De numeris geometricis. = Of the nature and proprieties of geometricall numbers. First written by Lazarus Schonerus, and now Englished, enlarged and illustrated with diuers and sundry tables and obseruations concerning the measuring of plaines and solids: all teaching the fabricke, demonstration and vse of a singular instrument, or rular, long since inuented and perfitted by Thomas Bedwell Esquire Bedwell, William, ca. 1561-1632. This text is an enriched version of the TCP digital transcription A11580 of text S116788 in the English Short Title Catalog (STC 21825). 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. 138 KB of XML-encoded text transcribed from 48 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2018 A11580 STC 21825 ESTC S116788 99852004 99852004 17301 2240904794

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Early English books online. (EEBO-TCP ; phase 1, no. A11580) Transcribed from: (Early English Books Online ; image set 17301) Images scanned from microfilm: (Early English books, 1475-1640 ; 1362:2) De numeris geometricis. = Of the nature and proprieties of geometricall numbers. First written by Lazarus Schonerus, and now Englished, enlarged and illustrated with diuers and sundry tables and obseruations concerning the measuring of plaines and solids: all teaching the fabricke, demonstration and vse of a singular instrument, or rular, long since inuented and perfitted by Thomas Bedwell Esquire Bedwell, William, ca. 1561-1632. Schöner, Lazarus, ca. 1543-1607. De numeris geometricis. [8], 82, [2] p., folded table Printed by Richard Field, London : 1614. A rewriting by William Bedwell, who signs the dedication, of: Schöner, Lazarus. De numeris geometricis. The folded table, signed "N", has title: Trigonum architectonicum. The last leaf is blank. Reproduction of the original in the Henry E. Huntington Library and Art Gallery. eng Bedwell, Thomas, d. 1595. Geometry -- Early works to 1800. Mensuration -- Early works to 1800. eebo De numeris geometricis. = Of the nature and proprieties of geometricall numbers. First written by Lazarus Schonerus, and now Englished, enlarged and i Bedwell, William, 1614 93 28413 43 0 52 0 0 0 33.44 C The rate of 33.44 defects per 10,000 words puts this text in the C category of texts with between 10 and 35 defects per 10,000 words. 2000-00 Assigned for keying and markup 2001-09 Keyed and coded from ProQuest page images 2002-06 Sampled and proofread 2002-06 Text and markup reviewed and edited 2002-07 Batch review (QC) and XML conversion

De Numeris Geometricis . OF THE NATVRE AND PROPRIETIES OF GEOMETRICALL NVMBERS .

First written by Lazarus Schonerus , and now Englished , enlarged and illustrated with diuers and sundry Tables and obseruations concerning the measuring of Plaines and Solids :

All teaching the fabricke , demonstration and vse of a singular Instrument , or Rular , long since inuented and perfitted by THOMAS BEDWELL Esquire .

LONDON , Printed by RICHARD FIELD . 1614.

TO THE HONORABLE KNIGHT , Sir Robert Coke . HONORABLE SIR ,

HIstorians report , that the Scythians were so far out of loue with learning , that it was a shame and disgrace for any man amongst them to be esteemed a scholler . The name of a Scythian , all Christians cannot choose but detest and abhorre . And yet is this their opinion , as we heare , maintained by the Nobilitie of some Nations . An argument , as I thinke , sufficient alone , if historians were silent , to shew from what race such men are descended . Barbarous , I am sure it is , & no way consonant to the nature of man . For as the Philosopher saith , Omnes homines naturâscire desiderant . And , Prudentia ( saith the Orator ) quae in vericognitione consistit , maxime naturam attingit humanam . The Nobilitie therefore of England , are in this most highly to be commended , who do generally so much esteeme of learning , that they thinke nothing more honorable then either to be accounted learned , or great patrons of learning : Knowing indeed that , as the Poet saith , In prudentia Nobilitas sita est . This generallitie first : and then next , your honorable affection and fauour to me in particular , hath emboldened me to offer this small Pamphlet vnto your patronage , vt aliquod saltem extaret meae inte obseruantiae & gratitudinis monumentum . Vale , V. O. ac nos tibi iampridem deuinctissimos , deuinctiores , si fieri potest , reddito . Londini Anglorum viij Kalend. Martias , an . MDCXIIII .

Your H. in all true affection , Wilham Bedwell .
TO THE COVRTEOVS READER .

IT is now , as farre as I remember , well neare fortie yeares since a Rular , for the measuring of Plaines and Solids , was first deuised by mine vnkle M. Thomas Bedwell . The vse of this , himselfe hath taught in a litle Treatise , which I haue seene long ago in his studie amongst other his writings of like nature and argument . The fabrick or maner of making it , he did not , to my knowledge , impart to any . That I suppose be reserued for a larger volume of diuerse other such inuentions of his , which he bad purposed to haue published , if God had spared him longer life . But these all being suppressed by some , who haue , as it seemeth , more respected their owne priuate gaine , then either his honour or the common good ; I did , about a dozen yeares ago , as some learned louers of these studies can testifie , labour to shadow out that worke of his . And at last hauing finished a Treatise , both of the construction and vse of the same , I offered it vnto some to haue it imprinted : But they seeing in the copie diuers figures and diagrammes , refused to meddle with it , except my selfe would beare a great part of the charge of that first impression . That I being vnwilling to do , I was forced to deferre the publishing of it vntill I might meete with some one that would of himselfe vndertake the businesse . In this interim , I found yet another scruple , which I thought necessarie first to be remoued . For as the construction of the first Author was meerely Geometricall , so that of ours was wholly Arithmeticall , and could not be done without the helpe of numbers : and therefore was such as might not easily be taught or conceiued , except first the nature and proprieties of such like numbers were well vnderstood . Whereupon , before that Treatise of the Rular , this of GEOMETRICALL NVMBERS was first to be published . Geometricall numbers I call them ; not Figurate or Cossicke numbers , as commonly they are called : For of those infinite sorts of this kinde of numbers , we haue to do with none but such as do represent and expresse some true Geometricall figure , such as the Quadrate , Parallelogramoblong , Prisma , and the Cubicke are . Now whereas haply some man may obiect and say , that these things haue bene often handled and sufficiently taught by others ; let him know , that although in generall we do confesse it to be true , yet no man , to our knowledge , hath applied those rules to that vse that we haue done : And therefore those Treatises of theirs , do in no waies fit our purpose . For first out of these rules , as afterward shall appeare , we haue framed certaine Tables for sundrie vses in measuring of Plaines and Solids . One of these , called TRIGONVM ARCHITECTONICVM , The Carpenters Squire , we caused for the publicke good , more then a twelue-moneth since , to be imprinted at our owne cost and charges . Moreouer , sundrie practises and workes of these rules more plaine and easie then commonly are vsed , are here shewed and discouered . Lastly , we haue here giuen thee a taste or sample of a forme and maner of resoluing any intricate demonstration mathematical , whereby it shall be made so plaine and easie , that it may be conceiued of the simplest . This we haue vsed these seuen or eight and twentie yeares ; and because we see it to be approued of the most learned Mathematicians of this kingdom , we shall ere long , God willing , more fully manifest it by many examples .

OF A RATIONALL FIGVRATE NVMBER .
CHAP. I. Of the Extraction of a Quadrate roote , or side .

1. A rationall figur ate number is , a number that is made by the multiplication of numbers betweene them-selues .

SVch a combination and affinitie there is between Arithmeticke and Geometry , that the whole nature & property of the one cannot well be taught and conceiued without the other . For so Ramus sayth , Geometria sui & generis & iuris magnam quidem partem est , neque aliter quàm geometricè tractabilis . Attamen parte quadam numeris associatur , & eis explicatur , numerique , geometricarum affectionum interpretes , geometricis vocabulis appellantur , vt Planus , Quadratus , Solidus , & Cubus , à geometrico Plano , Quadrato , Solido , Cubo ; quorum vmbr ae quoedam tales numers sunt . Item , Aristoteles affirmat gener aliser arithmeticam demonstrationem magnitudinum accidentibus conucnire , cum magnitadines numerifiunt : Et Proclus ait , Quicquid in Geometria 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 explicabile & cognobile sit , numeris explicari & cognosci . And indeed a great part of Euclides Elements are wholly arithmeticall , that is , that booke of Euclide , whose purpose is to teach the generall grounds of Geometry , hath notwithstanding many arithmeticall conclusions here and there intermedled among ; and to speake the truth , the vij , viij , ix , and x bookes of that his worke , are in a maner wholly arithmeticall . All this , I say , doth teath vs that Geometry connot be vnderstood without the knowledge and vse of Arithmeticke . Of this kinde most properly are those figurate numbers , which do participate of both natures . And therefore one defineth a figurate number thus ; Figuratus numerus est numerus qui per siguras & appellationes geometricas exprimitur , A figurate number is a number which is expressed by geometricall names and termes . Wherupon some haue called it a Geometricall number . Now a Figure , as Euclide sayth , is that which is contained on euery side with one or more bounds , 14 dj . Such an one is a circle , contained with one line , which they call a Peripherie . Item , a Triangle , which is bounded with three lines : or a Quadrate , with foure lines . Such in bodies is a Cube , bounded with sixe equall surfaces , & a Prisma , bounded with sixe vnequall surfaces , &c. Now a Rationall figure is a figure that is comprehended of the Base and Height rationall between themselues , 9 e iiij R : And the Base and Height are said to be rational one to another , when as the rate or reason of both may be expressed by a number of the same measure giuen ; by the 8 e j R. As for example , if the length of a quadrangle giuen be 14 inches , and the breadth 12 , it is said that that quadrangle is a rationall figure ; because the length , that is , the Base , and againe the breadth , or Height are both expressed by a number of the measure giuen , that is , by a certaine number of inches , to wit , that by 14 inches , this by 12. Now to be comprehended of the Base and Height , is when the length is multiplied by the breadth . For this geometricall comprehension , which here is vnderstood , is as it were a multiplication by numbers . So in the former example the Quadrangle before named is comprehended of the Base 14 , and of the Height 12. Therefore if thou shalt grant that the Base and Height are rationall betweene themselues , or that their rate may be expressed by a number of the measure assigned : then I affirme , that the numbers of those sides being multiplied the one by the other , shall shew the content of that figure . Item that the product of them , that is , 168 , is the figurate number expressing that content , as here thou seest .

2 The numbers thus multiplied , or the numbers which make the figurate number , are called Sides or Rootes : and the arte whereby the Sides of a figurate number giuen are found , is called The extraction of a roote .

That number which expresseth the area , or content of a rationall figure , is called , as we haue shewed before , a Figurate number : and the numbers representing or expressing the Height and Base , that is , the numbers thus multiplied , or making this figurate , are by the geometricall terme called Latera , Sides : but vulgarly of the Arithmeticians they are called Rootes . For as plants and tres do spring from their rootes : so these figurate numbers , whether plaines or solides , do arise and haue their beginning from their rootes . If then the sides be giuen , the figurate is easily found by multiplication . But to finde the sides , rootes , or numbers whereof any figurate number was made , is a matter not so easie : That arte or rule that teacheth to performe this , is called , The extraction of a roote ; of the Latines it is called , Analysis Lateris : as the multiplication aforesaid or manner of making a figurate number , is called of them Genesis figurati .

3 A figurate number is made by one multiplication , or by many : And either of them is equilater , or vnequilater .

4 An equilater figurate is made of equall numbers , or of one number multiplied by it selfe ; which multiplier is also specially called the Side or Roote : An vnequilater is that which is made of numbers which are vnequall between themselues .

As for example , the figurate 4 , is made by one multiplication of one number by it selfe , to wit , of the side 2 by it selfe ; Therefore 4 is a figurate equilater , and the side or roote of it is 2 : So 9 is an equilater , whose side is 3. This side of the equilater by the Arabians is called Radix , that is , The Roote , as Schoner testifieth . Item 6 is a figurate of vnequall sides , made , I say , of the multiplication of 2 by 3 , and therefore 6 is an vnequilater figurate .

Here obserue that an vnitie doth imitate euery kinde of equilater : For 1 , by multiplication increaseth not , neither yet doth it diminish any whit at all . It remaineth therefore that an vnity multiplying an vnity , maketh but an vnity , that is , it taketh vpon it the nature of an equilater .

5 Moreouer , an equilater figurate is twofold : either it is that whose true side is to be expressed by a number ; or such whose true side may not be expressed by any number .

6 The equilater whose side is to be expressed by a number , is that whose rate or reason vnto an 1 , may be shewed .

This diuision although it be not altogether proper to this place , yet because it is commonly vsed by the vulgar sort of Artists in this case , we would not omit it . The first sort they call 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , effabile , that is , such as may truly be pronounced or spoken by some arithmeticall number : As for example , 16 is an equilater , or a figurate of equall sides , and the true side is 4 , that is , a number which may be spoken , vttered and set downe by arithmeticall figures ; I meane whose quantity may be conceiued by the reason of it vnto 1. For in the quadrate or plaine equilater , as the reason of the number giuen is vnto the roote thereof : so is the same roote vnto an vnitie . For the roote or side is the meane proportionall between the figurate giuen , & 1. As in our example ; as 16 is vnto 4 , so is 4 vnto 1. And indeed al absolute numbers are conceiued & vnderstood by the reason they beare vnto an vnity ; and whatsoeuer they are , that they are sayd to be in respect and comparison of an vnity .

7 The equilater whose true side cannot be expressed , is a figurate , the reason of whose side vnto 1 , cannot be told or declared .

As for example , 3 may be conceiued to be an Eequilater , that is , to be a product made of two equall numbers , or of one and the same number multiplied by it selfe , which number thus multiplied is , by the former , the side of the said Figurate made . Now this side or number multiplied , is greater then 1. For 1 , multiplied by it self , by the 4 e , doth make the Equilater 1 ; which is lesser then 3. Againe the same side is lesser then 2. For 2 multiplied by 2 , doth make 4 , which is an Equilater greater then 3. Therefore the roote or side of 3 , doth fall to be some meane quantity between 1 , and 2. And yet what that number or difference is , or how specially it is to be vnderstood or conceiued , no man may possibly tell . Therfore such a like side is called arrheton , inexplicabile , irrationale , Surde , irrational , or not to be vttered , as the Arithmeticians do call it .

8 A figurate made by one multiplication , is called a Plaine : which is a figurate number made by the multiplication of two sides equall one to the other .

A figurate made by multiplication , as before is declared , is a number representing a parallelogram , and yet not any indifferently , but that onely which is right-angled . Now A Parallelogram is a Quadrangle whose opposite sides are parallel , 6 e x R. And A right-angled parallelogram is a parallelogram all whose angles are right-tangles , 2 e x j R. And because that all parallelograms are plaines ; and for that plaines haue but two dimensions , to wit Length & Breadth : therefore , by the 1 e , a plaine figurate is made by one multiplication only ; that is , by multiplying of the length by the breadth . As for example , 9 is a figurate made of two sides , to wit , of 3 multiplied by 3 ; and therefore 9 is a plaine figurate . Item 12 is a figurate made of two sides ; to wit , of 3 multiplyed by 4. Therefore 12 is plaine figurate , representing a right-angled prarallelogramme surface .

