Via regia ad geometriam. = The vvay to geometry Being necessary and usefull, for astronomers. Geographers. Land-meaters. Sea-men. Engineres. Architecks. Carpenters. Paynters. Carvers, &c. Written in Latine by Peter Ramus, and now translated and much enlarged by the learned Mr. William Bedvvell.

VIA REGIA AD GEOMETRIAM .

THE FIRST BOOKE OF Peter Ramus's Geometry , Which is of a Magnitude .

1. Geometry is the Art of measuring well .

THE end or scope of Geometry is to measure well : Therefore it is defined of the end , as generally all other Arts are . To measure well therefore is to consider the nature and affections of every thing that is to be measured : To compare such like things one with another : And to understand their reason and proportion and similitude . For all that is to measure well , whether it bee that by Congruency and application of some assigned measure : Or by Multiplication of the termes or bounds : Or by Division of the product made by multiplication : Or by any other way whatsoever the affection of the thing to be measured be considered .

But this end of Geometry will appeare much more beautifull and glorious in the use and geometricall workes and practise then by precepts , when thou shalt observe Astronomers , Geographers , Land-meaters , Sea-men , Enginers , Architects , Carpenters , Painters , and Carvers , in the description and measuring of the Starres , Countries , Lands , Engins , Seas , Buildings , Pictures , and Statues or Images to use the helpe of no other art but of Geometry . Wherefore here the name of this art commeth farre short of the thing meant by it . ( For Geometria , made of Gè , which in the Greeke language signifieth the Earth ; and Métron , a measure , importeth no more , but as one would say Land-measuring . And Geometra , is but Agrimensor , A land-meter : or as Tully calleth him Dec●mpedator , a Pole-man : or as Plautus , Finitor , a Marke-man . ) when as this Art teacheth not only how to measure the Land or the Earth , but the Water , and the Aire , yea and the whole World too , and in it all Bodies , Surfaces , Lines , and whatsoever else is to bee measured .

Now a Measure , as Aristotle doth determine it , in every thing to be measured , is some small thing conceived and set out by the measurer ; and of the Geometers it is called Mensura flamosa , a knowne measure . Which kinde of measures , were at first , as Vitruvius and Herodo teache us , taken from mans body : whereupon Protagoras sayd , That man was the measure of all things , which speech of his , Saint Iohn , Apoc. 21. 17. doth seeme to approve . True it is , that beside those , there are some other sorts of measures , especially greater ones , taken from other things , yet all of them generally made and defined by those . And because the stature and bignesse of men is greater in some places , then it is ordinarily in others , therefore the measures taken from them are greater in some countries , then they are in others . Behold here a catalogue , and description of such as are commonly either used amongst us , or some times mentioned in our stories and other bookes translated into our English tongue .

Granum hordei , a Barley corne , like as a wheat corne in weights , is no kinde of measure , but is quiddam minimum in mensura , some least thing in a measure , whereof it is , as it were , made , and whereby it is rectified .

Digitus , a Finger breadth , conteineth 2. barly cornes length , or foure layd side to side :

Pollex , a Thumbe breadth ; called otherwise Vucia , an ynch , 3. barley cornes in length :

Palmus , or Palmus minor , an Handbreadth , 4. fingers , or 3. ynches .

Spithama , or Palmus major , a Span , 3. hands breadth , or 9. ynches .

Cubitus , a Cubit , halfe a yard , from the elbow to the top of the middle finger , 6. hands breadth , or two spannes .

Vlna , from the top of the shoulder or arme-hole , to the top of the middle finger . It is two folde ; A yard and an Elne . A yard , containeth 2. cubites , or 3. foote : An Elne , one yard and a quarter , or 2. cubites and ½ .

Pes , a Foot , 4. hands breadth , or twelve ynches .

Gradus , or Passus minor , a Steppe , two foote and an halfe .

Passus , or Passus major , a Stride , two steppes , or five foote .

Pertica , a Pertch , Pole , Rod or Lugge , 5. yardes and an halfe .

Stadium , a Furlong ; after the Romans , 125. pases : the English , 40. rod.

Milliare , or Milliarium , that is mille passus , 1000. passes , or 8. furlongs .

Leuca , a League , 2. miles : used by the French , spaniards , and seamen .

Parasanga , about 4. miles : a Persian , & common Dutch mile ; 30. furlongs .

Schoenos , 40. furlongs : an Egyptian , or swedland mile .

Now for a confirmation of that which hath beene saide , heare the words of the Statute .

It is ordained , That 3. graines of Barley , dry and round , do make an Ynch : 12. ynches do make a Foote : 3. foote do make a Yard : 5. yardes and ½ doe make a Perch : And 40. perches in length , and 4. in breadth , doe make an Aker : 33. Edwar. 1. De terris mensurandis : & De compositione ulnarum & Perticarum .

Item , Bee it enacted by the authority aforesaid ; That a Mile shall be taken and reckoned in this manner , and no otherwise ; That is to say , a Mile to containe 8. furlongs : And every Furlong to containe 40. lugges or poles : And every Lugge or Pole to containe 16. and●½ ●½ . 25. Eliza. An Act for restraint of new building , &c.

These , as I said , are according to diverse countries , where they are used , much different one from another : which difference , in my judgment ; ariseth especially out of the difference of the Foote , by which generally they are all made , whether they be greater or lesser . For the Hand being as before hath beene taught , the fourth part of the foot whether greater or lesser : And the Ynch , the third part of the hand , whether greater or lesser .

Item , the Yard , containing 3. foote , whether greater or lesser : And the Rodde 5. yardes and ½ , whether greater or lesser , and so forth of the rest ; It must needes follow , that the Foote beeing in some places greater then it is in other some , these measures , the Hand , I meane , the Ynch , the Yard , the Rod , must needes be greater or lesser in some places then they are in other . Of this diversity therefore , and difference of the foot , in forreine countries , as farre as mine intelligence will informe me , because the place doth invite me , I will here adde these few lines following . For of the rest , because they are of more speciall use , I will God willing , as just occasion shall be administred , speake more plentifully hereafter .

Of this argument divers men have written somewhat , more or lesse : But none to my knowledge , more copiously and curiously , then Iames Capell , a Frenchman , and the learned Willebrand , Snellius , of Leiden in Holland , for they have compared , and that very diligently , many and sundry kinds of these measures one with another . The first as you may see in his treatise De mensuris intervallorum describeth these eleven following :

of which the greatest is Pes Babylonius , the Babylonian foote ; the least , Pes Toletanus , the foote used about Teledo in Spaine : And the meane betweene both , Pes Atticus , that used about Athens in Greece . For they are one unto another as 20. 15. and 12. are one unto another . Therefore if the Spanish foote , being the least , be devided into 12. ynches , and every inch againe into 10. partes , and so the whole foote into 120. the Atticke foote shall containe of those parts 150. and the Babylonian , 200. To this Atticke foote , of all other , doth ours come the neerest : For our English foote comprehendeth almost 152. such parts .

The other , to witt the learned Snellius , in his Eratosthenes Batavus , a booke which hee hath written of the true quantity of the compasse of the Earth , describeth many more , and that after a farre more exact and curious manner .

Here observe , that besides those by us here set downe , there are certaine others by him mentioned , which as hee writeth are found wholly to agree with some one or other of these . For Rheinlandicus , that of Rheinland or Leiden , which hee maketh his base , is all one with Romanus , the Italian or Roman foote . Lovaniensis , that of Lovane , with that of Antwerpe : Bremensis , that of Breme in Germany , with that of Hafnia , in Denmarke . Onely his Pes Arabicus , the Arabian foote , or that mentioned in Abulfada , and Nubiensis : the Geographers I have overpassed , because hee dareth not , for certeine , affirme what it was .

Looke of what parts Pes Tolitanus , the spanish foote , or that of Toledo in Spaine , conteineth 120. of such is the Pes. Heidelbergicus , that of Heidelberg , 137. Hetruscus , that of Tuscan , in Italie , 138. Sedanensis , of Sedan in France , 139. Romanus , that of Rome in Italy , 144. Atticus , of Athens in Greece , 150. Anglicus , of England , 152. Parisinus , of Paris in France , 160. Syriacus , of Syria , 166. AEgyptiacus , of Egypt , 171. Hebraicus , that of Iudaea , 180. Babylonius , that of Babylon , 200.

Looke of what parts Pes Romonus , the foote of Rome , ( which is all one with the foote of Rheinland ) is 1000. of such parts is the foote of Toledo , in Spaine , 864. Mechlin , in Brabant , 890. Strausburgh , in Germany , 891. Amsterdam , in Holland , 904. Antwerpe , in Brabant , 909. Bavaria , in Germany , 924. Coppen-haun , in Denmarke , 934. Goes , in Zeland , 954. Middleburge , in Zeland , 960. London , in England , 968. Noremberge , in Germany , 974. Ziriczee , in Zeland , 980. The ancient Greeke , 1042. Dort , in Holland , 1050. Paris , in France , 1055. Briel , in Holland , 1060. Venice , in Italy , 1101. Babylon , in Chaldaea , 1172. Alexandria , in Egypt , 1200. Antioch , in Syria , 1360.

Of all other therefore our English foote commeth neerest unto that used by the Greekes : And the learned Master Ro. Hues , was not much amisse , who in his booke or Treatise De Globis , thus writeth of it Pedem nostrum Angli cum Graecorum pedi aequalem invenimus , comparatione facta cum Graecorum pede , quem Agricola & alijex antiquis monumentis tradi derunt .

Now by any one of these knowne and compared with ours , to all English men well knowne the rest may easily be proportioned out .

2. The thing proposed to bee measured is a Magnitude .

Magnitudo , a Magnitude or Bignesse is the subject about which Geometry is busied . For every Art hath a proper subject about which it doth employ al his rules and precepts : And by this especially they doe differ one from another . So the subject of Grammar was speech ; of Logicke , reason ; of Arithmeticke , numbers ; and so now of Geometry it is a magnitude , all whose kindes , differences and affections , are hereafter to be declared .

3. A Magnitude is a continuall quantity .

A Magnitude is quantitas continua , a continued , or continuall quantity . A number is quantitas discreta , a disjoined quantity : As one , two , three , foure ; doe consist of one , two , three , foure unities , which are disjoyned and severed parts : whereas the parts of a Line , Surface , and Body are contained and continued without any manner of disjunction , separation , or distinction at all , as by and by shall better and more plainely appeare . Therefore a Magnitude is here understood to be that whereby every thing to be measured is said to bee great : As a Line from hence is said to be long , a Surface broade , a Body solid : Wherefore Length , Breadth , and solidity are Magnitudes .

4. That is continuum , continuall , whose parts are contained or held together by some common bound .

This definition of it selfe is somewhat obscure , and to be understand onely in a geometricall sense : And it dependeth especially of the common bounde . For the parts ( which here are so called ) are nothing in the whole , but in a potentia or powre : Neither indeede may the whole magnitude bee conceived , but as it is compact of his parts , which notwithstanding wee may in all places assume or take as conteined and continued with a common bound , which Aristotle nameth a Common limit ; but Euclide a Common section , as in a line , is a Point , in a surface , a Line : in a body , a Surface .

