreme , and also our generall reason ) to be for the Diameter of such a Sphear , ( as to a centesimal partition of the measure given ) 0.76 inch , which is but very little above 3/4 of an inch : and the Line for the Circumference , according to the common Tenent , to be 2.40 inches ferè , and according to our experiment , 2.39 inches ferè , ( so that here also one and the same Line of measure may indifferently serve in both ) which two Lines being divided as the former , and then the Diameter or Circumference be taken by their proper respective Line or Scale , and cubed , the same shal be the weight of the Sphear or Bullet in ounces and decimal parts immediatly : For so the Diameter of the foresaid bullet of 9li . and 2 oun. ferè avoirdupois , ( viz. 9.123626 li. ferè ) being 4 inches , and the Circumference , 12.57 ferè , wil be found each of them , by their proper cubatorie Line or Scale , for unciall gravity , ( being made 100 parts ) to be 5.27 ferè , which cubed , yields 146.363183 ferè , for the weight of the Bullet in ounces , which exceedeth the true weight being 145.978009 oun. by 0.385174 oun. ferè , which by conversion or reduction , yieldeth about 6 drams , and which in a thing of this nature is not considerable : But yet if we wil stand more precisely uppon the weight of this Bullet , if then we divide the two foresaid artificiall Lines of measure for this purpose into more parts , as 1000 ( for the naturall Line , or the Inch being so divided , the artificial Line wil be thereof , for the Diameter , 0.760 ferè , and for the Circumference , 2.387 ferè ) & so measure the Diameter & Circumference of this Sphear or bullet thereby , we shal find the same to be severally , 5.265 , which cubed , gives 145 . 946984625 ounces for the weight of the bullet , which wants of the true weight aforesaid , hardly half a dram , which in this kind of weight is as near as need be desired .

And so again in the other Bullet of 15li . & 12 1/4 oun. avoirdupoiz ( or 252.25 ounces ) weighed by us ; if the Diameter or Circumf . noted formerly , be taken by these Lines under a centesimall division , they wil be found each of them to be 6.32 ferè , whose Cube is 252.435968 oun. ferè , which exceedeth the true weight , only 0.185968 oun. which by conversion , gives near upon 3 drams , and which is not considerable : But being measured by the same Lines under a millesimall partition , they wil be found each of them , 6.318 , which cubed , affords , 252.196389 . &c. ounces , which now wants of the true weight , not fully one dram being but 0.86 dr.

And so also if the Diameter or Circumference of this Bullet be taken by their proper respective Lines of measure for librall weight beforegoing ( viz. 1.91 inch , for the Diameter , and 6.01 inches , for the Circumference ) the same wil be found severally ( according to a centesimal partition of the Lines ) 2.51 ferè , whose Cube is 15 . 813251 ferè for the weight of the bullet , ( which by reduction is 15li . 13 oun. and 0.19 dr. avoirdupois ) exceeding the true librall weight , viz. 15.765625 li. ( or 15 li. and 12.25 oun. ) only 0.047626 li. ferè , which by conversion into the proper parts of this weight , gives about 3/4 of an ounce , viz. 12.19 dr. But being measured by the said Lines in a millesimal partition , ( for the natural Line , or the Inch , being 1000 parts , the artificiall Line wil be thereof for the Diameter , 1.914 , and for the Circumference 6.014 ferè ) they wil be found severally , 2.507 , which cu bed , yields 15.756617 , &c. ( viz. 15li . 12 oun. 1.69 dr ) wanting now of the true weight only 2.3 drams .

Now if any shall upon occasion , make use of the Troy-weight in a Spheare of this The artificiall Lines of measure , for the most speedy discovering of the weight of a Spheare or Bullet of cast-Iron , in Troy-weight , both librall and unciall . metall ( though indeed this kind of weight is not usuall for any Metall besides Gold and Silver , as I have noted before ) then the artficial Lines of measure for the speedy discovering of the weight thereof by the diameter , wil be 1.794 inch for librall-weight , and by the Circumference , 5.636 inches : And the Lines for unciall weight , wil be for the Diameter , 0.784 inch ferè , and for the Circumference , 2.462 inches .

And now albeit no art or industry of man , can make a Spheare of metall or other matter , so very exact and precise indeed , as it ought to be ( according as I noted before from Ghetaldus ) Yet considering that the weight of this and the other coarser kind of Metals in any thing , is not so precisely stood upon as the weight of Gold and Silver , ( being pretious Metals ) So the exactnesse of a Sphericall body made of any of them for ordinary use ( as a Cannon-bullet , which is commonly made of cast-iron , as I noted before ) is not so much stood upon , so as it come somthing neer the same : and indeed there is hardly any Cannon-bullet , or other bullet for shooting , but is so round ( being cast in a Mould ) as that the Eye can hardly adjudge or discerne it to be not exactly orbicular or sphericall ; and so the weight thereof can be very litle mistaken , being obtained by the diameter or Circumference of the same , taken in a certaine set Measure , either naturall , as Inches , or the like ; or artificiall , as deduced therefrom , according as I have here shewed at large . And which severall dimensions with many other the like metrical Conclusions pertaining hereunto , and the Converse of the same ( and all cheifly in reference to Gunnery ) I will next expresse proportionally by Number , ( from our foregoing experiments ) according to the aforesaid Metall and Measure commonly used in this thing , and the Weight both Avoirdupoiz and Troy , and in each of these , both by the librall and unciall-weight together , which in the severall Sections of proportions following , are ( for brevity-sake ) noted by the letters of distinction , l , and u.

1. The Cube of the Diameter in measure , is to the Spheare it selfe in weight , as 1. to .1425566 . l. Avoirdupoiz . weight . 2.280906 . u. .17314487 . l. Troy-weight . 2.0777384 . u. Circumfer . .00459767 . l. Avoirdup . .07356273 . u. .0055841877 . l. Troy . .06701025 . u.

Conversly .

2. The Spheare in respect of weight , is to the Cube of its Diamet . in respect of measure , as 1. to 7.0147552 . l. Avoirdup . cubique measure . 0.4384222 . u. 5.7755104 . l. Troy . 0.4812925 . u. Circumf . 217.50144 . l. Avoirdup . 13.59384 . u. 179.07707 . l. Troy . 14.923089 . u

Hence ,

3. The Diamet . is to the side of the Cube equall to the Sphear in weight , as 1 to .522391 . l. Avoirdup . Linear measure . 1.316343 . u. .5573609 . l. Troy . 1.276038 . u. Circumf . .16628227 . l. Avoirdu . .41900507 . u. .1774135 . l. Troy . .4061755 . u.

Then for the speedy discovering of the weight of a sphericall or any other body whatsoever made of this Metall , by the solidity thereof , in the Measure aforesaid : and contrariwise the solid content by the weight thereof , both Avoirdupoiz and Troy , and in each of these both librall and unciall weight , the Proportions will be as followeth .

1. As 1. to .27226314 . l. Avoirdupois . Weight . 4.3562103 . u. .33068234 . l. Troy . 3.968188 . u.

Contrariwise .

2. As 1. to 3.672917 . l. Avoirdup . Solid measure . 0.229557 . u. 3.02405 . l. Troy . 0.25200418 . u.

And so likewise for discovering the gravity of any Body of fine or forged Iron by the magnitude , in the foresaid Measure , and contrá : the proportions will be from the forementioned experiment made upon this Metall , as followeth .

1. As 1. to 0.3474 . l. Troy . Gravity . 4.169 . u. 0.2861 . l. Avoirdupois . 4.5777 . u.

Conversly .

2 As 1. to 2.878 , l. Troy . Magnitude . 0.2399 ferè u. 3.495 . l. Avoirdupois . 0.218 . u.

And thus having shewed our artificiall Mensuration in regular Solids , aswell for gravity or ponde●osity , as for solid ( and superficiall ) measure ; I shall now close up this Section and Part , with our third theorematicall Proposition , ( answering to the third principall problematical or practicall Proposition in the first Part ) and the practicall demonstration thereof , in which our more particular , or speciall reason of the like dimension of all regular-like Solids ( as particularly of a Cylinder ) in the severall respects aforesaid ( in reference to the producing of the artificiall Lines of measure , for performing the same ) is contained .

THEOR. III. Expressing particularly , the artificiall Lines for the solid dimension of a Cylinder , from our particular or specia●l ground and reason formerly declared : And consequently , the Lines for the like dimension of a Cone , and all other regular-like Solids in generall .

IF the two proper Dimetients aforenamed , of a ( right or erect ) Cylinder , exactly adequate to the cubicall , ( composing or resolving ) Parts of the Rationall Line , shal be obtained according to an exact congruencie or congruity ; The same shall be the proper , respective artificiall Line of Cyhnoricall solidation , according to the Parts : And th●●e wil be of them to the correspondent or congruall ▪ Lines or d●m●nsion according to the whole Measure , the Reason that is of the Parts to the Whole ; which is as the Reason of their respective Cubes .

THe like reason holdeth for the Cylindricall Circumference conjunctly with the Dimetient of altitude : and also for both these in the Cone , in respect of its basiall Dimetient and Periphery with the Dimetient of altitude , for solid dimension , and with its Side for superficiary dimension ; and also for the basiall Dimetient of a Cylinder with its Side , in respect of superficiall , dimension : And so the like with these , for all Pyramids and Prismes constituted upon regular Bases , in respect both of solid and superficiary dimension , according to the several wayes formerly declared and demonstrated in all these Figures : only respect being had in all these , to the quadrate parts of the Rationall Line for superficiary dimension , as is here to the cubique parts for solid dimension .

And the like reason wil here hold in all these solid Figures , in respect of weight or gravity in any Metall whatsoever , as for their solid measure , according to what I shewed before in the Spheare , and consequently in the other re-Bodies , being made of Metall ( respect being here had to the parts of the weight proposed , as in the Theoreme it self , is to the parts of measure , as I shewed before in the like case upon the 2d Theorem ) so as that the like artificial Lines of Measure being extracted for these Figures severally in reference to weight , according to any Metall , Weight , and Measure proposed ; their gravities wil be thereby obtained in the same manner as their solid magnitudes or measures , according to the severall wayes formerly declared and demonstrated for the same .

And now because of the two several Dimetients ( or dimensional Lines ) here continually concurring , there occurreth some more variety , then in the two preceding Theoremes : Therefore I wil here give a full demonstration or illustration of this , in an Arithmetical manner ; by which those two ( with all the things necessarily depending on them ) as well as this , may be plainly understood , seing they be all grounded upon one and the same reason .

THerefore , let the Rationall Line ( in respect of its properly composing denominate parts ) be R 12. So the Cube thereof , CR , 1728 ; to which be conceived or constituted a Cylinder , exactly agreable for magnitude or dimension , & so , as that its two foresaid Dimetients , be in exact Congruencie , or Congruity ( such as the Greeks ( Euclid & Proclus ) cal 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ; the Latines , ( as Congruentia , or Congruitas : & so the greek Eucl. Axiom . com . 8. Ram. Geom. lib. 1. ●l . 9. & Schol. mathemat . lib. 8. prop. 4. words do generally signifie ) which therefore we are here next to enquire . Therefore seing then that like Solids in generall , do mutually hold in a triplicate or cubique reason of their homologall Terms by E 8. p. 19 & 27 ; and more particularly , E 11 , p. 33. and E 12. p. 8. and so in speciall , Cylinders of their basial Dimetients ( or other like Terms ) by E 12. p. 12. To this I assume another Cylinder , whose two Dimetients are already given in the same kind , with those supposed in the former Cylinder , which ( in the first Numbers from an unit , absolute , making the whole Cylinder absolute in all its dimensions , according to the more ancient and vulgar Cyclometricall terms ( in respect of the Base ) but not the newer ) let be ( under the generall notion of a Binomiall in relation to the Rational Line taken as before ) A = D R 12+2 ; whereupon wil arise the Cylinder , rationall and absolute ( according to the said vulgar Cyclometrical Terms , in respect of the Base , and binomially in reference to the Cube of the Rational Line taken as before ) CR , 1728+428 ; but according to the newer , and most approved ●etragonismall Terms , in respect of the base , and first in the more vulgar or Metian expression noted formerly ; for the performing of this operation after the more usuall or vulgar way of Numeration , ( and for that this expression is finite and limitted in it self , whereas the decimal expression runs infinitly ) it wil be irrational and inabsolute , CR , 1728+427 15/113 ( which is by decimal conversion of the fractionall termes , CR , +427.132743 , &c. very nearly agreeing with that which is produced by the proper decimall Cyclometricall termes , in respect of the base of the Cylinder ; and that both from themselves simply or naturally , and also artificially , from their ●ogarithme , as shal be shewed by and by ) then by E. 12. p. 12 a●oresaid , and also by the reason of ● . 7. p. 19. it holdeth in a triple or cubicall reason of the aforesaid Cylindrical Dimetients conjunctly , as the Terms homologall , thus ;

CR 1728+427 15/113. CR 1728 : : CR 1728+1016 ( viz. CA = D. R 12+2 in CR 1728+427 15/113 ) : CR 1728+472 38416/24353● or ( by their greatest common Divisour , or number of exact commensuration , 686 ) 472 56/355 in the least termes of the same reason ( viz. CA = D in CR 1728 ) which is decimally , CR 1728+472 . 157746 , &c. whose Root or Side irrationall or ineffable , ( and binomiall in reference to the Rational Line , taken as before , and according to a decimall extraction of the Cubicall Gnomon ) R 12+1 . 0062 , &c. is A = D in CR 1728 , being the thing first sought for ; or the artificial Line according to the parts of the natural Line of measure proposed : which from the most admirable invention of the Logarithmes , is most speedily performed by a simple Composition and Resolution of Numbers , together with a Trichotomie of the Cubicall Logarithme ; the first or greater Cylinder , being raised from the most exact Cyclometricall Logarithme , ( in respect of the Base ) answering to the most exact decimall Terms aforesaid .

Or again , seing that between two like Solids , there do necessarily intercede two mean Proportionals , by E. 8 , p. 12 , and 19 ; the two Meanes between these two Cylinders , wil be found ( binomially , in relation and comparison to the foresaid Rationall Cube , and according to a decimall enumeration of the cubick Gnomon ) CR 1728+274 . 15289 , &c. and CR 1728+132 . 03231 , &c. most compendiously and exactly , by Logarithmicall Computation , thus ;

CR 1728+427 . 13256 , &c. 3 , 3334739884 S CR 1728 3 , 2375437381 S Dif . 0 , 0959302503 A 1/3 dif . 0 , 0319767501 A CR 1728+132 . 0323 , &c. 3 , 2695204882 lesser Mean . CR 1728+274 . 152 , &c. 3 , 3014972383 greater Mean .

Or the same may be obtained more plainly , ( though not so readily ) in a dis-junct manner , at two distinct operations , by the Analogie which holds from the Cube of one Solid to the Cube of the mean Proportional falling next to it , as doth from that Solid to the other propounded , between which the two mean Proportionals are required .

Then seing the greater Mean serveth here our present purpose ; it followeth according to E 7. p. 19 thus ; CR 1728+427 . 13256 , &c. CR 1728+274 . 15289 , &c. : : A = D , R 12+2 : A = D , R 12+1 . 0062 , &c. as before : which appeareth plainly and briefly , by Logarithmicall or artificiall Numeration , thus ;

CR 1728+427 . 13256 , &c. 3 , 3334739884 S CR 1728+274 . 15289 , &c. 3 , 3014972383 A A = D , R 12+2 1 , 1461280357 A Sum. 4.4476252740 A = D , R 12+1 . 0262 , &c. 1 , 1141512856. Dif .

Agreeing exactly with the first Logarithmicall operation .

Or again , the said A = D in CR 1728 required ; found more briefly yet , after this manner .

CR 1728+427 . 13256 , &c. 3 , 3334739884 S CR 1728 3 , 2375437381 S Dif . 0 , 0959302503 1/3 dif . 0 , 0319767501 S A = D , R 12+2 given , 1 , 1461280357 S A = D , R 12+1 . 00622 , &c. inquired . 1 , 1141512856 Dif .

Which being found out these three severall waies ; it followeth lastly , according to the Consectarie of the third Theoreme , in this simple Analogie . R 12 : R 1 : : A = D , R 12+1 . 0062 , &c. A = D , R 1+ . 0838 , &c.

Which last found Number , is for the said congruall Dimetients of a Cylinder , according to the unity of measure ; and so for the artificial ( or second Rational ) Line according to the whole intire natural ( or prime Rationall ) Line in generall .

And what manner of working is used here , for the investigation of this artificiall Line in the Cylinder ; the same is to be used for the like in the Cone : and which ( though needlesse ) I wil here further illustrate in the same .