Therefore

9 If thou shalt deuide an inequilater parallelogramme by one side giuen , the quotient shall be the other side desired .

As for example , Suppose the figurate plaines were 12 , and the one side thereof giuen were 3 : here I say that the other side shal be 4 , to wit , the quotient of 12 , diuided by the same 3. This is plaine by the 1 e 6 j lib. Arithmet . Salignaci . The argument is thus concluded :

If a number be made of two numbers giuen , the one of the numbers giuen shall deuide the product by the other : and contrariwise . But a figurate plaine is made by the multiplication of two sides one by the other : 8 e. Ergo , The one side shall diuide the figurate plaine by the other .

This rule onely hath place in those examples where the one side of the vnequilater plaine is giuen : that which followeth is more generall .

10 All the rotes of the squares contained in a Figurate plaine giuen , which shall measure the said figurate , with their quotients , shal be all the sides of the said plaine giuen .

As for example ; Let the figurate plaine giuen , al whose sides are to be found , be 20 : the squares contained in the same , whose sides 1 , 2 , 4 , do measure 20 , let them be 1 , 4 , 16 : And the quotients by the same sides , let them be 20 , 10 & , 5 : I say that 1 , 2 , 4 , 5 , 10 , 20 , are all the sides measuring 20 , the figurate plaine giuen . For although 9 , be a quadrate also contained in the sayd figurate ; yet because that 3 , the side of the same , doth not measure the same figurate giuen , it is neglected as nothing pertaining to this our purpose . By this rule , as you see , may be performed that which the 2 verse of the 13 Chapter of the first booke of Salignacus his Arithmeticke , doth teach ; to wit , How to finde out all the measures of many , of any compound number giuen .

Let the compound number giuen , all whose measures of many are to be found , be 60 : Here al the squares contained in 60 , whose sides may measure the same , are 1 , 4 , 9 , 16 , 25 , 36 : And the quotients of 60 , by 1 , 2 , 3 , 4 , 5 , 6 , the sides of the said squares , are 60 , 30 , 20 , 15 , 12 , 10 : Therefore these twelue numbers 1 , 2 , 3 , 4 , 5 , 6 , 10 , 12 , 15 , 20 , 30 , 60 , are all the measures of 60 , the compound number giuen .

11 Like-plaines haue a doubled reason of their correspondent sides ; and one meane proportionall comprehended of the extreme , or meane proportionall sides . 20 p. v j : Item 11 & 18 p. viij E.

Figurates haue their denomination of Figures , ( as we haue shewed at the 1 e : ) And therefore their nature cannot otherwise be conceiued & taught but from the same . Like figures therfore , as Ramus at the 14 e iiij , teacheth , are figures whose corners are equall and proportionall to the shankes of the equall angles . Now plaines haue but two dimensions , and solides three . Wherefore they shall haue a doubled reason , these a trebled reason of the correspondent sides . Againe here , as Salignacus at the 2 e of the 14 Chap. of the ij booke of his Arithmeticke , teacheth , to double , treble , or quaduple a reason , is not to adde the same reason vnto it selfe twise , thrise , or foure-times : But to multiply it by it selfe , twise , thrise , or foure times . For example , let 8 and 18 be the two figurates giuen : And let their correspondent sides be 2 and 3 : Item 4 and 6 : those the Bases , these the Heights . I say , that the reason of 8 vnto 18 , is the reason of 2 vnto 3 , or of 4 to 6 , doubled : The reason is thus doubled , : or thus , . The reason then of the first figurate vnto the second , is the reason of 4 vnto 9 : or of 16 vnto 36 , that is , the reason of the first figurate vnto the second , is Subdupla-sesquiquarta . For the consequent being deuided by the antecedent , the quotient is 2¼ . Therefore in Plaines , where the dimensions are but two , to wit , Length and Breadth , the reason is onely doubled . This then is the reason of right-angled parallelogrammes , to which Plaine numbers do answer . And thus much of the first part of this proposition : The second part , of the meane proportionall , followeth . Like Plaines therefore , saith our Author , haue but one meane proportionall , comprehended of the proportionall sides . The cause is vnderstood out of the grounds of Arithmeticke : For if the reason of a number vnto a number be but doubled , then by the rule and nature of numbers continually proportionall , there can possibly be but one meane proportionall betweene them . The rule of this inuention , is thus , laid downe by Salignacus , at the 2 e 12 c ij booke of his Arithmeticke : If hauing two reasons giuen , the second bound of the first reason , do multiply both the bounds of the first reason : and the first bound of the second , do multiply the first bound of the first reason , the products shall be continually proportionall to the foure numbers giuen . In the two plaines before mentioned , 8 and 18 , the proportionall sides , or the two reasons giuen , were 2 , 3 : 4 , 6 : The meane continually proportionall , according to this rule , is thus : The meane plaine then continually proportionall , betweene the plaines giuen , is 12 : that is , as 2 is vnto 3 , so is 8 vnto 12 : and as 4 is vnto 6 , so is the same 12 vnto 18. This meane proportionall therefore now found , is made or comprehended of 4 , the length of the first figurate , and of 3 , the height of the second : or contrariwise of 2 , the height of the first figurate , and of 6 , the length of the second : For the product , is still the same : Therefore

12 If the sides of the two like Plaines be giuen , the meane proportionall of the same Plaines is also giuen .

This is manifest by the former : For there the meane proportionall was made either of the middle proportionals , or of the two extreames of the foure numbers giuen . As for example , let the two Plaines giuen be 12 and 48 : the length of the first , let it be 4 ; of the second , let it be 8 : The height or breadth of the first , 3 ; of the second , 6 : The product of 3 by 8 ; or of 4 by 6 , that is , 24 , is the meane proportionall desired . The example is thus : I say , as 12 is vnto 24 , so is the same 24 vnto 48. For the reason in both is subdupla . Thus much in generall of Figurate rationall numbers , or Figurate plaines : it followeth now in particular of their seuerall kindes .

CHAP. II. Of the Quadrate .

1 An equilater or equal-sided plaine is called a Quadrate , 18 d vij , & 3 e xij R. The vnequal-sided plaine is called an Oblong .

OF the diuers sorts of Plaines , handled by the Geometers , that onely is a Rationall plaine , all whose corners are equall , and opposite sides parallell . Such an one is the Right-angled parallelogramme , which alone of all the Geometricall plaines is comprehended of the base and height , as before is shewed . Now the figurate of a rationall right-angled parallelogramme , is called a Plaine . Admit for examples sake , that the Height of a right-angled parallelogramme were 3 , and the Base were 4 : Here if thou shalt multiply 4 by 3 , the product 12 , shall be the figurate , or content of the right-angled parallelogramme assigned . This product 12 , is called a Plaine ; and 3 and 4 , the numbers whereof it was made , are called Sides , by a Geometricall name . And indeed this maner of multiplication , as we haue taught before , is meerly Geometricall . Place 12 vnities , ( or , in stead of them , 12 ciphers ) in equall distances one from another , as here thou seest ; and it shall represent the figure , that is made of two right lines , whereof the one is 3 inches , the other 4. For if thou shalt first deuide thelines giuen into equall parts ; the one into 3 parts , the other into 4 : and againe shalt suppose , the Height to be erected perpendicularly vpon the end of the Length : And lastly , conceiue first the Height to be caried or mooued all along from the one end of the Length vnto the other : And in like maner , the Length to be mooued vpward , for all that whole plumme-line : The traces that are to be supposed those diuisions will make vpon the plaine , shall make 12 squares , within the Oblong thus : This kind of multiplication , I say , is geometricall : For here by this meanes , of lines are not made lines , ( as there of vnities do arise vnities onely , ) but a magnitude or bignesse , exceeding a line by one dimension , to wit , a surface . Before it was obserued , that the figure thus made , is a Right-angled parallelogramme . Now a right-angled parallelogramme , is either an Oblong , or a Quadrate . An Oblong is a right-angled parallelogramme of vnequall sides : A Quadrate or Square is a right-angled parallelogramme of equall sides : 1 & 2 e xij : The figurate of a Quadrate is called also a Quadrate : which is a figurate made of two equall numbers multiplied betweene themselues . This figurate aboue all other is to be accounted a Rationall : and yet euery Quadrate is not a Rationall number : For that onely is a Rationall , whose number is a true Quadrate ; that is , such a number is a rationall quadrate , whose side is to be expressed by a number . Of this kinde is that number onely that is made by the multiplication of one number by it selfe : c 3 e xij : Such are these nine following , made by the multiplication of the nine single figures , or digittes , as they call them , betweene themselues :

The Quadrates : 1 4 9 16 25 36 49 64 81 The Sides , or Rootes : 1 2 3 4 5 6 7 8 9

The Quadrate or Square number , is called of the Arabians , Zensus : of the Greekes Dynamis a power or valour , as Euclide , Diophantus , and Barlaam do testifie . Wherefore wheresoeuer , in these artes , thou shalt meete with these phrases , Potentia rectae est quadratum , The power of a right line is a square , or , Potentia numeri est quadratus , The power of a number is a Quadrate : there vnderstand that any number giuen , multiplied in it self , doth make a Quadrate . Item , Si basis trianguli subtendit rectum , aequè potest cruribus : 5 e xij R. If the base of a triangle be ouer against a right-corner , then the power of it is as much , as the power of both the other sides : that is , if of the three sides of a right-angled triangle thou shalt make three seuerall squares , the square of that side shall be equall to the other two squares , made of those sides which do enclose or containe the right-angle . Thus in Arithmeticke one number is said to be able to do as much as two other numbers , when as that number multiplied by it self , shall make as much as those two shal make , multiplied by themselues : thus 10 multiplied by 10 maketh 100. Item 6 by 6 , giueth for the product 36 : And 8 by 8 , yeeldeth 64. Now because that 36 × 64 , that is , 36 added to 64 , do make for the summe 100 : And that 100 , are equall to 100 ; Therfore I say that the power of 10 is as much as the powers , or possibilities of 6 , and of 8.

This kinde of multiplication of a number by it selfe , is called Quadratura numeri , or Genesis quadrati , the squaring of a number , or the making of a Quadrate : And contrariwise , the diuision of such a like square or quadrate , by the side whereof it was made , is called Analysis lateris quadrati , The extraction or inuention of a square roote . Now as euery Multiplication , and Diuision , so also this squaring of a number , and diuision or extraction of a square roote , is done either iointly by the whole , or seuerally by parts , at many operations . The first is to be done and performed by Pythagoras table , which is set downe a little before : The second is to be done by the prescript of the Rule next following . But first that Table is to be learned by heart .

2 If a number be diuided into two parts , the Square of the whole number shall be equall vnto the squares of the parts , with a double plaine made of them both .

This in Geometrie , spoken of a right-line , is the 8 e xij R : and is thus laid downe by Ramus : Si recta est secta in duo segmenta , quadratum totius aequatur quadratis segmentorum , cum duplici rectangulo vtriusque ; If a right line be cut into two portions , the square of the whole line , is equall to the squares of the portiós , with two right-angled parallelogrammes made of both the said portions . It is a consectarie drawne out of the 3 c II e x booke of Ramus his Geometrie . Suppose the right-line giuen were a c : let it be in b , deuided into two parts , a b , and b c. Now let the quadrate made of the whole line a c , be a c l d : and from the diuision b , drawne a line parallell to the side c l : Item , at the same distance , draw another parallell to a c ; and thou shalt deuide the quadrate a c l d , into two sorts of right-angled parallelogrammes , to wit , into c i , and d i , two diagonals : Item , into a i , and l i , two complements , as the Geometers call them . Now that these two diagonals , with the two complements , are equall vnto a c l d , the quadrate made of the whole line a c , it is most manifest . For the whole is equall to all his parts ioyntly taken . But the two Diagonals , and two Complements aforesaid , are all the parts of the Quadrate a c l d , as here thou seest . Therefore if a right-line be deuided &c. This rule the Arithmeticians , speaking of numbers , doe set downe in these tearmes : If a number , &c. Which Barlaam hath arithmetically demonstrated ; to whom I referre thee . Out of this rule is framed the Quadrature or Fabricke of the whole square , by the two segments or portions of the side ; and that in this maner : As for example , let the number giuen be 7 , and the quadrate of the whole number let it be 49. Then deuide the same into two portions , to wit , into 4 and 3. And let these two numbers multiply one another , the foure products 9 , 12 , 12 , and 16 added , shall make 49 , equall vnto the former product , made of 7 , the whole number . Againe , let the number giuen be 13 ; and the square of this whole number let it be 169. Now let 13 the whole line , be deuided into two parts , to wit , into 10 and 3 : let these multiply one another , and the seuerall products shall be foure , the summe of which added together , shall be equall vnto the quadrate or product made of the whole line 13 : the example is thus : Therefore

3 The side of the greater Quadrate , is the side of one of the Plaines ; and being doubled , it is the side of them both together : The other side of both the Plaines ioyntly taken , is the side of the lesser Quadrate .

It followeth necessarily of the former . This rule teacheth how to resolue a Quadrate giuen , or the maner of Extraction of a square or quadrate roote . The former rule taught that the true squaring of a number , diuided into parts , consisted of two squares and two plaines . So that the true resolution of a Quadrate so made , must be the inuention of the seuerall sides of the said particular squares and plaines , whereof the whole Quadrate giuen consisteth . For it is the same way from Cambridge to London , that it is from London to Cambridge . By the former , the side of the square 169 , was 13 : which side or number consisteth of two seueral numbers , to wit , of the article 10 , and the digit 3. Here , by the former rule , the greater square is 100 , and the side thereof is 10 : The one Complement or Plaine is 30 , made of two sides , where of the one is 10 , the side of the former square : Therefore 30 , is the side of 60 , the summe of both the plaines . The other side of the Plaine ( single or double ) is 3 , which is also the side of 9 , the lesser square . These grounds thus laid , the practise is to be performed according to the direction of this rule next following .

4 If the side of some greater square be sought , first beginning at the right-hand , distinguish the number giuen continually by paires , for so many particular squares : Then setting downe the side of the first particular square found within the Quotient , shalt double the side found , for the base of the first Complement . Lastly , the Complement deuided by the Base , the Quotient shall be the side of the next succeeding Square .