5. A bound is the outmost of a Magnitude .

Terminus , a Terme , or Bound is here understood to bee that which doth either bound , limite , or end actu , in deede ; as in the beginning and end of a magnitude : Or potentia , in powre or ability , as when it is the common bound of the continuall magnitude . Neither is the Bound a parte of the bounded magnitude : For the thing bounding is one thing , and the thing bounded is another : For the Bound is one distance , dimension , or degree , inferiour to the thing bounded : A Point is the bound of a line , and it is lesse then a line by one degree , because it cannot bee divided , which a line may . A Line is the bound of a surface , and it is also lesse then a surface by one distance or dimension , because it is only length , wheras a surface hath both length and breadth . A Surface is the bound of a body , and it is lesse likewise then it is by one dimension , because it is onely length and breadth , whereas as a body hath both length , breadth , and thickenesse .

Now every Magnitude actu , in deede , is terminate , bounded and finite , yet the geometer doth desire some time to have an infinite line granted him , but no otherwise infinite or farther to bee drawane out then may serve his turne .

6. A Magnitude is both infinitely made , and continued , and cut or divided by those things wherewith it is bounded .

A line , a surface , and a body are made gemetrically by the motion of a point , line , and surface : Item , they are conteined , continued , and cut or divided by a point , line , and surface . But a Line is bounded by a point : a surface , by a line : And a Body by a surface , as afterward by their severall kindes shall be understood .

Now that all magnitudes are cut or divided by the same wherewith they are bounded , is conceived out of the definition of Continuum , e. 4. For if the common band to containe and couple together the parts of a Line , surface , & Body , be a Point , Line , and Surface , it must needes bee that a section or division shall be made by those common bandes : And that to bee dissolved which they did containe and knitt together .

7. A point is an undivisible signe in a magnitude .

A Point , as here it is defined , is not naturall and to bee perceived by sense ; Because sense onely perceiveth that which is a body ; And if there be any thing lesse then other to be perceived by sense , that is called a Point . Wherefore a Point is no Magnitude : But it is onely that which in a Magnitude is conceived and imagined to bee undivisible . And although it be voide of all bignesse or Magnitude , yet is it the beginning of all magnitudes , the beginning I meane potentiâ , in powre .

8. Magnitudes commensurable , are those which one and the same measure doth measure : contrariwise , Magnitudes incommensurable are those , which the same measure cannot measure . 1 , 2. d. X.

Magnitudes compared betweene themselves in respect of numbers have Symmetry or commensurability , and Reason or rationality : Of themselves , Congruity and Adscription . But the measure of a magnitude is onely by supposition , and at the discretion of the Geometer , to take as pleaseth him , whether an ynch , an hand breadth , foote , or any other thing whatsoever , for a measure . Therefore two magnitudes , the one a foote long , the other two foote long , are commensurable ; because the magnitude of one foote doth measure them both , the first once , the second twice . But some magnitudes there are which have no common measure , as the Diagony of a quadrate and his side , 116. p. X. actu , in deede , are Asymmetra , incommensurable : And yet they are potentiâ , by power , symmetra , commensurable , to witt by their quadrates : For the quadrate of the diagony is double to the quadrate of the side .

9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given . Contrariwise they are irrationalls . 5. d. X.

Ratio , Reason , Rate , or Rationality , what it is our Authour ( and likewise Salignacus have taught us in the first Chapter of the second booke of their Arithmetickes : Thither therefore I referre thee .

Data mensura , a Measure given or assigned , is of Euclide called Rhetè , that is spoken , ( or which may be uttered ) definite , certaine , to witt which may bee expressed by some number , which is no other then that , which as we said , was called mensura famosa , a knowne or famous measure .

Therefore Irrationall magnitudes , on the contrary , are understood to be such whose reason or rate may not bee expressed by a number or a measure assigned : As the side of the side of a quadrate of 20. foote unto a magnitude of two foote ; of which kinde of magnitudes , thirteene sorts are mentioned in the tenth booke of Euclides Elements : such are the segments of a right line proportionally cutte , unto the whole line . The Diameter in a circle is rationall : But it is irrationall unto the side of an inscribed quinquangle : The Diagony of an ●cosahedron and Dodecahedron is irrationall unto the side .

10. Congruall or agreeable magnitudes are those , whose parts beeing applyed or laid one upon another doe fill an equall place .

Symmetria , Symmetry or Commensurability and Rate were from numbers : The next affections of Magnitudes are altogether geometricall .

Congruentia , Congruency , Agreeablenesse is of two magnitudes , when the first parts of the one doe agree to the first parts of the other , the meane to the meane , the extreames or ends to the ends , and lastly the parts of the one , in all respects to the parts , of the other : so Lines are congruall or agreeable , when the bounding , points of the one , applyed to the bounding points of the other , and the whole lengths to the whole lengthes , doe occupie or fill the same place . So Surfaces doe agree , when the bounding lines , with the bounding lines : And the plots bounded , with the plots bounded doe occupie the same place . Now bodies if they do agree , they do seeme only to agree by their surfaces . And by this kind of congruency do we measure the bodies of all both liquid and dry things , to witt , by filling an equall place . Thus also doe the moniers judge the monies and coines to be equall , by the equall weight of the plates in filling up of an equall place . But here note , that there is nothing that is onely a line , or a surface onely , that is naturall and sensible to the touch , but whatsoever is naturall , and thus to be discerned is corporeall .

Therefore

11. Congruall or agreeable Magnitudes are equall . 8. ax.j.

A lesser right line may agree to a part of a greater , but to so much of it , it is equall , with how much it doth agree : Neither is that axiome reciprocall or to be converted ; For neither in deede are Congruity and Equality reciprocall or convertible . For a Triangle may bee equall to a Parallelogramme , yet it cannot in all points agree to it : And so to a Circle there is sometimes sought an equall quadrate , although in congruall or not agreeing with it : Because those things which are of the like kinde doe onely agree .

12. Magnitudes are described betweene themselves , one with another , when the bounds of the one are bounded within the boundes of the other : That which is within , is called the inscript : and that which is without , the Circumscript .

Now followeth Adscription , whose kindes are Inscription and Circumscription ; That is when one figure is written or made within another : This when it is written or made about another figure .

Homogenea , Homogenealls or figures of the same kinde onely betweene themselves rectitermina , or right bounded , are properly adscribed betweene themselves , and with a round . Notwithstanding , at the 15. booke of Euclides Elements Heterogenea , Heterogenealls or figures of divers kindes are also adscribed , to witt the five ordinate plaine bodies betweene themselves : And a right line is inscribed within a periphery and a triangle .

But the use of adscription of a rectilineall and circle , shall hereafter manifest singular and notable mysteries by the reason and meanes of adscripts ; which adscription shall be the key whereby a way is opened unto that most excellent doctrine taught by the subtenses or inscripts of a circle as Ptolomey speakes , or Sines , as the latter writers call them .

The second Booke of Geometry . Of a Line .

1. A Magnitude is either a Line or a Lineate .

THe Common affections of a magnitude are hitherto declared : The Species or kindes doe follow : for other then this division our authour could not then meete withall .

2. A Line is a Magnitude onely long .

3. The bound of a line is a point .

4. A Line is either Right or Crooked .

This division is taken out of the 4 d j. of Euclide , where rectitude or straightnes is attributed to a line , as if from it both surfaces and bodies were to have it . And even so the rectitude of a solid figure , here-after shall be understood by a right line perpendicular from the toppe unto the center of the base . Wherefore rectitude is propper unto a line . And therefore also obliquity or crookednesse , from whence a surface is judged to be right or oblique , and a body right or oblique .

5. A right line is that which lyeth equally betweene his owne bounds : A crooked line lieth contrariwise . 4. d. j.

Therefore

6. A right line is the shortest betweene the same bounds .

Linea recta , a straight or right line is that , as Plato defineth it , whose middle points do hinder us from seeing both the extremes at once ; As in the eclipse of the Sunne , if a right line should be drawne from the Sunne , by the Moone , unto our eye , the body of the Moone beeing in the midst , would hinder our sight , and would take away the sight of the Sunne from u●● which is taken from the Opticks , in which we are taught , that we see by straight beames or rayes . Therfore to lye equally betweene the boundes , that is by an equall distance : to bee the shortest betweene the same bounds ; And that the middest doth hinder the sight of the extremes , is all one .

7. A crooked line is touch'd of a right or crooked line , when they both doe so meete , that being continued or drawne out farther they doe not cut one another .

Tactus , Touching is propper to a crooked line , compared either with a right line or crooked , as is manifest out of the 2. and 3. d 3. A right line is said to touch a circle , which touching the circle and drawne out farther , doth not cut the circle , 2 d 3. as here a e , the right line toucheth the periphery i o u. And a e. doth touch the helix or spirall . Circles are said to touch one another , when touching they doe not cutte one another , 3. d 3. as here the periphery doth a e j. doth touch the periphery o u y.

Therefore

8. Touching is but in one point onely . è 13. p 3.

This Consectary is immediatly conceived out of the definition ; for otherwise it were a cutting , not touching . So Aristotle in his Mechanickes saith ; That a round is easiliest mou'd and most swift ; Because it is least touch't of the plaine underneath it .

9. A crooked line is either a Periphery or an Helix . This also is such a division , as our Authour could then hitte on .

10. A Periphery is a crooked line , which is equally distant from the middest of the space comprehended .

Therefore

11. A Periphery is made by the turning about of a line , the one end thereof standing still , and the other drawing the line .

Now the line that is turned about , may in a plaine , bee either a right line or a crooked line : In a sphericall it is onely a crooked line ; But in a conicall or Cylindraceall it may bee a right line , as is the side of a Cone and Cylinder . Therefore in the conversion or turning about of a line making a periphery , there is considered onely the distance ; yea two points , one in the center , the other in the toppe , which therefore Aristotle nameth Rotundi principia , the principles or beginnings of a round .

12. An Helix is a crooked line which is unequally distant from the middest of the space , howsoever inclosed .

13. Lines are right one unto another , whereof the one falling upon the other , lyeth equally : Contrariwise they are oblique . è 10. dj .

Hitherto straightnesse and crookednesse have beene the affections of one sole line onely : The affections of two lines compared one with another are Perpendiculum , Perpendicularity and Parallelismus , Parallell equality ; Which affections are common both to right and crooked lines . Perpendicularity is first generally defined thus :

Therefore ,

14. If a right line be perpendicular unto a right line , it is from the same bound , and on the same side , one onely . ê 13. p. xj .

15. Parallell lines they are , which are every where equally distant . è 35. dj .

Parallelismus , Parallell-equality doth now follow : And this also is common to crooked lines and right lines : As heere thou seest in these examples following .

Parallell-equality is derived from perpendicularity , and is of neere affinity to it . Therefore Posidonius did define it by a common perpendicle or plum-line : yea and in deed our definition intimateth asmuch . Parallell-equality of bodies is no where mentioned in Euclides● Elements : and yet they may also bee parallells , and are often used in the Optickes , Mechanickes , Painting and Architecture .

Therefore ,

16. Lines which are parallell to one and the same line , are also parallell one to another .

The third Booke of Geometry . Of an Angle .