Therefore , suppose here the foresaid Rationall Line ( taken in its composing , denominate parts ) R 12. and to its Cube , C R 1728 , a ( right , or Isoskelan ) Cone exactly adequate ( as the Cylinder before ) having ●ts two like proper Dimetient-lines exactly congruall ( as those of the Cylinder ) in the investigation or inquisition wherof conjunctly , consisteth the first operation . Therefore seing then , that as Cylinders ( and other like Solids ) so Gones , are respectively one to another , in a triple or cubicall Proportion of their Basiall Dimetients ( or other correspondent Termes of dimension ) by E 12. p. 12 aforesaid : To this be supposed another like Cone , of given or known Dimetients , and exactly congruall , as the former ; which let he the same with those of the foregoing Cylinder , included in the Binomie , A = D , R 12+2 ; whereupon this Cone wil be subtriple that Cylinder , viz. according to the vulgar Numeration , from the Metian Cyclometricall-Terms , as to the composition of the Base ( under the common acception of an Apotome , in reference to the foregoing Rationall Cube ) irrationall or unabsolute , C R 1728-1009 211/339 ( which is by decimall conversion of the ●raction-part , 1728-1009 . 622418 , &c. and which very nearly agrees with that which is produced by the most exact ( decimall ) Cyclometrical termes in regard of the Base , and that both naturally or simply from themselves , and also artificially from their correspondent Logarithme ; being C R 1728-1009 . 622480 . as wil appear in the Logarithmicall operation following ) Then by E 12. p. 12 , and also by the reason of E 7. p. 19 before cited , it holdeth in a triplicate or cubique reason of the aforesaid Conicall Dimetients conjunctly , as the Termes homologall , thus ;

C R 1728-1009 211/339 : C R 1728 : : C R 1728+1016 ( viz C A = D , R 12+2 in C R 1728-1009 211/339 ) : C R 1728+4872 115248/24353● ; or rather in the least homologall fractional termes ( by their greatest common Measure , or Number of exact Symmetrie , 686 ) C R 1728+4872 16●/35● , viz. C A = D in C R 1728 ; which by decimall di●umeration , is C R 1728+4872 . 473239 , &c. whose Root or side cubicall , and irrationall or inexplicable , is ( binomially in reference to the foresaid Rational Line , and according to a decimall eradication , or resolution of the cubique Gnomon , R 12+6 . 7723 , &c. for A = D in C R 1728 , being the thing first inqu●red ; or the artificial Line of measure , according to the parts of the natural Line proposed : And which may most easily and readily be obtained by Logarithmicall or artificiall Numeration , by a simple Composition and Resolution , or Prosthaphereticall suppu●ation only ; and that the severall waies which I shewed before in the Cylinder ; whereas the working out of these things by the natural Numbers ( or the vulgar way of Numbring ) is most laborious , and intricate , by reason of the many large and tedious Multiplications , Divisions , and Radicall extractions , both quadrate and cubique , arising from ; which notwithstanding I have also done both in this of the Cone , and that of the Cylinder , and likewise in other Figures , where the same might be conveniently performed after this manner , as in the Circle and Sphear , for the finding out of their respective artificiall Lines of Mensuration before set forth .

But now to the operation of our present question in the Cone , by the artificiall or Logarithmicall Logistique ; according to a cubicall proportion of the aforesaid homologall Terms , thus ;

Which differeth from the number found by the naturall operation , in the decimall fraction , but very little , that giving it .7723 &c. this .7582 &c. which breedeth no sensible difference in the Conclusion of the worke ; but however the number produced by the Logarithmicall operation is the truest ; the second assumed Cone ( though the first in place here ) being produced from the exactest Cyclometricall Logarithme , as to the composition of its Base , upon which it is raised and constituted .

Or againe secondly , by finding out the two Mean Proprotionals between these two Cones , which is most speedily and accuratly performed also by virtue of the Logarithmes , as before in the Cylinder ; and so they will be found ( under the two common Apotomies , in reference to the Rational Cube , and according to a decimall Analysis of the cubicall Gnomon ) C R 1728-765 . 465285 &c. for the lesser Meane ; and 1728-438 .3256268 &c. for the greater Meane , thus ,

Or the same may be found more disjunctly , according to the Analogy before declared in the Cylinder , for the finding of the two Means ; the reason being the same here .

Now the lesser Meane serving here our purpose ; it followeth thence Analogically thus : C R 1728-1009 , &c. : C R 1728-765 . 465 , &c. : : A = D in C R 1728-1009 , &c. viz. R 12+2 : A = D in C R 1728 , viz. R 12+6 . 7582 , &c. as before ; as appeareth briefly and plainly , by this subsequent Logarithmeticall Calculation .

Exactly agreeing with the first Logarithmicall operation .

Or again thirdly and lastly , the same Dimetients in the Cone , C R 1728 , found out conjunctly and congrually , most briefly of all , thus ,

C R 1728-1009 . 6224 &c. 2,8563527 S C R 1728-3,2375437 S Dif . 0,3811910 1/3 0,1270637 A A = D , R 12+2 given 1,1461280 A A = D , R 12+6 . 7852 , inquired 1,2731917 . aggreg .

Which being had these severall waies ; it followeth lastly thereupon , according to the reason of the Consectary of the foregoing Theor●me , by way of Analogy , in these Termes ; viz ,

R 12 : R 1 : : A = D , R 13+6 . 7852 &c : A = D , R 1+ . 56318 &c. Or R 12 : R 12+6 . 7852 &c. : : R 1 : R 1+ . 5632 ferè . Which last found Number is for A = D in the Cone answering to C R 1 ; and so for our artificiall Line of rectangle parallelepipedation ( according to an exact quadrate Base ) or parallelepipedall consolidation of a Cone desired , in reference to the whole intire natural Line of measure in generall .

And after the same manner may be found cut the like artificiall Lines for the consolidation of a Cone , and Cylinder , by their basiall Peripheries and altitudinary Dimetients together , as hath been here done , by their basiall and altitudinary dimetients together : ( which basiall Dimetient and Periphery do continue the same throughout the Cylinder , it being of equall crassitude , but continually alter in the Cone ) And these Lines we shewed before in the practicall demonstration and use of the other two Lines of solidation pertaining to these two Bodies , to be for the Cylinder , of the Rationall Line , 2+ . 3266 &c. and for the Cone 3+ . 3531 &c. and these by the naturall way of working , from the Metian Cyclometricall termes , in respect of the Base of the Cylinder and Cone : and by the artificiall or Logarithmeticall way , I find the same to be for the Cylinder 2+ . 32489 &c. ( very little differing from the other ) and for the Cone , exactly agreeing with the former : And therefore I shall not need ( I conceive ) to make any more adoe about the demonstration hereof : And the same way holdeth for the extraction of the like artificiall Lines for the Superficiall dimension of the Cylinder and Cone , and also for the dimension both solid and superficiall of all regular-based Pyramids and Prisms : And for all these Figures , not only in respect of solid and superficiall measure , but also of gravity or weight , according to any Metall proposed ; as I have noted before upon the 3d. Theorme .

And all these things , together with the like in the other Figures , before-mentioned , may be yet more readily obtained , and that according to our generall ground and reason of this artificiall Dimension ( or of our artificiall metricall Lines ) which was formerly said to consist in unity alone . For as in our particular , partiall , or speciall Reason thereof , every particular Measure propounded ( as the prime or naturall Rational Line ) being considered according to its Parts , ( composing or constituting , or otherwise dividing ) every particular kind of Figure , or figurate Magnitude , measurable in this way , was consequently said to be considered in the first or second Power of the same ( as in the powers of Number ) according to the nature and kind of the Dimension ; for the producing of the artificiall Lines , ( according to the three foregoing Theoremes ) So here in our generall Reason , or Reason of the Whole , or of Unity ; every Measure being considered in it self simply and absolutely , ( as was formerly declared ) every such figurate Magnitude ( as aforesaid ) is to be considered accordingly , in the like Powers of Unity ( as of the whole intire Measure in generall ) according to the nature of the Dimension . And so again , as in the performance of these Dimensions , by the naturall Measure proposed ( or the prime Rationall Line in generall ) in an Arithmeticall manner , by way of Proportion ( according as hath been shewed in the severall dimensions beforegoing , among the dimensionall Proportions ) for the more immediate producing of the sides of the Squares and Cubes , &c. equal to the Figures proposed to be measured , and which are immediatly given by our artificiall Lines of measure ; we begin with Unity simple , and proceed from that to Number simple or linear , being some Root of a Figurate Number superficial or solid ; & so to the figurate Number it self ; as from some dimensional line of a Figure ( by which the same is proposed to be measured in a quadratary or cubatory manner , according to the nature of the Dimension ; or by which the side of the equal Quadrat or Cube of the Figure is to be produced , either naturally in an arithmetical analogicall manner , as aforesaid ; or artificially , by a Line of measure convenient for the purpose ) to the side of the equal Quadrator Cube ; and so come to the Content of the Figure it self ? Or we come to the same more immediatly , and also most naturally and properly ; proceeding from Unity taken in the power thereof , prime or second , ( according to the nature of the dimension propounded ) to Number figurate , and then from that to the correspondent Root thereof ; as from some dimensional Line of a Figure , in the aforesaid powers thereof , to the Figure it self ; and then from that to the side of the equall Quadrat or Cube . So in this ( for the immediate producing of the artificiall Lines ) we proceed from Unity figurate in the first or second power thereof , according to the nature of the Dimension , to a Number figurate or potentiall , accordingly , and from that to the correspondent Root thereof ; as from every kind of Figurate Magnitude in general , falling under our artificiall Dimension , to the like powers of some one of its dimensional lines by which it is proposed to be thus measured ( or squared and cubed , or otherwise artificially measured , according to the nature and kind of the Magnitude propounded , and of the Dimension it self ) and so from thence to the said dimensionall line it self of the Figure , for the artificiall Line of measure required : which as in all exactly ordinate or regular Figures , it is but one such line simply ; so in all regular like-Figures it is of two together according as we have lately shewed . And so we might hereupon raise a 4th . and generall Theoreme , if it were needfull ; but that which I have already said concerning the same , is very sufficient .

And what hath bin here spoken for solid Figures , in reference to their solid measure , is to be understood in the same accordingly , for gravity or weight ( according to any Metall , or other like ponderous matter proposed ) For as before , according to our particular or speciall reason , every Integer of weight whatsoever proposed , was said to be considered in its parts taken simply in number ; and so every kind of metalline , or other like ponderous ( regular and regular-like ) Body , was accordingly to be considered in the same , as in the like parts of Unity : So here the Integer of weight propounded , is to be considered simply absolutely and intirely , as in the nature and reason of unity it self ; and so every kind of metalline ( or other like ponderous ) regular and regular-like body , is to be considered accordingly , for the producing of the artificiall Lines of measure therefrom , for the speedy discovering of the gravity or ponderosity of any such Body proposed , in any magnitude whatsoever .

PART III. Containing that kind of Dimension , or metricall practice , which is commonly called , the Gauging of Vessells ; after a most artificiall , exact , and expeditionall manner .
SECT. I. Concerning the measuring or gauging of Vessells , in generall .

ANd what hath be●● here done for the solid Dimension of a Cylinder ( or the dimension of a solid Cylinder ) by artificiall Lines found out for the same , according to any Measure appointed : the like may be done for the liquid dimension of a concave Cylinder ( or Cylindricall Vessell ) for the more speedy finding out of the liquid Content , according to any Liquour and Measure given . As our Vessells for Wine and Ale , or Beer ( which two , most commonly come to be measured ) and other liquid things , which though they be not absolute Cylinders in themselves , yet may be ( and commonly are ) conceived or supposed as Cylinders , ( being reduceable by Art thereunto ) for the better measuring of the same , by finding out and taking the Meane between the Diameter at the Head , and the Bung of the Vessell ( or otherwise the Meane between those two Circles ) and so obtaining the liquid Content thereof ; which is commonly called Gauging of Vessells ; the Measure by which these Vessells are thus valued or estimated , being usually a Gallon , and which is the greatest of our liquid Measures , and but the beginning , as it were , of Vessell-Measure ; but in Wine , and Ale or Beere , holdeth not one and the same ; but differeth very sensibly , as is commonly known : And therefore we will next briefly shew the measuring or gauging of these Cylindricall Vessells ( or Spheroidall , as some will have it , this kind of Vessel being more commonly taken for a Sphaeroides , having the two ends equally cut off ; though for mine own part , I conceive this kind of Vessel may more properly be termed a Cylindroides , by the same reason that a Sphaeroides and Conoides are so called ; this having the same similitude or resemblance to an exact Cylinder , that those have to an exact Spheare and Cone ) by Lines of measure peculiarly appropriated and applied to them ( as we did before in the Cylinder in general for solid measure ) according to the different kind and quality of the Liquour , and so the different quantity or magnitude of the liquid Measure given , in relation to solid Measure in inches .

SECT. II. Setting forth the Quantities of the Wine and Ale-gallons , in reference to the gauging of Vessels .

IT is generally holden by Artists about the City of London , that a Wine-Gallon containeth in its concave Capacity , 231 cubicall or solid inches , or is insensibly differing therefrom : But for the Ale or Beer-Gallon , I finde the same to be as generally controverted among them . Mr. William Oughtred , a reverend Divine , and most eminent Mathematician , before-named , after some experiments made by him to find out the solid content of this Gallon in inches , besides the experiments of some others which came to his sight , finding some difficulty therein , in regard both of the irregularity which he observed to be usually in the severall Standard-Gallons which he met with , and also their disagreement one from another in their Contents , as himself confesseth and declareth in his foresaid book of the Circles of Proportion , Part 1. chap. 9. looketh back there to the first ground and principle of our English Measuring from Barley-corns : and so at length he commeth to a rationall conjecture of the Ale-gallon ( and that very neatly , and prety nearly also , as I shal straightway shew ) in cubique inches , according to the number of the square-parts or Feet in the common Statute-Pertch or Pole , viz. 272 1/4 , as you may see in the place fore-cited .

But Mr. John Reynolds aforenamed also , ( who seemeth to have been as industrious in this , as in many other mathematicall experiments ) wil have this Gallon to contain 288 3/4 cubique inches ; holding that the Wine-gallon ( which he strongly affirmeth to be 231 inches ) is to the Ale-gallon in such proportion precisely , as 4 to 5 ; or rather for the reduction of Wine-measure to Ale-measure , as 5 to 4 ; which is according to Mr. John Goodwyn long agoe , in his little Tract entituled , A Table of gauging , published above 50 years since , and dedicated to the then Lord Major and Aldermen of the City of London : wherein he shewing how to reduce Wine-measure into Ale-measure , & contrà ; saith , that 5 gallons of wine-measure make but 4 gallons of Ale-measure : with which very nearly agrees the opinion ( not certain experiment perhaps ) of some others , who wil This Mr. Goodwyn was Master in the Mathematicks to Mr. Reynolds , as himself hath told me . make this Gallon to contain just 288 inches upon this ground , that a cube-Foot should hold in its concave capacity just 6 Ale-gallons , and so consequently one ale-gallon must contain just 288 inches , which that learned gentleman Mr. Edm. Wingate , a Barrester of Grayes-Inne ( a man eminent for his mathematicall abilities ) first declared to me by word of mouth , and soon after I found the same noted in his book of the use of his Rule of Proportion , chap. 10. And these two jump so nearly together , as if one were borrowed from the other : but I declaring this to Mr. Reynolds at my first seing of him , he said that he had not observed this thing .

Now as for Mr. Oughtred's Ale-gallon of 272 1/4 inches , the said Mr. Reynolds indeed alloweth of such a Gallon-measure , but not for any liquid thing , but for drie things , as Corne , Coals , Salt , and other dry things measurable by this kind of Measure ; and so calleth it the drie Gallon-measure : And thereupon he wil have to be 3 severall Gallons ( or other like Measures ) one for Wines , ( which also serveth for oiles , strong-waters and the like ) Another for Ale and Beer , and a third for Corne , Coales , and the like ; and this he maketh lesser then the Ale-measure , whereas surely it should rather be greater , if there be any difference at all between them : And these three severall Gallonmeasures , he compareth together , or differenceth by these three Numbers , viz. 28 , 33 , 35. as to shew their proportions one to another : viz. the Wine-Gallon ( 231 ) to the dry Gallon-measure ( 272.25 ) as 28 to 33. which is so in the least terms , rationall or absolute ; but otherwise in the least proportionall termes , irrationall or unabsolute , and finite or limited , I find them to be as 7 to 8.25 . And the said Wine-Gallon to his Ale-Gallon ( 288.75 ) as 28 to 35 , which in the least rationall termes , is indeed as 4 to 5 ; but otherwise in the least termes irrationall ( but finite or limitted ) I find them to be as 1. to 1.25 . and which again is integrally rationall , 100 to 125 : And the dry-Measure to the Ale-measure , as 33 to 35 , which cannot be abbreviated in terms rational .

And surely , evil Custome seemeth to have brought up three such distinct measures ( and which the foresaid Mr. Wingate hath also expressed to me ) For at the Guild-Hall in London , where is generally holden to be the true Standard for these Measures , and so from which all others of the like kind throughout the Kingdome , are usually derived , there are but two such distinct Measures only ( as we have been there informed for a certain truth ) viz. one for Wines ( and so for strong-waters , oiles , and the like ) and the other for Ale , Beer , and drie things , as Corn , Coales , Salt , and the like ; which latter is commonly called the Winchester Measure , and from this are taken the bigger drie Measures , as the half-Peck and Peck , and so on to the Bushell , which is the greatest of our drie Measures : Which said Standard-Measures at the Guild-hall , the foresaid Mr. Reynolds confessed to me ( going to him on purpose to receive some satisfaction from him about the Wine and Ale or Beer-measures ( which was in June 1646 , and then he gave me in writing under his hand , the solid content of the Ale-gallon to be 288 3/4 inches , and so its Proportion to the Wine-gallon , to be exactly as 5 to 4 ( or for the reduction of Ale-measure to wine-measure , as 4 to 5 ) that he had never made any triall of them , ( neither could I find that any other had , or if they had , it was surely to small purpose ) but only of those Measures in the Tower of London ( which he pleads for to be the most ancient and true standard Measures ) and at Cowpers-Hall , and some other such places , which seem to be but some particular Customary measures , differing from the generally received Standard-measures at the Guild-hall .