As in 144 , the example so oft mentioned , first beginning at the right-hand , or first figures , I distinguish it thus , I , 44 ; wherby I vnderstand that the particular squares , whereof the whole side is comprehended , are two . This done , I seeke amongst the squares of the single figures , at the 1 e , for the side of 1 , the first square , & I finde the side to be also 1 : This side I place in the quotient . This square I subtract from 1 , the last period , and there remaineth nothing . Secondarily , I double the same quotient , or side found , and I make 20 ( for indeed the first diagonall's side is 10 ; and the side of both together iointly taken is 20 ; as we saw at the 3e . ) By this I deuide 44 , the whole period , or number remaining , and I finde the quotient to be 2 , for the other side of the Complements , and the roote of the lesser Diagonall : Therefore I multiplie first this quotient last found by itself , and I make 4 , for the lesser Diagonall or square . Againe I multiplie the same quotient by 20 the Deuisor , and I make 40 , for the doubled Complement . Lastly , I subtract 44 , the summe of the said doubled complement , and lesser diagonall , from 44 , the number remaining , and there remaineth nothing : Therefore I say that , 144 the figurate giuen , is a true Quadrate ; And the side or Roote thereof is 12 , which was desired .

Item , Let the side of the Quadrate 9604 be sought . First , beginning at the right hand , I distinguish the quadrate giuen continually into payres of degrees , thus 96 , 04 ; and I finde the number to containe two such paires : And therfore I conclude that the whole side of the quadrate giuen doth consist of two single figures . This done , I first seeke amongst the squares of the single figures , at the 1 e , for the square 96 ; which number , because I finde it not amongst them , I say is no true square . Now the greatest square conteined in 96 I finde to be 81 , and the side or Roote of it to be 9. This side I place within the quotient . Then multiplying 9 by it selfe , I make 81 , for the greatest Diagonall , which placed vnder 96 , and subtracted from the same , there do remaine ouer the head thereof 15 : Now cancelling 04 , I place them also ouer the head , as high as the said 15. Secondarily , doubling 9 my quotient nowe found , I make 18 ( or , for the reason before recited , 180 ) for the Base of the Complement : Then by this base I deuide 1504 , the number remaining , and I find the quotient 8. This therfore I likewise set downe by 9 , for the other side of the Plaines , as also for the side of the lesser Quadrate . For the proofe of this later worke , I multiplie first the said quotient by it selfe : Then I multiply the same quotient by the doubled Complement , that is , by the Deuisor : Lastly I subtract the summe of these two products from the number remaining . As in this our example , I multiply 8 , the quotient , by it self , and I make 64 , for the second or lesser quadrate : Then I multiply 180 , my deuisour , or base of the complements , by the same quotient 8 , and I make 1440 , for the said double Complement : Now 1440 × 64 , or the summe of 1440 and 64 , is 1504. This I subduct from 1504 , the number remaining , and there is left nothing : Therefore I say that 9604 , the number giuen , is a true Quadrate , and the side or Roote therof is 98 , which was desired . The example is thus : Therefore if thou shalt multiply 98 in it selfe , thou shalt make 9604 , the number giuen . For the Quadrates of the Segments 90 , and 8 , with their Plaines or Complements , are the parts of the Quadrate of the whole number 98. The genesis or making of this Quadrate , after our prescript , is thus :

5 Hauing found the Quotient of two , or more figures , if yet the whole side of the Quadrate giuen be not found , thou shalt double the whole quotient already found : And then shalt in all things else whatsoeuer , obserue the same order as was prescribed in the former .

And in this example : The side of 15129 is desired : First , hauing deuided it , as in the former we haue taught , continually into paires , thus , 1 , 51 , 29 , I finde 1 to be a Quadrate ; and 1 also to be the side thereof . This side I write in the Quotient . The quadrate of this quotient I subtract from 1 , and there remaineth nothing . Then I double the said quotient 1 , and I make 2 , ( which in respect of the next period I cal 20 ) for the base of the doubled Complement : By this I deuide 51 , the next period , and I find the quotient 2 , for the height of the said Complement , and side of the lesser quadrate . This latter quotient therfore I multiply first in it selfe , and I make the quadrate 4 : Then again by the same quotient I multiply my deuisor 20 , and I make 40 , for the doubled complement : Now 40 × 4 , are 44 , which I subtract from 51 , and there remaine 7 : Therefore the doubled Complement , with the second Quadrate , is 729. Lastly by this rule , I double the whole quotient 12 , and I make 24 , or in respect of the period following 240 , for the base of the doubled Complement : now the quotient of 729 , by 240 , is 3 , for the side of the lesser quadrate , and height of the said Complements . This done I multiply 3 , the last quotient found by it self , and I make 9 , for the lesser quadrate . Item I multiply by the same quotient 3 , the deuisour 240 , and I make 720 , for the doubled Complement . Now 720 × 9 , that is , 729 , I subtract from 729 , and there remaineth nothing . Therefore I conclude that 15129 the number giuen , is a quadrate : And the side thereof is 123 , consisting of three figures . The whole worke is thus : In greater examples there is yet greater variety to be obserued . Let therefore the practioner resolue this one great Quadrate , 61929672906515252224 , such as are often to be resolued in the making of those tables of Sines , Secants , and Tangents , of so great and wonderful vse in many businesses , where the vse of these Artes are required .

From hence do follow many particular Consectaries , First the Single figures of any Quadrate giuen cannot exceed the double of the number of the single figures of the side thereof . The reason is from the 1 e ij : to wit , because the product of the greatest single figure by it self , consisteth not of more then two figures .

And 2 If the number giuen beginning at an vnity , and so increasing according to the natural order of numbers vnto the middlemost , shall from the same , in the same maner decrease vnto an vnity , the side of the sayd quadrate shall consist of vnities : And the same middle number shall shew the number of them . As for example : Suppose the number giuen were 1234321 , here because it beginneth at an vnitie , and so increaseth vnto the middlemost , and from the same middlemost decreaseth in like maner continually vnto an vnity : And for that the same middlemost number is 4 : I say that the side or roote of the quadrate giuen consisteth altogether of vnities : And that the number of them is but 4 , thus 1111. Item , the side of this quadrate following , 12345678987654321 , is 111111111. Therefore on the contrary , If a number to be multiplied by it selfe , do consist altogether of vnities , vnder the number of tenne , the product shall from an vnity increase and decrease , as aforesayd .

6 The difference of two vnequal quadrates giuen , is the quadrate of the difference of their sides , with a double plaine made of the same difference , and the lesser side . This is called the Gnomon , or squire , 2 d xij ; or 12 e x R.

As for example , Let the two Quadrates giuen be 144 , and 100 ; And their Difference let it be 44. Againe , Let their Sides be 12 and 10 ; And their difference let it be 2. Nowe let the Quadrate of 2 , the Difference of the sides be 4. Againe , Let the Plaine made of 10 , the lesser side , & of 2 , the Difference of the sides , be 20. Now 20 × 20 , are 40. And 40 × 4 , are 44 , the Gnomon or difference , whereby the two Quadrates 144 , and 100 , do differ one from another .

Therefore

7 If the number giuen be not a true Quadrate , the remaine ( to be added vnto the side of the greatest Quadrate conteined in the sayd number giuen ) shal be denominated of the Gnomon , or difference of that said greatest Quadrate , and that which is next aboue it .

Let the number giuen be 148 , which by the 5 e , is no true Quadrate : and therefore the true side thereof can neuer be found . But the side , somewhat neare vnto the true side , is thus found . The greatest Quadrate contained in 148 , the number giuen , 144 , whose side by the said 5 e , is 12. Item , the Remaine or difference of 144 , & 148 , is 4 , for the Numerator of the parts sought . Againe , the Quadrate next greater then 144 , is 169 , and the side thereof is 13. Lastly , the Gnomon or Difference betweene 169 , and 144 , is 25 , the denominator sought . Therefore 4/25 , are the parts desired . The side then of 148 , neare vnto the true side , is 12 4/25. Briefly , ¹ The Denominator of the parts sought , is the difference of the greatest square contained in the number giuen , and the next greater aboue it . Let the number giuen be 11 : the greater Quadrate contained in it is 9 , whose side is 3. Now 11 — 9 , is 2 , for the Nummerator desired . Item , the next Square greater then 9 , or 11 , is 16 , and the side thereof is 4. Now 16 — 9 , is 7 , for the Denominator of the said parts . The side of 11 , neare vnto the true side , is , 3 2/7. Item , ² The said Denominator is the double of the side of the greatest Quadrate encreased by an vnitie . In the last example , the side of the greatest quadrate contained in 11 , is 3 ; whose double is 6. Therefore 6 + 1 , that is 7 , is the Denominator sought , and the parts are 2/7. Againe , 3 The denominator of such like parts shall be the summe of the side found , and of the side of the next greater quadrate aboue it . In the former example , the side of 9 , the greatest Quadrate contained in 11 , is 3 : and the side of 16 , the quadrate next aboue 9 , is 4. Now 4 × 3 , that is 7 , is the Denominator desired : and the side sought is 3 2/7.

8 The product made of two like-plaines , is a Quadrate : and the side thereof is the product of the base of the one , by the height of the other .

Like-plaines , are plaines whose sides are proportionall , 20 d vij . Let the Like-plaines giuen be 6 , and 24 : whose sides , 3 , 2 , 6 , 4 , are proportionall : that is , as 3 , the base of 6 , is vnto 2 , the height of the same : so let 6 , the base of 24 , be vnto 4 , the height thereof . Here first I say , that 144 , the product of 24 , by 6 , is a quadrate . Againe I say , that 12 , the product of 3 , the base of 6 , the first plaine , by 4 , the height of 24 , the second plaine : or contrariwise , the product of 2 , the height of 6 , the first plaine ; by 6 , the base of 24 , the second plaine , shall be the roote or side of the said Quadrate . The example is thus : The Quadrate . The side or roote thereof .

9 If three numbers be continually proportionall , the product of the first and the last shal be equal to the quadrate of the middle numbers ; And contrariwise , If the quadrate of the middle number be equall to the product of the first and the last , the three numbers giuen shal be continually proportionall .

Let the three numbers giuen continually proportionall , be 4 , 6 , 9 , the product of 4 , by 9 , the first and the last , shall be 36 : Item the square or the product of 6 by it selfe shall also be 36 : On the contrary , if the quadrate , square , or product of the middlemost be equall vnto the product of the first and the last , the three numbers giuen are continually proportionall : As Euclid doth demonstrate and teach at the 16 p vj , and 20 p vij , of his Elements . Item , Ramus handleth this proposition in three seuerall places ; to wit at the 15 eij of his Arithmeticke : At the ij consect . of the 14 ex , and againe more specially at the 4 e x ij , of his Geom. Yet in deed , saith Schoner , there is something in this more then may be performed by that 15 chap. of the second booke of his Arithmeticke . For diuision cannot alwayes finde out the meane proportionall : But in diuerse cases there is required , to the performance of this , the extraction of a square roote ( of which we haue spoken before at the 4 and 5 e : ) by the which also the Deuisour is sought . As for example , suppose the meane proportionall betweene 16 , and 64 , the extreames giuen , were to be sought . Here the product of 64 by 16 , is 1024 ; and the side of 1024 , by the 4 e , is 32 , the meane proportionall desired .

A proportion requireth foure bounds or numbers . If the second and third bound be the same , as in this our example , it is called a Continuall proportion : If they be diuerse , it is named a Disioyned proportion . Now generally , whether the proportion be disioyned or continuall , this rule is true : If foure numbers be proportionall , the product of two middle numbers shal be equal to the product of the first and the last . As for example , As 12 is to 4 , so let 6 be to 2. I say the product of 6 by 4 , shal be equall to the product of 12 by 2. For let the product of 12 by 6 , be 72. Here therefore the product of 6 by 12 , shal be also 72 , by the 3 c iiij c j Saligna .

If two numbers be proportional to one and the same number , they are equall betweene themselues : 16 e v c j Salig . If one number do multiply many numbers , the products shal be proportionall vnto the numbers multiplied : 5 e vj. But 12 doth multiply 6 ( and maketh 72 by the construction : ) Item it multiplieth 2. Therefore as 72 is vnto the product of 12 by 2 : so is 6 vnto 2 : and so , by the grant , is 12 vnto 4. If one number do multiply many , &c. 5 e vj c j Salign . But 6 doth multiply 12 , and 4. Therefore as 72 ( which is the product of 6 by 12 , as before was manifest ) is vnto the product of 6 by 4 , so is 12 vnto 4. Therefore the products of 12 by 2 , and of 6 by 4 , are equall . q. e. d.

This proposition is commonly called the Golden rule : and indeed truth it is that the Mathematical arts do daily reape from hence fruites in value more worth then gold .

10 If the three numbers continually proportionall giuen , be simple between themselues , the first & the last shal be quadrate or square numbers : Item , If of three numbers continuallie proportionall giuen , the first be a quadrate , the third also shal be a quadrate .

As for example , Let the three numbers continually proportionall , and simple betweene themselues giuen , be 4 , 6 , 9 ; the first and the last , to wit 4 and 9 , are quadrate numbers .

Again , if 16 , 32 , 64 , be continually proportionall , & 16 , the first bound be a quadrate ; 64 also , the third bound shal be a quadrate . These both are demonstrated by Euclid , and his Expositours at the 2 , and againe at the 22 prop. of his viij booke .

11 If a quadrate do multiply a quadrate ( or one many continually ) the product shal be a quadrate ; and the side of the quadrate so made , shal be the product made by a continuall multiplication of the seuer all sides of the particular quadrates giuen .

As for example ; Let the quadrate 4 , multiply the quadrate 9 , and let the product thus made be 36 : Here first I say , the product of 36 is also a quadrate . Item I say , that 6 , the product of 2 by 3 , the sides of 4 & 9 , the particular quadrates giuen , shal be the side of 36 , the compound quadrate .

Item , 25401600 , the product continually made of 4 , 9 , 16 , 25 , 36 , 49 , is a compound quadrate : And the side thereof is 5040 , which is the product made of 2 , 3 , 4 , 5 , 6 , 7 , continually multiplied between themselues .

12 If a Quadrate doe deuide a number assigned by a Quadrate , the assigned shall be a Quadrate .

This also ariseth from the same fountaine . As for example , Let 4 , a quadrate deuide 64 , the number assigned by 16 , a quadrate . I say that 64 , the number assigned is likewise a quadrate .

13 If the product of two numbers assigned be a quadrate , the side of that quadrate shal be the meane proportionall betweene the assigned .

As for example , Let the nuumber assigned , between which we desire the meane proportionall , be the quadrates 144 , and 64 ; and let the product of them be 9216 : I say that 96 , the side of the said compound quadrate made , shal be the meane proportionall desired , that is , as 144 is vnto 96 : So 96 is vnto 64. The demonstration is built vpon the 9 evj cj Salign . which is a special consectary drawn from the 2 c 14 e , of the x booke of Ramus his Geometry : The argument is thus framed

If the product of the two middlemost numbers be equal to the product of the first and the last , the foure numbers giuen are proportionall betweene themselues : 9 e vj cj Sal. The side is a number , which multiplied in it selfe , doth make a quadratc . But 96 is the side of 9216 , by the construction and grant . Therefore 9216 is made of 96 , multiplied by it selfe . Therefore 144,96,96,64 , the foure numbers giuen , are proportionall betweene themselues : and so the meane proportionall betweene 144 , and 64 , the two numbers giuen , is found , q. e. f.