1. A lineate is a Magnitude more then long .

A New forme of doctrine hath forced our Authour to use oft times new words , especially in dividing , that the logicall lawes and rules of more perfect division by a dichotomy , that is into two kindes , might bee held and observed . Therefore a Magnitude was divided into two kindes , to witt into a Line and a Lineate : And a Lineate is made the genus of a surface and a Body . Hitherto a Line , which of all bignesses is the first and most simple , hath been described : Now followeth a Lineate , the other kinde of magnitude opposed as you see to a line , followeth next in order . Lineatum therefore a Lineate , or Lineamentum , a Lineament , ( as by the authority of our Authour himselfe , the learned Bernhard Salignacus , who was his Scholler , hath corrected it ) is that Magnitude in which there are lines : Or which is made of lines , or as our Authour here , which is more then long : Therefore lines may be drawne in a surface , which is the proper soile or plots of lines : They may also be drawne in a body , as the Diameter in a Prisma : the axis in a spheare ; and generally all lines falling from aloft : And therfore Proclus maketh some plaine , other solid lines . So Conicall lines , as the Ellipsis , Hyperbole , and Parabole , are called solid lines because they do arise from the cutting of a body

2. To a Lineate belongeth an Angle and a Figure .

The common affections of a Magnitude were to be bounded , cutt , jointly measured , and ascribed : Then of a line to be right , crooked , touch'd , turn'd about , and wreathed : All which are in a lineate by meanes of a line . Now the common affections of a Lineate are to bee Angled and Figured . And surely an Angle and a Figure in all Geometricall businesses doe fill almost both sides of the leafe . And therefore both of them are diligently to be considered .

3. An Angle is a lineate in the common section of the bounds .

So Angulus superficiarius , a superficiall Angle , is a surface consisting in the common section of two lines : So angulus solidus , a solid angle , in the common section of three surfaces at the least .

4. The shankes of an angle are the bounds compreding the angle .

Scèle or Crura , the Shankes , Legges , H. are the bounds insisting or standing upon the base of the angle , which in the Isosceles only or Equic●urall triangle are so named of Euclide , otherwise he nameth them La●era , sides . So in the examples aforesaid , e a. and e i. are the shankes of the superficiary angle e● And so are the three surfaces a e i. i e e o. and a e o. the shankes of the said angle o. Therefore the shankes making the angle are either Lines or Surfaces : And the lineates formed or made into Angles , are either Surfaces or Bodies .

5. Angles homogeneall , are angles of the same kinde , both in respect of their shankes , as also in the maner of meeting of the same : [ Heterogeneall , are those which differ one from another in one , or both these conditions . ]

6. Angels congruall in shankes are equall .

This is drawne out of the 10. e j. For if twice two shanks doe agree , they are not foure , but two shankes , neither are they two equall angles , but one angle . And this is that which Proclus speaketh of , at the 4. p j. when hee saith , that a right lined angle is equall to a right lined angle , when one of the shankes of the one put upon one of the shankes of the other , the other two doe agree : when that other shanke fall without , the angle of the out-falling shanke is the greater : when it falleth within , it is lesser : For there is comprehendeth ; here it is comprehended .

The same Lunular also may bee equall to an obtusangle and Acutangle , as the same argument will demonstrate .

Therefore ,

7. If an angle being equicrurall to an other angle , be also equall to it in base , it is equall : And if an angle having equall shankes with another , bee equall to it in the angle , it is also equall to it in base . è 8. & 4. pj.

For such angles shall be congruall or agreeable in shanks , and also congruall in bases . Angulus isosceles , or Angulus oequicrurus , is a triangle having equall shankes unto another .

8. And if an angle equall in base to another , be also equall to it in shankes , it is equall to it .

And

9. If an angle equicrurall to another angle , be greater then it in base , it is greater : And if it be greater , it is greater in base : è 52 & 24. pj.

And

10. If an angle equall in base , be lesse in the inner shankes , it is greater .

Or as the learned Master T. Hood doth paraphrastically translate it . If being equall in the base , it bee lesser in the feete ( the feete being conteined within the feete of the other angle ) it is the greater angle . [ That is , if one angle enscribed within another angle , be equall in base , the angle of the inscribed shall be greater then the angle of the circumscribed . ]

Therefore ,

11. If unto the shankes of an angle given , homogeneall shankes , from a point assigned , bee made equall upon an equall base , they shall comprehend an angle equall to the angle given . è 23. p j. & 26. p xj .

[ This consectary teacheth how unto a point given , to make an angle equall to an Angle given . To the effecting and doing of each three things are required ; First , that the shankes be homogeneall , that is in each place , either straight or crooked : Secondly , that the shankes bee made equall , that is of like or equall bignesse : Thirdly , that the bases be equall : which three conditions if they doe meete , it must needes be that both the angles shall bee equall : but if one of them be wanting , of necessity againe they must be unequall . ]

This shall hereafter be declared and made plaine by many and sundry practises : and therefore here we bring no example of it .

12. An angle is either right or oblique .

Thus much of the Affections of an angle ; the division into his kindes followeth . An angle is either Right or Oblique : as afore , at the 4 e ij . a line was right or straight , and oblique or crooked .

13. A right angle is an angle whose shankes are right ( that is perpendicular ) one unto another : An Oblique angle is contrary to this .

Therefore ,

14. All straight-shanked right angles are equall .

[ That is , they are alike , and agreeable , or they doe fill the same place ; as here are a i o. and e i o. And yet againe on the contrary : All straight shanked equall angles , are not right-angles . ]

Angles therfore homogeneall and recticrurall , that is whose shankes are right , as are right lines , as plaine surfaces ( For let us so take the word ) are equall right angles . So are the above written rectilineall right angles equall : so are plaine solid right angles , as in a cube , equall . The axiome may therefore generally be spoken of solid angles , so they be recticruralls ; Because all semicircular right angles are not equall to all semicircular right angles : As here , when the diameter is continued it is perpendicular , and maketh twice two angles , within and without , the outter equall betweene themselves , and inner equall betweene themselves : But the outer unequall to the inner : And the angle of a greater semicircle is greater , then the angle of a lesser . Neither is this affection any way reciprocall , That all equall angles should bee right angles . For oblique angles may bee equall betweene themselves : And an oblique angle may bee made equall to a right angle , as a Lunular to a rectilineall right angle , as was manifest , at the 6 e.

The definition of an oblique is understood by the obliquity of the shankes : whereupon also it appeareth ; That an oblique angle is unequall to an homogeneall right angle : Neither indeed may oblique angles be made equall by any lawe or rule : Because obliquity may infinitly bee both increased and diminished .

15. An oblique angle is either Obtuse or Acute .

One difference of Obliquity wee had before at the 9 e ij . in a line , to witt of a periphery and an helix ; Here there is another dichotomy of it into obtuse and acute : which difference is proper to angles , from whence it is translated or conferred upon other things and metaphorically used , as Ingenium-obtusum , acutum ; A dull , and quicke witte , and such like .

16. An obtuse-angle is an oblique angle greater then a right angle . 11. dj .

17. An acutangle is an oblique angle lesser then a right angle . 12. dj .

The fourth Booke , which is of a Figure .

1. A figure is a lineate bounded on all parts .

SO the triangle a e i. is a figure ; Because it is a plaine bounded on all parts with three sides . So a circle is a figure : Because it is a plaine every way bounded with one periphery .

2. The center is the middle point in a figure .

In some part of a figure the Center , Perimeter , Radius , Diameter and Altitude are to be considered . The Center therefore is a point in the midst of the figure ; so in the triangle , quadrate , and circle , the center is , a e i.

Centrum gravitatis , the center of weight , in every plaine magnitude is said to bee that , by the which it is handled or held up parallell to the horizon : Or it is that point whereby the weight being suspended doth rest , when it is caried . Therefore if any plate should in all places be alike heavie , the center of magnitude and weight would be one and the same .

3. The perimeter is the compasse of the figure .

4. The Radius is a right line drawne from the center to the perimeter .

5. The Diameter is a right line inscribed within the figure by his center .

Therefore ,

6. The diameters in the same figure are infinite .

Although of an infinite number of unequall lines that be only the diameter , which passeth by or through the center notwithstanding by the center there may be divers and sundry . In a circle the thing is most apparent : as in the Astrolabe the index may be put up and downe by all the points of the periphery . So in a speare and all rounds the thing is more easie to be conceived , where the diameters are equall : yet notwithstanding in other figures the thing is the same . Because the diameter is a right line inscribed by the center , whether from corner to corner , or side to side , the matter skilleth not . Therefore that there are in the same figure infinite diameters , it issueth out of the difinition of a diameter .

And

7. The center of the figure is in the diameter .

This consectary , saith the learned Rod. Snellius , is as it were a kinde of invention of the center . For where the diameters doe meete and cutt one another , there must the center needes bee . The cause of this is for that in every figure there is but one center only : And all the diameters , as before was said , must needes passe by that center .

And

8. It is in the meeting of the diameters .

9. The Altitude is a perpendicular line falling from the toppe of the figure to the base .

10. An ordinate figure , is a figure whose bounds are equall and angles equall .

In plaines the Equilater triangle is onely an ordinate figure , the rest are all inordinate : In quadrangles , the Quadrate is ordinate , all other of that sort are inordinate : In every sort of Multangles , or many cornered figures one may be an ordinate . In crooked lined figures the Circle is ordinate , because it is conteined with equall bounds , ( one bound alwaies equall to it selfe being taken for infinite many , ) because it is equ●angled , seeing ( although in deede there be in it no angle ) the inclination notwithstanding is every where alike and equall , and as it were the angle of the perphery be alwaies alike unto it selfe : whereupon of Plato and Plutarch a circle is said to be Polygonia , a multangle ; and of Aristotle Holegonia , a totangle , nothing else but one whole angle . In mingled-lined figures there is nothing that is ordinate : In solid bodies , and pyramids the Tetrahedrum is ordinate : Of Prismas , the Cube : of Polyhedrum's , three onely are ordinate , the octahedrum , the Dodecahedrum , and the Icosahedrum . In oblique-lined bodies , the spheare is concluded to be ordinate , by the same argument that a circle was made to bee ordinate .

11. A prime or first figure , is a figure which cannot be divided into any other figures more simple then it selfe .

So in plaines the triangle is a prime figure , because it cannot be divided into any other more simple figure although it may be cut many waies : And in solids , the Pyramis is a first figure : Because it cannot be divided into a more simple solid figure , although it may be divided into an infinite sort of other figures : Of the Triangle all plaines are made ; as of a Pyramis all bodies or solids are compounded● such are a e i. and a e i o.

12. A rationall figure is that which is comprehended of a base and height rationall betweene themselves .

So Euclide , at the 1. d. ij . saith , that a rightangled parallelogramme is comprehended of two right lines perpendicular one to another , videlicet one multiplied by the other . For Geometricall comprehension is sometimes as it were in numbers a multiplication : Therefore if yee shall grant the base and height to bee rationalls betweene themselves , that their reason I meane may be expressed by a number of the assigned measure , then the numbers of their ●ides being multiplyed one by another , the bignesse of the figure shall be expressed . Therefore a Rationall figure is made by the multiplying of two rationall sides betweene themselves .

Therefore ,

13. The number of a rationall figure , is called a Figurate number : And the numbers of which it is made , the Sides of the figurate .

As if a Right angled parallelogramme be comprehended of the base foure , and the height three , the Rationall made shall be 12. which wee here call the figurate : and 4. and 3. of which it was made , we name sides .

14. Isoperimetrall figures , are figures of equall perimeter .

This is nothing else but an interpretation of the Greeke word ; So a triangle of 16. foote about , is a isoperimeter to a triangle 16. foote about , to a quadrate 16. foote about , and to a circle 16. foote about .

15. Of isoperimetralls homogenealls that which is most ordinate , is greatest : Of ordinate isoperimetralls heterogenealls , that is greatest , which hath most bounds .