And therefore to be fully satisfied in this point , concerning the true Wine , and Ale or Beer Measures , according to the common Standards , ( and more especially about the Ale or Beer-measure , finding such a diversity of opinions concerning the same , and in so vast a difference , as that between Mr. Oughtred's and Mr. Reynold's Ale-gallon , being 16 1/2 cube-inches ) My self and one Mr. Baptist Sutton , ( a man well known in the City among artists ) did agree to go together to the Guild-hall , where he was wel acquainted with the keeper of the Standard-measures and Weights , who otherwise I found to be very nice and scrupulous in shewing of them ; and for our further satisfaction herein , we made known our intention to the foresaid Mr. Wingate , who much approved of the same , expressing his desire also of it : And so August 9th . 1645 ; we repaired together to the Guild-hall , carrying along with us two large square glasse-vials , which we first weighed in a Gold-smiths Ballance by Troy-weights , ( as being the best ) which were supoosed to be exact enough ; and afterwards filling the two brasse-standard-Gallons for Wine , and Ale or Beer , with fair water from the Cisterne , and that with all possible precisenesse , we powred forth the same with the like accuratenesse , into the said two glasse vials , and then weighed the Glasses with the water in them by the same weights : and so comparing the weight of each Glasse alone , with the weight of the glasse and water together , we found the Wine-Gallon of water to weigh 117 ●/4 ounces-Troy , and the Ale-gallon of water 140 9/16 ounces , ( which last , according to the common division of the Ounce-troy by penny-weights , is 140 ounces , 11 penny-weights , and a quarter ) which do hold in proportion ( from the lesser to the greater ) as 10000 11937 , which comes very near , as 5 to 6. Then seing that Weight and solid measure do hold in proportion one to another , so as that one may be deduced from the other , as I have shewed before ; if we compare these two Gallon-weights of water , with severall experiments made by my self , and Mr. Reynolds , severally ( and conferred together ) for the finding out of the true weight of water in relation to its solid measure in inches , ( or for the comparing of its gravity and magnitude together , which thing is most admirable and excellent use , as I shall shew more afterwards ) we shal thereby discover the solid capacity of the said two Gallon-vessels in inch-measure ; which is the very ground-work of Gauging .

Now as to the foresaid experiments ; the said , Mr. Reynolds did first ( amongst other things to this purpose ) cause a Vessel to be made of Wood , by an exact * * Mr. John Thomps●● in Hosiar Lane . Workman , in the forme of an oblong rectangle Parallelepipedum , ( or long Cube as some term it , though improperly , as they call an oblong rectangle Parallelogram a long Square ) whose Base was 4 1/2 inches square , and the heighth , depth , or length , ( which you wil ) 14 inches , and so the solid capacity thereof , 283 1/2 inches ; and which was closed up at both the ends or bases , saving that in the middle of one end , was made an hole for the powring in of water , and which was no bigger , then that he might guesse in the filling thereof to a drop or two of water , more or lesse : which Vessell therefore being precisely filled with fair setled Rain-water ( as being the fittest , as I shal shew afterwards ) and then as precisely weighed by Troy-weights , he found the water thereof alone to weigh 12 li. and 5 1/4 oun. or 149 1/4 ounces troy . And he not being contented with this own experiment , he caused such another Vessel to be made , every way like and equall in its dimensions with the former , and that by the same Workman ; which he filling with the like water , found it to agree in weight exactly with the former . But yet he not resting fully satisfied with these two experiments , he procured such another Vessel to be made , by another Workman , of the very same Dimensions with the former ; which he filling with the like water as aforesaid , found the weight of the said water alone to be 12 li. and very near 6 oun. Troy , ( or 12 1/2 pounds-Troy ferè ) exceeding the former weight about 3/4 of an ounce , and which he conceived to be the truer , ( notwithstanding the exact agreement between the two former experiments ) by comparing these experiments with some other of the like kind , which had been made before by himself , or some other body ; And this difference of weight seemeth to proceed chiefly from some difference of measure in the Inch , by which the two first Vessels , and the last were made , being done by two severall Workmen . And therefore ( considering the difficulty in a work of this nature , in respect of the nicety and curiosity of the experiment ) he comparing these with some observations which I had then made by the bye , to this purpose ; we concluded together at length , that the nearest and indifferentest weight of the water exactly filling up the foresaid Vessel of 283 1/2 cubick or solid inches , would be 12 li. and 5 1/2 oun. Troy ( or 149 1/2 ounces ) and this to stand good .

And then after this , I got an exact cubical Vessel to be made of throughly seasoned wood , with all the accuratnesse & precisenesse that could be , being 6 inches the in-side , ( or the base thereof exactly 6 inches square ) and so the whole Cube in its concave capacity , exactly 216 inches ; and which then , to keep it from sucking in water in any part , or any water to soak into it , was well primed all within , with a thin oile-colour ( yet of a sufficient body ) having afterwards a Cover put on it , with a little hole in the middle thereof , about 3/4 of an inch wide , as the foresaid Vessels of Mr. Reynolds had : And which cubicall Vessell I then filling with all the exactnesse and precisenesse that might be , with fair setled Rain-water , at Gold-smiths-Hall ; and so having the same as exactly weighed by the Standard Troy-weights ; I found the weight of the water alone ( deducting the weight of the empty Vessel it selfe first of all had , from the weight of the vessel and water together ) to be upon the very point of 114 ounces , or 9 li. & 1/2. without any considerable difference therefrom : and thus I found it to be , at two several trials . Now according to the two first observations of Mr. Reynolds , aforesaid ; 216 inches of the forenamed water , should weigh 113.7 oun. Troy , viz. 113 oun. and 14 p. w. which comes short of our observation , by about a quarter of an ounce ; and according to his last observation the same should weigh 114.3 oun. ferè , viz. 114 ounces , and neer upon 6 p. w. which exceeds thè weight found by us , just so much as the other wants of the same ; So that the weight of this cubical body of water produced by our experiment , falleth directly in the middle between the severall weights of the same deduced from his foresaid experiments , upon one and the same kind of vessel . And according to this our most exact observation ; the weight of 283.5 cubick or solid inches of the foresaid water , ( being the content of each of Mr. Reynolds his th●●e foresaid vessels ) will be 149.625 ounces troy exactly , which is 149 oun. 12p . w. and an half . And this is the very arithmetical Mean beween his two first observations , agreeing one with another , being 149.25 oun. and his last , being 150 oun.

And so now according to this experimental Conclusion of mine own ; I shall proceed exactly in the subsequent operations upon the Wine and Ale-Gallons : For so the weights of the two severall Gallons of water aforesaid , being compared severally with this experiment , the solid capacity of the Wine-gallon , wil be found 223.105 inches , and of the Ale or Beer-gallon , 266.329 inches .

But I not resting fully satisfied with this one experiment in the said standard-Gallons ( though we conceived the same to be performed with as much care and diligence as might be ) and so desirous to trie the same thing over again , to see how nearly two severall trials would agree to confirm the matter ; knowing that two testimonies upon any thing are much better then one ; I again moved the said M. W●●gate and Mr. Sutton ( whom I still desired as witnesses to what was done ) for another triall of this thing , and that divers times ; but could not accomplish my desires herein , til about two yeares after : And so in July 1647 , I and Mr. Sutton went together again 2d. experiment upon the wine & ale-gallons . to the Guild-hall ( Mr. Wingate having promised to go along with us , but was hindred by other occasions ) carrying along with us two other great glasse-Vials like the former , into wh●ch first powring the foresaid Standard-gallons of water exactly filled , and then weighing the said Glasses with the water in them severally , by the great standard-Ballance there , with Avoirdupois-weights , and afterwards the empty Glasses severally ( being wel dried first ) by the same weights , and so comparing them together as before ; we found the weight of the Wine-gallon of water alone , to be 8 li. 1 oun. 3 dr. avoirdupois ( or 129 3/16 ounces avoirdupois ) and of the Ale-gallon of water , to be 9 li. 9 oun. 12 dr. ( or 153 3/4 ounces avoirdupois ) which are in Troy-weight ( according to the most exact Proportions of the Avoirdupois weight to the Troy-weight before noted , viz. 14 to 17. and 56 to 51 ) 9 li. 9 oun. and 5.223 dr. or 13 p. w ( which is 117.65 oun. troy ) the Wine-gallon ; wanting of the first observation or experiment ( viz. 9li . and 9.75 oun. or 117.75 oun. ) only 0.1 oun. troy , which is 2 p. w. or 4/5 of a dram-troy , which difference is of no moment . And for the Ale-gallon , 11li . 8 oun. and 5/28 of a dram-troy , or 25/56 of a p. w , viz. 10 5/7 gr . ( which is 140 5/224 oun. or 140.0223 oun. troy ) wanting also of the first experiment ( viz. 11li . 8 oun. and 11 1/4 p. w. or 4 1/2 dr. or 140 9/16 oun. or 140.5625 oun. troy , ( about half an ounce-troy , and which difference is of small moment in the matter of gauging ; but yet this latter experiment is the truer , as more nearly agreeing with the other experiments and observations following .

Now these two Gallons of water in this second experiment , are in proportion ( from the lesser to the greater ) as 100000 to 119013 , which comes neer the former proportion : And being compared with the foresaid experimentall Conclusion made by me , ( for the weight of water in reference to solid inch-measure ) will give the solid content of the Wine-gallon , 222.9 inches , and of the Ale-gallon , 265.3 inches ; which wanteth of the former solid measure in the Wine-gallon , not fully 1/5 of an inch , and in the Ale-gallon , about 1 inch . Whith difference between these two experiments , especially in the Ale-gallon , though in the matter of Gauging , the same can breed no sensible errour or difference , as wil afterwards plainly appear , when we come to shew our gauging-Lines : yet for my further and fuller satisfaction in this nice and curious piece of art , so much handled and controverted by Artists , as I said before ; and that I might come as near the matter as possibly might be ; I urged again for another triall : A●d thereupon in November next following , I again 3d. experiment upon the Wine & Ale-gallons . repaired to the Guildhall , carrying ●●●h me two other large glasse-vials , ( ●iffering much in form from the other before used , though indeed this be nothing materiall to the purpose , ) which I first caused to be weighed severally by the standard - avoirdupois weights there ; and then ( with the help of the keeper of the Standards ) filled both the standard-Gallons with fair water from the Cistern , with all the accuratnesse that might be , as before ; and which with the like accuratenesse being poured out into the said two Glasses : I caused the Glasses with the water in them , to be weighed severally by the same weights and Ballance , and that as exactly as might be , and thereupon found ( by the Comparison aforesaid ) the Wine-Gallon of water to weigh alone , 8li . 1 oun. 11 dr. avoirdupois ( or 129 11/16 oun. avoirdup . ) and the Ale-gallon of water , to weigh 9li . 9 oun. 15 dr. avoirdupois , ( or 153 15/16 ounces ) which do exceed the second observation , in the weight of the wine-Gallon 8 dr. or half an ounce avoirdupois , and in the weight of the Ale-gallon , only 3 dr. or 3/16 of an ounce avoirdup . And these two last Gallon-weights of water , do hold in proportion ( from the lesse to the greater ) as 100000 to 118699 ferè , which is very little lesse then the former proportions . And these also being collated with our foresaid experiment of 216 cube-inches of water , to weigh 114 ountroy ( or 9li . and an half ) which is in Avoirdupois-weight according to the nearest proportions of the Troy-weight to the Avoirdupois , before declared and demonstrated , 125 9/51 oun. ( or 7 14/17 li. ) do give the solid capacity of the Wine-Gallon , 223.784 inches , and of the Ale-gallon 265.629 inches ; which do exceed the first experimentall observation , in the Wine-gallon , by 0.679 inch only ; & the second , by 0.86 inch ; and doth want of the first observation in the Ale-gallon , only 0.7 inch ; and exceeds the second , by 0.3 inch only ; which last is very little .

But I being desirous to be further satisfied in the weight of these two last Gallons of water : So soon as I had performed the same at the Guild-hall by the Avoirdupoiz weights there ; I caused the said Glasse-Vials with the Gallons of water in them , to be streightway carried unto Gold-smiths-Hall , to be tried by the great Standard-Ballance of Troy-weight there ( as being the most exact kind of weight ) where first weighing each Glasse together with its water , and afterwards each Glasse alone ( being throughly drie ) I found ( by comparing the one with the other as before ) the weight of the Wine-gallon of water alone , to be 118 1/16 oun. troy , viz. 118 oun. and 1/2 dr. or 1 1/4 p. w. ( which make 9li . 10 oun. 0.5 dr. or 1.25 p. w. ) and the weight of the Ale-gallon of water alone , to be 140 oun. and 4 1/4 p. w. or 1.7 dr. ( which is 11li . 8 oun. and 4.25 p. w. or 1.7 dr. Troy ) And these two Gallon-weights do hold in proportion ( from the lesse to the greater ) as 100000 to 118761 , which exceeds that which was produced by the avoirdupois-weight , ( viz. 118699 ferè ) by 62 parts of 100000. And being conferted with our foresaid experiment for finding the proportion between the ponderall and dimensionall quantity of water , or its gravity and magnitude ; will give the solid content of the Wine-gallon , 223.697 inches , and of the Ale or Beer-gallon , 265.666 inches : which exceeds the first experiment , in the wine-gallon , by 0.59 inch ; and the second by 0.776 inch ; and wants of the third experiment in the same , from the avoirdupois-weight by the ballance , only 0.087 inch . And it wants of the first observation in the Ale-gallon , 0.66 . inch ; and exceeds the second by 0.36 inch only ; and this third by avoirdupois-weight from the ballance , only by 0.04 inch ferè , which is as much as nothing .

And here having a fit occasion and opportunity , I shal ( by way of Digression ) The foresaid severall Proportions between the Troy and Avoirdupois Weights , compared again together , & examined by the Ballance upon another Experiment . speak somwhat more concerning the Proportions between the Troy and Avoirdupois weights , for a further confirmation & verification of what was said forme●ly , and also demonstrated upon a Cannon-bullet concerning the same : And therefore the weight of these two last Gallons of water , taken first by Avoirdupois-weight at the Guild-hall , ( viz. 8li . 1 oun. 11 dr. or 129 11/16 oun. the Wi●e-gallon , and 9li . 9 oun. 15 dr. or 153 15/16 oun. the Ale-gallon ) being converted into Troy-weight , and that first , by the more common termes of proportion , viz. 60 to 73 for pound-weight , or 80 to 73 for ounce-weight , wil give the Wine-gallon , 9li . 10 oun. 6 p. w. and 19.125 grains exactly , or 118 oun. 6 p. w. and 19.125 gr. or 118 oun. 2 dr. and 43.125 gr. troy : & the Ale-gallon , 11li . 8 oun. 9 p. w. 8.62 gr. or 140 ounces , 9 p. w. 8.62 gr. or 1●0 oun. 3 dr. & 44.62 gr. troy , which do exceed the true weight taken from the Ballance at Goldsmiths-hall , by 5 p. w. and 13.125 gr. or 2 dr. and 13.125 gr. in the Wine-gallon , and by 5 p. w. and 2.62 gr. or 2 dr. and 2.62 gr. in the Ale-gallon . And then by the other terms of proportion , viz. 14 to 17 for librall weight , or 56 to 51 , for unciall-weight , the Wine-gallon wil be 9li . 10 oun. 2 p. w. and 3.96 gr. or 118 oun. 2 p. w. and 3.96 gr. or 118 oun. 0.5 dr. and 21.96 gr. troy : and the Ale-gallon , 11li . 8 oun. 3 p. w. and 20.7 gr. or 140 oun. 3 p. w. and 20.7 gr. or 140 oun. 1 dr. and 32.7 gr. troy ; which differ from the true weight of the Ballance , ( by way of excesse ) in the Wine-gallon , but 21.96 gr. or 22 gr. ferè ; and ( by way of defect ) in the Ale-gallon , only about 9 gr. which differences are very inconsiderable , the greatest of them not amounting to a penny-weight , whereas the least of the differences produced by the vulgar proportions , is above a quarter of an ounce-troy . And hereby it is again most manifest , that these latter Proportions between these two kinds of weight , are much the truer , and surely the nearest and truest that may be found , and are therefore generally to be received .

And these our experiments beforegoing for the discovevering of the solid Contents of the foresaid Wine and Ale-gallons at the Guild-hall , may be confirmed by some other experiments which I afterwards made upon the same . For before my second observation of this thing , by the weight of the Gallons of water , I caused a concave Cube to be made of Brasse , of 4 inches the Side exactly , and so the whole Cube , 64 inches ; 4th . experiment upon the Wine and Ale-gallon . into which ( being set level ) my selfe , and Mr. Sutton aforenamed , together pouring out the two Gallons of water in the Glasses which we had from the Standard-Vessels at the Guild-hall , as aforesaid , with all the accuratenesse and precisenesse that might be ( at severall times ) we found first the Wine-gallon to fill the Cube three times , and then halfe the Cube ( as ne●rly as we could possibly measure it ) which being computed , doe make 224 cube-inches exactly , for the solid content of the Wine-gallon ; And then we found the Ale-gallon to fill the Cube 4 times , and moreover to arise to such an height of the said Cube , ( viz. 0.7 inch ) as being computed , did make 11.2 cube-inches ; all which together do give 267.2 cube-inches for the solid capacity of the Ale-gallon . Both which do so neerly agree with the former experiments ( especially in the Wine-gallon ) as that this experiment may sufficiently confirme the former ; this being as plaine and demonstrative an experiment as ●ay be . And if it had been performed by a cubicall Vessell so large , as might have received into its concave capacity each of the Gallons of water wholly at once ( which indeed I afterwards wished had been done , but my forementioned Cubicall Vessell of wood was made long after the triall of this Experiment ) then the same might probably have yet come neerer the truth ; for that the of●en filling of this small Cube , might cause some small errour , which by a more large or capacious Vessel , might have been avoided , wherein the solid content of each Gallon might have been had at once , by the heighth ( or depth ) of its water ; though indeed the difference between this which we tried , and some of our former experiments ( in the Wine-gallon ) is in a manner insensible , and between all of them , in respect of both Gallons , is altogether inconsiderable . And this experiment by the brass Cube , I afterwards tried again by my self , and found it to differ as much as nothing from the former . But indeed to have the solid Contents of the Wine and Ale-gallons so very exactly and precisely by their liquid Contents , as can be imagined according to the str●ctnesse of Art , is ( I may say ) impossible , unlesse the Standard-Vessels were so narrow-mouthed , as that in the filling thereof , one might be able to guesse at a few drops of water , which in the Standard Vessels at the Guild-hall ( and I thinke in-all others ) cannot be done , they being so wide at the mouthes or tops , as that a spoonfull of water more or lesse in the filling thereof is hardly discernable , or so much more , as might breed the difference of halfe an ounce more or lesse , in the weight of the water , and so consequently of one inch more or lesse , in the solid content ; seing that one inch cubick or solid of water weigheth ( as wee shall shew anon ) halfe an ounce-troy , at least : And yet in all these severall experiments and observations compared together , the greatest difference of solid measure is but about one inch and an halfe , and that in the Ale-gallon : but yet setting aside the experiment made by the brass Cube , ( which is too large ; ) the greatest difference in the same wil be but one inch solid , which wil breed no sensible errour in the matter of gauging , as I , said before . And which may shew how very near the matter we have come , for the discovering of the true Contents of the common standard-Gallons for Wine and Ale or Beer , and surely as near the truth as can wel be gone .