Therefore

14 If a number shall by it selfe deuide a number made of any two numbers giuen , the quotient shal be the meane proportionall between the two assigned : 10 e v j c i Salig .

The rule is generall thus : If one number shall deuide the product of two assigned , the deuisour and the quotient shall be the meane continually proportionall , betweexe the assigned . As for example , let 6 , multiplying 4 , make 24 : And let 8 deuide 24 by 3 : I say , as 6 is to 8 , so is 3 to 4. But the speciall consectarie is more fitter for our purpose . Therefore for example , Let the product 144 by 64 , be 9216 , and let 96 diuide 9216 , the said product , by 96 : I say , that 96 is the meane proportionall betweene 144 , and 64 , the two numbers giuen : that is , as 144 is vnto 96 , so is 96 vnto 64.

CHAP. III. Of the making and use of certaine Tables and Instruments , deuised for the more exact and speedie measuring of all sorts of Plaines .

1 PLaines , according to their diuers nature and qualitie , are measured by diuers and sundry kindes of measures : For some are measured by the Foote , others by the Yard or Elne , othersome by the Rodde or Perch , and such like . Now these measures being defined by Act of Parliament , it shall not be amisse to set downe the words of the Statute , so farre foorth as it shall concerne this argument . It is ordained , saith the Statute , that three graines of barley drie and round , doe make an Inch ; twelues inches do make a Foote ; three foote do make a Yard ; fiue yards and an halfe , doe make a Perch ; and fortie Perches in length , and foure in breadth , doe make an Akre . 33. Edw. I , De terris mensurandis . Item , De compositione Vlnarum & Perticarum . See also more of this hereafter .

By the Foote we measure Boord and Glasse . A Foote therefore of flat measure , is a right-angled square , 12 inches long , and 12 inches broade , that is , a foote of boord is a plaine , containing 144 square inches : For such is the product of 12 by 12. Here , as also in the other which follow , obserue , That the Breadth is alwaies giuen ; the Length is desired . If therefore the breadth giuen be 12 inches , it is plaine by the former definition , that 12 inches of length doth make a foot of flat measure . But if the breadth giuen , be either greater or lesser then 12 , the length desired is not so easily found . For here some art is oft times required . This then is to be conceiued and done by the 14 c of the former Chapter : For there we learned how to equall plaines of diuers breadths . Admit now a boord to be measured were but 9 inches broade ; here , as those rules did teach vs , I diuide 144 , by 9 the breadth giuen , and I finde in the quotient 16 , for the length desired . For as 12 is to 9 , so is 16 to 12 ; that is , 16 inches in length , of the breadth of 9 , are equall to 12 inches of length , of the breadth of 12. Thus you may make a Table or Instrument to serue readily at all times , for the more speedie and exact measuring of all sorts of plaines by the foote square . For , If you shall diuide 144 , by the seuer all breadths giuen , the quotients shall shew the length desired , answering to their seuer all diuisors or breadths giuen . Now where you do beginne ( I meane whether at the greater breadth , and so descending shall end at the lesser , or contrariwise ) the matter is not great . Againe , the Carpenters or workemens instrument which they vse in this case , being but two foote in length , it shall not be necessarie to beginne at any breadth greater then 24 , as shall be manifest hereafter , when we come to shew the vse of this Instrument or Rular . Beginning then at 24 , and so descending to the lesser , the Table is thus : The vse of this Table , to him that vnderstandeth the former , is plaine and easie . For hauing any breadth giuen , betweene 24 and 1 , I seeke it amongst the breadths in that columne vpon the left hand , and in the other columne vpon the right hand oueragainst it , I find the length desired .

1 As for example ; A boord to be measured is 18 inches broade , I desire to know what length of that breadth shall make a foote . R In the second columne , oueragainst 18 , in the first columne , I finde 8. Therefore I say , euerie 8 inches in length , of that boord that is 18 inches broade , shall make a foote of flat measure .

2 Againe , suppose the breadth of a peece of Glasse to be measured , were 5 inches broad . Here I finde , that euery 28 inches , and ⅘ of one inch , doth make a foote of that kinde of measure .

3 But what if the breadth giuen be greater , or lesser then any breadth between 24 and 1 ? I answer , the question is answered with as great facility . For first admit the breadth giuen were 36 , which is greater then any breadth in our Table , here if I take some such knowne parte of 36 ( partem aliquotam , a knowne parte , the Arithmeticians call it ) as may be founde in our Table , the length desired shall answeare vnto it in the like proportion : As for example , if I take 18 the ½ of 36 : then it is manifest , the length desired shal be but 4 inches , the one halfe of 8 , which is the length answering to the breadth 18. Or , if I shall amongst the Breadths take 12 , which is but the third part of 36 , my breadth giuen : then also shall 4 , which is the ⅓ of 12 , the length found , be the length desired , answerable to 36 the breadth giuen : Or which is all one , euery X2 inches in length , of that boord that is 36 inches broade , shall containe three foote of flat measure .

4 Againe , suppose the breadth giuen were an 100 inches . R Here 20 , is but the ⅕ part of 100. Therefore I say that 7 ● in length , of that plaine that is 100 inches broad , shall containe fiue foote of flat measure .

5 Lastly , Admitte that the breadth giuen were but ½ inch , which is lesser then any one of those set downe in our Table . R , Here I double , treble , or quadruple the number giuen , vntill I may finde it in the Table , and then it is manifest , that the length found shall be answearable to the breadth giuen in the like proportion . As for example , If I double ½ , I make 1. Now to the breadth of 1 inch , in the Table there answeareth 144 inches of length : Therefore the double of 144 , that is , 288 shal be the length desired answearing to ½ inch of breadth . Or , which is all one , 144 inches of length of that breadth , that is but ½ inch broade , shal be but ½ foote of flat measure . The reason of this is manifest , out of the 14 element of the tenth booke of Ramus his Geometry .

By this Table you may make a Ruler or instrument , whereby the same may be performed yet with greater speed and more facilitie . For this instrument is none other then that which Attificers do vse in this case . It is , I say , no other but a Rule , as they call it , of two foote long ( more or lesse , as you shall thinke most conuenient for your vse ) and of what breadth you please , diuided , as the maner is , into inches and partes of inches , with a thwarting or beuelling line drawne from side to side , beating the seuerall lengths aboue mentioned in our Table . That , for thy better vnderstanding , we do more particularly thus describe . First with the iage I strike two parallell lines , the one as neare vnto the one edge , the other vnto the other , as shall be thought most conuenient : These are the breadths , 24 and 6. Now to the breadth 24 , in the Table , there answereth 6 inches for the length : Therefore in the vpper line beginning at the right-hand , I account toward the left , 6 inches : There I make a marke . Again , to the breadth 6 , in the same Table , there answereth 24 inches for the length : Therefore in the neather line counting as afore , I make a mark at the 24 inch . From these two markes I draw a line ouerthwart the Rular . In this beuelling or thwarting line , by the helpe of a squire , I note all the lengths noted in my Table , betweene 24 and 6. As for example , To the breadth 23 , in the Table , there answereth for the length 6 inches , and 6/23 partes of one inch . This length , I note , beginning mine account as before , in one of the parallel lines . Then by the help of the squire I note the same vpon the beuelling line . In like maner I set all the rest of the breadths from this , vnto 6. This being done the Rular is perfected , and fit for vse in all cases , as before is prescribed in the vse of the Table .

The vse of this Rular is plaine and easie , if either that of the table , or the maner of making this be wel vnderstood . For , If you shall seeke out the breadth giuen in the beuelling line , the inches & parts of inches from thence vnto the end of the Rular toward the right-hand , shall be the length desired . I. As for example , Suppose the breadth giuen were 12 inches : Here I finde the breadth 12 noted at the twelfth inch from the beginning of the Rular : Therefore I say , Euery 12 inches in length , of that plaine that is 12 inches broad , shall be a foote of flat measure . II. Againe , Admitte the Breadth were 9 inches . This is noted in the thwarting line at 16 inches from the sayd end . Therefore I say , That 16 in length of this breadth , shall make a foote of flat measure . III. Admit the breadth giuen were 64 inches . Here because I haue none so great , I take 16 , the ¼ of it . Now the breadth 16 , is noted on the beuelling line , 9 inches from the sayd end . Therefore I say that euery 9 inches in length , of that plaine that is 64 inches broad , doth contain 4 foote of flat measure . IV. If the breadth giuen beleffer then any vpon the beuelling line , then double , treble , or quadruple , &c. the same , as afore is taught ; and the length found shall be but the halfe , third part , or fourth part &c. of the length desired . As for example , Admitte the breadth giuen were 2 inches : Here because vpon the beuelling line I finde no number lesse then 6 : Therefore I treble 2 , and I make 6. Now 6 being placed vpon the line of breadths , or beuelling line , 24 inches from the said fore-end , I say , That 24 inches of length , of that plaine that is but 2 inches broad , is but ⅓ part of a foote of flat measure . V. If the breadth giuen , besides the whole inches , do containe also some part or parts of an inch , then that is to be proportioned out between the whole number giuen , and the next greater aboue it . As for example , Admit the breadth giuen were 9 inches & ½ : Here the length desired will fall out proportionally between 9 inches of breadth and 10 , aboute 15 inches and 3/19 , from the sayd end .

Of the measuring of Cloth , Wainscot , Painting , Pauing &c. by the yard .

2 By the Yard we do measure Cloth , Wainscot , Painting , Pauing , &c. The yard , as is aforesaid , doth containe in length 3 foote , or 36 inches . Therfore the yard-square , or a yard of flat measure doth containe 9 foote , or 1296 inches . This kinde of measure they commonly diuide into 4 quarters : And euery quarter into 4 nayles .

Vpon the former grounds the like Table and instrument for this kinde of measure , may be framed as was for that of boord , if any man shall thinke it worth the while . But it shall not be much losse of time or labor , to shew in a word or two how it may be done . In a yard-square there is 16 quarters of a yard ; or 64 nayles : Therfore if you shall diuide 16 , or 64 , by the breadth giuen , the quotient shall be the length desired . The Table then for this kinde of measure is thus :

The vse is plaine . I. Admit a peece of Wainscotte were three quarters of a yard broade , what length of that breadth shall make a yard of flatte measure ? R. one yard , one quarter , one naile and halfe , or thereabout . The proportion , as afore , is thus : As 4 is to 3 , so is 5⅓ vnto 4.

2 A peece of painting is 2 quarters , that is , halfe a yard broade ; what length shall make a yard square ? R. two yards in length .

3 But the vse of this is better seene in buying and selling of cloth : For here oft times the skilfull Taylour , although he do well know how much cloth or stuffe of this or that breadth wil serue to make such a garment , is cleane to seeke how much shall make the like garment of stuffe of any other kinde of breadth : Onely this aduantage he hath , that he wil be sure to aske enough , which shall be no losse to himselfe .

1 Admit 4 yards of cloth of 1 yard broade did make a garment : how much stuffe of ¾ broade shall make the like ? R. The Table for the breadth 1 y , giueth 1 y , for the length of 1 yard square : Therfore 4 yards of this breadth , do make 4 yards square . Secondarily , the same Table for ¾ breadth , giueth the length 1 y 1 q 1½n . Therefore foure times 1 y 1 q 1½n , that is , 5 y 1 q 1 n shall make the same garment .

2 Againe , 12 yards of stuffe of ¾ broade , did make a gowne ; how much cloth of 1 y 3 q broade shall make the like ? R. The Table for ¾ breath , doth give the length 1 y 1 q 1½n : This length I seeke how many times I may finde in 12 yards : that is , I seek how many square yards 12 yards of ¾ broad doth containe . Againe , the same Table for 1 y 3 q breadth , giueth 2 q ½n . This length so many times taken as you found the former length in 12 , is the number of yards required .

Of measuring of Land by the Aker .

3 BY the Rodde we measure Land , Mcdowes , Wood , Grasse , Corne , &c. The Rodde , Lugge , Perch , or Pole doth contain , as we haue heard , 16½ foot , or 5½ yards . This kind of measure did vary according to their customs with the countrey : But now by Act of Parliament that varietie is taken away . The words of which Statute , entituled , An Act for restrains of new buildings , &c. in and neare to the cities of London and Westminster , made in the xxv yeare of the reigne of Queene Elizabeth , are these : Be it enacted by the authoritie aforesaid , that a Mile shall be taken and reckoned in this manner , and no otherwise : that is to say , a Mile to containe eight Furlongs ; and euery Furlong to containe fortie Lugges or Poles ; and euery Lugge or Pole to containe sixteene foote and an halfe . Thus farre that Statute . Now an Aker of land , as we haue shewed before , is a plot of ground containing 40 Roddes in length , and 4 in breadth ; or , which is all one , an Aker of land containeth 160 square Roddes or Poles of ground . If therefore you shall deuide 160 by all the seuer all breadths betweene 40 and 1 , the products shall shew the lengths desired , answering to those seuer all breadths giuen . The Table then of Land-measure is thus : The vse of this Table is in all respects like vnto that of boord-measure : Therefore one or two examples shal be sufficient to make it knowne vnto the simplest .

I Suppose a piece of ground to be measured were 32 pole in Length : I demand how much in breadth is required to make an Aker . R. To 32 , the length in the first columne answereth in the next columne 5 : Therefore I say , euery 5 rods in breadth , shall be equall to that field or plot which is 40 pole long , and 4 pole broad .

II Admitte the field to be measured were 23 pole long . R. To 23 in the first columne I finde answering in the second columne 6 22/33 , that is , 6 pole , and 22 parts of a pole diuided into 23 parts . Therefore I say euery 6 22/33 poles , or 7 pole ferè , is equall vnto that which is 40 long , and 4 broad , that is , to 160 square poles .

III But what if the length be greater then any number found in the first columne ? R. Here as before , I take some partem aliquotam , as one halfe , one third , one fourth &c. As for example , suppose the field to be measured were 100 poles in length , here because I finde no number so great as 100 , therefore I take 20 , the fifth part of 100 : To which 20 I finde 8 pole for the breadth . Wherefore the fifth part of 8 roddes , that is , 1 rodde and ⅗ partes of one rodde of that length shall be equall to that plot which is 40 long , and 4 pole broade . Or , which is the same in effect , Euery 8 pole in breadth , of that field which is 100 pole in length shall containe 5 Akers of land .

IV Admit it were 84 pole in length . R. Here the fourth part is 21. Now to 21 pole of length , there do answer 7 poles and 13/21 parts of one pole in breadth . Therefore euery 7 poles , and about two third parts of a pole in breadth , of that length , shall make 4 Akers . For if it were but 21 pole long , then euery 7 pole and 13/21 of breadth , shall make one Aker . But 84 containeth the length 21 , foure times : Therefore 7 13/21 of the whole length 84 , shall containe 4 Akers .