So an equilater triangle shall bee greater then an isoperimeter inequilater triangle ; and an equicrurall , greater then an unequicrurall : so in quadrangles , the quadrate is greater then that which is not a quadrate : so an oblong more ordinate , is greater then an oblong lesse ordinate . So of those figures which are heterogeneall ordinates , the quadrate is greater then the Triangle : And the Circle , then the Quadrate .

16. If prime figures be of equall height , they are in reason one unto another , as their bases are : And contrariwise .

Therefore ,

17. If prime figures of equall heighth have also equall bases , they are equall .

[ The reason is , because then those two figures compared , have equall sides , which doe make them equall betweene themselves ; For the parts of the one applyed or laid unto the parts of the other , doe fill an equall place , as was taught at the 10. e. j. Sn. ] So Triangles , so Parallelogrammes , and so other figures proposed are equalled upon an equall base .

18. If prime figures be reciprocall in base and height , they are equall : And contrariwise .

19. Like figures are equiangled figures , and proportionall in the shankes of the equall angles .

First like figures are defined , then are they compared one with another , similitude of figures is not onely of prime figures , and of such as are compounded of prime figures , but generally of all other whatsoever . This similitude consisteth in two things , to witt in the equality of their angles , and proportion of their shankes●

Therefore ,

20. Like figures have answerable bounds subtended against their equall angles : and equall if they themselves be equall .

Or thus , They have their termes subtended to the equall angles correspondently proportionall : And equall if the figures themselves be equall ; H. This is a consectary out of the former definition .

And

21. Like figures are situate alike , when the proportionall bounds doe answer one another in like situation .

The second consectary is of situation and place . And this like situation is then said to be when the upper parts of the one figure doe agree with the upper parts of the other , the lower , with the lower , and so the other differences of places . Sn.

And

22. Those figures that are like unto the same , are like betweene themselves .

This third consectary is manifest out of the definition of like figures . For the similitude of two figures doth conclude both the same equality in angles and proportion of sides betweene themselves .

And

23. If unto the parts of a figure given , like parts and alike situate , be placed upon a bound given , a like figure and likely situate unto the figure given , shall bee made accordingly .

This fourth consectary teacheth out of the said definition , the fabricke and manner of making of a figure alike and likely situate unto a figure given . Sn.

24. Like figures have a reason of their homologallor correspondent sides equally manifold unto their dimensions : and a meane proportionall lesse by one .

Thus farre of the first part of this element : The second , that like figurs have a meane , proportional lesse by one , then are their dimensions , shall be declared by few words . For plaines having but two dimensions , have but one meane proportionall , solids having three dimensions , have two meane proportionalls . The ca●se is onely Arithmeticall , as afore . For where the bounds are but 4. as they are in two plaines , there can be found no more but one meane proportionall , as in the former example of 8. and 18. where the homologall or correspondent sides are 2. 3. and 4. 6.

Therefore ,

Againe by the same ru●e , where the bounds are 6. as they are in two solids , there may bee found no more but two meane proportionalls : as in the former solids 30. and 240. where the homologall or correspondent sides are 2. 4. 3. 6. 5. 10.

Therefore ,

Therefore , ) 25. If right lines be continually proportionall , more by one then are the d●mensions of like figures likelily situate unto the first and second , it shall be as the first right line is unto the last , so the first figure shall be unto the second : And contrariwise .

Out of the similitude of figures two consectaries doe arise , in part only , as is their axiome , rationall and expressable by numbers . If three right lines be continually proportionall , it shall be as the first is unto the third : So the rectineall figure made upon the first , shall be unto the rectilineall figure made upon the second , alike and likelily situate . This may in some part be conceived and understood by numbers . As for example , Let the lines given , be 2. foot , 4. foote . and 8 foote . And upon the first and second , let there be made like figures , of 6. foote and 24. foote ; So I meane , that 2. and 4. be the bases of them . Here as 2. the first line , is unto 8. the third line : So is 6. the first figure , unto 24. the second figure , as here thou seest .

Againe , let foure lines continually proportionall , be 1. 2. 4. 8. And let there bee two like solids made upon the first and second : vpon the first , of the sides 1. 3. and 2. lee it be 6. Vpon the second , of the sides 2. 6. and 4. let it be 48. As the first right line 1. is unto the fourth 8. So is the figure 6. unto the second 48. as is manifest by division . The examples are thus .

And

26. If foure right lines bee proportionall betweene themselves : Like figures likelily situate upon them , shall be also proportionall betweene themselves : And contrariwise , out of the 22. pvj. and 37. pxj.

The proportion may also here in part bee expressed by numbers : And yet a continuall is not required , as it was in the former .

In Plaines let the first example be , as followeth .

The cause of proportionall figures , for that twice two figures have the same reason doubled .

In Solids let this bee the second example . And yet here the figures are not proportionall unto the right lines , as before figures of equall heighth were unto their bases● but they themselves are proportionall one to another . And yet are they not proportionall in the same kinde of proportion .

The cause also is here the same , that was before : To witt , because twice two figures have the same reason trebled .

27. Figures filling a place , are those which being any way set about the same point , doe leave no voide roome .

This was the definition of the ancient Geometers , as appeareth out of Simplicius , in his commentaries upon the 8. chapter of Aristotle's iij. booke of Heaven : which kinde of figures Aristotle in the same place deemeth to bee onely ordinate , and yet not all of that kind● But only three among the Plaines , to witt a Triangle , a Quadrate , and a Sexangle : amongst Solids , two ; the Pyramis , and the Cube . But if the filling of a place bee judged by right angles , 4. in a Plaine , and 8. in a Solid , the Oblong of plaines , and the Octahedrum of Solids shall ( as shall appeare in their places ) fill a place ; And yet is not this Geometrie of Aristotle accurate enough . But right angles doe determine this sentence , and so doth Euclide out of the angles demonstrate , That there are onely five ordinate solids ; And so doth Potamon the Geometer , as Simplicus testifieth , demonstrate the question of figu●es filling a place . Lastly , if figures , by laying of their corners together , doe make in a Plaine 4. right angles , or in a Solid 8. they doe fill a place .

Of this probleme the ancient geometers have written , as we heard even now : And of the latter writers , Regiomontanus is said to have written accurately ; And of this argument Maucolycus hath promised a treatise , neither of which as yet it hath beene our good hap to see .

Neither of these are figures of this nature , as in their due places shall be proved and demonstrated .

28. A round figure is that , all whose raies are equall .

Rotundum , a Roundle , let it be here used for Rotunda figura , a round figure . And in deede Thomas Finkius or Finche , as we would call him , a learned Dane , sequestring this argument from the rest of the body of Geometry , hath intituled that his worke De Geometria rotundi , Of the Geometry of the Round or roundle .

29. The diameters of a roundle are cut in two by equall raies .

The reason is , because the halfes of the diameters , are the raies . Or because the diameter is nothing else but a doubled ray : Therefore if thou shalt cut off from the diameter so much , as is the radius or ray , it followeth that so much shall still remaine , as thou hast cutte of , to witt one ray , which is the other halfe of the diameter . Sn.

And here observe , That Bisecare , doth here , and in other places following , signifie to cutte a thing into two equall parts or portions● And so Bisegmentum , to be one such portion● And Bisectio , such a like cutting or division .

30. Rounds of equall diameters are equall . Out of the 1. d. i●● .

Circles and Spheares are equall , which have equall diameters . For the raies , which doe measure the space betweene the Center and Perimeter , are equall , of which , bei●g doubled , the Diameter doth consist . Sn.

The fifth Booke , of Ramus his Geometry , which is of Lines and Angles in a plaine Surface .

1. A lineate is either a Surface or a Body .

LIneatum , ( or Lineamentum ) a magnitude made of lines , as was defined at 1. e. iij. is here divided into two kindes : which is easily conceived out of the said definition there , in which a line is excluded , and a Surface & a body are comprehended . And from hence arose the division of the arte Metriall into Geometry , of a surface , and Stereometry , of a body , after which maner Plato in his vij . booke of his Common-wealth , and Aristotle in the 7. chapter of the first booke of his Posteriorums , doe di●tinguish betweene Geometry and Stereometry : And yet the name of Geometry is used to signifie the whole arte of measuring in generall .

2. A Surface is a lineate only broade . 5. dj .

Epiphania , the Greeke word , which importeth onely the outter appearance of a thing , is here more significant , because of a Magnitude there is nothing visible or to bee seene , but the surface .

3. The bound of a surface is a line . 6. dj .

The matter in Plaines is manifest . For a three cornered surface is bounded with 3. lines : A foure cornered su●face , with foure li●es , and so forth : A Circle is bounded with one line . But in a Sphearicall surface the matter is not so plaine : For it being whole , seemeth not to be bounded with a line . Yet if the manner of making of a Sphearicall surface , by the conversion or turning about of a semiperiphery , the beginning of it , as also the end , shal be a line , to wit a semiperiphery : And as a point doth not only actu , or indeede bound and end a line : But is potentia , or in power , the middest of it : So also a line boundeth a Surface actu , and an innumerable company of lines may be taken or supposed to be throughout the whole surface . A Surface therefore is made by the motion of a line , as a Line was made by the motion of a point .

4. A surface is either Plaine or Bowed .

The difference of a Surface , doth answer to the difference of a Line● in straightnesse and obliquity or crookednesse .

Obliquum , oblique , there signified crooked ; Not righ● or straight : Here , uneven or bowed , either upward or downeward . Sn.

5. A plaine surface is a surface , which lyeth ●qually betweene his bounds , out of the 7. dj .

Planum , a Plaine , is taken and used for a plaine surface : as before Rotundum , a Round , was used for a round figure .

Therefore ,

6. From a point unto a point we may , in a plaine surface , draw a right line . 1 and 2. post . j.

Three things are from the former ground begg'd : The first is of a Right line . A right line and a periphery were in the ij . booke defined : But the fabricke or making of them both , is here said to bee properly in a plaine .

Now the Geometricall instrument for the drawing of a right plaine is called Amussis , & by Petolemey , in the 2. chapter of his first booke of his Musicke , Regula , a Rular , such as heere thou seest .

And from a point unto a point is this justly demanded to be done , not unto points ; For neither doe all points fall in a right line : But many doe fall out to be in a crooked line . And in a Spheare , a Cone & Cylinder● a Ruler may be applyed , but it must be a sphearicall , Conicall , or Cylindraceall . But by the example of a right line doth Vitellio , 2 p j. demaund that betweene two lines a surface may be extended : And so may it seeme in the Elements , of many figures both plaine and solids , by Euclide to be demanded ; That a figure may be described , at the 7. and 8. e ij . Item that a figure may be made vp , at the 8. 14. 16. 23.28 p. vj . which are of Plaines . Item at the 25. 31. 33. 34. 36. 49. p.xj. which are of Solids . Yet notwithstanding a plaine surface , and a plaine body doe measure their rectitude by a right line , so that jus postulandi , this right of begging to have a thing granted may seeme primarily to bee in a right plaine line .

Now the Continuation of a right line is nothing else , but the drawing out farther of a line now drawne , and that from a point unto a point , as we may continue the right line a e. unto i. wherefore the first and second Petitions of Eu●lde do agree in one .

And ] 7. To set at a point assigned a Right line equall to another right line given : And from a greater , to cut off a part equall to a lesser . 2. and 3. pj.