But yet that there might no likely way be left unatrempted , for the discovery of this thing , I wil add to the former , one experiment more , 5th . exper●iment in the Ale-gallon alone , by taking it's Dimensions . being very demonstrative , which I made last of all in the foresaid brasse standard Ale-gallon at the Guild-hall , by taking the proper linear Dimensions thereof , it being indeed an exact segment of a Cone ( or Calathoidall ) the internall or concave superficies from the top to the bottome , being very straight and smooth , aswell as the externall or convexe Supe●ficies , and also exactly circular throughout ; only a little shelving or arching at its meeting or connexion with the bottome , and this not precisely plain , but rather a little hollowish , yet not so much as to make any sensible errour in giving the solid capacity of the Vessel , as wil straightway appear , by comparing the same with the observations before-going , and also one other observation following after . Which standard-gallon Vessel , as it is some what like in form to one of those which Mr. Oughtred speaketh of in the ●orecited place of his Circles of Proportion , concerning Gauging ( as being shewed Circles of Proportion . Part 1 chap. 9. Sect. 4. unto him , and also the measures thereof first given him , by that great Antiqua●y Mr. William Twine of Oxford , whom he faith to have undergone great pains and charge in finding out the true Contents of our English Measures ( and whom I wel knew at Oxford , being of the same House with him ) So also it comes very near it in all its dimensions : For by the Diameters of the top and bottome , and the height of that Vessell which they together measured , ( and which you may see in the fore-mentioned book ) they found , that the same would contain in its concave capacity , 268.85 cubick inches , which exceedeth the measure of our Gallon-Vessel , produced by the experiment of the brasse Cube aforesaid , but little more then one inch and a half ; and the measure deduced from the first experiment by weight , about two inches and an half ; and that which was deduced from the second experiment , by weight , ( being the least of all ) by about three inches and an half : But indeed , beside that the sides of that standard Ale-gallon were a little arching ( as Mr. Oughtred saith ) he observed divers other irregularities in the said Vessell , which might wel hinder the discovering of the true Content thereof , by some few inches .

Now the linear dimensions of the Standard Ale-Gallon at the Guild-hall ( which I took as exactly as I could , and I believe insensibly differing from the truth ) I found to exceed those of Mr. Twine's Vessell ( taken both by him and Mr. Oughtred together , as I noted before ) in the Diameter of the top , but 0.33 inch ; and in the Diameter of the bottome , but 0.1 inch ; and to want of that in the perpendicular height ( or the depth ) 0.8 inch ; by which I find the solid dimension of this Conicall Vessel ( or of this decurtate or detruncate Cone ) to be 265.5 inches , according to a multiple or conjunct composition of the aggregate of the two Bases , and their meane Proportionall ( produced most exactly by Logarithmicall supputation ) with a trient of the Altitude , thus ,

Which comes very neer the solid Content of this Vessel produced by all our former experiments ; especially the second and third , from which it differs as much as nothing ; the one of them , giving 265.3 inches , and the other 265.6 inches :

And all these our experiments in the Ale-gallon , 6th . Experiment in the Ale-gallon , taken from the half-Peck : and that from the Bushell . may yet be further confirmed by another observation or experiment being taken from the half-Peck , which they hold at the Guild-hall , to be equall in Content to the Ale-gallon , as being taken therefrom ( and so I find it to be ) and which I shall deduce from the Bushell , according to the Dimensions thereof , established by an Act of the common-Councell of the City of London , and yearly published by authority of the Lord Major , which ordains the breadth or widenesse of a Bushell to be 19 inches , and the depth 7 2/2 inches ; which being cast up according to a Cylindrical dimension , wil be fonnd to contain in in its concave capacity , 2126.5 cubick or solid inches , whose 1/8 for the half-Peck , is 265.8 inches , from which the solid Measure of the Ale-gallon , found by most of our former experiments , doth insensibly differ in a manner , especially that of the third experiment ( by weight both avoirdupoiz and Troy , from the ballance ) which gave the same , 265.6 inches ; or somewhat nearlier , that from Troy-weight , 265.7 ferè . And if we shal mediate between that of the first observation , being the greatest ( except that of the brasse Cube , which is too large ) viz. 266.3 inches , and that of the second , being the least , viz. 265.3 inches , ( as is usually done in such like cases , where severall observations or experiments made upon one and the same thing , do a little differ , and as they for the most part will , let all the art and industry be used that may be ) the Mean between them , wil be just 265.8 inches , for the solid capacity of the Ale-gallon , exactly agreeing with that of the half-Peck .

And nearly agreing with the foresaid Bushell , I found the Content of a standard-Bushell of Queen Elizabeth , which I was informed to be in the hands of the City-Founder , which was made of Brasse , in the last year but one of her reign viz. Anno 1601 having her Inscription or Title about it ; and somwhat resembling a segment of a Cone , it being wider at the top then at the bottome , whose dimensions there fore I took with all the accuratnesse that might be , and found the Diameter of the top ( or upper base ) to be one foot , and 7.5 inches , or 19.5 inches : the Diameter of the bottom ( or lower base ) one Foot and 5 inches , or 17 inches : and the depth ( or height ) 8 inches ( or more accuratly , 8.1 inches ) which being cast up , according as the Ale-gallon beforegoing ; the solid capacity thereof wil be found , 2122.165 inches ; thus ,

Which wanteth of the London-Bushell , ( being 3126.465 inches ) 4.3 inches : But indeed this Bushel was made a litle turning or winding outwards near the edg or top therof , ( which was made very sharp or thin ) which might give so much more in the solid content thereof , as to make it equall with the City-bushell ; and so I suppose this Londonbushell was first intended to be made equall with that of Queen Elizabeth , as nearly as might be ; And indeed the true Diameter of the top ( from edge to edge ) I found to be 19.6 inches , according to which if the solid capacity of the Bushell should be computed , the same would be found , 2134 inches , but that is too large : and the other comes nearer the truth : And the 8th . part of this last , for the half-peck , and so for the Ale-gallon , wil be but 266 3/4 inches .

Another measure of an Ale-gallo● , Mr. Oughtred There came into the hands of the City-Founder , together with the foresaid standard-Bushell of Queen Elizabeth , a standard A●e-gallon of the same Queen , made of Brasse in the very same Yea ●f her Raigne , having also her Epigraph about i● : & which , I went with an intent chiefly to have measured ; but indeed before my comming thither , it was sold away to a Town in Yorkshire called Whitby : But the same having been compared with the standard Ale-gallon at the Guild-hall , by the measure of water ; was found to agree there with , without any considerable difference , both as one of the Founders men , and also the under-keeper of the Standards for the City , told me , who were present at the triall thereof : which may be a means to confirm that at the Guild-hall : And these two standard Gallons , as they were both of a like capacity , so also of a like form . makes mention of in the place fore-cited , as being presented to him also by the foresaid Mr. Twine , it being ( as he calleth ●t ) a Standard-Gallon of Queen Elizabeth , which the said Mr. Twine had tried by another Vessell made of brasse , in manner of a Parallelepipedum , whose base was exactly six inches square and the Sides divided into inches , and twentieth parts : into which he pouring out the said standard-Gallon filled with water , found it to arise un●o such an height therein ( viz. 7.6 inches ) as being computed , wouldgive 273.6 cubique inches for the solid content of the said Ale-gallon . And hereabout Mr. Oughtred conceives might be the true Content of the other Ale-gallon measured by him and Mr. Twine .

All which experiments and observations aforegoing , ( with some others made by Mr. Oughtred ) may sufficiently demonstrate , the true Ale-gallon not to be of so large a capacity , as to contain 288 cubick inches , and upwards , as Mr. Reynolds , ( and some others ) will have it .

As for the discovering of the solid measure of the Standard Wine-Gallon at the Guild hall , in a geometricall manner , by the Linear dimensions thereof , as before of the Ale-gallon ; the truth is , I attempted not the same , in regard of the irregularity which I found in that Vessel , by the much arching or curvity of its Sides ( whereby it is much like to that Ale-gallon which Mr. Oughtred and Mr. Twine thus measured ; only this is wider at the bottome then at the top , whereas that was wider at the top then at the bottome , as the Ale-gallon at the Guild-hall ) which makes it to differ from a segment of an exact Cone , and so may rather be taken for a ●egment of a Conoid .

But having by our two last experiments ( especially the 5th . ) most plainly and manifestly discovered in a geometricall manner , the dimensionall quantity of the Ale-gallon in inch-measure , as neerly as may be ( which two most plain and demonstrative experiments do not considerably differ , the difference between them being but .3 of an inch-cubique ) and then by the 2d. and 3d. experiments the ponderall quantity thereof , in respect of the weight of the water exactly filling the same , ( which two experiments also do not considerably differ ; the difference being but 3dr . avoirdupois , as we there shewed , or about 1 1/3 dr. troy , being neer 1/6 of an ounce-troy ) especially the 3d. experiment ( which I take to be the most exact for this Gallon , in this respect ) we may thereby ( collating them together ) be able to discover that which was experimented by me , from my large cubicall Vessell of 216 inches : and also confirme the same ; and so consequently likewise confirme the solid Contents of the Wine and Ale-gallon , first of all discovered or produced thereby : For the solid measure of this Ale-gallon , being found most plainly and demonstratively by our two last observations or experiments , to be very nigh 266 inches : and the weight of the water exactly contained in it , found most neerly ( by the 3d. experiment ) to be ( in Troy-weight ) 11 li. 8 oun. and neer about 4 p. w. or 140.2 ounces : the weight of the water exactly contained by our foresaid Cubicall Vessel , will be found thereby ( according to geometricall proportionality ) 113.86 oun. which wants of that found by us , only 0.14 oun. which is but about 3. p. w. And so likewise may the same be very neerly confirmed by the Wine-Gallon , according to our 4th . and most plain demonstrative experiment for the same ( by the brasse concave Cube ) by which the solid measure thereof being found 224 inches ; and by the three former experiments ( conferred together ) the weight of the water contained exactly in the said Gallon-vessel , very neer about 9 li. and 10 oun. Troy , or 118 oun. the weight of the water exactly contained in the foresaid large cubicall Vessel of 216 inches , will be found thereby ( according to the foresaid proportionality ) very nearly as before , viz. 113.78 oun. which wants of the true weight found by us , but 0.22 oun. and which is only about 4 p. w. or 1/5 of an ounce-troy .

Moreover for a further confirmation of this thing , and that very nearly , I shal produce another experiment , which I made upon the foresaid brasse-Cubicall Vessel of 64 inches : For so soon as I and Mr. Sutton aforenamed , had measured out the Standard Wine and Ale-Gallons thereby , as aforesaid ; we first weighed the said concave Cube by a small Ballance of his , with Troy-weights , and then again filled the same with fair water , as precisely as possibly we could ; which we also weighing together , found the additional or differentiall weight ( take it which way you wil ) to be ( as nearly as we could possibly guesse ) 33.5 ounces , for the water alone ; And afterwards I being desirous to make a further and exacter triall of this experiment , I carried this Cube unto Goldsmiths-hall , where I procured the same to be weighed first alone , by the exact Standard Troy-weights , with all the accuratnesse that might be ; and then filling the Cube with fair setled water , as precisely as possibly I could , had the same weighed together , with the like accuratnesse ; and so by collating these two severall weights together , found the weight of the 64 inches of water alone , to be exactly in a manner as before , viz. 33.5 oun. ( or but some few grains over ) and this the Assay-master judged to be the nearest weight thereof that could be given , considering the widenesse of the Vessel , to be such , ( and that fully open on one side ) as that some few drops of water more or lesse in the filling thereof , were not very discernable . Now according to our former experiments , the weight of this cubicall body of water , wil be found , 33.7 oun. troy ; ( or rather 33.8 ferè ) which comes prety near the other : but indeed this last being deduced from our experiments made by Vessels of a much greater capacity ( or by a greater quantity of water ) must needs be the truest ; so that the other is somewhat wanting of the true weight . And these our experimentall Conclusions upon the weight of water in reference to its solid measure ; or the comparing of its quantity ponderall with its quantity dimensionall , I shall also afterwards as nearly confirm by a manifest experiment made last of all by me , upon a solid body of a known magnitude in inch-measure , in respect of its gravity taken both in the aire and in the water , and the same compared together .

SECT. III. Containing the practice of Gauging , according to our artificiall way of measuring : together with the naturall Dimension ( by way of comparison ) for a confirmation of the same .

HAving by the severall waies and means before-going , discovered the nearest quantities of the common standard-Gallons for Wine and Ale or Beer , in solid inches ; and that of the Wine-Gallon , to be at the most but 224 inches , and of the Ale or Beer-gallon , but 266 ( which two are in proportion very nearly as 5 to 6 ; being as 5 to 5 15/1● ) We shall now proceed to the practice of Gauging it self ; or the discovering of the liquid Contents of Vessels for Wine and Ale or Beer in Gallon-measure ; and that in the most easy and speedy manner that may be , according to our artificiall way of measuring , being here , as the solid dimension of a Cylinder , before declared and demonstrated , ( as I have already noted ) by the like artificiall Lines of measure : though indeed the reason thereof cannot be exactly referred to the 3d. Theoreme before-going , which is for Cylindricall Dimension in generall ; nor yet to our other , or generall Reason of measure , fully expressed soon after that ; according as the prime Integer of Vessel-measure ( a Gallon ) is here considered ; to wit , in the nature of a solid ( Cylindricall ) body , expressed in some certain solid measure , according to the capacity thereof : whereas else the same being considered either simply , absolutely , and intirely in it self , as such a liquid Measure , or in its Parts compounding , &c. ( as being by themselves apart taken for other lesser or inferior liquid Measures ) like as the Integer of measure ( or of Weight ) in generall , was formerly ; then wil the Reason of this Dimension ( as in reference to the quantities of measure in the artificiall gauging-Lines ) be the same with that of Measure ( or Weight ) in generall ; especially according to our first or generall acception or consideration thereof ; but wil exacty agree with that of weight , in both the respects of Reasons aforesaid . And thus the quantity of the artificiall gauging-Line for Wine-measure ( according to the Gallon of 224 inches ) wil be 6.58 inches ; and for Ale or Beer measure , ( according to the Gallon of 266 inches ) 6.97 inches . But yet considering the losse that happens in Wine by the dregs or lees thereof , and much more in Ale and Beer , by the frothing and otherwise : therefore we have thought it meet and convenient , to allow one inch more at least in the measure of the Wine-gallon , and 4 inches in the Measure of the Ale or Beer-gallon ; and so make the Wine-gallon to be 225 inches , and the Ale-gallon 270 inches ; which we conceive to be the most neer and indifferent measures ( in a generall respect , ) for these two Gallons , that may be : And these two do hold in proportion , exactly as 5 to 6 : And in this proportion we might suppose they were first intended , considering how very neer the same , the true Contents thereof do come : And therefore according to these two measures , we shall proceed in the worke of Gauging . And the length of our artificiall gauging-Line serving hereunto , will be for Wine-measure , 6.59 inches , and for Ale or Beere-measure , 7.00 inches : which being divided decimally ( as all the former Lines ) into 10 , or rather 100 parts , or more , ( and so turned over severall times upon a Rod or Rular of a convenient length ) and then the Diameters and lengths of Vessels taken thereby ; their Contents shall be obtained in Gallon-measure immediatly , as the solid Contents of Cylinders were formerly had , according to any Measure appointed .

But seeing these Vessels are not exactly Cylindricall ( as I said at first ) but Cylindroidall ( or as others will have them , Spheroidall ) and so the Diameter altereth between the middle and the end of them , or the bung-hole , and the head ; Therefore must a way be first had for the finding out of such a mean Diameter , as may reduce this unequall Body into an absolute Cylinder , or so neer the same as possibly may be ; that so the true Contents thereof may be had , or very neer the same : By which mean Diameter , must not here be understood a Meane either arithmetical or geometricall , but a third sort of Meane differing from them both : the discovering of which , is the first and main thing in the practise of Gauging , next after the discovering of the Contents of the Wine and Ale-gallons in solid inch-measure . And which we shall shew , most easily and speedily how to performe , by this first Rule next ensuing .

RULE I. How to find the meane or equated Diameter of any Vessell .

Augment the difference of the two Diameters , by .7 , and add the Product to the Diameter at the Head : the aggregrate shall be the meane or equated Diameter .

But Mr. Oughtred neglecting , and indeed wholly rejecting a meane Diameter , hath regard to the meane Circle ( as it were ) between that at the Bung , and that at the Head , being composed of 2/3 of the Circle at the Bung , and 1/3 of the Circle at the Head , and thereby reduceth the Vessell to a Cylinder ; and See Circ. of proport . part. 1. chap. 9. which two portions of those two Circles , he sheweth how to find most readily , by this twofold Analogie , viz.

1. As 1. is to 0.5236 : So is the Square of the Diameter at the Bung , to 2/3 of the Circle at the Bung. 2. As 1. to 0.2618 : So the Quadrat of the Diameter at the Head , to 1/3 of the Circle at the Head .

Or , for the more exactnesse in working , these Proportions might better be somewhat extended in the Terms ; And so the first of them , will be , as 1. to . 523599 ferè : And the second , as 1. to . 261799.