Hauing finished this Treatise , and it being altogether ready for the Presse , turning ouer the Statutes , to see what I might finde more for this our purpose , I light vpon the very like Table , made many yeares since , by Act of Parliament , in the time , as I take it , of Edward the first , intituled De terris mensurandis . See the latter Abridgement of the Statutes of Rastall or Poulton , in the title of Weights and Measures .

A very speedie and readie way or maner of Diuision in this case of surueying .

4 SVrueyours , which are often employed in measuring of whole Farmes , Lordships and Townes , are at last to giue an estimate of the grosse summe or number of Akers contained in some very great summe or number of Roddes , after multiplication of the sides , and addition of many seuerall parts , &c. This vsually they performe and finde by diuiding of the product or summe by 160 , the number of poles that one Aker doth containe . Now here because the Diuisor consisteth of many seuerall or single figures , if especially the product or diuidend be any thing great , the worke , as euery one meanly practised in Arishmeticke doth know , must needs be long , and peraduenture ( except great heed be taken ) not without some error . And againe because , Faciliùs & expeditiùs numer amus numeros paruos , quàm magnos , we do more readily , and with greater facilitie number small numbers , then great : If any man shall teach how this Diuisor may be reduced to a small number , consisting of one single figure , he shall much shorten the worke , and deserue much thanke of all Surueyors or others delighted in these studies . Ingenui pudoris est , saith Plinie , fateri per quos profeceris . I confesse the inuention is not mine : but I am the first , if I be not deceiued , that hath made it publici iuris , common to all . Thus then this matter is to be done . If you shall diuide the number giuen , and likewise the quotient now found , continually by 20 foure times , the last quotlent ( with the parts , if any do remaine ) shall shew the Akens , and parts of an Aker contained in the number giuen : or , if you shall like better , shall diuide the number giuen , by 40 twise . Let the examples be those following .

I. Suppose a field being measured and cast vp by multiplication , the product or number of perches were 1280 , I would know how many Akers it doth containe . Here note out of an abridgement of diuision , because my Diuisor 20 , hath for his last figure a cipher or nought : therefore I cut off the last figure of my Diuidend , thus 128 ( 0. Or thus ,

R. I answer by the last quotient of both formes , that that field which containeth 1280 square perches , doth containe 8 Akres .

II. Item , Suppose the product or grosse summe of some peece of land surueyed were 102400 perches . Or thus , Here I say , that 102400 perches , do make 640 akers of land .

These examples may suffice in such cases where no fraction shall remaine : but if in any of the workes of diuision , any parts shall remaine , the value of those parts shall be esteemed by these rules following .

1 If the first figure ( the last , some peraduenture would call it ) of the number giuen be a significant figure , then it is manifest , that ouer and aboue certaine akers , &c. there are some odde perches .

2 Secondarily , if diuiding by 20 , there do remaine ought in the first diuision , that remainder shall be ¼ of 2 Roode or 10 perches : if in the second , ½ Roode or 20 perches : if in the third , ¼ , of an Aker , or 40 perches : if in the fourth , ½ of an aker , or 80 perches .

3 Lastly , if diuiding by 40 , there shall ought remaine at the first diuision , those parts shall be parts of a Roode : if at the second , they shall be parts of an aker . The examples following , shall make all plaine and easie . Or thus by 40 : Therefore that field that containeth 939 perches , doth containe 5 Akers , 3 Roodes , and 19 Perches .

II Or thus : The number 13593 , containeth in land-measure , 84A , 3R , 33P . or 84 Akers , 3 Roodes , and 33 Perches . The like may be done in Boord-measure , and Timber-measure , if any man shall finde it of any such vse , as it is in this kinde .

CHAP. IIII. Of the Extraction of a Cubicke roote .

1 A Cube is a right-angled solid , comprehended of equall sides : 25 d xj E.

HItherto we haue spoken of the Quadrate , and of such proprieties and corrollaries as did belong vnto this our purpose . Now it remaineth that we in like maner do handle the Cube and Cubicke number , and that with as few words , and as briefly as we may . For as the vses of the Quadrate , and of the extraction of his Roote were many : So the Cube , being of a more excellent nature , cannot yeeld fewer , if not more , and those also of more and greater worth . For as Vitruuius , in the preface to his fift book of Architecture , writeth , Pythagoras was so much delighted with the Cube , that he wrote all his precepts in cubicall numbers . His words are these : Etiamque Pythagora , hisque qui eius haeresim fuerunt secuti , placuit Cubisis rationibus pracepta in voluminibus scribere , constitueruntque Cubum 216 versuum , cosque non plus quàm tres in vna conscriptione oportere esse pusarunt , &c. Moreouer also Pythagoras , and those which followed his faction , were much delighted to write their precepts and rules of Philosophie in a kinde of cubicall proportion , making a Cube of 216 verses ; deeming that there ought not to be aboue three in one staffe . Now a Cube is a square bodie , consisting of sixe sides , plaine and equall one to another . This kinde of bodie , when it is cast at an aduenture out of the hand , vpon which side soeuer it pitcheth , no man touching it , standeth firme and constant . Such are the dice , which gamesters that play at Tables do vse . The Greeke comicall Poets also , which in the midst of their playes do cause the Quiristers to sing a song , haue so diuided the pauses of their comedies , that making the parts in a kinde of Cubicall proportion , they much , by such rest , do ease and helpe the pronunciation of their actors . Thus farre Vitruuius . This Cube or Cubicall number of 216 , hath for his side 6. For 6 times 6 , do make 36 , for the one side , or square plaine including the Cube . And 6 times 36 , are 216 , the Cubicke number of the Cube here mentioned . The mysterie hereof conceiued by Pythagoras and his schollers , I leaue to others to vnfold . That pertaineth not to the Mathematician : that other also belongeth to the Poets , whereof our age doth affoord plentie . That which the same author in another place hath , of the answer of the oracle of Apollo : Item , that of Eratosthenes vnto Ptolomey king of Egypt , of Glaucus his tombe , do more concerne our businesse : And therefore hereafter , in their place , we shall handle them at full , if God permit .

Hereyou see how the definition of the Cube doth answer to the definition of the Quadrate : for each of them is a right-angled and straight-bounded figure : And as the general differences of bodies or solids , were drawne from the generall differences of surfaces , plaine and oblique : So here the particular differences are taken from the speciall differences of the same surfaces . A plaine solid is that which is comprehended of plaine surfaces : this is generall of what kinde soeuer those plaines are of : but a Cube , or cubicall bodie , is that which is comprehended of square or quadrate surfaces . The nature then of those surfaces which do comprehend them , is that they must be plaine , not oblique or vneuen : Secondarily , they must be squares or Quadrata , that is , Right-angled , and of equall sides . This is Euclids definition , and meaning . The number of these surfaces Vitruuius doth tell : Cubus est corpus ex sex lateribus aequali letitudine planitierum quadratum : that is , A Cube is a square bodie comprehended of sixe plaine sides of equall breadth . Item , Martianus Capella in these words : Solida figura Quadrati sex superficies habet : The solid square ( so he calleth the Cube ) hath six surfaces . From hence is it that they attribute stabilitie or constancie vnto this kinde of bodie . For 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , corpus quadratum , is such a bodie , if I mistake not , that consisteth of such an equall temperature of the humors , ( eucrasia , ) that it is not subiect to that alteration and change that others are . And Aristotle calleth a good man 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , virum quadratum , that is , Cubicum , as I vnderstand him , meaning such a man as is constant , and not easily moued vpon euery chance and misfortune , that in this world doth happen to mortall men . Such an one'as they report Bias , one of the seuen wise men of Greece to haue bene . For Cubus cum est iactus , saith Vitruuius , quam in partem incubuit , dum est intactus , immotam habet stabilitatem : vti sunt etiam tessera , quas in aluso ludentes iaciunt . A Cube when it is cast out of the hand at an aduenture , looke vpon what side soeuer it lighteth , it standeth fast and stable ; like as those dice also do , which gamesters vse that play at Tables . Therefore , If sixe equall squares be ioyned with solid corners , they shall comprehend a Cube . Item , the sides of a Cube ( hedrae ) are sixe : the edges ( latera ) are twelue : the plaine-angles , twentie foure : the solid-angles are eight ; as the author of the Scholium vpon the xv booke of Euclids elements , hath taught .

2 The power of the Diagonall line of the Cube , is thrise so much as the power of the side of the Square comprehending it .

Harmonia , saith our Author , est musica literaetura obscura & difficilis ; maximè quidem quibus Graecae literae non sunt notae , quam si volumus explicare-necesse est etiam Graecis verbis vti , quod nonnulla eorum Latinas non habent adpellationes : that is , The theory of Musicke is very hard and difficult : especially to those which are ignorant of the Greeke toongue : For whosoeuer shall take vpon him to write of this argument , in what language soeuer , he shall be forced to vse many Greeke words , because many of them haue no termes answerable to them in other languages . The same may I say of Geometrie , where the most termes or words of art are meerly Greeke , or at leastwise fained in imitation of , and that oft times not very fitly . For surely I doubt not but at the first those words seemed harsh enough , euen to the first inuentors of them : but time and vse haue made them familiar and pleasing . Of this sort is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , Potentia , power , we call it : and posse , to be able , to be of value and power . Both which do signifie in this place or this art , nought else but a Geometricall multiplication , as they are oft vsed by Euclid the great Geometer , and Diophantus the ancient Algebraist , and that by the testimonie of Aristotle the Prince of Philosophers . For of this rule , Potentia rectae est quadratum ; or as Diophantus speaketh , Appellatur quadratus Facultas , that is , the power of a right-line is a square : the meaning is , that if a right line be multipled in it selfe geometrically , it shall make a Quadratum or square surface . Likewise then in this place this proposition , Diagonius Cubi potest triplum lateris , The power of a Diagonall line of a Cube , is thrise so much as is the power of the side ; is thus to be vnderstood : If the Diagonall of a Cube be multiplied by it selfe geometrically ; and the side of that square which comprehendeth the same Cube , be also multiplied by it selfe , the square that is made of the Diagonall line , shall be three times so great as is that which is made of the quadrate including the same Cube .

Item , againe here obserue , that that right line which crossing a plaine from side to side by the center , is called the Diameter : in a solid , the Axis or axeltree ; in both , if it passe from corner to corner , is properly termed the Diagonius , or the Diagonall line . The power of this line , saith our author , is thrise so great as is the power of the said side or edge , as I call it . That we thus demonstrate :

That , whose power is as much as is the power of any thing single , and of that whose power is doubled to the power of the same single , ioyntly : is of treble power vnto the single . Axio . Logicum . The power of that side in a right-angled triangle , which is opposite to the right-angle , is equall to the single powers of the two other sides . 5 e xij R. But the Diagonall in the Cube , the side of the square , with the Diagonal of the same square , do make a right-angled triangle ; and this Diagonall of the Cube , is opposite to the right-angle , ex thesi . Therefore the power of the Diagonall of the Cube , is as much as is the power of the side , and diagonall of the square . Therefore the power of the Diagonall line of the Cube , is thrise so much as is the power of the side of the square . q. e. d.

3 If of foure right-lines continually proportional , the First be the halfe of the Fourth ; the Cube of the first , shal be the halfe of the Cube of the second .

This proposition is a consequent or corrollary drawne out of the 15 e iiij R , which teacheth that , if certaine right-lines be proportionall ( to wit , more in number by one then are the dimensions of the like figures , alike situated vnto the First and Second ) it shall be as the First right-line is vnto the Last , so the First figure shall be vnto the Second : and contrariwise . Now a Cube , by the I e xxj R , is a figure of three dimensions ; because it is a bodie , which hath length , breadth and thicknesse . Therefore here the lines compared in the proportion must be foure . The truth of this rule will easily appeare by an example in numbers . But here obserue , that you must not expect that our example shall altogether directly answer to this rule of doubling the Cube ; but to some other . For truth it is , that Arithmeticke , as hereafter shall more plainly appeare , cannot double the Cube ; that is , Arithmeticke although it can tell what shall be the double of the Cube giuen , yet it can by no meanes tell thee in numbers , what the Latus or side of that Cube shall be . Let the foure numbers continually proportionall giuen , be 2 , 4 , 8 , 16 : that is , as 2 is vnto 4 , so let 4 be to 8 : and as 4 is to 8 , so let 8 be to 16. Here I say by that 15 e iiij R , as 2 the first number is vnto 16 the last : so shall the Cube of 2 the first number , be vnto the Cube of 4 the second . But 2 is but the eighth part of 16 : Therefore the Cube of 2 shall be but the eighth part of the Cube of 4. The Cube of 2 , is 8. The Cube of 4 is 64. Now and

Here you see , that the quotient of 16 , the last number , by 2 , the first : Item , the quotient of 64 , the Cube of the second , by 8 , the Cube of the first , to be alike , or the same : And therefore the proportion is the same . q. e. o.