Therefore ,

8. One right line , or two cutting one another , are in the same plaine , out of the 1. and 2. p xj .

One Right line may bee the common section of two plaines : yet all or the whole in the same plaine is one : And all the whole is in the same other : And so the whole is the same plaine . Two Right lines cutting one another , may bee in two plaines cutting one of another ; But then a plain● may be drawne by them : Therefore both of them shall be in the same plaine . And this plaine is geometrically to be conceived : Because the same plaine is not alwaies made the ground whereupon one oblique line , or two cutting one another are drawne , when a periphery is in a sphearicall : Neither may all peripheries cutting one another be possibly in one plaine .

And

9. With a right line given to describe a peripherie .

Talus , the nephew of Daedalus by his sister , is said in the viij . booke of Ovids Metamorphosis , to have beene the inventour of this instrument : For there he thus writeth of him and this matter : — Et ex uno duo ferrea brachia nodo : Iunxit , ut aequali spatio distantibus ipsis : Altera pars staret , pars altera duce●et orbem .

Therfore

10. The rai●s of the same , or of an equall periphery , are equall .

The reason is , because the same right line is every where converted or turned about . But here by the Ray of the ●eriphery , must bee understood the Ray the figure contained within the periphery .

11. If two equall perip●eries , from the ends of equall shankes of an assigned rectilineall angle , doe meete before it , a right line drawne from the meeting of them unto the toppe or point of the angle , shall cut it into two equall parts . 9. pj.

Hitherto we have spoken of plaine lines : Their affection followeth , and first in the Bisection or dividing of an Angle into two equall parts .

12. If two equall peripheries from the ends of a right line given , doe meete on each side of the same , a right line drawne from those meetings , shall divide the right line given into two equall parts . 10. pj.

13. If a right line doe stand perpendicular upon another right line , it maketh on each side right angles : And contrary wise .

The Rular , for the making of straight lines on a plaine , was the first Geometricall instrument : The Compasses , for the describing of a Circle , was the second : The Norma or Square for the true ●recting of a right line in the same plaine upon another right line , and then of a surface and body , upon a surface or body , is the third . The figure therefore is thus .

Therefore

14. If a right line do stand upon a right line , it maketh the angles on each side equall to two right angles : and contrariwise out of the 13. and 14. pj.

And

15. If two right lines doe cut one another , they doe make the angles at the top equall and all equall to foure right angles . 15. pj.

And

16. If two right lines cut with one right line , doe make the inner angles on the same side greater then two right angles , those on the other side against them shall be lesser then two right angles .

17. If from ●●oint assigned of an infinite right line given , two equall parts be on each side cut off : and then from the points of those sections two equall circles doe meete , a right line drawne from their meeting unto the point assigned , shall bee perpendicular unto the line given . 11. pj.

18. If a part of an infin●te right line , bee by a periphery from a point given without , cut off a right line from the said point , cutting in two the said part , shall bee perpendicular upon the line given . 12. pj.

19. If two right lines drawne at l●ngth in the same plaine doe never meete , they are parallell● . è 35. dj .

Therefore

20. If an infinite right line doe cut one of the infinite right parallell lines , it shall also cut the other .

As in the same example u y. cutting a e. it shall also cu● i o. Otherwise , if it should not cut it , it should be parallell unto it , by the 18 e. And that against the grant .

21. If right lines cut with a right line be pararellells , they doe make the inner angles on the same side equall to two right angles : And also the alterne angles equall betweene themselves : And the outter , to the inner opposite to it : And contrariwise , 29,28,27 . p 1.

The cause of this threefold propriety is from the perpendicular or plumb-line , which falling upon the parallells breedeth and discovereth all this variety : As here they are right angles which are the inner on the same part or side : Item , the alterne angles : Item the inner and the outter : And therefore they are equall , both , I meane , the two inner to two right angles : and the alterne angles between themselvs : And the outter to the inner opposite to it .

If so be that the cutting line be oblique , that is , fall not upon them plumbe or perpendicularly , the same shall on the contrary befall the parallels . For by that same obliqua●ion or slanting , the right lines remaining and the angles unaltered , in like manner both one of the inner , to wit , e u y , is made obtuse , the other , to wi● , u y o , is made acute : And the alterne angles are made acute and obtuse : As also the outter and inner opposite are likewise made acute and obtuse .

The same impossibility shall be concluded , if they shall be sayd , to be lesser than two right angles●

The second and third parts may be concluded out of the first . The second is thus : Twise two angles are equall to two right angles o y u , and e u y , by the former part : Item , a u y , and e u y , by the 14 e. Therefore they are equall betweene themselves . Now from the equall , Take away e u y , the common angle , And the remainders , the alterne angles , at u , and y shall be least equall .

The third is thus ; The angles e u y , and o y s , are equall to the same u y i , by the second propriety , and by the 15 e. Therefore they are equall betweene themselves .

If they be oblique angles , as here , the lines one slanting or liquely crossing one another , the angles on one side will grow lesse , on the other side greater . Therefore they would not be equall to two right angles , against the graunt .

From hence the second and third parts may be concluded . The second is thus : The alterne angles at u and y , are equall to the foresayd inner angles , by the 14 e : Because both of them are equall to the two right angles : And so by the first part the second is concluded .

The third is therefore by the second demonstrated , because the outter o y s , is equall to the verticall or opposite angle at the top , by the 15 e. Therefore seeing the outter and inner opposite are equall , the alterne also are equall .

Wherefore as Parallelismus , parallell-equality argueth a three-fold equality of angels : So the threefold equality of angles doth argue the same parallel-equality .

Therefore ,

22. If right lines knit together with a right line , doe make the inner angles on the same side lesser than two right Angles , they being on that side drawne out at length , will meete .

And

23. A right line knitting together parallell right lines , is in the same plaine with them . 7 p xj .

And

24. If a right line from a point given doe with a right line given make an angle , the other shanke of the angle equalled and alterne to the angle made , shall be parallell unto the assigned right line . 31 pj.

An angle , I confesse , may bee made equall by the first propriety : And so indeed commonly the Architects and Carpenters doe make it , by erecting of a perpendicular . It may also againe in like manner be made by the outter angle : Any man may at his pleasure use which hee shall thinke good ; But that here taught we take to be the best .

And

25. The angles of shanks alternly parallell , are equall . Or Thus , The angles whose altenate feete are parallells , are equall . H.

And

26 If parallels doe bound parallels , the opposite lines are equall è 34 p.j. Or thus : If parallels doe inclose parallels , the opposite parallels are equall . H.

And

27. If right lines doe joyntly bound on the same side equall and parallell lines , they are also equall and parallell .

On the same part or side it is sayd , least any man might understand right lines knit together by opposite bounds as here .

28. If right lines be cut joyntly by many parallell right lines , the segments betweene those lines shall bee proportionall one to another , out of the 2 p vj and 17 p x j.

Thus much of the Perpendicle , and parallell equality of plaine right lines : Their Proportion is the last thing to be considered of them .

If the lines cut be not parallels , but doe leane one toward another , the portions cut or intercepted betweene them will not be equall , yet shall they be proportionall one to another . And looke how much greater the line thus cut is : so much greater shall the intersegments or portions intercepted be . And contrariwise , Looke how much lesse : so much lesser shall they be .

The third parallell in the toppe is not expressed , yet must it be understood .

This element is very fruitfull : For from hence doe arise and issue , First the manner of cutting a line according to any rate or proportion assigned : And then the invention or way to finde out both the third and fourth proportionalls .

29. If a right line making an angle with another right line , be cut according to any reason [ or proportion ] assigned , parallels drawne from the ends of the segments , unto the end of the sayd right line given and unto some contingent point in the same , shall cut the line given according to the reason given .

Schoner hath altered this Consectary , and delivereth it thus : If a right making an angle with a right line given , and 〈◊〉 it unto it with a base , be cut according to any rate assigned , a parallell to the base from the ends of the segments , shall cut the line given according to the rate assigned . 9 and 10 p v j.

Punctum contingens , A contingent point , that is falling or lighting in some place at al adventurs , not given or assigned

This is a marvelous generall consectary , serving indifferently for any manner of section of a right line , whether it be to be cut into two parts , or three parts , or into as many patts , as you shall thinke good , or generally after what manner of way soever thou shalt command or desire a line to be cut or divided .

Now 〈◊〉 be cut into three parts● 〈◊〉 which the first let it bee the halfe of the second : And the second , the halfe of the third : And the conter minall or right line making an angle with the sayd assigned line , let it be cut one part a o : Then double this in o u : Lastly let u i be taken double to o u , and let the whole diagramme be made up with three parallels y● and os , The fourth parallell in the toppe , as a fore-sayd , shall be understood . Therefore that section which was made in the conterminall line , by the 28 e , shall be in the assigned line : Because the segments or portions intercepted are betweene the parallels .

And

30. If two right lines given , making an angle , be continued , the first equally to the second , the second infinitly , parallels drawne from the ends of the first continuation , unto the beginning of the second , and some contingent point in the same , shall intercept betweene them the third proportionall . 11. p v j.

And

31. If of three right lines given , the first and the third making an angle be continued , the first equally to the second , and the third infinitly ; parallels drawne from the ends of the first continuation , unto the beginning of the second , and some contingent point , the same shall intercept betweene them the fourth proportionall . 12. p vj.

Let the lines given be these : The first a e , the second e i , the third a o , and let the whole diagramme be made up according to the prescript of the consectary . Here by 28. e , as a e , is to e i so is a o , to o u. Thus farre Ramus .

Lazarus Schonerus , who , about some 25. yeares since , did revise and augment this worke of our Authour , hath not onely altered the forme of these two next precedent consectaries : but he hath also changed their order , and that which is here the second , is in his edition the third : and the third here , is in him the second . And to the former declaration of them , hee addeth these words : From hence , having three lines given , is the invention of the fourth proportionall ; and out of that , having two lines given , ariseth the invention of the third proportionall .

2 Having three right lines given , if the first and the third making an angle , and knit together with a base , be continued , the first equally to the second ; the third infinitly ; a parallel from the end of the second , unto the continuation of the third , shall intercept the fourth proportionall . 12. pvj.

The Diagramme , and demonstration is the same with our 31. e or 3 c of Ramus .

3 If two right lines given making an angle , and knit together with a base , be continued , the first equally to the second , the second infinitly ; a parallell to the base from the end of the first continuation unto the second , shall intercept the third proportionall . 11. p v j.

The Diagramme here also , and demonstration is in all respects the same with our 30 e , or 2 c of Ramus .

Thus farre Ramus : And here by the judgement of the learned Finkius , two elements of Ptolomey are to be adjoyned .

32 If two right lines cutting one another , be againe cut with many parallels , the parallels are proportionall unto their next segments .

The same demonstation shall serve , if the lines do crosse one another , or doe vertically cut one another , as in the same diagramme appeareth . For if the assigned a i , and u s , doe cut one another vertically in o , let them be cut with the parallels a u , and s i : the precedent fabricke or figure being made up , it shall be by 28. e. as a u , is unto a o , the segment next unto it : so a y , that is , i s , shall be unto o i , his next segment .

The 28. e teacheth how to finde out the third and fourth proportionall : This affordeth us a meanes how to find out the continually meane proportionall single or double .

Thefore

33. If two right lines given be continued into one , a perpendicular from the point of continuation unto the angle of the squire , including the continued line with the continuation , is the meane proportionall betweene the two right lines given .