And so accordingly in this our Example , the Square of the Diameter at the Bung , being 1085.7025 ; the 2/3 of the Circle at the Bung , wil be 568.47 , &c : and the Square of the Diameter at the Head , being 694.8496 ; the 1/3 of the Circle at the Head , wil be 181.91 , &c. and which wil also be had , by finding the whole Circles at the Bung and Head , according to their Diameters : For the whole Circle at the Bung , wil be found , 852.7087 , whose 2/3 is 568.4725 : and the whole Circle of the Head , wil be found 545.7336 , whose 1/3 is 181.9112 as before : which two portions or sections of these two Circles being added together , do make , as it were , the mean Circle , between them , 750.3837 which must stand for the equated Base , as it were , of the Cylindricall Vessel , supposed to be reduced by this means to an absolute Cylinder : the Diameter of which Circle therefore , may not unfitly be called the mean or equated Diameter ; and which wil be found to differ but very little from ours , it being 30.910 ferè , and ours is 30.973 , and so the difference only about 0.06 inch , which in my slender judgment is scarcely considerable : And therefore I see no just cause why Mr. Oughtred should so much inveigh against Mr. Gunter , for his mean Diameter , as he doth , in the forecited place of his Circles of Proportion : though I must needs confesse this way proposed by Mr. Oughtred , for finding a mean Circle , is the most ●ationall and demonstrative . And which way of Dolial Dimension , is ( amongst others ) laid down as the most sacile , though a little in another manner , by that great Geometrician , of famous memory . Mr. Henry Briggs , in his Arithme●ica Logarithmica , ( first written and published by him in Latine ) cap. 24. viz. by finding the Circle at the Bung ( or of the middle crassitude of the Vessell , as he cals it ) and at the Head , by the Diameters thereof ; & so raising from them and the heighth , ( or length ) of the Vessel , two exact Cylinders , and then taking the difference between them , whose 2/3 being added to the lesser Cylinder ; or the 1/3 subtracted from the greater ; shall give the solid capacity of the Vessell ; as I shall shew by and by , among the other operations of this kind .

Now the arithmeticall Mean between the Diameter at the Bung , and the Head , in this our example , is 29.655 . and the geometricall Mean ( or the mean Proportionall ) is 29.47 , which is lesse then the other , and that is lesse then the aforesaid mean Diameter .

Having thus shewed most briefly how to find out the mean or equated Diameter ; I shall next shew as briefly how to find out the Content of any Vessell in Gallons , either of Wine or Ale and Beer , according to the common or naturall measure by Inches , and that by a twofold Aualogisme , in reference to the two aforesaid Gallon-measures .

RULE II. How to find the Content of any Vessel in Gallons .

As 286.5 ferè ( if Wine-measure ) or 343.8 ferè ( if Ale or The like Analogisme for Wine-measure , according to the Gallon of 231 inches , will be by the Number 294 : and for Ale-measure , according to the Gallon of 272 1/4 inches , by the number 346.6 . And here from our Gallonmeasures , the gauge point ( according to M. Gunter ) would be for Wine-measure , 16.93 inches ; and for Ale-measure , 18.54 inches : But with these points we meddle not . Beer-measure ) to the length of the Vessell in inches : So is the Square of the equared Diameter , to the Content in Gallons .

The reason of these two Numbers , or of this Analogisme , is deduced from that of a Circle to its circumscribing or Diametral Quadrat , which is vulgarly , as 11 to 14 ( but more accurately , 1000000 to 1273239 , or in the nearest rationall termes , according to Metius , as 223 to 284 ; but more truly , according to his Cycloperimetricall termes , it will be in termes irrationall , as 88.75 to 113 ) Wherefore if the cubick inches contained in a Gallon , be augmented by the consequent terme , and the product be resolved by the antecedent , the Quotient shall yield the first analogismall terme .

Now for a triall of this Rule : Suppose a Vessell to be in length , 39.54 inches ; the Diameters at the Bung and Head as before : the Square of the mean or equated Diameter ( found by our way ) is 959.326729 , which multiplied in the length , gives 37931.77886 , &c. cubique or solid inches , which divided by 286.5 ( if Wine-measure ) giveth in the Quotient , 132.40688 , &c. gallons of Wine : or being divided by 343.8 , ( if Ale or Beer-measure ) giveth 110.339 gallons of Ale or Beer . For the mean or equated Diameter , being by us , 30.973 inches , the Circle answering there●o ( for the equated base of the Vessell ) is 753.45345 square or superficiall inches , which multiplied into the length , gives the solid content of the Vessell ( as if it were a just Cylinder ) 29791.54945 , &c. cubique or solid inches ; which being divided by 225 ( if Wine-measure ) gives in the Quotient , 132.4068 , &c. gallons of wine , as before ; or by 270 ( if Ale-measure ) gives 110.339 gallons of Ale or Beer , as before also . But now according to Mr. Oughtred , the mean Circle between that at the Bung & that at the Head ( for the eq●●ted base of the Vessel to be reduced to a Cylinder ) being but 750.3837 inches , the same multiplied by the length of the Vessell , wil produce the solid Content thereof , but 29670.17149 inches . And which wil also be produced by Mr. Briggs his way before declared : For so the Circle at the Bung , or middle of the Vessell ( as the greatest Circle of the Vessel ) viz. 852.7087 . &c. inches , being multiplied by the length of the Vessell , 39.54 inches , wil produce the solid content of the greater of the two Cylinders before-mentioned , 33716●10396 , &c. inches . And the Circle of the Head , or end of the Vessel ( as the least Circle of the Vessell ) viz. 545.7335 , &c. inches , being multiplied by the same length of the Vessell , will produce the solidity of the lesser Cylinder aforesaid , 21578.30654 &c. inches . All which are most accurately and easily produced by the Logarithmical Logistique , and not otherwise ; or else not without a great deale of pains and trouble of Calculation ; and either of which here to shew , is altogether needlesse and superfluous . Now the difference of these two Cylinders , is 12137.79742 , &c. inches , whose 1/3 viz. 4045.93247 , &c. being deducted out of the greater Cylinder , or 2/3 thereof , viz. 8091.86494 , &c. being added to the lesser Cylinder ; there will either of these wayes , result 29670.17149 inches , for the solid capacity of the Vessel , as before : and which you may see performed all together , in this subsequent operation ;

Which divided by 225 , ( for Wine-measure ) affoordeth 131.8674 , &c. gallons of Wine : or by 270 ( for Ale-measure ) yieldeth 109.8895 gallons of Ale or Beer , which our accompt exceedeth in the Wine-measure , about 0.5395 gallon , which is somewhat more then half a gallon : and in the Ale-measure , 0.4495 gallon , which by conversion or reduction into the proper parts of liquid Measure , is about 3 pintes , and an half , and which differences in so big a Vessel , arescarcely considerable .

But now to apply these naturall operations in vessel-measure , to our artificiall way of Gauging , that thereby the use of our gauging-Lines may be the better understood , and the verity thereof demonstrated ; I wil lay down the severall dimensions of the fore-supposed Vessell , according to the same , delivering first a brief Rule for the use thereof , which is thus ;

First , take the diameter at the Bung and Head of the Vessel , by these Lines , and thereby get the mean Diameter as before is shewed : Then , multiply the Square thereof by the length of the Vessell , taken by the same Lines ; and the Product shall be the Content in Gallons .

1 Example , in Wine-measure .

Which fraction above 132 gallons , gives somewhat more then halfe a gallon , viz. 4.32 pintes exceeding our former measure of 13● . 40688 gallons , by about one pinte : and the measure of 131.867 gallons , ( deduced from the solid content of the Vessell in inches , found by Mr. Oughtred's , and Mr. Brigg's way ; ) by somewhat more then 5 pintes . But indeed the Diameters and the length of this Vessel , here set down from the artificial gauging-Line , are a little larger then what they really should be ; according to the quantities of the said dimensional lines of the Vessel , laid down before from the natural or vulgar Measure , being arithmetically compared with the said artificial Line taken in its true quantity from the naturall Measure , according to a more ample or numerous partition thereof : and so they give the liquid content of the Vessel in gallon-measure , somewhat more then otherwise they would . But these falling so very neerly upon whole parts , or Integers of the artificiall gauging-Line ; therefore for more plainness and readiness of working , I thought good to expresse them artificially in whole Numbers only .

And the liquid content of this Vessel , would be according to the solid content of the Wine Gallon , commonly taken to be * * The artificiall gauging-Line will be according to this Gallon , 6.65 inches . 231 inches ; but 128.4423 , gallons ; which is 128 gallons , and somewhat more then 3 1/2 pintes : and which wanteth of the true measure , ( being 131.8674 gal . according to the foresaid Standaru-Gallon of 225 inches ) 3.425 gal . which by reduction of the parts , is 3 gal . and 3 2/5 pintes .

2 Example , in Ale or Beere-measure .

Which parts above 109 gallons , do yield 7 pintes , and neer upon 1/5 of a pinte : and which doth want of our former measure of 110.339 gallons , about 3 pintes and an halfe ; but exceedeth the measure of 109.8895 gallons ( deduced from the foresaid solid content of the Vessel found by Mr. Brigg's and Mr. Oughtred's way ) only about 0.08 pinte , which is altogether inconsiderable . But yet the more parts these Lines are divided into , the more exactly still will be produced thereby , the content of any Vessel in gallon-measure , according to what was said and demonstrated in all the other Dimensions beforegoing , upon the severall artificiall Lines of measure : So that this kind of Gauging-Line , is as exact as any whatsoever .

And as I formerly shewed in the solid dimension of a Cylinder , how the same might be performed artificially , not by the Diameter & the height ( or length ) thereof together , b● also by the Circumference with the height : So here in like manner , may the Content of any Vessell be had in Gallon-measure , not only by the mean Diameter with the length thereof ; but also by finding a mean Circumference between that 〈◊〉 the Bu●g and that at the Head ; whose Square bei●g ●●●●iplied in the length of the Vessel , shall g●●● the liquid capacity thereof immediatly in gallons : And the quantity of the artificiall Line of measure , serving hereunto , wil be for Wine-measure ( ●ccording to to our foresaid Gallon ) 14.14 inches , and for Ale or Beer-measure , 15.03 inches . But because the usuall way , by the Diameters , is the easies● and readiest , and also that the Circumferences of the in-sides or Cōcaves of Vessels ( especially at the middle or bung ) can hardly be taken ; therefore I wil use no more words about this thing .

But here it may perhaps be expected , that I should shew the ready manner of taking the Diameters of Vessels at the Bung and Head , and theirlengths , by our gauging-Lines or Rods ; but this being a thing easily understood by the pregnant Practitioner , and the same also fully shewed in books particularly for Gauging , long since published ; I shall passe them by , having said sufficient for the practice of Gauging ; it being not mine intent and purpose , here to set down every particular Circumstance pertaining to Gauging ; but briefly to shew the making and use of these new artificiall gauing-Lines or Scales , and that according to our new experiments and observations for the measure of Wine and Ale , or Beer .

A-TACTOMETRIA . OR An Appendix , for the dimension , ( or the discovering of the solid quantity or capacity , ) of any irregular kind of Body whatsoever ; after the most exquisite manner that may be .

HAving now fully finished the Dimension of regular and , regular-like Magnitudes in generall , according to our artificiall way of measuring , ( or more artificiall kinde of practicall Geometry ) at first proposed ; together with the natural or vulgar , by way of metricall comparison : and of the Solids both regular and regular-like , aswell in respect of gravity , or quantity ponderal ( according to any Metall assigned ) as of magnitude , or quantity mensurall : And then in the last Part , the like dimension of that irregular kind of concave Body , ( or Vessell ) which is chiefly used for the containing or keeping of our most common potable Liquours , viz. Wine , Ale , and Beer ; I shall now at last ( by way of an Appendix ) add somewhat , for the most easie and exact discovering or obtaining of the solid capacity , or dimensionall qnantity , of any other kind of Body whatsoever , how irregular soever it be ; whose Dimensions can in no wise be taken by a Line of measure ( especially so as it be but of a reasonable bignesse ) and so whose solid content cannot be obtained in any certain Measure proposed , after the manner of geometricall Mensuration . And this I shall shew from our forementioned experiments made upon the most common liquid Body , for the comparing of its gravity and magnitude , or ponderall and dimensionall quantity together ; drawing first from thence some apt and brief Notes or Conclusions , for the more easie and ready performing of this kind of Dimension . And so from thence we gather , the nearest and truest weight of one Inch cubique or solid of clear or fair setled water in generall , to be 0.527 oun. troy , or rather , 0.528 ferè , being 0.52777 , &c. which is but little more then halfe an ounce , it being ( by conversion or reduction into the proper parts of this weight ) 4 dr. or 10 penny-weight and 13 gr. precisely in a manner . And yet afterwards , for further curiosity in this thing , I caused to be made of brasse , a concave Cube of one inch , open on one side , but having a lid or Cover made very exact , to clap close on it , with an hole in the middle thereof , no bigger , then to receive through it a small drop of water , that so I might be able to fill the same precisely to the least drop : ( but indeed this Cube was made too scanty , and so wanted of it's true magnitude , as I found by comparing it with other exact cube-inches ; and it being filled with fair water , and weighed by a very curions Ballance at Goldsmiths-hall , as precisely as might be ; the weight of the water alone , was but half an ounce troy , & about 2 gr. over , and this I tried twice over : though yet for all this , I wel knew that the true weight of one inch of water ( or of any other humid or liquid body ) could not be so exactly deduced from a Vessel or Body of one inch only , as from a Vessel or body containing many inches ; in regard of the exceeding nicety and curiosity in the making of so small a Vessel , and then in the filling & weighing of the same ; over there is in a larger Vessel : & for that in the filling of so smal a Vessel , the least drop of liquour more or less , is of some moment , wheras in a large Vessel , a few drops more or lesse are not considerable , so as to make any sensible or considerable difference in the weight of one inch being deduced therefrom . ( Nor is our foresaid cubical Vessel of 64 inches , so very sufficient for this purpose , as to ground so nice & curious an experimentall Conclusion thereupon . ) And therefore we must proceed herein , à majori ad minus , from a greater quantity to a lesse , both for the discovering of the weight by the measure ( as before ) and also of the Measure by the weight : and not à minori ad majus , from a lesse to a greater ; for that wil not hold good in this thing . But indeed in a solid Body , of what matter soever it be , the thing wil hold just contrary to the former , in both the foresaid respects : For herein , we must rather go à minori ad majus , then otherwise : And so the weight of one inch cubick or solid , may more exactly be had from a Body of one inch cubick only ( being exactly made , and which I find is hard to do ) then from a Body of many inches ; and consequently , the gravity or ponderosity of any bigger body ( of the same matter ) may be had most exactly and speedily therefrom : For thus Ghetaldus in his experiments before-mentioned , for the comparing of sundry Metals together , in gravity and magnitude , and so to determine exact proportions between them ; began upon smal quantities , and from thence deduceth greater quantities : And so in particular , for finding the gravities of spherical Metalline bodies by their Diametrall magnitudes , hee first began with a Sphear of a small magnitude ; Or rather , because hee could not have such a body exactly made ( as I noted formerly ) he procured a metalline Cylinder to be made , ( which might be more exactly done ) of equall Diameter and altitude ; and the same to be but two inches from the Roman Foot ; and thereby he obtained the gravity of a Sphear , of the same metal , whose Diameter was equal with the Diameter , and altitude of the Cylinder , &c. and from this he deduceth the gravity of any other Sphear of the same Metal , of what magnitude soever , as I formerly shewed . And so likewise have I seen our Countrey-man Mr. Reynolds beforenamed , in making the like experiments , for finding the proportions of gravity , &c. in Metals , perform the same by Bodies of one Inch cubique only , holding it a surer way , so to obtain the weight of one inch cubick or solid of any Metall ( or other solid substance ) then by a body of a greater magnitude : Though yet ( for mine own part ) I conceive there may not be altogether so much curiosity or difficulty in this , as in the former ; but that the weight of a solid body of one inch in magnitude , may as ( if not more ) nearly be deduced from the weight of a body of a greater magnitude or dimension , ( of the same matter ) as the weight of a liquid or humid body of one inch in magnitude or bulk , can be deduced from the weight of a liquid body ( of the same kind ) of a greater bulk or crassitude ; by how much , the weight of any solid body in generall , may be had more exactly then the weight of any liquid body ; sith that a solid body is weighed immediatly by it self in the Ballance or Scale , without the help of any other thing to contain it ; whereas a liquid body ( especially one so thin and fluid as water is ) cannot wel be weighed by it self ( unlesse it be in a smal quantity ) but by means of some Vessel to hold it , whose weight must also be had first or last by it self ( though indeed , it is not absolutely materiall or requisite to take notice of the same , as to any certain , known , regular , denominate weights , unlesse the Vessel be last of all weighed by it self , for otherwise any irregular indenominate weights wil serve to poize it , if it be first weighed alone , before it receive the liquid Body whose weight is required . But this by the way .

Now one inch of water weighing , 0.5277 , &c. oun. Troy , viz. 0.528 oun. ferè the same wil be found in Avoirdupois-weight , to be 0.5795 oun. ( which are in librall or pound-weight , 0.04398 li. troy , and 0.03622 li. avoirdupoiz ) And The dimensionall and ponderall quantity of water compared together severall ways . from the same experiment I find , that a body of water of 40 inches , will weigh 21 oun. troy almost exactly , and 23 oun. avoirdupois almost as nearly ; and that 36 inches of water wil weigh 19 oun. troy exactly , but which wil be in avoirdupois-weight , 20 44/5● oun . or 20.86 , &c. And so contrariwise , arguing from the gravity or ponderall quantity of water , to its magnitude or dimensionall quantity ; the solid measure of one ounce - troy , will be ( in unciall or inch-measure ) 1.8947 inch ; and of one ounce avoirdupois , 1.72556 inch ; ( and consequently in Pedall or Foot-measure , 0.001096 , for Troy-weight ; and 0.00099859 for Avoirdupois-weight . ) And so the solid measure of one pound-troy of water , wil be , Uncially , 22.7368 ; and of one pound-avoirdupois , wil be Uncially 27.609 : and the same wil thereupon , be Pedally , 0.013158 ferè , for troy-weight , and 0.015977 , for avoirdupois-weight . And so from our foresaid experiment made by the large cubicall Vessel , which was an Octant of a Foot cubique ; the weight of a Body of water of a just Foot in magnitude , ( or the weight of a cube-Foot of water ) wil be eactly , 912 oun. Troy , and which is just 76 li. Troy : And this wil be in Avoirdupois-weight 1001 21/51 oun. or 1001.41176 oun. and so 62 10/1● li. or 62.588 li.