By this rule an answer was made vnto that great and strange question , moued by the oracle of Apollo at Delphos . The inuention of this proposition , that is , of the first answer to that problem , by some is attributed to Plato the diuine Philosopher , a though certaine it is , that Eratosthenes , in an epistle of this argument written to Ptolomey king of Egypt , doth ascribe it to Hippocrates Chius . The historie is briefly touched by Vitruuius in these words : Alius enim alia ratione explicare curauit , quod Delo imperauerat responsis Apollo , vti arae eius quantum haberet quadratorum id duplicaretur , & it afore , vt hi qui essent in ea insula tunc religione liber arentur . For each of them ( meaning Architas Tarentinus , and Eratosthenes Cyrenaeus ) labored by sundry wayes to performe that which Apollo at Delos had giuen in charge in his answer , whereby he commanded , that looke how much soeuer the altar that was before him should containe in square feete , it should be doubled , and so it should come to passe , that those in that Iland , which were sicke of the plague , should be freed from that curse . But Eratosthenes before mentioned , layeth downe the historie more largely and plainly , thus : To king Ptolomaeus , Eratosthenes sendeth greeting ; It is reported that one of the ancient tragedians bringeth in Minos , purposing to reedifie Glaucus his tombe . And being giuen to vnderstand , that it was euery way an hundred foote , said , That is too little for the sepulcher of so great a king . Let it be therefore doubled . It seemeth that Minos vnderstood not what he said : For if you shall double the sides , the plaine shall be foure times so great , and the solid eight times . It was demanded therefore of the Geometricians , How the solid giuen , might be made iust as big againe as it was at the first , and yet the forme and nature of the figure to remaine the same still . This proposition is called , The doubling of the Cube . For making the question of the Cube , they laboured to make that double or as big againe . All men therefore doubting and studying a long time how this should be done , at length Hippocrates Chius found that , If vnto two right-lines giuen , whereof the greater should be double vnto the lesser , two meane proportionals might be found , then the Cube might be doubled . But thus this doubt was to be resolued by another thing , as difficult and as hard to be done as that other . Within a while after , they report that the citizens of Delos being commanded by the Oracle to double a certaine altar , and falling into the same doubt and difficultie , were constrained to seeke vnto the Geometricians which were with Plato in the Vniuersitie , that they would at their request , finde out some way how to performe the same . Now these students diligently conferring vpon the matter , and laying their heads together to finde out how , vnto two right-lines giuen , two meane proportionals might be found , it is said that Architas Tarentinus found them by semicylinders , and Eudoxus by those lines that are called crooked lines , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , curuae , &c. Thus farre Eratosthenes . How then to satisfie the demaund of the citizens of Delos , to performe , I say , the command of Apollo and Minos , was demonstrated by Plato , or Hippocrates Chius , to wit , by the finding out of two meane proportionals , betweene two right-lines giuen , whereof the one is double vnto the other . But now againe here ariseth ther question , or as Eratosthenes saith , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , that this doubt of Apollo was to be resolued by another doubt , no lesse difficult then it selfe . For how this was to be done , had not as then by any before that time bene taught . But vpon these occasions giuen ( moued also partly with glorie ) many fine wits in sundrie ages haue found out diuers and sundrie wayes how to performe the same , as Eutocius Askalonita , Archimedes his learned interpreter , hath at large set downe in his commentarie vpon his second booke De cylindro . These authors , as there you shall finde , are thus and in this order named by him : Plato , Heron , Philo Bizantius , Apollonius , Diocles , Pappus , Sporus , Menechmus , Architas Tarentinus , Eratosthenes , Nicomedes , and Eudoxus Cnidius , whose inuention he reiecteth , as not answering to the purpose of his owne proposition . Of all these P. Ramus , at the 8 e xiij of his Geometrie , describeth that of Heron onely , neglecting all the rest . Hinc existit , faith he , Mesographus Heronis Mechanici seu Mesolabus , dictus ab inuentione duarum continue mediarum proportionalium inter duas datas : vnde existit problema Deliacum quod Apollinem ipsum exercuit . Mesographus autem Heronis est infinita regula , quae sistitur cochleato vnco per cauum mobili . Est verò , vt Pappus ait initio libritertij , Mesographus iste architectis aptissimus , multoque promptior Platonis Mesographo . Mesographi mechanica est apud Eutocium secundt de sphaera , sed paulò faciliùs à nobis ita proponctur : Si duas dat as rectas comprehendentes rectangulum , & infinitè continuat as , mesographus tangens oppositum angulum angulo datarum intersecet aequidistantèr à centro , intersegmenta erunt media continuè proportionalia datis : that is , From hence ( he meaneth out of the 7 e of that booke ) ariseth the Mesographus of Heron the inginer , which otherwise is called a Mesolabus : For the vse of it is to finde out two lines continually proportionall betweene two other lines giuen . By this meanes was found how to answer that problem of Delos , which did much trouble Apollo himselfe . Now this Mesographus of Heron , is an infinite tight-line which is fastened with a screw-pin , that is to be slid vp and downe in a riddie . This Mesographus , as Pappus Alexandrinus in the beginning of his third booke writeth , is a maruellous commodious instrument for Architects and Carpenters , and is much more conuenient and readie then the Mesographus of Plato . A mechanicall description and vse of this Mesographus , is set downe by Eutocius in his commentarie vpon the ij booke of Archimedes De sphaera & cylindro , but is somewhat more plainly set out by vs in this maner : 8 If ( a right-angled parallelogramme being made of the two lines giuen , the same also being continued as sarre as need shall require ) the Mesographus touching the angle , that is opposite vnto the angle contained of the lines giuen , shall cut those continued lines equally distant from the center , the portions of those continuations thus cut , shall be the middle lines continually proportionall betweene the two lines giuen . Thus farre Ramus , vnto whom I referre thee for further satisfaction . But whereas P. Ramus , as we haue shewed before , doth affirme that the Mesographus of Heron , as Pappus Alexandrinus writeth in the beginning of his third book , is a maruellous commodious instrument for architects , and is much more readie and conuenient then the Mesographus of Plato ; it is so to be vnderstood , that those words , Multoque promptior Platonis mesographe , be not supposed to be spoken by Pappus : For indeed in that place he doth not once name Platoes mesographus . His words at the 4 p iij , as Commandinus , his interpreter hath expressed them , are these : Exponemus igitur quatuor eius constructiones , vnà cum quadam nostra tractatione ; quarum prima quidem est Eratosthenis , secunda Nicomedis , tertia Heronis maximè ad manuum operationem accommodata , ijs qui architecti esse volunt . Of those many and sundry instruments , that haue bene found out by diuers men for the inuention of two proportionals , &c. we will describe the vse and making of foure , adding thereto a certaine treatise of our owne . The first is Eratosthenes his way , the second is Nicomedes , the third is Herons , most commodious for all handy-crafts , or mechanicks , and such as are desirous to be architects . Here you see is no mention of Platoes inuention . But howsoeuer , that of Platoes , in mine opinion , is more proper to this place and purpose of ours ; because it is to be done with the Carpenters tooles , which are alwayes at his hand : And therefore we wil here also , out of the same Eutocius , describe that more fully . This Mesographus , or instrument deuised by Plato , is described by Eutocius to haue bene a right-angled parallelogramme , consisting of foure straight rulars , so contriued and put together , that they might be put nearer or farther off , as occasion should require . We haue vsed two Carpenters squires , so placing the side of the one squire vpon the side of the other , that the other two sides might continually be parallell , and containe right-angles betweene them . Or if any man so please , one squire may serue the turne with the rule which they vse in measuring . The instrument now according to these directions , is thus to be vsed : If ( the two lines giuen comprchending a right-angle , and from the angle infinitely continued ) the instrument be so applied , that when in each corner the infinite 〈◊〉 do fall , the sides of the same instrument do touch the ends of the lines giuen ; the portions of the continued lines intercepted between the said corner , and the corners of the instrument , shall be the two middle lines continually proportionall between the two lines giuen . Concerning the Diagramme , I must referre thee either to Eutocius vpon Archimedes , or Daniel Barbarus vpon Vitruuius : Let the instrument be f m , the squire : The ruler to be moued vp and downe , let it be n o. The vse according to our rule is thus : Let the two lines , whereof the one is double to the other , be e b , and b g ; the one falling vpon the other perpendicularly , that is , they both containing the right-angle e b g. Again , let e b , be drawne out infinitely toward c : and g b , in like maner toward d. Now apply the instrument to this figure in this maner , that when by mouing n o , the moueable side , vp and downe , the two continued lines falling precisely in the corners m , n ; the two parallels sides m , and n o , may at the same instant touch e and g , the ends of the lines giuen . I say that b d , and b e , the portions of the continued lines , cut or intercepted betweene the corner b , and n , m , the corners of the instrument , are the two middle lines continually proportionall betweene e b , and b g , the two lines giuen . The words of Daniel Barbarus at the 3 chap. of the ix booke , are these : Coniungantur in b , ad rectum angulum duae rectae , inter quas duae comparabiles , ac ratione pendentes medias vis inuenire . Esto b g , maior : e b , minor : vtrag , verò extra angulum b producatur : Maior , ad d : minor , ad c : & ad duos rectos angulos , vnum in c ; alterum in d : in suis respondentibus lineis . Estoque angulus vnus , g c d ; alter c de : Aio inter duas datas c b , & b g , esse duas medias comparabiles inuentas : bd , scilicet , & b c. Quoniam posuimus angulum e d c rectum : &c d aequedistantem ipsi c g ; Ideo sequitur , ex 29 e j , angulum g c d , rectum esse : & aequalem angulo c d e , quem similiter rectum esse posuimus . Sed a b , ex constructione supra c b e , ad rectos cadit , similiter c b perpendicularis est ipsid bg . Ex corollario itaque 8 e vj , b d est comparabilis illi quae cadit inter c b , & b c : Pari quoque ratione b e , est media inter b d , & bg . &c. Vide pag. 276 & 277.

Continua proportio est , in qua idem terminus pro secundo & tertio sumitur . A continuall proportion is that where the same bound , is taken for the second and third : 8 e 2 c ij Salig . A plum-line falling from a right-angle vpon the base , is the meane proportional betweene the portions so made : 1 c 4 c viij R. But e d c , being a right-angle ( ex thesi & fabrica ) and d b falling plum vpon e c , cutteth the portions e b , and b c. Therefore d b is the meane proportionall betweene e b , and b c : that is , as e b , is to db , so is bd , to bc . A plum-line falling , &c. 1 c 4 e viij R. But d c g being a right-angle ; and c b falling vpon d g , doth make the portions d b , and b g. Therefore c b , is the meane proportionall betweene d b , and b g : that is , as d b , is to b e : so is b e , vnto b g. Therefore d b , and b e , are continually proportionall betweene c b , and b g , the two lines giuen . That is , as c b , is to d b ; so is d b , vnto b e : And as b d , is vnto b c ; so is b c , vnto b g. quod erat demonstrandum .

Thus you see , that that which was then so difficult , how easily it is now to be done , and that by many and sundrie maner of waies ; and yet neither of these do satisfie the Geometrician : The reason is , for that , as Pappus Alexandrinus saith , those authors quod natura solidum est , Geometrica ratione innixi , construere non potuerunt . Instrumentis enim tantùm ipsum in operationem manualem , & commodam , aptamque constructionem mirabilitèr traduxerunt . Because , saith he , those authors , although they haue all shewed how to do this wonderfully fitly and well ; yet all of them haue done it but mechanically onely , by the helpe of certaine instruments , none of them yet hauing found how it may be done geometrically .

4 The Solid number of a Cube , is called a Cubicke .

5 A Cubicke number is that which is made by multiplying of any number giuen into his quadrate .

A Quadrate number is a number that is made by the multiplication of a number in it selfe . Therfore a Cube is a number which is made by the multiplication of three equall numbers ; or , a Cubicke number is that which is made by the multiplication of any number by it selfe , and againe , by the multiplication of himselfe into the said product . As for example , 2 by 2 , do make the quadrate 4 : Now 2 by the said 4 , do make 8 , a Cubicke . Of this number , Martianus Capella thus writeth : Octonarius numerus , primus cubus est , & perfectus , Vulcano dicatus : Nam ex primo motu , id est , Diade , quae Iuro est , constat . Nam dias per diadem , facit tetradem : At bis ducta , facit octadem . Perfectus item , quòd à septenario tegitur . Omnis enim Cubus sex superficies habet . Item , Ex imparibus consecutis impletur . Nam primus imparium trias , fecundum pentas , ambo octadem faciunt . Item , Cubum , qui à triade venit , id est , 27 , sequentes impares reddunt , id est , heptas , enneas , & 11 : qui omnes faciunt 27. Item , Tertius Cubus , qui à tetrade venit , id est , 64. Nam quater quaterni sunt sedecim . Hoc quater , 64 fit . Et hic ex imparibus quatuor , qui supcriores sequuntur , id est , 13 , 15 , 17 , 19 , fiunt simul 64. Et sic omnes Cubi per imparium incrementa inueniuntur , sui duntaxat numeri . Sanè hic octonarius Cubus , it a omnium Cuborum primus est , vt monas omnium numerorum . Cubus autem omnis etiam matri Deum tribuitur : Nam ideò Cybele nominatur . Thus farre Capella . This rule is conceiued in few words thus : Digestis à ternario imparibus , si duo priores ; postea tres ; deinde quatuor , &c. coniungantur , Cubos proferunt : that is , If odde numbers digested according to their naturall order , be added , ( an vnitie onely , which is of it selfe a Cube , excepted ) first two ; then three ; then foure , &c. they shall make the Cubes naturally following one another , thus : 1Therfore Therfore looke of how many vnities the radix or side of any Cube doth consist , of so many odde numbers is the same Cube composed and made . Item , ² If the first figure ( I meane that on the right hand ) of a Cube giuen , be an odde number , the number of odde numbers whereof it was made , is odde : If it be an euen number , the number of odde numbers whereof it consisteth , is euen . Item , 3 If a Cube do multiply a Cube , the product shall be a Cube ; and the side of that Cube shall be the product made by the multiplication of the sides of the Cubes giuen . As for example , Let the two Cubes giuen be 8 and 27 ; whose sides let them be 2 and 3 : Now let 8 multiply 27 , and let the product be 216 : I say the side of the Cube 216 , is 6 , which is the product of 2 by 3. Thus haue you an easie way , by the helpe of the xiiij Chapter of the second book of Salignacus his Arithmetick , how to finde out the cube of any place , and the fide thereof , without any farther adoe , onely by the helpe of one multiplication . Now the sides of cubes lesse then 1000 , being single figures , to wit , what cube is made of euery single figure multiplied in it selfe cubically , and what the side of euery such cube is , must first be knowne before we go any farther : That is done by this table :

6 If a right-line be cut into two portions , the Cube of the whole line shal be equall to the Cubes of the segments , and a double solid thrise comprehended of the quadrate of his portion , and the other portion .

The generall inuention of a Cube , both Geometricall and Arithmeticall , was shewed at the former proposition : The speciall or particular inuention of the same is diuers and manifold by numbers continually proportionall and out of the cubes themselues , as Euclid teacheth , of whom thou mayst learne , if thou shalt thinke the fruite of that knowledge may counteruaile thy trauell . Analysis quadrati lateris , The extraction of a square roote , as they call it , hath in Euclids Geometrie a proper element , theoreme , or rule , teaching how to performe it ; but the extraction of a cubicke roote hath not any at all : Notwithstanding according to that of the square by him laid downe , it is not difficult by analogie to make one , whereby the roote also of the Cube may be found . The proposition therefore according to that analogie , teaching this skill , is thus layd downe by Ramus : If a right-line be cut into , &c.

As for example , Let the side or roote 12 , be cut into two portions , 10 and 2 : I say the Cube 1728 , made of 12 , the whole line , is equall to 1000 , and 8 , the cubes made of 10 and 2 , the said portions : and two diuers solids , whereof the first is 600 , comprehended thrise of 100 , the square of 10 , his segment ; and of 2 , the other segment : The second , 120 , which is thrise comprehended of 4 , the square of his segment 2 , and of 10 the other segment . But the frame and making of the whole cube , will make the matter more plaine and easie in one example , to wit , how the outter and meane solids are made . Let therefore a cube be made of 12 , 12 , and 12 , three equall sides : and first let the second side be multiplied by the first , thus :

Let not the products 24 , and 120 , be added together ; but let the other side multiply them seuerally , and then adde the seuerall degrees by themselues , thus : Or thus ,

This is the making the Cube of the whole line : Now the making of the same according to the former diuision of the said line , is thus :

But in this maner which followeth , of our inuention , the particular solids , in this kinde of making the cube of the seuerall segments , do more plainly appeare , then in that practised by Ramus : as these examples do shew .

Now that we may apply this example vnto our rule , adde the solids of the same kinde and qualitie together , that is , 40 , 40 and 40 , together : Item , 200 , 200 and 200 together , and adde the summes of them , 120 , and 600 , vnto 1000 , and 8 , the cubes of the segments , and the sum 1728 , shall be equall vnto 1728 , the cube of the whole line .