A squire ( Norma , Gnomon , or Canon ) is an instrument consisting of two shankes , including a right angle . Of this we heard before at the 13 . e . By the meanes of this a meane proportionall unto two lines given is easily found : whereupon it may also be called a Mesolabium , or Mesographus simplex , or single meane finder .

And

34 If two assigned right lines joyned together by their ends right anglewise , be continued vertically ; a square falling with one of his shankes , and another to it parallell and moveable upon the ends of the assigned , with the angles upon the continued lines , shall cut betweene them from the continued two meanes continually proportionall to the assigned .

The former consectary was of a single mesolabium ; this is of a double , whose use in making of solids , to this or that bignesse desired is notable .

And thus wee have the composition and use , both of the single and double Mesolabium .

35. If of foure right lines , two doe make an angle , the other reflected or turned backe upon themselves , from the ends of these , doe cut the former ; the reason of the one unto his owne ●egment , or of the segments betweene themselves , is made of the reason of the so joyntly bounded , that the first of the makers be joyntly bounded with the beginning of the antecedent made ; the second of this consequent joyntly bounded with the end ; doe end in the end of the consequent made .

Let therefore the two right lines be ● e , and a i : and from the ends of these other two reflected , be i u , and e o , cutting themselves in y ; and the two former in u , and o. The reason of the particular right lines made shall be as the draught following doth manifest . In which the antecedents of the makers are in the upper place : the consequents are set under neathe their owne antecedents .

The businesse is the same in the two other , whether you doe crosse the bounds or invert them .

Here for demonstrations sake we crave no more , but that from the beginning of an antecedent made a parallell be drawne to the second consequent of the makers , unto one of the assigned infinitely continued : then the multiplied proportions shall be ,

The Antecedent , the Consequent ; the Antecedent , the Consequent of the second of the makers ; every way the reason or rate is of Equallity .

The Antecedent the Consequent of the first of the makers ; the Parallel ; the Antecedent of the second of the makers , by the 32. e. Therefore by multiplication of proportions , the reason of the Parallell , unto the Consequent of the second of the makers , that is , by the fabricke or construction , and the 32. e. the reason of the Antecedent of the Product , unto the Consequent , is made of the reason , &c. after the manner above written .

Againe , I say , that the reason of e y , unto y o , is compounded of the reason of e u , unto u a , and of a i , unto i ● .

Theon here draweth a parallell from o , unto u i. By the generall fabricke it may be drawne out of e , unto o i.

Therefore the reason of e n , unto i o , that is of e y , unto y o , shall be made of the foresaid reasons .

Of the segments of divers right lines● the Arabians have much under the name of The rule of sixe quantities . And the Theoremes of Althin●us , concerning this matter , are in many mens hands . And Regiomontanus in his Algorithmus : and Maurolycus upon the 1 piij. of Menelaus , doe make mention of them ; but they containe nothing , which may not ; by any man skilfull in Arithmeticke , be performed by the multiplication of proportions . For all those wayes of theirs are no more but speciall examples of that kinde of multiplication .

Of Geometry , the sixt Booke , of a Triangle .

1 Like plaines have a double reason of their hom●logall sides , and one proportionall meane , out of 20 p vj. and xj . and 18. p viij .

OR thus ; Like plaines have the proportion of their corespondent proportionall sides doubled , & one meane proportionall : Hitherto wee have spoken of plaine lines and their affections : Plaine figures and their kindes doe follow in the next place . And first , there is premised a common corollary drawne out of the 24. e , iiij . because in plaines there are but two dimensions .

2 A plaine surface is either rectilineall or obliquelineall , [ or rightlined , or crookedlined . H. ]

Straightnesse , and crookednesse , was the difference of lines at the 4. e , i j. From thence is it here repeated and attributed to a surface , which is geometrically made of lines . That made of right lines , is rectileniall : that which is made of crooked lines , is Obliquilineall .

3. A rectilineall surface , is that which is comprehended of right lines .

4 A rightilineall doth make all his angles equall to right angles ; the inner ones generally to paires from two forward : the outter alwayes to foure .

Or thus : A right lined plaine maketh his angles equall unto right angles : Namely the inward angles generally , are equall unto the even numbers from two forward , but the outward angles are equall but to 4. right angles . H.

5 A rectilineall is either a Triangle or a Triangulate .

As before of a line was made a lineate : so here in like manner of a triangle is made a triangulate .

6 A triangle is a rectilineall figure comprehended of three rightlines . 21. dj .

Therefore

7 A triangle is the prime figure of rectilineals .

A triangle or threesided figure is the prime or most simple figure of all rectilineals . For amongst rectilineall figures there is none of two sides : For two right lines cannot inclose a figure . What is meant by a prime figure , was taught at the 7. e. iiij .

And

8 If an infinite right line doe cut the angle of a triangle , it doth also cut the base of the same : Vitell. 29. t j.

9 Any two sides of a triangle are greater than the other .

Let the triangle be a e i ; I say , the side a i , is shorter , than the two sides a e , and e i , because by the 6. e ij , a right line is betweene the same bounds the shortest .

Therefore

10 If of three right lines given , any two of them be greater than the other , and peripheries described upon the ends of the one , at the distances of the other two , shall meete , the rayes from that meeting unto the said ends , shall make a triangle of the lines given .

And

11 If two equall peripheries , from the ends of a right line given , and at his distance , doe meete , li●es drawne from the meeting , unto the said ends , shall make an equilater triangle upon the line given . 1 p.j.

12 If a right line in a triangle be parallell to the base , it doth cut the shankes proportionally : And contrariwise . 2 p v j.

As here in the triangle a e i , let o u , be parallell to the base ; and let a third parallel be understood to be in the toppe a ; therefore , by the 28. e.v. the intersegments are proportionall .

The converse is forced out of the antecedent : because otherwise the whole should be lesse than the part . For if o u , be not parallell to the base ei , then yu , is : Here by the grant , and by the antecedent , seeing a o , o e , a y , y e , are proportionall : and the first a o , is lesser than a y , the third : o e , the second must be lesser than y e , the fourth , that is the whole then the part .

13 The three angles of a triangle , are equall to two right angles . 32. p j.

Therefore

14. Any two angles of a triangle are lesse than two right angles .

For if three angles be equall to two right angles , then are two lesser than two right angles .

And

15 The one side of any triangle being continued or drawne out , the outter angle shall be equall to the two inner opposite angles .

Therefore

16 The said outter angle is greater than either of the inner opposite angles . 16. p j.

This is a consectary following necessarily upon the next former consectary .

17 If a triangle be equicrurall , the angles at the base are equall : and contrariwise , 5. and 6. p.j.

Therefore

18 If the equall shankes of a triangle be continued or drawne out , the angles under the base shall be equall betweene themselves .

And

19 If a triangle be an equilater , it is also an equiangle : And contrariwise .

And

20 The angle of an equilater triangle doth countervaile two third parts of a right angle . Regio . 23. p j.

For seeing that 3. angles are equall to 2. 1. must needs be equall to ⅔

And

21 Sixe equilater triangles doe fill a place .

22 The greatest side of a triangle subtendeth the greatest angle ; and the greatest angle is subtended of the greatest side . 19. and 18. p j.

The converse is manifest by the same figure : As let the angle a e i , be greater than the angle a i e. Therefore by the same , 9 e iij. it is greater in base . For what is there spoken of angles in generall , are here assumed specially of the angles in a triangle .

23 If a right line in a triangle , doe cut the angle in two equall parts , it shall cut the base according to the reason of the shankes ; and contrariwise . 3. p v j.

The mingled proportion of the sides and angles doth now remaine to be handled in the last place .

The Converse likewise is demonstrated in the same figure . For as e a , is to a i ; so is e o , to o i : And so is e a , to a u , by the 12 e : therefore a i , and a u , are equall , Item the angles e a o , and o a i , are equall to the angles at u , and i , by the 21. e v● which are equall betweene themselves by the 17. e.

Of Geometry , the seventh Booke , Of the comparison of Triangles .

1 Equilater triangles are equiangles . 8. p.j.

Thus forre of the Geometry , or affections and reason of one triangle ; the comparison of two triangles one with another doth follow . And first of their rate or reason , out of their sides and angles : Whereupon triangles betweene themselves are said to be equilaters and equiangles . First out of the equality of the sides , is drawne also the equalitie of the angles .

Triangles therefore are here jointly called equilaters , whose sides are severally equall , the first to the first , the second , to the second , the third to the third ; although every severall triangle be inequilaterall . Therefore the equality of the sides doth argue the equality of the angles , by the 7. e iij. As here .

2 If two triangles be equall in angles , either the two equicrurals , or two of equall either shanke , or base of two angles , they are equilaters , 4. and 26. p j.

Oh thus ; If two triangles be equall in their angles , either in two angles contained under equall feet , or in two angles , whose side or base of both is equall , those angles are equilater . H.

This element hath three parts , or it doth conclude two triangles to be equilaters three wayes . 1. The first part is apparent thus : Let the two triangles be a e i , and o u y ; because the equall angles at a , and o , are equicrurall , therefore they are equall in base , by the 7. e iij.

3 The third part is thus forced : In the triangles a e i , and o u y , let the angles at e , and i , and u , and y , be equall , as afore : And a e. the base of the angle at i , be equall to o u , the base of angle at y : I say that the two triangles given are equilaters . For if the side e i , be greater than the side u y , let e s , be cut off equall to it , and draw the right line a s. Therefore by the antecedent , the two triangles , a e s , and o u y , equall in the angle of their equall shankes are equiangle ; And the angle a s e , is equall to the angle o y u , which is equall by the grant unto the angle a i e. Therefore a s e , is equall to a i e , the outter to the inner , contrary to the 15. e v. j. Therefore the base e i , is not unequall to the base u y , but equall . And therefore as above was said , the two triangles a e i , and o u y , equall in the angle of their equall shankes , are equilaters .

3. Triangles are equall in their three angles .

And yet notwithstanding it is not therefore to be thought to be equiangle to it : For Triangles are then equiangles , when the severall angles of the one , are equall to the severall angles of the other : Not when all joyntly are equall to all .

Therefore

4. If two angles of two triangles given be equall , the other also are equall .

All the three angles , are equall betweene themselves● by the 3 e. Therefore if from equall you take away equall , those which shall remaine shall be equall .

5. If a right triangle equicrurall to a triangle be greater in base , it is greater in angle : And contrariwise . 25. and 24. pj.

6. If a triangle placed upon the same base , with another triangle , be lesser in the inner shankes , it is greater in the angle of the shankes .

This is a consectary drawne also out of the 10 e iij. As here in the triangle a e i , and a o i , within it and upon the same base . Or thus : If a triangle placed upon the same ba●e with another triangle , be lesse then the other triangle , in regard of his feet , ( those feete being conteined within the feete of the other triangle ) in regard of the angle conteined under those feete , it is greater : H.

7. Triangles of equall heighth , are one to another as their bases are one to another .

Thus farre of the Reason or rate of triangles : The proportion of triangles doth follow ; And first of a right line with the bases . It is a consectary out of the 16 e iiij .

Therefore

8. Vpon an equall base , they are equall .

9. If a right line drawne from the toppe of a triangle , doe cut the base into two equall parts , it doth also cut the triangle into two equall parts : and it is the diameter of the triangle .

10. If a right line be drawne from the toppe of a triangle , unto a point given in the base ( so it be not in the middest of it ) and a parallell be drawne from the middest of the base unto the side , a right line drawne from the toppe of the sayd parallell unto the sayd point , shall cut the triangle into two equall parts .