But indeed , of all these termes of proportion or comparison , between the gravity and magnitude , or the ponderall and dimensionall quantity of water , those of ●1 ounces , to 40 inches ( in respect of Troy-weight , which is here the best ) are the efittest for a generalluse : for that I find them to answer indifferently to all the severall experiments that have been made by me , and Mr. Reynolds severally , for the foresaid purpose ; which said severall experiments , though in the weight of one Inch of water , and so in the solid measure of one Ounce , they may make some small difference , so as som what to alter the decimal numbers , by which the same are expressed ( the said numbers being produced or extended beyond two or three figures in the fraction ; and in which all our best experiments do concurre , viz. as to make the weight of one cubick inch of water , 0.527 oun. troy , or very near thereabout ; and so the solid measure of one ounce-troy of water , about 1.895 cube-inch ) and so to make some litle difference in the operations performed thereby , especially the greater that the quantity of water used , is ; yet they all agree in this , as to give to 40 inches of water the weight of 21 oun. troy , ( and so contrarily , to 21 oun. troy of water the solid measure of 40 inches , ) without any considerable difference : And then besides that these two numbers thus answering each other reciprocally , they may also more easily be borne in memory , then the decimall numbers , which denote the gravity of one inch of water , and the magnitude of one ounce of water : And the like wil be for 23 ounces to 40 inches , where the operation is performed by Avoirdupoiz-weights , for want of Troy-weights . But yet again after all this , considering the foresaid experiment of the large cubick Vessel , made by me , was as exact , ( I may be bold to say ) as wel may be , or as need be desired ; then the foregoing quantities of 19 oun. troy of water to 36 inches of the same , ( which are those of 114 oun. and 216 inches , in the least Terms ) may be generally used , ( being as easily born in memory as those of 40 inches and 21 oun. ) to expresse the Proportion between the gravity and magnitude , or the ponderall and dimensionall quantity of water ; and so for the producing or discovering of the solid measure of any irregular body , as I shal next of all shew .

Having therefore thus discovered the weight of water in generall , in relation to its solid measure , &c. ( and that as nearly we conceive , as may be , or need to be ) or the proportion between the gravity or ponderall quantity , and the magnitude or dimensionall quantity thereof , and the same laid down here in all the fittest terms that may be , for a generall and speedy use ; we may thereby be able easily and exactly to discover the solid content of any solid Body whatsoever , how inordinate How to find out the solid capacity of any irregular Body , by the weight of water . or irregular soever it be ; by conferring it with the foresaid common liquid Body . according to magnitude ; and that is , by finding out or discovering such a body or quantity of water , as is equal in bulk , magnitude or crassitude , which the solid body , whose measure is required : Which water being exactly weighed , and then the dimensionall quantity or solid measure thereof , deduced from its weight , by means of the foregoing terms of proportion for that purpose : the same shall be also the solid measure of the solid body required . And this may be performed by any irregular kind of Vessel whatsoever , by filling the same exactly with fair water , How to finde the exact quantity of water ( or other liquid body ) which is equall in magnitude or dimension to any solid Body given . and then sinking in the solid body , so low at least , as wholly to cover it : for so the water forced out of the Vessel by the solid body , shal be equall in bulk or magnitude thereunto ; the solid body now taking up the same room or space in the Vessel , which that did . And the quantity of water so forced out of the Vessel , may be found , either by first weighing the Vessel full of water , and then the Vessel with the remainer of the water left therein , after the taking out of the solid body , ( for here the weight of the Vessell alone need not be had ) whose difference of weight shal be the weight of the water effluxed : or else ( which is much easier and exacter , especially if the Vessel be so great , as that it cannot conveniently be weighed ) by making a small hole near the top of the Vessel , or a nick or notch in the very edge , brim , or top thereof ; and so filling the Vessel exactly up to the said hole or notch , through which the water forced out by the solid body may run , and so be received into some other smaller Vessel ; which may be done exactly to a drop , if there be a quill or some such like things as a pipe or spout , closely fastened in the said hole or notch , whereby to convey the water cleary away into the other Vessel , without spilling any ; for so the weight of this water ( which is equall in magnitude to the solid body ) being taken ; and then the solid dimension thereof produced thereby , ( according to some of the foresaid proportional Conclusions for this purpose ) the same shal be the solidity of the body , given to be measured . And herein , it maketh no matter , whether the solid body be sunk into the Vessel of water , any lower then just to cover it , or not ; for it comes all to one passe , seing that it still takes up but the same room or place in the water ; and so consequently causeth but the same quantity of water to flow out . And so may the solid measure of any part assigned , of any solid body be had , by sinking that part only into the Vessel of water , and then observing the quantity of the water effluxed thereby ; for this shal be of equall bulk or magnitude with that part of the solid body ; and therefore the solid content thereof being found out by its weight ( as before ) the same shal be the solid content of the said assigned part of the solid body , as was required .

And thus also may the capacity of any irregular concave Body or Vessel whatsoever , be discovered in solid measure , by the weight of the water exactly filling the same ( or the solid measure of any part thereof assigned , by the weight of the water exactly filling that part ) if the bignesse thereof hinder not the convenient weighing of it ; according as I formerly shewed for discovering of the concave capacities of the City-Standard-Gallons for Wine and Ale or Beer , in solid inch measure ; and both which Vessels were of an irregular forme , especially the Wine-Gallon , whose dimensions could not be rightly taken by a Line of measure , as I then noted .

Or again , the weight of the water equall in magnitude to any irregular solid body , whose measure How , to find the gravity of the Water ( or other liquid body ) which is equall in magnitude to solid body given , by the gravity of the solid body only . And so the manner of weighing a solid body in Water . is disired , may be had by the weight of the solid body only ; the same being first exactly weighed in the Ballance or Scale , after the usuall manner ; and afterwards out of the Scale , in water ; which is ( according as * * Archimed . promot . post exempl. 1 prob . seu 8 prop. Ghetaldus teacheth ) by hanging the solid body by one of the Scales of the Ballance , into the water , so as it may hang freely therein , being covered therewith , and that by an horse-haire , ( which hesaith to be , in a manner , of equall weight with water , and so neither addeth to , nor taketh from the gravity in the Body to be weighed ) or many such haires joyned together , if the gravity of the Body require them ; but with due consideration had of such part thereof ( in respect of weight ) as hangs out of the water , that is ( as he saith ) as hangs out of the Scale to which the Body is fastened , even to the Body it selfe ; by putting the like quantity of haire in the other Scale where the weights must be put ; l●st that the many haires together , should add some weight to the Body , ( and so neglecting that part of the haire which is wound about the body , or with which the same is bound up , because that is supposed to be equally weighty with the water , as being with the Body in the water , and so to have no gravity therein ) for so the Scales shall become equiponderant ; and then the Body hanging freely in the water ( and so as neither of the Scales touch the water ) the same shall be ponderated or poised , as if it hung in the aire ; and will be in the water so much lighter then it is in the aire , as is the gravity of that portion of water , which is equall in magnitude or bulke to the solid body : and which Archimedes demonstrateth , and from him Ghetaldus in the Archimed . lib. 1. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ( 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ) seu de iis quae insident aut vehuntur in humido , sive liquido , ( Vel 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 in aqua , ut vulgò dicitur ) prop. 7. forecited prop. of his Archimed . promot . And therefore the difference betweene the gravity of the solid body taken in the aire , ( which is the true gravity thereof ) and the gravity in the water , shall be the gravity of so much water , as is equall in magnitude to the solid body . But this must be understood of such a solid body or Magnitude , as is heavier then water ; that is , being put therein , will sinke , or be carried down into the same , of its own accord : Whereas else , if it be a solid body which is lighter then the water , that is , being put into the same , will not sink , or descend down of it self ; then the gravity of such a portion of the water , as is equall in magnitude to this solid body , must ( in this way ) be had by the adjection or apposition of some other solid body , which is weightier then water , ( whose gravity both in the aire and water are first had , as also the gravity of the lighter body in the aire ) so as these two several Solids being joyn'd together , may make up ( as it were ) one Body ; which being let down into the water , may voluntarily sink or descend : And both which Solids thus put together in one , wil be lighter in the water , then the heavier Solid alone , ( as Ghetaldus sheweth in the fore-named place ) And wil be also lighter therein , then they are in the air , ( both their true weights being taken together ) by so much , as is the gravity of the water equall in magnitude to them both , ( according to that of a solid body alone which is heavier then water ) And this shall exceed the gravity of the water equall in magnitude to the lighter body alone , by so much , as is the gravity of the water equall in magnitude to the heavier body ; and therefore the gravity of the water equall in magnitude to the lighter body ( which was required ) is thereby immediatly obtained . And what hath been here spoken concerning Water , the like understand of any other liquid or humid body , in which a solid body may be thus weighed . And indeed this latter way for the finding of the gravity of the water ( or other liquid body ) which is equal in magnitude to a solid body given , by the gravity of the solid body only ( & therby to discover the quātity or capacity of the solid body , according to a certain measure appointed , as before ; which is in my judgment , the mostexcellent and beneficiall use that can be made of it ) is much more neat , then the former , but withall much more curious and difficult in the performance ; and therefore I shal rather refer the Practicioner to the first way , as being very plain , easie , and exact .

But now , that I might be found not more wanting in the practice and experience of these things , then of the other before going ( for with the bare speculation of things I could never yet well content my selfe , though they were never so likely in themselves , ( as many men do , and so take up all things upon trust ) I wil here produce a manifest experiment made by my self , of both these waies . under one , in reference to the discovering A double experiment upon the foregoing Conclusions . Apr. 12.1649 . of the dimensionall quantity of a solid body , by such a body of water as is equall in magnitude thereto , whereby the verity of all these mathematicall Conclusions may plainly appear . And therefore to this end and purpose , I sought to get some solid body of Metall or Stone , of a reasonable bignesse , which might not suck or receive water into it , and be also wrought into some such regular form , as that the solid content thereof might be exactly obtained by a Line of measure , according to the plain and usuall way of measuring , and also such an one , as might easily be hanged in a string ; without any fear of slipping out when it was in the water ; as some Prismall or Cylindricall body , or the like ; but which ( though with much seeking up and down ) I could not meet with ready to my hand , nor otherwise easily obtain : And so after all this adoe , I could procure none that might any way serve the turn , but one which was of a sphericall form , and that was a Marble-bullet , made as round and smooth as might wel be imagined , ( and which therefore was the most troublesome for the performance of this experiment , that could be , being the most difficultly hung in a string , of any other kind of body ) whose diameter I therefore first took most exactly by a Callaper Compasse , and applying the same to a Diagonal Scale of an Inch , I found it to be 4.95 inches , and from thence the solid Content , 63.5 inches : which being done ; I immediatly carried the Bullet to Goldsmiths-hall ( where I was to make my experiment ) and there got the same exactly weighed in the Scale ; and so found the weight thereof , 89 oun. 2 p. w. ( which are 7 li. 5.1 oun. troy ) Then for the discovering of the weight thereof in water , as also for the performing of the other Conclusion , for the finding out of the quantity of water equall in magnitude thereunto , both under one : I got presently an earthen Vessell , very broad and deepe , which held about a good Paile-full and an half , or more ; causing first a small hole to be drilled through it neer the top , and then a pipe of Tinne to be fitted thereto , and fastened therein so close , as that no water might passe through the hole beside the pipe ; and then I next prepared severall small lines or strings of horse-taile haires twisted together , with which I girted the Bullet several times about , till it would hang firmly , and not slip out ( and which I found very tedious and troublesome to doe , by reason of the smoothnesse and weightinesse of the body , and the slipperinesse of the horse-haire together ) continuing the said strings or lines out together from the bullet to a convenient length , and tying them together , whereby the bullet might hang down in the water from the Ballance : which done ; I fastened the said strings ( not to one of the Scales , as Ghetaldus seemes to intimate , but ) to one end of the Beame , where the Scale is hung , which I conceived to be the best way ; especially considering the weightinesse of the Body : And then because in that respect , it required a strong Beame whereon to hang , and that the Scales belonging to the great Beame would be troublesome , and an impediment in the way , in respect of their largenesse : therefore I , and the Assay-master of Gold-smiths-hall together , conceived it best to put in their place a small paire of Scales , which might be sufficient to hold the Weights ; and so the Body might hang freely down by the Scale without interruption ; which being done ; I set the foresaid earthen Vessel under that end of the Beame on which the Body hung , and then filled the same with faire water exactly up to the hole therein , till the water entred the pipe , ( letting it run a few drops till it staied of it selfe , for so I was sure of the exact filling thereof to a due height ) under which I then set a glasse-viall with a little Funnell in it , which might receive the water forced out of the Vessell by the bullet , to a drop ; and then sinking the said Body gently into the water , ( affording it so much string therein , as might keep it fully covered all the time ) I had the Weights put into the other Scale , together with neer about such a quantity of horse-haire , as was contained in the several small strings joyned together , by which the bullet hung ( neglecting that part thereof which was wound or wrapped about the bullet in the water , for the reason before alledged ) and so the bullet hanging freely in the water , we found it to weigh 55 ounces and 3 quarters , Troy ( which is 4 li. and 7.75 oun. ) and thus I let it hang in the water duly poized , till all the water was run out into the Glasse , which it could make to flow out ; for this water thus effluxed , was of equal magnitude with the said Bullet . And this being finished I repeated one part of my experiment the next day , by filling up the Vessel againe to the hole thereof exactly as before ; & then setting another Glasse with the Funnell in it , under the pipe , I sunk the said bullet into the water , letting itabide therein , till it had forced out all the water that it could , to the last drop ( not one drop running beside the pipe ) & then afterwards weighing the said several Glasses of water effluxed out of the Vessel by meanes of the bullet , I found them to differ insensibly ; the weight of the water alone being 33 ounces , and 8 p. w. or 33.4 oun. troy , ( which nearly agrees with our experiment made upon the brasse Cubicall Vessel of 64 inches aforesaid , whose water we found , to weigh 33.5 oun. troy ) Now the difference between the weight of the Bullet in the air , or the Scale , 89.1 oun. ( or 89 oun. 2 p. w ) and its weight in the water , 55.75 oun. ( or 55 oun. and 15 p. w. ) is 33.35 oun. ( or 33 oun. 7 p. w. ) for the weight of the water equall in magnitude with the bullet ; which you see differs ( defectively ) from the weight of the water forced out of the Vessel by the bullet , only 1 p. w. which is not considerable : And yet had I been so curious in weighing the bullet in water , as to have put in the Scale where the weights were , the same quantity of horse-hair , ●s precisely to an hair , as were contained in the lines or s●rings by which the bullet hung , except those which were about it in the water ( according as Ghetaldus directeth ) then haply might the weight of the water equall in magnitude with the bullet , produced thereby , have exactly agreed with the weight of that which was forced out of the Vessel by the bullet ; but inded I think there wanted about so much hair in the weight-Scale as might have made the bullet to weigh in the water , 1 p. w. lesse : But I conceived there was no need of so great curiosity in such an experiment as that was , especially upon so great a body ; neither do I conceive it absolutely necessary & requisite to perform the same by horse-hair , but that silk or thread might serve the turn , putting so much in the Scale with the weights , as is used about the body to be weighed , and this can breed no sensible errour , unlesse it be for the performing of some very nice and curious experiment indeed ; as to find the exact difference and proportion between the weight of a Metall ( as Gold or Silver especially ) other thing , in the aire , and in the water , and the like ; then indeed Ghetaldus in A●chimed . p●omot . post prop. 19. speaking of his new artifice , whereby he would find out the quality of Gold from the gravity only which it hath in the air , and in the water ; saith , that the weight of pure Gold , which is in the air 19 , wil be in the water 18. And the weight of Silver , which in the air is 31 , will be in the water 28 : And so the weight of air , which is in the air 9 , wil be in the water 8. to use horse-haire , and in that strictnesse and preciseness , as Ghetaldus speakes of , will be altogether requisite , for the reason before del●vered : And now you see how these two experimentall Conclusions do manifestly confirm one another , & so both of them do confirm our former experiments for the weight of water in relation to its solid measure , ( and so for the solid measure thereof in reference to its weight ) for thereby the weight of 63.5 inches of water , ( the same as the solidity of the Marble-Sphear aforesaid ) wil be upon the point of 33.5 ounces troy : And so ( to come to the point in hand , which is the discovering of the solid capacity of this spherical body in inch-measure , by the weight of the water which is of equal magnitude with the same ) the solid measure of 33.4 oun. troy of water ( the quantity of water agreeing exactly in magnitude with the Bullet ) wil be thereby 63.3 inches , for the solid content of the bullet , which wanteth of the solidity found at first by the Diameter 63.5 inches , only 1/5 of an inch . And indeed the solid Content thus found by water , I may adjudge to be the truer measure , in regard there was a smal flotting in one place of the bullet , which might well make the solid content thereof lesser by so much then it was , being taken according to a full rotundity or sphericalnesse every way , without any flatting in the same , and according to which the Diameter was taken . And therefore in any solid body , of how regular a form so ever it be , where there is any such flatting , dent , or hollowishnesse , or other like defection in any part of the superficies thereof , which may diminish somewhat of the true solid dimension which it naturally obtaines and ought to have , according to such a forme ( and which yet will be fully deduced from its linear Dimensions , according to a plain way of measuring , as if there were no such defection ) there the solid Content ( in the state the body then is ) wil be most truly discovered by water ( as I have here shewed two several waies ) for that the water wil search out the true quantity of those defections , which a Line of measure cannot , and so give the solid content of the Body accordingly .