If any man shall thinke this rule to be true onely in this case where the line is diuided according to the nature of the number , thus consisting of two digits , as they call them , he is deceiued : For the same effect will fall out , howsoeuer the line shall be deuided . Let therefore the same number first be diuided into two other segments vnequall betweene themselues , to wit , into 8 and 4.

Secondarily let the fame line be cut into two equall segments , to wit , 6 , and 6. Therefore , if the right-line giuen be cut into two equall segments , the seuerall solids of the segments shal be equal vnto the Cubes that are also equall betweene themselues , that is , there shall be eight solids all equall one vnto another : or , that which is all one , the solids are the same with the cubes .

Therefore

7 The side of the first particular Cube , is the one side of the second solid , and the square of the same side is the other side of the first solid , whose other side is the side of the scond Cube ; and the square of the same other side , is the other side of the second solid .

In this equalling then of foure solids with one solid , there is to be obserued a singular kinde of frame and composition : First , that the last cube be made of 2 , the last segment : Then , that the second solid made of 4 , the square of 2 his segment , and of 10 , the other segment , be thrise taken . Againe , that the first solid made of 100 , the square of 10 his owne segment , and of 2 the other segment , be also thrise taken . Lastly , that the cube 1000 , be made of 10 , the greater segment . Out of this frame or making of a Cube , the contrary Analysis or resolution of the same is deriued , out of the mutuall combination of the Cubes with the solids , such as we haue before shewed in the Analysis of a square . For here , although a solid be named onely , yet there are two sides to be considered : because that the one is compound and plaine . Therefore 10 , the side of 1000 , the first cube , is the key for the opening of the two solids and cube following . For it is the one side of the second solid , to wit , thrise comprehended of it and of the square of the second segment . Againe , 100 , the square of the same side 10 , is the one side of 600 , the first solid , to wit , thrise comprehended of this quadrate 100 , and the other segment 2. Item 2 , the other side of the first solid , is the side of the next cube . And lastly , the square of 2 , the same side , is the other side of the second solid . By this meanes then , the great varietie and difficultie of this businesse is vnfolded , like as was done in the square . For when you haue found the sides of the seuerall cubes , then haue you withall also found the side of the whole cube : For although the whole Cube be greater then the cubes of the parts ; yet the whole side is equall to the sides of the seuerall cubes : For we vse the solids that are considered betweene the cubes , onely as a meanes to finde out the side of the following cube . Thus much concerning the true forme of analysing or resoluing of a Cube : But because this may seeme somewhat difficult and hard vnto a learner , out of this proposition we haue framed another , which doth as it were more distinctly expresse and point at euery particular in this practise .

8 If the side of any greater Cube shall be sought , thou shalt from the right-hand toward the left distinguish the number giuen into perfect periods , for so many particular Cubes : Then hauing found the side of the first particular Cube , thou shalt set it downe within the quotient : Againe , hauing squared this side now found , thou shalt treble the product for the base of the first solid ; but for the height or other side of the second solid , thou shalt onely square it . Lastly , the first solid diuided by his base , the quotient shall be the side of the next following Cube .

As for example ; Let the side of the cube 1728 , be sought . I. Here first beginning at the right-hand , I distinguish it into perfect periods , that is , into three degrees thus : 1 , 728. Now because after this distinction it appeareth , that our number giuen is a compound period , composed of two single periods ; therefore by the former rule I say that the whole consisteth of two particular cubes . II. Secondarily , the first particular cube being 1 , I seeke his side amongst the cubes of single figures , at the 5 e , and I finde the side to be 1 : that I set downe therefore in my quotient . Now , as the maner is in diuision , I set downe 1 , the cube of this quotient or side , vnderneath the first figure of the number giuen : Item , subtracting the one out of the other , I cancell all the figures of the number giuen , setting the remaine , or the whole next period , aboue the head right ouer their degrees . III. Thirdly , I square 1 , the side found , and I make 1 , which I treble for the base of the first solid , and I make 3 : by this base , or treble 3 , I diuide 7 , the first solid or first complement ; the quotient 2 , I place in the quotient for the side of the succedent cube : By this quotient 2 , I multiply 3 , the Diuisour , and I make 6 for the first solid : Then by 4 , the square of the same quotient , I multiply 3 , the treble before reserued ; and I make 12 for the second solid : Thirdly , I multiply 2 , the quotient cubically , and I make 8 : Fourthly , placing all these products in their true places , iust one degree behinde another , so that the first solid be placed in the first degree ; the second solid one degree farther toward the right-hand : the cube one degree farther then that , or next of all to the right-hand , in this manner :

Again , they being thus placed , I adde them , and I find the summe 728 : Now this summe I subtract from the remaine 728 : Lastly , because after subtraction there remaineth nothing , I say , that 1728 , the number giuen , is a cubicke , and the side of it is 12. The example of this practise is thus : But here obserue , for that the first solid ( or complement ) is the third degree from the second cube ; and the second solid , the second or next vnto the same ; therefore vnto the first solid I adde two siphers , and vnto the second but one : For these figures do so guide the practitioner , that he cannot easily erre in the addition of them . Or thus , as Salignacus an excellent artist , setteth it downe . Let the side of the cubicke 389017 , be found . Here first , because that 389 is not found amongst the cubes at the 5 e , that is , because that 389 is not a cubick number , therefore I take 343 , the next lesser cubicke before it , whose side 7 I place within the quotient . II. Then taking 343 out of 389 , there do remaine 46 : Therefore here the two complements with the second cube , are 46017. III. This done , I square 7 the side found , and I make it 49. IV. This square I treble , and I make 147 , for the base of the first complement . V. Againe , I treble the side found , and I make 21 for the height of the second complement . VI. Lastly , diuiding 460 , the first complement by 147 his base , I find the quotient 3 , which I set downe in the quotient , for the side of the second cubicke .

For the proofe of this worke or practise , First we multiply the quotient in it selfe cubically : Secondly , we multiply the height of the cube by the base of the first complement : Thirdly , we multiply the base of the same cube , by the height of the second complement : Fourthly , we adde all these products together : Lastly , we subtract the summe found , out of the vpper number . As for example : I cube 3 , or multiply it in it selfe cubically , and I make 27. Now the height of this cube is 3 ; the base is 9 : Therefore I multiply 3 , this height of the cube , by 147 , the base of the first complement , and I make 441. Againe , I multiply 9 , the base of the said cube , by 21 , the height of the second complement , and I make 189. Now I adde all these products , and the summe is 46017 : which summe subtracted out of 46017 , the vpper number , nothing remaineth . Therefore if you shall square 73 , the quotient ; and then shall multiply this square by the same 73 , thou shalt make the cubick 389017. For the cubes of the segments 70 , and 3 , with the two complements now found , are the parts of the cube that is made of the whole 73 ; as may easily be tried by that which we haue taught in the former . Briefly then , and in this order is this whole worke to be done . Let the example be the cube 1728. I. By the 5 e , I finde the side of the cubick 1 , to be also 1 , which I set downe in the quotient . II. This quotient now found , I treble , and I make 3. III. Againe , this treble I multiply by the sayd quotient , and I finde 3 also for the Diuisor . By this Diuisor 3 , I diuide 7 , the first complement or solid : and I finde the quotient 2 for the side . IV. Now by this quotient I multiply the Diuisor 3 , and I make 6 , for the first complement : Therefore to it I adde two cyphers , and I make 600. V. Againe , I square the quotient 2 , that is , I multiply it by it selfe , and I make 4 : This 4 , I multiply by the treble 3 , and I make 12 , for the second complement : Therefore I adde to it one cypher , and I make 120. VI. This done , I multiply the same quotient 2 , in it selfe cubically , and I make 8. VII . Lastly , placing all these orderly one vnder another , I adde them , and I finde the summe 728 , which subtracted from 728 the number remaining , nothing is left : Therefore I say that 1728 is a cubicke .

9 If , hauing found the quotient of two or more figures , as yet the whole side of the Cubicke giuen be not found , then for the finding out of the quotient following , thou shalt square the whole quotient now found for the base of the next complement ; and shalt treble the same square ; and then shalt in all things follow the prescript of the former rule , the whole side shall be found .

Suppose the side of 34 , 012 , 224 , were to be sought : Item of 320 , 013 , 504 : The examples would be thus :

Somtimes after that you haue found the cubicall side , I meane , hauing subtracted the cube of the first period , there will remaine in the next places neither solid nor cube . Therefore in this case vnto the side found , you shall adioyne a cipher , as in this cube 8120601 ( 201.

If the second side of the first complement be greater then the side of the cube following , then the solid diuided did containe a part of the second complement : And therfore that side must be diminished , as in the cube 17 , 576 , the first side shall be 2 ; then if thou shalt make the second side 7 , as it seemeth at the first view it should be , thou shalt make the second side of the first complement greater then the side of the cube following . Therefore that side must be taken lesse , and for 7 I take 6 , and so thou shalt finde the true side of the cubicke giuen to be 26. The example is thus :

This is the generall and common way of finding of the side of any cubick whatsoeuer , though neuer so great . Thus also parts , whose bounds are cubicall numbers , may be resolued into lesser bounds , by finding out of the sides of the same bounds ; As 8/27 by this meanes are resolued into ⅔ , And 1782/2197 , into 12/13.

In parts ( fractions they commonly call them ) there is also a kinde of cube . For if fo be the number giuen be not a cubick , then it hath no side that may be expressed in numbers ; and yet the true side of the greatest cube contained in any number giuen , may be found : As in this number 17 , 616 , which is not a cubicke , the greatest cubicke is 17 , 576 , and the side of it is 26 , and there do remaine , ouer and aboue the cubicke , 40. Therefore , there cannot possibly any side of a number that is not a cubicke , be found so neare , but it is possible to finde one more neare ; as is also before taught of the Quadrate . There are also here two maner of wayes of finding out a cubicall side in such like numbers very neare to the true side , such as were before shewed in the quadrate . The first subducteth the two complements and the last cubick , contained in the next greater cubicke aboue it ; whereby may be vnderstood the difference of two continuall cubickes , like as before the difference of two quadrates was vnderstood . For here there is to be vnderstood a certaine cubicall Gnomon or squire , which you may conceiue to be made of three plaines or sides of the cube , as before you conceiued the squire or gnomon of the quadrate to be made of two sides of the quadrate . So in this example 17 , 616 , where the side is 26 , and there do remaine 40 , of the cube next following , thou shalt diuide 27 , the next greater cubicke , into 26 and 1 , and , as before thou hast learned , thou shalt make two complements and one solid of them . The first complement 676 , made of the square of 26 , the one segment ; and of 1 the other segment ; which being thrise taken , shall be 2028. The second complement is 26 , comprehended of the square of 1 , the one segment ; and of 26 , the other segment : which being thrise taken , do make 78. Now the cube of 1 , is 1. This done , adde all these , and the summe 2107 , shall be the denominator of the fraction sought . The parts therefore to be added vnto 26 , the former side , shall be 40/2107. So that the whole side of 17 , 616 , the number giuen , somewhat neare vnto the true side , is 26 40/2107. Wherby it is to be vnderstood as aboue in the quadrate , that looke how much the numerator 40 , doth want of 2107 , so much the cubicke giuen doth differ from the next cubick aboue it . And therefore if you subtract 40 the numerator , from 2107 the denominator ; and shall adde the remaine 2067 , vnto 17,616 , the cubicke giuen , the whole shall be 19,686 , the cubicke of the side 27.

The second way is by parts of some great denomination , so as it be vnderstood that they be cubicall parts , that their side may be certainly knowne before . As for example , the same number 17,616 , reduced vnto one hundred cubicall parts , that is , vnto 1,000,000 , do make 17,616,000,000 , for the numerator . The parts then are thus ; 17,616,000,000/1,000000 . Now the side of the numerator is 2,601 , for the numerator of the one hundred parts giuen : ( For the former denominator being made by the multiplication of 100 , by it selfe cubically ; the roote or side of it must needs be 100 , for the denominator of the parts sought . ) Therefore the parts desired are 2,601/100 , that is , by reduction 26 1/100 , and besides that , there do remaine 19,712,199 , which cannot adde so much as 1/100 part vnto the side found ; because that the difference of this cubicke from the next greater aboue it , is greater then this remaine : And therefore that remaine is neglected , as not of any moment . Thus farre of the extraction or finding out of a cubicke side or roote .

10 The two complements are the two meanes continually proportionall betweene the two cubes : that is , as the greater cube is vnto the greater complement ; so is the greater complement vnto the lesser : and so is the lesser complement vnto the lesser cube .

As for example , Let the whole cubicke be 1728. Here the two seuerall cubes , let them be 1000 , and 8. The greater complement let it be 200 , and the lesser 40 , as we haue shewed at the 6. p. Here I say 200 and 40 are the meane proportionals betweene the two cubes 1000 and 8 : That is , as 1000 are to 200 , so are 200 to 40 ; and so are 40 to 8. The cause of this is manifest by the 5 e xxij R , which teacheth , that , Solids that are alike , haue a trebled reason of their correspondent sides ; And also they haue two meane proportionals comprehended of the crosse multiplication of the base and height of the extremes : 19 p viij E. But the two complements here are contained of the base and height of the extremes crossely multiplied . Therefore the two complements are the meane proportionals betweene the two cubes , that is , the two extremes .

11 If foure numbers be continually proportionall , the products of the extremes by the each other squares , shall be the cubes of the middle numbers : to wit , the greater of the greater ; and the lesser of the lesser . Iordanus 57 p 6 , of his Arithmeticke .

Let 16 , 24 , 36 , and 54 , be foure numbers continually proportionall : And let the quadrate of 16 , be 256 ; of 54 , be 2916. Againe , let the product of 16 , the lesser extreme , by 2916 , the square of 54 , the greater number , be 46,656 . Item , let the product of 54 the greater extreme , by 256 , the square of 16 the lesser extreame , be 13,824 . Here I say , the side of 46,656 , the greater cubicke , shal be 36 : and the side of 13,824 , the lesser , shall be 24 , as may be proued for experience , by the former . The example is thus : The foure numbers giuen , 16 24 36 54 The squares of the extremes : The product of the greaters square by the lesser extreme , The product of the lessers square by the greater extreame :

Now the side of 46 , 656 , the greater , is 36 ; of 13 , 824 , the lesser , it is 24. But 24 , and 36 , are the middle numbers of the foure number : Therefore , If foure numbers be continually proportional , the products of the extremes by their each other squares , are the cubes of the meane .

This rule then , as you see , is a kinde of Mesolabium , that is , a way to finde out the middle numbers , or meane proportionals : whereby I meane , the extremes being giuen , the meane proportionals are easily found .

Any two numbers are thus in the maner of solids alike betweene themselues , conceiued , as oft as between them two meane continually proportional , are sought . But many times these middle numbers , the cubicall sides of the products of the extreames by the each other quadrates or squares , are surd numbers , as they call them , that is , such as cannot be expressed by arithmeticall numbers , but by such characters as are deuised by the Algebraists .