11 If equiangled triangles be reciprocall in the shankes of the equall angle , they are equall : And contrariwise . 15. p. vj. Or thus , as the learned M● . Brigges hath conceived it : If two triangles , having one angle , are reciprocall , &c.

The converse , is concluded by the same sorites , but by saying all backward . For u a unto a e is , as u a o is unto o a e , by the 7 e : And as e a i , by the grant : Because they are equall : And as i a is unto a o , by the same . Wherefore u a is unto a e , as i a is unto a o.

12 If two triangles be equiangles , they are proportionall in Shankes : And contrariwise : 4 and 5. p. vj.

Therefore ,

13. If a right line in a triangle be parallell to the base , it doth cut off from it a triangle equiangle to the ●hole● but lesse in base .

14. If two trangles be proportionall in the shankes of the equall angle , they are equiangles : 6 p vj.

15 If triangles proportionall in shankes , and al●ernly parallell , doe make an angle betweene them , their bases are but one right line continued . 32 p. vj.

Or thus : If being proportionall in their feet , and alternately parallels , they make an angle in the midst betweene them , they have their bases continued in a right line : H.

The cause is out of the 14 e v. For they shall make on each side , with the falling line a i , two angles equall to two right angles .

16 If two triangles have one angle equall , another proportionall in shankes , the third homogeneall , they are equiangles . 7. p. v. j.

Of Geometry the eight Booke , of the diverse kindes of Triangles .

1 A triangle is either right angled , or obliquangled .

The division of a triangle , taken from the angles , out of their common differences , I meane , doth now follow . But here first a speciall division , and that of great moment , as hereafter shall be in quadrangles and prismes .

2 A right angled triangle is that which hath one right angle : An obliquangled is that which hath none . 27. d j.

A right angled triangle in Geometry is of speciall use and force ; and of the best Mathematicians it is called Magister matheseos , the master of the Mathematickes .

Therefore

3 If two perpendicular lines be knit together , they shall make a right angled triangle .

4 If the angle of a triangle at the base , be a right angle , a perpendicular from the toppe shall be the other shanke● [ and contrariwise Schon . ]

As is manifest in the same example .

5 If a right angled triangle be equicrurall , each of the angles at the base is the hal●e of a right angle : And contrariwise .

Therefore

6 If one angle of a triangle be equall to the other two , it is a right angle [ And contrariwise Schon . ]

Because it is equall to the halfe of two right angles , by the 13. e v.j.

And

7 If a right line from the toppe of a triangle cutting the base into ●wo equall parts be equall to the bisegment , or halfe of the base , the angle at the toppe is a right angle : [ And contrariwise Schon . ]

8 A perpendicular in a triangle from the right angle to the base , doth cut it into two triangles , like unto the whole and betweene themselves , 8. p v. j. [ And contrariwise Schon . ]

Therefore

9 The perpendicular is the meane proportionall betweene the segments or portions of the base .

As in the said example , as i o , is to o a : so is o a , to o e , because the shankes of equall angles are proportionall , by the 8 . e . From hence was Platoes Mesographus invented .

And

10 Either of the shankes is proportionall betweene the base , and the segment of the base next adjoyning .

For as e i , is unto i a , in the whole triangle , so is a i , to i o , in the greater . For so they are homologall sides , which doe subtend equall angles , by the 23 . e , iiij . Item , as i e , is to e a ; in the whole triangle , so is a e , to e o , in the lesser triangle .

Either of the shankes is proportionall betweene the summe , and the difference of the base and the other shanke . And contrariwise . If one side be proportionall betweene the summe and the difference of the others , the triangle given is a rectangle . M. H. Brigges .

This is a consectary arising likewise out of the 4 e. of very great use .

In the triangle e a d , the shanke a d , 12. is the meane proportionall betweene b d , 18. ( the summe of the base a e , 13. and the shanke e d , 5. ) and 8. the difference of the said base and shanke : For if thou shalt draw the right lines b a , and a c , the angle b ac , shall be by the 6 . e , a rectangle ; ( because it is equall to the angles at b , and c , seeing that the triangles b e a , and e a c , are equicrurall . ) And by the 9 e , b d , d a , and d e , are continually proportionall .

If a quadrate of a number , given for the first shanke , be divided of another , the halfe of the difference of the divisour , and quotient shall be the other shanke , and the halfe of the summe shall be the base . Or thus , The side of divided number doubled , and the difference of the divisour and quotient , shall be the two shankes , and the summe of them shall be the base .

Let the number given for the first shanke be 4. And let 8. divide 16. the quadrate of 4. by 2. The halfe of 8 — 2 , that is 3. shall be the other shanke : And the halfe of 8 — 2 , that is 5. shall be the base .

Therefore

If any one number shall divide the quadrate of another , the side of the divided , and the halfe of the difference of the divisour and the quotient , shall be the two s●ankes of a rectangled triangle , and the halfe of the summe of them shall be the base thereof .

Let the two numbers given be 4. and 6. The square of 6. let it be 36. and the quotient of 36. by 4. be 9 : And the side it 6. for the one shanke . Now 9 — 4. that is , 5. is the difference of the divisour and quotient , whose halfe 2 . ½ , is the other shanke . And 9 — 4. that is 13. is the summe the said devisour and quotient , whose halfe 6 . ½ , is the base .

Againe let 4. and 8. be given . The quadrate of 8. is 64. And the quient of 64 is 16. and the side of 64. is 8. for the one shanke . The halfe 16 — 4. that is 6. is the other shanke . And the halfe of 16 — 4. that is 10 , is the base .

11. If the base of a triangle doe subtend a right-angle , the rectilineall fitted to it , shall be equall to the like rectilinealls in like manner fitted to the shankes thereof : And contrariwise , out of the 31. p. v j.

Or thus : If the base of a triangle doe subtend a right angle , the right lined figure made upon the base , is equall to the right lined figures like , and in like manner situate upon the feete : H.

The Converse is thus proved : Let the triangle be a e i : And let the perpendicular e o , be erected upon a e , equall to e i : And draw a right line from o to a : Here by the former , the rectilinealls situate at o e , and e a , that is by the construction , at a e , and i e , are equall to the rightilineall at a o , made alike and situate a like : And by the graunt they are equall , to the rectilineall at a i , made a like and situated alike . Therefore seeing the like rectilineals at a o , and a i , are equall ; they have by the 20 e iiij , their homologall sides equall : And the two triangles are equiliters : And by the 1 e vij , equiangles . But a e o , is a right angle , by the construction : And a e i , is proved to be equall to the same a e o : Therefore , by the 13 e v. a e i , also is a right angle .

12 An obliquangled triangle is either Obtusangled or Acutangled .

The division of an obliquangled triangle is taken from the speciall differences of an oblique angle . For at the 15 e iij , we were taught that an oblique angle was either obtuse or acute : Therefore an obliquangled triangle is an obtuseangle , and an Acutangle .

13 An obtusangle is that triangle which hath one blunt corner , 28. di .

There can be but one right angle in a triangle , by the 2 e. Therefore also in it there can be but one blunt angle .

Therefore

14. If the obtuse or blunt angle be at the base of the triangle given , a perpendicular drawne from the toppe of the triangle , shall fall without the figure : And contrarywise .

As here in a e i , the perpendicular i o , falleth without : This is manifest by the 4 e.

And

15. If one angle of a triangle be greater than both the other two , it is an obtuse angle : And contrariwise .

This is plaine by the 6 e.

And

16. If a right line drawne from the toppe of the triangle cutting the base into two equall parts , be lesse than one of those halfes , the angle at the toppe is a blunt-angle . And contrariwise .

17. An acutangled triangle is that which hath all the angles acute . 29 dj .

Therefore

18 A perpendicular drawne from the top falleth with out figure : And contrariwise .

As in a e i , the perpendicular a o falleth without as is plaine by the 4 e.

And

19. If any one angle of triangle be lesse then the other two , it is acute : And contrariwise .

As is manifest by the 6 e.

And

20. If a right line drawne from the toppe of the triangle ; cutting the base into two equall parts , be lesse than either of those portions , the angle at the toppe is an acute angle : And contrariwise .

The ninth Booke , of P. Ramus Geometry , which intreateth of the measuring of right lines by like right-angled triangles .

THe Geometry of like right-angled triangles , amongst many other uses that it hath , it doth especially afford us the geodaesy or measuring of right lines : And that mastery , which before ( at the 2 e viij ) attributed the right angled triangles , shall here be found to be a true mastery indeed . For it shall containe the geodesy of right lines ; and afterward the geodesy of plaines and solides , by the measuring of their sides , which are right lines .

1. For the measuring of right lines ; we will use the Iacobs staffe , which is a squire of unequall shankes .

Radius , commonly called Baculus Iacob , Iacobs staffe , as if it had beene long since invented and practised by that holy Patriarke , is a very auncient instrument , and of all other Geometricall instruments , commonly used , the best and fittest for this use . Archimedes in his book of the Number of the sand , seemeth to mention some such thing : And Hipparchus , with an instrument not much unlike this , boldly attempted an haynous matter in the sight of God , as Pliny thinketh , namely to deliver unto posterity the number of the starres , and to assigne of fixe them in their true places by the Norma , the squire or Iacobs staffe . And indeed true it is that the Radius is not onely used for the measuring of the earth and land : But especially for the defining or limiting of the starres in their places and order : And for the describing and setting out of all the regions and waies of the heavenly city . Yea and Virgill the famous Poet , in his 3 Ecloge , Ecquis fuit alter , Descripsit radio totum , qui gentibus orbem ? and againe afterward in the 6 of his Eneiades , hath noted both these uses . Coeliquè meatus . Describent radio & surgentia sidera dicent . Long after this the Iewes and Arabians , as Rabbi Levi ; But in these latter daies , the Germaines especially , as Regiomontanus ; Werner , Schoner , and Appian have grac'd it : But above all other the learned Gemma Phrisius in a severall worke of that argument onely , hath illustrated and taught the use of it plainely and fully .

The Iacobs staffe therefore according to his owne , and those Geometricall parts , shall here be described ( The astronomicall distribution wee reserve to his time and place . ) And that done , the use of it shall be shewed in the measuring of lines .

2 The shankes of the staffe are the Index and the Transome .

3 The Index is the double and one tenth part of the transome .

Or thus : The Index is to the transversary double and 1/10 part thereof . H. As here thou seest .

4 The Transome is that which rideth upon the Index , and is to be slid higher or lower at pleasure .

Or , The transversary is to be moved upon the Index , sometimes higher , sometimes lower : H. This proportion in defining and making of the shankes of the instrument is perpetually to be observed : as if the transome be 10. parts , the Index must be 21. If that be 189. this shall be 90. or if it be 2000. this shall be 4200. Neither doth it skill what the numbers be , so this be their proportion . More than this , That the greater the numbers be , that is the lesser that the divisions be , the better will it be in the use . And because the Index must beare , and the transome is to be borne ; let the index be thicker , and the transome the thinner .

But of what matter each part of the staffe be made , whether of brasse or wood it skilleth not , so it be firme , and will not cast or warpe . Notwithstanding , the transome will more conveniently be moved up and downe by brasen pipes , both by it selfe , and upon the Index higher or lower right angle wise , so touching one another , that the alterne mouth of the one may touch the side of the other . The thrid pipe is to be moved or slid up and downe , from one end of the transome to the other ; and therefore it may be called the Cursor . The fourth and fifth pipes , fixed and immoveable , are set upon the ends of the transome , are unto the third and second of equall height with ●innes , to restraine when neede is , the opticke line , and as it were , with certaine points to define it in the transome .