And this way for discovering the measure of any irregular Solid , must needs be generally better then that which Mr. Diggs hath delivered in his * * Pantomet . lib. 3. cap. 14. Stereometry ; to be performed by an exact Cubical , or other rectangular prismal or parallelepipedal Vessel , with water ; and that in the usual way of measuring ; by taking the Dimensions of the Vessel , as to the finding of the superficiall content of it's rectangle Base , taken according to the concavity thereof ; and then noting the ascent of the water , in the Vessel , when the solid body lies covered therein , and also when it is out , whose difference being infolded with the superficicies of the water ( or of the foresaid rectangle Base of the Vessel ) gives the solid measure of the water equal in bulk or magnitude with the said Body , and so withall of the Body it self ; Clav. Geom. pract. l. 5. c. 11. and which way Clavius also speaketh of , as being the way vulgarly noted and used by Artists : whereas the way delivered here by us , may be performed by any kind of Vessel whatsoever , how irregular soever it be ( as was the Vessel by which I performed my foregoing experiment upon the Marble-Sphear or Bullet , for the discovering of its solid measure by water ) and not only for the discovering of the Content of any irregular solid or grosse body , but also of any irregular concave body , according to a solid dimension , as I have both said and also experimentally shewed before . But however that common way delivered by Mr. Diggs and also by Clavius and some other Latin Authours , need not to be tied or confined wholly to such a kind of Vessel as they speak of , but the same may be as wel perform'd by any other kind of prismal , or any Cylindrical Vessel , such as a Pail or other circular Vessel , being exactly made : but that indeed the Bases of these Vessels , wil not be altogether so readily obtained as those of an exact cubicall , or other rectangular Vessel , unlesse it be by our artificiall way of measuring , where the base of the Vessel is equilateral or exactly circular . And of this way I would have produced an experimentall example , aswell as of the other , and that by the same regular body , to have compared them together , if I could have met with ever a Vessel fit for the purpose : but however the thing being so very plain and perspicuous of it self , there needs no example either from experiment or otherwise , to illustrate it .

But now whereas after all this , it may Objection . concerning the difference of gravity in Water , in reference to the foregoing Dimēsion of Bodies . be objected , that all water weigheth not alike ; but that different kinds of water ( as Rain-water , Fountain-water and River-water ) are of a different gravity ; and therefore our foregoing experimental Conclusions for the weight of common water in general , in relation to its solid measure ( or for the comparing of its gravity and magnitude , or ponderal and dimensional quantity together ) cannot hold generally true : To this I answer ; that albeit these several waters Answer . do usually differ somewhat in gravity , ( which I cannot deny , but must needs acknowledge , and that not only from natural reason it selfe , but also from my owne experience , which I shall now come to shew , ) yet not so much , as to make any notable , or considerable difference in the solid content of a Body produced severally thereby ( especially Rain and River-water ) for which end those Conclusions were aymedat , and intended by us ; but that any one of them may be indifferently used in this thing , without the committing any considerable errour , as I shal shew by and by , in discovering the gravities of several waters and comparing them together . And Marinus Ghetaldus in all his several experiments made upon Water , for the comparing of it and divers other bodies both solid and l●quid , together , in respect of gravity and magnitude , speaketh only of water in generall , without any difference or distinction .

Now the water by which Mr. Reynolds made his experiments upon his 3 severall parallelepipedall Vessels of 283 1/2 inches , formerly mentioned , to find the true weight thereof , was ( as he told me ) fair setled Rain-water ; And such was also the water , by which I made my experiment upon the cubicall Vessel of 216 inches , as I noted formerly , it being the most simple kind of water , and so generally the best for that purpose : for so that learned Mathematician W. Snellius going about to discover the true weight of the Rhynland-Foot , in its cubicall capacity ( which he wil have to be exactly equall to the old * * Suel . in Eratosth . Batav . l. 2. c. 2. And where he pitcheth upon the same Roman Foot ( by name , ) for the truest , which Mr. Greaves and most others do ; namely , Pes Colotianus aforementioned : but differeth in the magnitude thereof from Mr. Greaves ; for that he makes it greater then the English Foot ; whereas Mr. Greaves makes it lesser . For he saith that the London-Foot , which is generally used throughout England , and whose measure was taken from the Iron Standard of 3 Feet , kept in the Guild-hall ; and so transmitted to him , is 968 parts , such as the Leyden-Foot , vulgarly called there the Rbynland-Foot , ( which he makes exactly equall with the foresaid Roman-Foot ) is 1000 , Eratosth . But. l. 2. c. 1. whereas Mr. Greaves makes the very same measure of the English-Foot to be 1034.13 , such as Pes Colot . is 1000 : and the Rhy●land-Foot ( or that of Snellius ) to be 1068.25 of the same parts . And so this Foot to be 1033 such parts as the said english Foot contains 1000 ( and the foresaid Roman-Foot to be but 967 such parts as was formerly shewed ) and consequently the English Foot will be from thence , 968 such 〈◊〉 ●s the Rhynland Foot is 1000 , which ●●tees exactly with the observation of Snellius . But indeed Snellius takes the dimension of Pes Colot . chiefly from the bare description and de●ineation thereof made by Ph●l●nder in his Commentaries upon Vitruoius , Architect . l. 3. c. 3. ( who there faith , that he found this Foot ●o agree with that of Statisius ) presuming upon that typograp●icall ssenent , which he had received about the contraction of letters and Lines upon the paper , after their impression , by a 60th part . For so saith he in Eratosth . Bat. l. 2. c. 1. before-cited , Charta uda dum praelo subiicitur & typum patitur , ipsá pressurá & humore quem anteà imbiberat , non nihil extenditur & seipsā fit amplier , qua post modùm siccate , iterùm contrabitur , & simùl linearum measuras quas receperat , jūsto exhibet minores . Pars enim sexagesima typorum & formoiuns longitudini excusis decedit , quemadmodùm à diligentibus & peritis Typographis sciscitando edoctus sum . Which last must needs be erronious and uncertaine , and cannot hold , generally true , as reason it selfe ( besides the experience of my selfe and others ) may plainly demonstrare , according to what I formerly : said concerning this point . Roman-Foot ) in water , that thereby he might ( as it seemed to him ) transmit the measure thereof so much the more certainly and easily to posterity , ( like as many eminent men together had done before at Rome , for the determining and establishing of the exact measure or quantity of the said Roman Foot ( as Ciaconius out of Lati●us Lati●ius his * observations of the Roman Foot reporteth , who was one of the eight that made the experiment together at Rome ) and so two others , by name , Lucas Paeius , and Villalpandus , by two other Vessels ; conceiving it to be a surer way to discover ( or recover ) the same , then by haires , grains of Corne , digits , palms , and the like ) did above all other waters , chuse Rain-water , because that ( saith he ) being fallen from the See M. Greaves his Discourse of the Roman Foot . pag. 12. 13. heaven , brings down with it no taincture of any earthy dregs ; and moreover for that it seemed to be alike in a manner , in all places ; and this hee used after many daies setling , being thereby made very pure and cleare .

And then , the water with which I measured the Standard - Wine and Ale-gallon at the Guild-hall , to find the weight thereof , and thereby the solid Content in inches , was from the new River of Ware ( commonly called Middleton's River , and which water is in a manner of equall weight with Rain-water , as I shall shew by and by ) as also that at Goldsmithe-hall , with which I measured my two brasse concave Cubes , the one of 64 inches , and the other of 1 inch , to find the weight thereof : ( and the lesse Cube also another time with rain-water ) and also by which I performed my last experiment there , upon the weighing of a solid body in water , &c.

As for Rain-water compared with Fountain , Spring , or Well-water , Est in aequali mole ratio aquae plaviae ad distillatam , quemadmodùm 1000000 ad 997065 pluviae au●èm ad putealem , ut 1000000 ad 1007522. Eratosth . Bat. l. 2. c. 5. The last of which I find indeed to be so , according to the Weights of those two waters set down by him , or more compleatly , as 10005000 to 10075216. But the first of these I find according to his weights of the waters in the Vess●ll by which he made the experiment , ( and so of his cube-Foot of the same waters , which I exactly deduced from thence ) to be as 1000000 to 997117. being more compleatly , as 10000000 to 9971167. in respect of gravity ; Snellius observed the proportion to be , as 1000000 to 1007522 ; and of Rain-water naturall , to the same artificiall , or distilled , to be as 1000000 to 997065 : For that he might have his Rain-water exactly defecated , or clensed & purified from all earthy dregs or grownds , it seemed good to him ( as he saith ) to use for the experiment of the exact weight of a cubicall Foot of water , chiefly that which might again be collected into it self from vapours and exhalations , and so betooke himself to Chymicall distillation , according to that form , which they usually call by a technicall or artificiall term or expression , Balneum Mariae , seu Maris , because that is in no wise forcible or violent , but very gentle ; And then moreover hee used pure Fountain-water , to trie what his cubicall Foot might alter in gravity , in these three waters : And this experiment he performed by a Cylindricall Vessel made of brasse with all the accuratenesse that might be , having its altitude and Diameter equally semi-pedall , from whence he deduced the weight of a cube-Foot in Water .

But if we wil be so very curious concerning the difference of gravity in severall waters ; then we may as well question , whether all water of the same kind ; be of the same gravity or not ( except Rain and Snow-water , which we conceive to hold alike in all places , without any sensible difference ) as whether all Fountain , and all River-water do weigh alike , especially the first of these two , but that it may alter and differ in gravity as well as in other things , according to the different nature and quality of the Earth where it is engendred , and of the veines and passages thereof , through which ( as it were through Channels ) it runs , and so the various matter wherewith it is mixed ; and more especially , if one water be a meer simple water , and another be a minerall ( for so Naturalists do usually distinguish waters ) for then these wil more sensibly differ in gravity ; and so wil severall minerall waters among themselves , being of a different nature : And indeed most of our River-water seems to come from Fountaines or Springs ; and also for that Aristotle gives to them both , the same originall of generation , Aristos . l. 1. Meteor . c. 13. to wit , from vapours and fumes ( or air ) in the caves and passages of the earth , condensed and concreted into water by the coldnesse of the Earth . And therefore seing that the generation Magir. Physiol . peripatet . l. 3. c. 4. Com. & l. 4. c. 7. of fumes and vapours . ( and thence of water ) under the earth , is continuall ; it followeth , that the flowing , and the course of Rivers is perpetuall : so that a River is , as it were , no other thing then the water of a perpetuall Fountain or Spring , continually running on in a great body or bulk . Keckerm . System . physic . l. 2. c. 15.

But now as for the difference of gravity in waters homogeneall , ( or of the like kind and denomination ) I shall prove the same by experiment , from four severall Fountains , or Springs ( and partly from two Rivers ) in comparing them with Rain-water severall times : and which is as much as ( if not much more then ) any other hath done in this kind , that I could ever yet hear of : And the waters which I first tried by way of ponderall comparison ; were first , Rain-water , as being the Base of the Experiment , in regard of what hath been said there of before ; and so to which all the other waters are here compared ; and which I received into a clean Vessel as it fell from the Clouds , that so I might have it as pure as might be , without the mixture of any earthly matter : And the Fountain-waters which I now used , were from the Conduit in Grayes-Inne Fields , commonly called by the name of Lambs-Condait , which is a pure Spring of it self ; and from the Standard in Cheap-side , which is conveighed under ground by pipes , from a Spring at Padington , being a Village about 3 miles distant from London to the N-W . ( as also is the water of the other two Conduits at the two ends of the said Street , from the same Spring-head ) and yet is received as pure and cle●r at the said Conduit , as if were taken at the Spring-head it self : And then the River-waters were from the Thamès , and from the new River of Ware , or Midleton's River aforesaid . And with these I tried Snow-water , ( which was pure Snow as it fell , having never touched the ground nor other thing , beforeit came in the Vessel where it was put , and there dissolved . ) Which other severall Waters aforesaid , I took the pains to see always fetched from the right places , thereby to avoid all errours and mistakes which might happen by trusting to any messenger alone ; and so had them carried to Goldsmiths-hall ( where I was to make this experiment also ) leting them stand there a setling for 2 dayes : ( though indeed the Fountain-waters being so very pure and clear , needed not so much setling ) And the Vessel , by which ( as a Standard or Gage ) I first tried these six severall waters , was the same by which Mr. Roynolds had formerly made the like experiments ( though not upon so many waters , unlesse aritificiall waters , as distilled strong-waters , and wines , & some other liquid substances , as he told me ) and that was a glasse-Phiall which held almost 3 quarters of a Wine-pinte , having the neck thereof done about with lead or pewter , and a top or Cover of the same metall made to screw on , which upon the proposall of my intention to him concerning this experiment , and his good liking and desire of the same , he courteously offered to lend me , and I as courteously accepted ; for I could not then meet with one so fit for that purpose as that was , save those which were too small ; for this was at the smallest . And so the 21th of March 164 8/9 , I made my experiment at Goldsmiths-hall by the said Glasse with all the exactness that possible I could , I experiment for the gravities of severall waters , and the same compared with Rain-water . Mar. 21. 1648/9 . both in filling the said Glasse with the severall waters one after another as they were weighed ( beginning with Rain-water ) still screwing on the said top , to keep in the water from falling out , as nearly as I could , & then in the weighing of them ; & thereby I found first the Snow-water to 2 graines lesse then the Rain-water , and the two River-waters to weigh each of them I gr . more then Rain-water ; and so also the Standard-water in Ch●ap-side , ( and thereupon these 3 to be of equal weight ) & the Lambs-Conduit-water to weigh 4 gr. more then Rain-water . But indeed finding this Glasse to be very uncertaine for the performing of this so nice and curious an experiment , in regard both that as I still screwed on the top or Cover thereof after the filling it , to keep in the water that none might fly out , there still issued forth some water , and besides also that the mouth of the Glasse was somewhat too wide , ( considering the smalnesse of the Glasse ) for the filling of it with every several water exactly alike to the least drop as was requisite to doe in do : small a Vessel ; sith that I observed , that one ordinary drop of water more or lesse would alter the weightfull 2 graines more or lesse ; which made me continually to iterate the experiment by the said Glasse upon every several water , by filling up the Glasse againe , and then drying the out-side thereof before I put it againe into the Scale● . But however being doubtful of the same , and that I might make this experiment with all the exactnesse that might be ; I did thereupon for my further satisfaction , repair to the Glasse-house in Broad-street , and there cau●●d a Glasse-Phiall to be presently blown before me , which might hold a Wine-pinte , as neer as could be guessed I for this I conceived would be a convenient bignesse , so ●s it being filled with water , might be conveniently weighed in a small Ballance that would turne upon some smal part of weight , as that was which we had used for the other Glasse , being a very nimble and subtile Ballance , that would turne upon the 4th . or 5th . part of a graine , which was as much as could well be expected from a ballance to weigh a Vessel of this capacity being filled with water or other liquour , ) and was also blown round , somewhat after the manner of an Urinall , but made flat at the bottome , so as to stand upright , and very smooth and even at the top , and with so smal a mouth or orifice , as that it might be alwaies filled alike to the least drop ; for that , when I came to fill the same , not one drop could be shaken out , till a little was first suckt out with a quill , ( insomuch as that one might have carried it full of water in their pocket , with the mouth downwards , and not have spilt one drop therein : ) and which being thus exactly made for the purpose ; I weighed the very same waters therein , which I had done before by the other Glasse , having let them stand all the while in their several Glasses , with an addition of two other Fountainwaters , which in the mean space I had been told of , as being generally accounted the two finest waters abo●t London ; the one neer the Post●rne-gate on Tower-hill , called the Postern-Spring ; the other on the backside of St. Giles-Church at Cripple-gate , called there by the name of Crowders Well ; both which are commonly accounted exceeding good for all manner of soare eyes ; and have a very plesant taste , like that of new-milke , ( especially that of * * Upon the weighing of this water , the Assay-master of Gold-smiths-Hall , by name , Mr. Alexander Jackson ( who was pleased with much ccurtesie , humanity , and patience , to assist me in these and all other the experiments which I th●re made ; as to the work of the Ballance , or the matter of weighing , and which he performed with his own hands , with all the accuratnesse that might be ) told me that he once knew an ancient man in this City , who whensoever he was sick , would drink plentifully of this water , and was thereby immediatly made well : and so , being overcome with Drink ( as he often would be ) would presently drink of this water to make him sober , as finding it to be the most speedy remedy . Crowders-Well , which much exceeds the other for sapour and gravity ) and which therefore for their vertue and gravity above the rest , may be taken for mineral . And so by this new Glasse , and the foresaid 2d . Experiment , for the gravities of severall waters &c. Mar. 31. 1649. Ballance , I found first the Snow-water to weigh one grain lighter then Rain-water ; as also the new River-water ( and so these two to weigh alike ) and the Thames-water not to differ sensibly in gravity from Rain-water , and the Conduit-water in Cheap-side to be 2 gr. heavier then Rain ; and and the Lambs conduit-water to be 3 gr. heavier then Rain ; and then , the Postern-Spring-water , to weigh 7 gr. more then Rain ; and St. Giles-water , ( or that of Crowders-Well ) to weigh 12 gr. or halfe a pennny-weight more then Rain-water ; and so to be weightier then the Postern-Spring-water by 5 gr. And so as I weighed each water , I continually iterated my experiment upon the same , by putting out a few drops , and then filling up the Glasse againe ; and so afterwards still drying the Glasse throughly on the out-side , I againe committed the same to the Ballance , and so found the several waters to weigh as before ; save only the Thames water now weighed one gr. more then Rain-water , which whether it was in the filling of the Glasse , or in the weighing , I cannot justly say : But soon after this it happening to rain , I took some pure fresh rain-water , and withall , such other of the aforesaid Waters as were neer the place of observation , fresh againe , viz. the Conduit-water in Cheapside and the two foresaid River-waters , and ( after due setling ) tried them over againe by the same Glasse and Ballance ; as also the Snow-water 3d. Experiment for the gravities of severall waters . Apr. 7th . which I had used before ( for fresher I could not then get ) and found them all to weigh exactly as at first by this Glasse ; and so the Thames water to be of equal gravity with Rain-water .

And then I having a smal Glasse standing by me , whose mouth was rather lesse then that of the pinte-glasse and held little more then 2 ounces-troy and an halfe ; I thought good to make an experiment by the same , upon some of the foresaid waters , 4th . Experiment for the gravities of severall waters . Apr. 21. viz. Rain & Snow-water , & the two River-waters : ( because these did neerliest agree one with another ) and thereby found the Snow-water to be half a grain lighter then Rain-water ; and the two River-waters , to be each of them of equal weight with the Rain-water as before . And this was performed by a very smal and subtil Ballance , which would easily be turned with one mite , or the 20th . part of a Grain .