Many other rules might hither be added , but we especially in this place regard such as may any kinde of way help , either to the vnderstanding of this present argument of extraction of the square and cubicke roote ; or may be of vse for the making of our Mesolabium architectonicum , or Carpenters rular .

12 The product made continuall of three numbers continually proportionall , shall be the cube of the meane or middlemost number . 36 p 11 E.

Let 4 , 6 , 9 , be three numbers continually proportionall : And let them be multiplied continually betweene themselues : 216 , the product so made , shall be the cube of 6 , the middle number . This proposition , saith Schoner , is a kinde of golden rule in solids , by which hauing the one of the extremes of the three proportionals giuen , with the solid made of them all multiplied betweene themselues continually , the rest are also giuen . As for example , Let 216 be the solid giuen , made after this order of three proportionall numbers , and let 4 be the one extreme giuen . Here I say , the middle number by this rule shall be 6. For the Cubicke roote of 216 , is 6. Now the product of 4 by 6 , is 24 ; and the quotient of 216 , by 24 , is 9 , the third or greatest extreame . Or , the product of 6 by it selfe , is 36. Now 36 diuided by 4 , doth yeeld 9 , for the other extreame . For , if three numbers be continually proportionall , the product of the middlemost by it selfe , shall be equall to the product of the two extreames by themselues , by the rule of proportion . Therefore , if one of the extreames giuen shall diuide the product of the middlemost by it selfe , the quotient shall be the other extreame that was desired .

13 Hauing found the one of the meane proportionals , the other also shall be found , if the meane proportionall giuen be multiplied by the extreame that is farthest from it , the roote of the product shall be the meane proportionall desired .

As for example , Let 32 , 16 , 8 , 4 , be foure proportionals giuen ; and suppose that the third bound were not knowne . Here therefore I multiply 16 by 4 , the extreame that is farthest from it , and I make 64 : Now the square side of 64 , by the 5 ej , is 8 , the third proportionall sought . Item , suppose that 16 , the first of the two meane proportionals were vnknowne ; Here 8 shall multiply 32 , the first bound ; not 4 , the last ; because that is farthest off from it : The roote of 256 , the product , by the ej , is 16. This roote is the first meane proportionall of the two , betweene 32 and 4 , the extreames giuen .

CHAP. V. Of the measuring of Timber by the Foote .

BY the Foote we do also measure Timber : but Timber being a solid body of three dimensions , to wit , length , thicknesse and breadth ; by a foote of timber we vnderstand here a cube of 12 inches square , as they call it : ( For here they abuse the word , as also some euen of the learned haue done , as before we haue shewed : ) That is , a foote of timber doth containe 1728 square inches . Here therefore commonly two dimensions are giuen , to wit , breadth and thicknesse : the length is sought . If then the square timber to be measured be 12 inches thicke , and 12 inches broade ; there is no question of the length : For euery 12 inches of that peece shall make a foote of timber . But if the breadth and thicknesse do varie neuer so little from these two cases nominated by the statute , although they varie no whit one from another , that is , although the breadth be equall to the thicknesse , here presently riseth a question , what the length should be ( according to that breadth and thicknesse ) that must make a foote square , or that must be equall to that peece which is 12 in length , 12 in breadth , and 12 in thicknesse . Yet this euery one can doe , that knoweth ought in this businesse : For the Carpenters haue vpon their Rulars , or vpon some peece of paper or parchment , all the square measures set downe , from 1 inch square , vnto 24 , 36 , or 40 inches square . But if the breadth and thicknesse be different one from another whether little or much , this is not onely troublesome and vnreadie , as the former , but very false and erronious , as some of their owne companie haue truly noted . For that of the squares , was made , as seemeth , by some skilfull in the Mathematicks ; but their practise in the latter case , is deceitfull , false , and wholy against the rules of Geometrie , and can no way be iustified Now because that all men that haue occasion to vse this skill of measuring , do not vnderstand how these tables or rules are made : as also for that the same being copied out oft times by vnskilfull men ; I thinke it not amisse here to set downe by the former grounds , as we haue done before for boord-measure and others , the maner of calculating and making of the same : and that especially because that these tables tend directly to the making of our Rular , which is the chiefe cause that first moued me to vndertake this labour , or to write of this argument . The Rule therefore whereby this is performed , is thus : If by the product of the breadth and thicknesse giuen , thou shalt diuide the cube of 12 , ( that is , 1728 ) the quotient shall shew the length required to make a foote of timber . The formes of timber which commonly are to be measured of Carpenters , are long squares ( such as the Geometers call Parallellepipida oblonga ) or Rounds , otherwayes called by them Cylinders , which the vulgar sort of carpenters do also account , as we haue shewed , amongst the number of squares , whose breadth and thicknesse are the same . A foot square ( a cubicall foote I meane ) saith our statute , as we haue before taught , must containe 1728 square inches . If therefore the breadth and thicknesse be greater or lesse then 12 , the length shall be found by the prescript of the former rule . Suppose , for example , that a peece of squared timber were to be measured , whose breadth and thicknesse are equall , yet greater then 12 as namely 16 inches : Here I multiply the breadth by the thicknesse , that is , 16 by 16 ; and I make 256. By this product I diuide 1728 , and I find the quotient 6¾ : Therefore I say , that euery 6 inches , and three quarters of one inch , shall , of that breadth and thicknesse , make a foote of timber , according to the intention of that Statute .

Item , suppose the peece to be measured , whose breadth and thicknesse is equall , were 6 inches square , and it were demanded , how much of that in length would make a foot of timber ? R. The product of 6 by 6 , is 36 : Now the quotient of 1728 by 36 , is 48. Therefore I say , that 48 inches of that sticke are required to make a foote of timber .

Now suppose the breadth and thicknesse were vnequall ; as for example , first let the breadth be 18 , and the thicknesse 12 : here the product of 18 by 12 , is 216 ; and the quotient of 1728 by 216 , is 8 : Therefore I say , euery 8 inches of that peece of timber shall be equall to a foote of solid measure .

Admit a planke or table were to be measured after the maner of timber measure , whose breadth is 36 inches , and thicknesse 4 : how much in length fhall make a foote of solid timber ? R. The product of 36 by 4 , is 144. And the quotient of 1728 , by 144 , is 12 , the desired length . Thus then a Table for solid or timber measure is to be made , like vnto those which we haue before shewed for boord and land-measure . Onely this is to be obserued , that this table is onely of such peeces as are square , that is , of such whose breadth and thicknesse are equal ; although it may also be extended & made for all , if need so require .

If beginning at an vnitie , and so ascending vpward vnto the greatest , thou shalt deuide 1728 by the square of any number asigned , and so forth , the seuerall quotients placed against the number giuen , shall make the table of timber measure . As for example , The square of 1 is 13 and the quotient of 1728 by 1 , is 1728. Item , the square of 2 is 4 ; and the quotient of 1728 by 4 , is 432. Lastly , the square of 12 is 144 , and the quotient of 1728 by 144 , is 12 , for the length desired . The Table then for square measure , that is , for the measuring of such timber where the breadth and thicknesse are equall , is thus .

The vse of this Table is easie out of the former : onely this difference is to be obserued , that because plaines haue but two dimensions , therfore one dimension giuen there was sufficient for the finding out of the other vnknowne . But here , for that solids haue three dimensions , two ( to wit , breadth and thicknesse ) are required for the finding out of the third desired . Yet now againe , seeing that here the said two dimensions giuen , are equall one to another ; if you shall with either of the dimensions giuen , enter the first columne of the Table , the columne on the right-hand shall yeeld the length desired . As for example , Admit a timber sticke to be measured were 4 inches square , that is , were 4 inches thicke , and 4 inches broade ; I demand , how much of that sticke in length , shall be required to make a foote of solid measure . R. Here because 4 is equall vnto 4 , that is , for that the breadth and thicknesse are equall , I enter the columne on the left-hand with 4 , and I finde answering to it , in the columne on the right-hand , 9 ᶠ : Therefore I say , euery 9 foote in length , of that breadth and thicknesse , shall make a foote of solid measure ; The product , I say , made continually of 4 , 4 , and 108 , shall be equall to the product continually made of 12 , 12 , and 12 ; that is , the product continually made of 4 , 4 , and 108 , shall be 1728 , thus :

Item , suppose a timber sticke to be measured were 16 inches square . The columne on the right-hand answering to 16 , giueth 6¾ ⁱ : Therefore I say , euery 6 inches , and 3 quarters of an inch in length , of that breadth and thicknesse , shall make a foote of solid measure .

Againe , admit it were 48 inches square : what length shall be required to make a foot of solid measure : R. Here 48 is greater then any number contained in the columne of breadth , on the left-hand : Therefore I take 24 , the halfe of 48 , and with that entring the Table , I finde 3 j to answer for the length desired . But here obserue , 24 as it is but the halfe of 48 i , the breadth : so is it also but the halfe of the thicknes , which is supposed likewise to be 48i. Therefore 24 is but the one quarter of both the assigned : Wherfore I say that 3 i in length , of that breadth and thicknes , shall containe 4 foote of solid measure .

Lastly , admit a pillar were but halfe an inch square : Here I finde no number amongst the breadths of the left-hand so litle as ½ : Therefore I enter the Table with 1 i , the double of ½ , & I find for the length desired 144f. But here because 1 i , the breadth , is twise so much as ½ ⁱ , the breadth giuen ; and is also twise so great as the thicknesse , which is supposed to be equall to the breadth . Therefore 144 ᶠ , or 1728 i in length , shall containe but ¼ of a foot of solid measure : or euery 576 ᶠ in length , of that breadth and thicknes , shall be equall to the cube of 12i.

Hauing finished the former Table for timber-measure , & knowing it to be seruiceable in those cases onely where the two dimensions giuen are equall one to another , I bethought me of a Table more generall , for the measuring of all sorts of squared timber whatsoeuer . That this might be done , I doubted not ; but that it might be done in few words , or within so little a compasse as here thou seest , I knew not vntill I had made triall : And hauing contriued it , in the maner of Pythagoras tables , into a triangular forme , like vnto that musicall instrument which Swidas , as I remember , calleth Trigonum musicum , I thought good to name it of the forme and vse , Trigonum architectonicum , The Carpenters squire . Now that by this Table all those foule and grosse errors are auoided , which in measuring of timber are by the common sort of workmen committed , that is , that by this our Table the Carpenter may not onely measure all sorts of squared timber most exactly and truly , but also more easily and speedily then by any way commonly practised or published ; besides that , it is demonstrable out of the grounds of these arts , euery man of meane vnderstanding , that shall please to compare them , shall be able to testifie with me .

The making of this Table is in a maner the same with that of the former : the difference onely is this , that there 1728 , the cube of 12 , was diuided onely by the square of euery seuerall number giuen : Here the same cube is to be diuided , not onely by the square , that is , by the product of euery such number by it selfe , but also by the products of euery such seuerall number by any other assigned whatsoeuer . Now it is manifest , that the quotient thus found ( which answereth vnto the third dimension sought ) is to be placed in the common angle or meeting of the columnes of the two dimensions giuen , that is , of those two numbers , by whose product this quotient was found .

First therefore hauing made a right-angled triangle , I place the numbers giuen vpon the sides without the same : These two ranks of numbers answering ( as we haue said ) vnto the two dimensions giuen , the one of them at the top beginneth with the greatest of the numbers giuen , and endeth at the right-corner with an vnitie : The other running along vnderneath the triangle , beginneth at the same right corner with the greatest , and endeth with the least . These numbers also , for the better helpe and guide vnto the eye , may also be placed vpon the hypotenusa , or slanting side of the triangle which is opposite vnto the said right-corner . This being done , I begin either at the greatest or least numbers , it skilleth not whether , and I finde the numbers for the third dimension , as before is taught , which I place within the triangle in their seuerall columnes , as the factors shall appoint . As for example , beginning at the greatest , I multiply 24 by 24 ; and with 576 the product , I diuide 1728 , and I finde 3 , which I place within the triangle in the first columne against 24 , &c. Againe , I multiply the same 24 by 23 , and with the product 552 , I diuide 1728 , and the quotient 3 72/552 ( or 3 3/13 , being reduced vnto parts of the least denomination ) I set in the same first columne against 23. In like maner I multiply 23 by 23 ; and by the product 519 , I deuide 1728 , and I finde 3 ● , which I place in the second columne from the right-hand , against the same 23 on the outside . Item , thus I multiply 22 , first by 24 ; then by 23 ; and lastly by 22 ; by the products still diuiding the same 1728 , I set the quotients in their seuerall places . Thus dealing with all the numbers on the right-hand of the Triangle , vntill thou comest vnto 1 , the lowest on that side , the Table shall be made .

The vse of the Table in Timber-measure .

This Table , as the Title sheweth , serueth generally for the measuring both of Plaines and Solids by the foote . Now Solids , such as Timber and Stone are , haue three dimensions , to wit , breadth , thicknesse and length . Of these , two , breadth and thicknesse , are giuen : The third , I meane length , is sought . Here therefore , Seeke the one of the dimensions giuen amongst those numbers on the outside of the Triangle , which at the top begin at 24 , and so descend to 1 : The other seek amongst those at the base vnderneath the same : The space within the Triangle , to which these two do point , shall shew the length desired . As for example , Suppose a peece of squared timber were 12 inches broade , and 12 inches thicke : In the common angle , or meeting of these two numbers , next to the beuelling or slanting line of the Triangle , I finde 12 for the length desired . Againe , suppose the breadth of some stone to be measured were 18 inches , and the thicknesse 6. These numbers do shew the length desired to be 16 inches : Not 12 , as the vulgar rules do teach .

In Boord-measure .

In boord : measure , onely breadth and length are considered : Therefore here , The breadth giuen , is sought sometimes amongst those numbers on the outside vpon the right-hand : Sometimes amongst those vnderneath the same , as occasion shall serve : For it must so be taken , that it may meet continually with 12. An example or two shall-make all plaine : Suppose a boord were 9 inches broade ; here I seeke 9 amongst those numbers on the right hand : The common angle or meeting of this with 12 , of those numbers at the bottome , doth shew that 16 inches is the length desired . Againe , suppose it were 18 inches broade ; here no number on the right hand , greater then 12 , can meete with 12 , in the vnder-ranke : Therefore I seeke 18 amongst those of the lower ranke ; And I finde the common angle of 18 of this rank , and of 12 , of the numbers amongst those on the right hand , to giue 18 inches for the length desired .

If any man shall desire that the fractions should be made more seruiceable for the vse of practitioners , it may easily be done if any man will take the paines .

FINIS .
TRIGONVM ARCHITECTONICVM : OR , THE CARPENTERS SQVIRE :

that is , A Table seruing for the measuring of Boord , Glasse , Stone , Timber , and such like Plaines and Solids by the Foote .