And thus having discovered the difference of gravity in severall waters by smaller Vessels , ( or quantities of water ) I conceived it very convenient after all this , to make one experiment or observation more upon all the foresaid severall waters , by a much larger quantity ; considering that these smaller quantities were not so sufficient to discover the difference of gravity , as to ground or determine Proportions of gravity thereupon ; and so that the greater the quantity of the waters was , the greater & more apparent still would be their difference of gravity ; and so to see how this would agree in Proportion with the other : And to this end I got the largest vitreall Vessel or Viall that I could meet with fit for the purpose ; which was one that held near about five wine-pintes and an half , having a very small neck , done about with lead or pewter , and a ●op or Cover to screw on very close ( like the Glasse by which I made my first observation in this kind ) and which in the screwing on , would not force the least drop of water out of the Glasse , being exactly filled ; and the mouth of the Glasse not being half an inch wide : And so having provided all the foresaid eight waters fresh again ( except the Snow-water , which could not be had fresher then that which I used before ) and the same duly setled ; I found by this Glasse ( from the great 5th . experiment upon severall waters , in respect of gravity . May 3d. Standard-ballance , which would sensibly turne opon one grain ) first the Snow-water to weigh 8 gr. lesse then Rain-water ; and the two River-waters to be equi-ponderant with the Rain-water : and the Conduit-water in Cheap-side , to weigh 14 gr. more then Rain ; and the Lambs-conduit-water to weigh 24 gr. ( or 1 p. w. ) more then Rain : and the Postern-Spring water to be heavier then Rain-water , by 2 p. w. and 9 gr. ( or 57 gr. ) and lastly , the water of Crowders-Well , to be weightier then Rain-water , by 3 1/2 p. w. or 84 gr. and so to exceed that of the Postern-Spring in grauity , by 1 p. w. and 3 gr. viz. 27 gr.

So that now from these severall exact experiments and observations , it is manifest , that all waters homogeneall ( or of the same kind ) are not of the same gravity , but do sensibly differ therein , as we have here proved from four severall Fountain or Spring-waters , all of them differing in gravity one from another ; and that in the very same continued order and proportion in a manner , from all the severall experiments by which they were tried : as that of the Conduits in Cheapside ( from Padington-Spring ) to be the lightest ; and the next above that , the water of Lambs Conduit ( in Graies-Inn fields ) and then the next to exceed that , the Postern-Spring water ( on Tower-hill ) and then the heaviest of all , the water of Crowders-Well at St. Giles . Cripplegate ; and which thing I was very desirous to demonstrate ▪ And therefore consequently , that no certain , positive Proportion can be determined between Rain and Fountain , or Pluviall and Puteall water , as Snellius hath done ; and which I cannot but wonder , that he ( so intelligent an artist ) should do , from one single experiment or observation only , having used but one Fountain-water ( by what I can perceive from him ) as if he took all Fountain or Puteall waters to be of the same gra●ity . And that which he used in that experiment , seemeth to be much heavier then the heaviest of those fountain-waters which we have here made use of ; as appeareth by his comparing it with Rain-water the proportion being ( as I shewed before from him ) as 1000000 to 1007522 ; whereas the proportion of Rain-water to the weightiest of , the Fountain-waters which we have here experimented , wil be ( by our l●st and largest experiment , which is the best for that purpose ) but as 1000000 to 1002104 ferè , and from which the Proportion between these two waters , deduced from the observation made by my new-pint-Glasse , wil but little differ , ( and that by way of defect ) it holding there from Rain-water , to St. Giles-Well-water , as 1000000 to 1001925 ferè , both which by a millenary contraction , wil be in a manner the same , viz. 1000 to 1002 ; and by a decu-millenary contraction or abbreviation , the one wil be as 10000 to 10019 , and the other ( and better ) as 10000 to 10021. and both which by arithmeticall mediation , wil be as 10000 to 10020 ; and so between 1001925 and 1002104 , the intermedian proportion arithmeticall , will be as 1000000 to 1002014.

But indeed , if we should have yet further tried the same by a larger quantity of the waters : then probably might the proportion have still risen higher ; and so have come somewhat neerer that of Snellius , according as we have still observed from our several experiments ; that as the vessel or quantity of each particular water hath been greater , so the higher hath risen the proportion of gravity betwen them , ( though very little ) according as the difference of gravity hath become somewhat greater then perhaps it should : For that surely , whether the same be experimented by a lesser or larger quantity ; yet the same proportion of gravity should arise , according as the the difference of gravity should be proportionably the same ; and which hath neerly happened in these several observations by several quantities of waters , except those of the first experimene●●● the imperfect Glasse . And which would continually happen , in case the gravity of each particular water could still be taken in every several Vessel , or Quantity , so very precisely to the least part of weight , as might be imagined : but which , considering that every Ballance ( as an Instrument or Organ , consisting of divers parts ) is continually subject to mutation upon every smal occasion , ( especially those smaller , nice , nimble , and subtil Ballances , such as we used in most of our experiments of this kind , which by the ordinary breath of one's mouth or nostril ( any thing neer at hand ) or the least motion of the aire , or noise , or sound , wil be sensibly diverted from their due course and positure , to which they tend ) and so not infallible ; therefore the same cannot well be expected , though notwithstanding we here continually used all the sedulity and sollicitude that could be ; taking continually the gravity of each particular water , upon an exact equilibrity or equiponderancy of the Scales , according to the most precise perpendicularity or rectitude of the Index , or Tongue of the Ballance ; as neerly , as by the sight might possibly be adjudged ; insomuch as that we could not perceive any sensible errour or mistake to be committed therein . Aud moreover for a further avoiding of errours herein ; I performed each particular experiment upon the severall waters ( according as they are set down orderly before ) stil at one time , according to one and the same setting , or rectifying of the Ballance ; conceiving it to be more convenient and sure so to do , then at sundry times apart , for that there are hardly any Scales to be met with , but at severall times , wil require a several setting or rectification .

But for the determining of a certain , positive Proportion between Rain and any other 〈◊〉 , ( or between any other two particular waters ) it is best surely to use as large a quantity of those waters , as can conveniently be weighed , ( consideration being had of the bignesse and ponderousnesse of the Vessel to contain them ) But the vitreall Vessel or Viall , by which I made my last observation , would hold welnigh as much as the aereall Cylindricall Vessel by which Snellius made his observation , as I have computed it , by comparing his Weight and Measure with ours .

And by most of our severall observations , we find the two River-waters aforesaid to be of like gravity with Rain-water , without any sensible difference , and so to be equiponderant in themselves . And for Snow-water , we find the same by every particular observation ( even from the least vessel or quantity of water , where the difference of gravity was least discernable ) to be sensibly lighter then Rain-water , & so the lightest of all ; & in which , our experimentall observations wil agree with naturall reason it self , which sheweth Snow to be a much lighter substance then Rain ; and in which all * * Zanard . Disput. de Vniverso el●memari , part. 3. qua●st 22. Aliara dispositionem habet nubes nivis ab ea quae est aquae : nam nubes pluviae habet , quod sit magis densa , magis obscura , & magis unita in anum locum : Nivis autem nubes est tenuis & subtilis , & cùm multum de aereo continet , est clara & quasi alba , & per aerem sparsa , &c. Philosophers do agree ; that the Cloud or matter of which Snow is engendred , is more hot and drie then that of which Rain is , and also more aery , and so the humour or moisture resolved out of Snow , is aery , and very light , and as it were a froth or some , whereupon it so nourisheth and cherisheth the Earth : And because it contains much of aire , it behooveth that it should contain also much of warmth and moisture , although yet it is drier then water . And so Aristotle likeneth Snow to Froth or Foame ( & also calleth it so ) in respect of its whitenesse , which he saith to arise chiefly by means of those parts thereof which are more aery ; for that for

Tenet multum de calido aereo nubes nivosa &c. Zanard . ibidèm .

Aristor . lib. 2. de generat . animal . cap. 2.

Plin. nat. hist. lib. 17. cap. 2. & Keckerm . Syst. phys. lib. 6. cap. 9. Theor. 1. & 2. de nive .

Colleg. Conimbr . in Aristot . meteor . Tract. 7. cap. 5.

Magir. physiolog . p●ripat . lib. 4. cap. 6.

this reason also , Froth ( saith he ) is white ; and so water having oile mingled with it . And so also Plinie calleth Snow aquarum coelestium spumam , which his english Translatour Dr. Holland interpreth , the fome or froth of Rain-water from Heaven , concerning which see Keckerman . And the Conimbricensians ( according to Aristotle , lib. 1 Meteor . cap. 11. and lib. de mundo , c●p. 4 ) say , that Snow is a Cloud conglaciated , or frozen together in a friable density ; and which Cloud obtaines so much a greater ficcity then that which is changed into water , by how much it congeales or grows together by the power or efficacy of the more prevalent Cold ; sith that the Cold while it bindeth , doth expresse , or force out the moisture , &c.

Seeing therefore that the matter of Snow , is more hot , and dry , and more aery thin and subtile then that of Rain ; and consequently more light ; and that Snow-water is no other then Snow dissolved , and so still retaines the nature of Snow : therefore also will Snow-water be necessarily lighter then Rain-water .

As for the Proportion between these two Waters , I may from all our several The Proportion of Rain-water to Snow-water . observations and experiments beforegoing , conclude the same to hold generally from the heavier to the lighter , as 10000 to 9998. For by 2 or 3 several observations made by the new Glasse of almost a wine-pinte , where the gravities of these two waters ( and so the difference of gravity ) were found still the same ; the proportion will be as 100000 to 99984 , ferè : and by the last observation , being made by the great Glasse of about 5 1/2 wine-pintes , holding about 7 pounds-troy of water , it will be as 100000 to 99980 ferè ( and which I take to be the truer , ) and if we will take the intermediate proportion arithmetical , the same will be 100000 to 99982 ferè . Or the former proportion will neerly happen , if we shall mediate between that which ▪ will be produced from the least Glasse of all , ( which was that of about 2 1/2 oun. troy ) being as 10000 to 9996 compleatly , and that which we have here produced from the greatest Glasse of all , viz. 10000 to 9998 in a manner compleatly ; which will be 10000 to 9997.

And now from these our severall experim̄ents before-going , for discovering , for discovering the gravities of the several kinds of common , simple , or naturall Waters , as in reference to the finding out of the solid quantities or capacities of Bodies altogether inordinate or i●●egular , which wil not in themselves admit of an ordinary or regular kind of Dimension , but their contents must be obtained by some extraordinary or unusuall kind of way , as we have lately shewed ; it is manifest , that the way here propounded by us for the same , may be performed by any such kind of Water in generall , as we have here tried , according to the experiments formerly laid down by us for that purpose , from Rain-water , ( and so upon which this our Atactometricall practice is grounded , as being the most indifferent water for a generall use in this kind : ) for that in our last observation , which was made by the greatest quantity of the severall Waters asore-named , weighing about 84 oun. or 7li. Troy , the difference of gravity between Rain-water and the heaviest of the Fountain or Spring-waters , was but 3 1/2 p. w. or 84 gr. And which said quantity of water is much more then double the quantity of that which was found to be of equall magnitude with the foresaid sphericall stone-body , and so from whose gravity we obtained the content of the said solid body , in the measure propounded ; & therefore had the same been thus inquired from both those Waters , that is , the lightest ( except S●owwater ) and the heaviest , ( though the Water by which it was done , we have shewed to be of like gravity with Rain-water ) there could have been no considerable difference therein .

THE CONCLUSION . For the more spèedy absolving or expediting of all the foregoing Dimensions in generall .

ANd now for the more easie and speedy performance still of all the Dimensions , and metrical Conclusions contained and mentioned in this Book , I shall subnect this one thing as a Coronis to the whole Work , by way of advertisement and advice to the practical Reader that is not yet acquainted with the most compendious waies of arithmetical Calculation , which is , that he would use continually all along with his geometrical Lines , or Lines of measure , ( whether natural or artificial ) that most excellent artificial arithmetical Line ( as I may so terme it ) or Line of Numbers ( so called by the authour and contriver thereof , Mr. Edm. Gunter , being deduced by See also Mr. Edm. Wingates Rule or Scale of proportion , of the like kind . And , Mr. Oughtred's Circles of Proportion ; or rather the same converted into ● Spirall Line . him from the Logarithmical or artificial numbers , and so being , as it were , a Line or Scale of Logarithmes ) by which all Mathematical Questions and Conclusions to be wrought by Proportion , ( as indeed what are there , which may not be reduced to a way of proportion ) are cast up , or computed in a most compendious manner ( as it were geometrically or mechanically ) meerly by Scale and Compasse , without the labour of the pen : For all those our artificial Dimensions , which consist meerly upon the squaring of some one number , or of one simple quadrature only , ( such as are of all ordinate or regular Planes or Superficies , and whether taken simply in themselves alone as Figures , or many of one kind conjunctly , as constituting the total Superficieties of regular Solids , ) will be absolved by one extent of the Compasses only , being doubled , or once repeated upon the said Line , according to the Analogism of Multiplication . And that will be , as from an Unit upon the said Line of numbers , to the linear number or term taken by the artificial Line of measure , or Line of quadrature , ( for some dimensional line of a Figure whose superficial Content is required therefrom , as representing artificially the side of the equal Square ) so from that , to the quadrate number . for the superficies of the Figure .

And all those artificiall Dimensions which depend meerly upon the cubing of some one number or term ( such as are of all ordinate or regular Solids , for their solid measure ) wil be absolved by one and the same extent of the Compasses being trebled , or twice repeated upon the said Line of artificiall Numbers , according to the foresaid Analogisme ( in a compound or double Multiplication ) which wil be , as from an Unit , to the linear number or term taken by the artificiall Line of measure , or Line of Cubature , ( for some dimensionall line of a regular Solid given to be measured thereby , it representing artificially the side of the equall Cube ) so from thence to the Square thereof ; and from that to the Cube , for the solid content .

And all those artificiall Dimensions which consist not of cubing , but yet of a twofold Multiplication ; the one whereof is a Quadrature , ( such are of all regular-like Solids , as Cylinders , Cones , and the like ) wil be absolved by two severall extents of the Compasses , each of them being doubled or once iterated upon the said Line of numbers ; which wil be first , as from an Unit to the quadratary term taken by the artificiall Line of measure , representing some dimensionall line of the ordinate Base ; So from that , to the Quadrat thereof , for the artificiall Base : Then , as from an Unit to that Quadrat ; So from the other number or term taken by the same artificiall Line of measure , for the Axis or Altitude of the solid Figure to be measured ; to the solid area of the Figure . Or it wil be , As from an Unit , to the number for the Axis or altitude ; So from the quadrate number for the Base , to the rectangular solid number , for the whole solid Figure it self . And here , if the two foresaid lines of a regular-like Solid , to wit , the Axis , & the basiall line ( whatsoever it be ) do happen to be equall ( as they very rarely do ) then wil the Dimension be cubicall , as that of an exactly ordinate Solid by some one of its dimensional lines ; and so be absolved upon the Line of Numbers accordingly . Then lastly , for the superficiall Dimension of these regular-like Solids , consisting of one Multiplication only , and that commonly of two unequall Terms , and therefore not an exact Quadrature ; the same wil be absolved by one extent of the Compasses , once repeated upon the said Logarithmeticall Line , or Line of Proportion , according to the Analogism of Multiplication ; which wil be , as from an Unit to one of the two linear Numbers or terms taken by the artificiall Line of measure ( which do represent some one dimensionall line of the Base , and the side , or other line upon the superficies of the solid Figure , according to the nature thereof , ) So from the other , to the rectangle superficiary Number ▪ for the superficiall Area of the solid Figure ( the Base ▪ thereof being secluded ) And here also , if the said two dimensionall lines of the regular-like Solid , do happen at any time to be equall ( as they v●ry seldome wil ) then this Dimension wil be exactly quadratary , as the superficiall dimension of an exact regular Solid , by some one of its dimensionall lines .

And the like with these , understand for the more spèedy computing of the gravities , or ponderall quantities of regular and regular-like metalline bodies , being inquired out artificially in the same manner as their solid measures : and therefore , as there the last proportionall number or term upon the said Line or Scale of Numbers , denoteth solid measure , here it wil denote gravity or ponderosity .

And so likewise in the work of Gauging , for the speedier computing of the liquid Contents of Vessels in Wine or Beer ( as was of the solid content of a Cylinder , or any other regular-like Solid , from our artificiall way ) where , after the mean Diameter of the Vessel being had , according to the artificiall Gauging-Lines ; it wil hold upon the foresaid Logarithmical Line ( or Line of Numbers ) by a twofold extent or opening of the Cōpasses , thus ; viz. first , as from an Unit to the mean Diameter , So from that to the Square thereof ; Then , as from an Unit to that Square ; So from the length ( or heighth ) of the Vessel to the liquid content thereof in Gallon-measure . Or as from an Vnit to the length of the Vessel , so from the Quadrat of the mean or equated Diameter , to the liquid measure aforesaid .

And so with the like expedition by this Line , understand all Atactometrical operations to be absolved , in computing the solid quantities of irregular bodies from the gravity of the equal body of water , according to the Terms of Proportion or comparison between the gravity and magnit●de , or the ponderall and dimensional ▪ quantity of this liquid Body , delivered by us in our Atactometrical Discourse .

Aud so all metrical operations arising here ●●ō the natural Measure , wil be thus expedited , according to the dimensional Proportions delivered in this Book , both for measure and weight ( and all others of the like kind , not here particularly expressed ) according to what I said before . And thus I put a Period to these my mathematicall Contemplations and Exercitations .

Soli Deo universipotenti , qui omnia ( ut loquitur Sapiens ) mensurâ , numero , & pondere disposuit , sit gloria , honos & laus , in saecula , & omnem sempiternitatem . AMEN .