The three first pipes may , as occasion shall require , be fastened or staied with brasen scrues . With these pipes therefore the transome may be made as great , as need shall require , as here thou seest .

The fabricke or manner of making the instrument hath hitherto beene taught , the use thereof followeth : unto which in generall is required : First , a just distance . For the sight is not infinite . Secondly , that one eye be closed : For the optick faculty conveighed from both the eyes into one , doth aime more certainely ; and the instrument is more fitly applied and set to the cheeke bone , then to any other place . For here the eye is as it were the center of the circle , into which the transome is inscribed . Thirdly , the hands must be steady ; for if they shake , the proportion of the Geodesy must needes be troubled and uncertaine . Lastly , the place of the station is from the midst of the foote .

5 If the sight doe passe from the beginning of one shanke , it passeth by the end of the other : And the one shanke is perpendicular unto the magnitude to be measured , the other parallell .

These common and generall things are premised . That the sight is from the beginning of the Index by the end of the transome : Or contrariwise , From the beginning of the transome , unto the end of the Index . And that the Index is right , that is , perpendicular to the line to be measured , the transome parallell . Or contrariwise . Now the perpendicularity of the Index , in measurings of lengthts , may be tried by a plummet of lead appendent● But in heights and breadths , the eye must be trusted ; although a little varying of the plummet can make no sensible errour . By the end of the transome , understand that which is made by the line visuall , whether it be the outmost finne , or the Cursour in any other place whatsoever .

6 Length and Altitude have a threefold measure ; The first and second kinde of measure require but one distance , and that by granting a dimension of one of them , for the third proportionall : The third two distances , and such onely is the dimension of Latitude .

Geodesy of right lines is two fold ; of one distance , or of two . Geodesy of one distance is when the measurer for the finding of the desired dimension doth not change his place or standing . Geodesy of two distances is when the measurer by reason of some impediment lying in the way betweene him and the magnitude to be measured , is constrained to change his place , and make a double standing .

Here observe , That length and heighth , may be joyntly measured both with one , and with a double station : But breadth may not be measured otherwise than with two .

7 If the sight be from the beginning of the Index r●ght or plumbe unto the length , and unto the father end of the same , as the segment of the Index is , unto the segment of the transome , so is the heighth of the measurer unto the length .

The same manner of measuring shall be used form an higher place ; as out of y , the segment of the Index is 5. parts ; the segment of the transome 6 : and then the height be 10 foote : the same Length shall be found to bee 12 foote .

Neither is it any matter at all , whether the length in a plaine or levell underneath : Or in an ascent or descent of a mountaine , as in the figure under written .

Thus mayest thou measure the breadths of Rivers , Valleys , and Ditches . For the Length is alwayes after this manner , so that one may measure the distance of shippes on the Sea , as also Thales Milesius , in Proclus at the 26 pj , did measure them . An example thou hast here .

Hereafter in the measuring of Longitude and Altitude , fight is unto the toppe of the heighth . Which here I doe now forewarne thee of , least afterward it should in vaine be reitered often .

The second manner of measuring a Length is thus :

8. If the sight be from the beginning of the index parallell to the length to be measured , as the segment of the transome is , unto the segment of the index , so shall the heighth given be to the length .

As if the segment of the Transome be 120 parts : the height given 400-foote : The segment of the Index 210 parts : The length , by the golden rule shall be 700 foote . The figure is thus . And the demonstration is like unto the former ; or indeed more easier . For the triangles are equiangles , as afore . Therefore as o u is to u a : so is e i to i a.

This is the first and second kinde of measuring of a Longitude , by one single distance or station : The third which is by a double distance doth now follow . Here the transome , if there be roome enough for the measurer to goe farre enough backe , must be put lower , in the second distance .

9. If the sight be from the beginning of the transverie parallell to the length to be measured , as in the index the difference of the greater segment is unto the lesser ; so is the difference of the second station unto the lenth .

This kinde of Geodaesy is somewhat more subtile than the former were . The figure is thus ; in which let the first ayming be from a , the beginning of the transome , and out of a i the length sought by o , the end of the Index , unto e , the toppe of the heighth : And let the segment of the Index be o u : The second ayming let it be from y , the beginning of the transome , out of a greater distance by s , the end of the Index , unto e , the same note of the heighth : And let the segment of the Index be s r.

Here the measuring performed , is the taking of the difference betweene o u and s r. The rest are faigned onely for demonstrations sake . Therefore in the first station let a m l , be from the beginning of the transome , be parallell to the e. Here first m u , is equall to s r. For the triangle● m u a , and s r y , are equall in their shankes u a , and r y , by the grant ( Because the transome standeth still in his owne place : ) And the angles at m u a , u a m , are equall to the angles : And all right angles are equall , by the 14 e iij. These are the outter and inner opposite one to another : And such are equall by the 1. e. v. ) Therefore they are equilaters , by the 2 e vij ; And o m , is the difference of the segments of the Index . Then as o m is to m u , so is e l , to l i ; as the equation of three degrees doth shew . For , by the 12 e vij , as o m is to m a : so is e l to l a : And as m a is to m u ; so is l a , to l i. Therefore by right , as o m , is to m u : so is e l , to l i : And by the 12 e v j , so is y a , to a i : As if the difference of the first segment be 36 parts : The second segment be 72 parts : The difference of the second s●ation 40 foote . The length sought shall be 80 foote . And here indeed is no heighth definitely given , that may make any bound of the principall proportion . Notwithstanding the Heighth , although it be of an unknowne measure , is the bound of the length sought : And therefore it is an helpe and meanes to argue the question . Because it is conceived to stand plumbe upon the outmost end of the length .

Therefore that third kinde of measuring of length is oftentimes necessary , when by neither of the former waye● the length may possibly be taken , by reason of some impediment in the way , to wit of a wall , or tree , or house , or mountaine , whereby the end of the length may not be seene , which was the first way : Nor an height next adjoyning to the end of the length is given , which is the second way .

Hitherto we have spoken of the threefold measure of longitude , the first and second out of an heighth given the third out of a double distance : The measuring of heighth followeth next , and that is also threefold . Now heighth is a perpendicular line falling from the toppe of the magnitude , unto the ground or plaine whereon the measurer doth stand , after which manner Altitude on heighth was defined at the 9 e iiij . The first geodesy or manner of measuring of heighths is thus .

10. If the sight be from the beginning of the transome perpendicular unto the height to be measured , as the segment of the transome , is unto the segment of the Index , so shall the length given be to the height .

Let the segment of the transome be 60 parts : the segment of the Index 36 : the Length given 120 foote : the height sought shall be , by the golden rule , 72 foote .

The Figure is thus : And the demonstration is by the 12 e vij , as afore : but here is to be added the height of the measurer ; which if it be 4 foot , the whole height shall be 76 foote .

Therefore in an eversed altitude

11. If the sight be from the beginning of the Index parallell to the height , as the segment of the transome is , unto the segment of the index , so shall the length given be , unto the height sought .

Eversa altitudo , An eversed altitude ( Reversed , H ; ) is that which we call depth , which indeed is nothing else , in the Geometers sense , but heighth turned topsie turvie , as we say , or with the heeles upward . For out of the heighth concluded by subducting that which is above ground , the heighth or depth of a Well shall remaine .

12. If the sight be from the beginning of the Index perpendicular to the heighth to be measured , as the segment of the Index is unto the segment of the Transome , so shall the length given be to the heighth .

Therefore

13. If the sight be from the beginning of the Index ( perpendicular to the magnitude to be measured ) by the names of the transome , unto the ends of some known part of the height , as the distance of the Names is , unto the rest of the transome above them , so shall the known part be unto the part sought .

Or thus : If the sight passe from the beginning of the Index being right , by the vanes of the transversary , to the tearmes of some parts ; as the distance of the vanes is unto the rest of the transversary above the index , so is the part knowne unto the remainder : H.

This is a consectary of a knowne part of an height , from whence the rest may be knowne , as in the figure .

The first and second kinde of measuring of heights is thus : The third followeth .

14 If the sight be from the beginning of the Index perpendicular to the heighth , as in the Index the difference of the segmeut , is unto the difference of the distance or station ; so is the segment of the transome unto the heighth .

Hitherto you must recall that subtilty , which was used in the third manner of measuring of lengths .

Let the first aime be taken from a , the beginning of the Index perpendicular unto the height to be measured : And from an unknowne length a i , by o , the end of the transome , unto e , the toppe of the height e i : And let the segment of the Index be u a. The second ayme , let it be taken from y , the beginning of the same Index ; and out of a greater distance , by s , the end of the transome , unto the same toppe e. And the segment of the Index let it be r l.

Here , as afore , the measuring is performed and done , by the taking of the difference of the said y r , above a u : Now the demonstration is concluded , as in the former was taught . Let the parallel l s m , be erected against a o e.

Here first the triangles o u a , & s r l , are equilaters , by the 2 e vij . ; ( seeing that the angles at a , and l , the externall and internall , are equall in bases o u , and s r : for the segment in each distance is the same still : ) Therefore u a , is equall to r l. Now the rest is concluded by a sorites of foure degrees : As y r , is unto y u : so by the 12. e vij . is s r , that is , o u , unto e i : And as o u , is unto e i , so is a u , that is , l r , unto a i. Therefore the remainder y l , unto the remainder y a ; shall be as y r , is unto the whole y i , and therefore from the first unto the last , as s r , is to e i.

Therefore let the difference of the Index be 23 : parts● The difference of the distance 30. foote : The segment of the transome 23. parts : The height shall be 57. 9/23 . or foote .

Therefore

15 Out of the Geodesy of heights , the difference of two heights is manifest .

Or thus : By the measure of one altitude , we may know the difference of two altitudes : H.

For when thou hast taken or found both of them , by some one of the former wayes , take the lesser out of the greater ; and the remaine shall be the heighth desired . From hence therefore by one of the towers of unequall heighth , you may measure the heighth of the other . First out of the lesser , let the length be taken by the first way : Because the height of the lesser , wherein thou art , is easie to be taken , either by a plumbe-line , let fall from the toppe to the bottom , or by some one of the former waies . Then measure the heighth , which is above the lesser : And adde that to the lesser , and thou shalt have the whole heighth , by the first or second way . The figure is thus , and the demonstration is out of the 12. e vij . For as a e , is to e i , so is a o , to o u. Contrariwise out of an higher Tower , one may measure a lesser .

16 If the sight be first from the toppe , then againe from the base or middle place of the greater , by the vanes of the transome unto the toppe of the lesser heighth ; as the said parts of the yards are unto the part of the first yard ; so the heighth betweene the stations shall be unto his excesse above the heighth desired .

For let the parts of the yard be 12. and 6. and the summe of them 18. Now as 18. is 12. so is the whole altitude u y , 190. foote , unto the excesse 126⅔ foote . The remainder therefore 63⅓ foote , shall be a s , the lesser heighth sought .

The second station might have beene in o , the end of the perpendicular from a. But by taking the ayme out of the toppe of the lesser altitude , the demonstration shall be yet againe more easie and short , by the two triangles at the yard a e i , and a e f , resembling the two whole triangles a o u , and a o y , in like situation , the parts of the shanke cut , are on each side the segments of the transome .

One may againe also out of the toppe of a Turret measure the distance of two turrets one from another : For it is the first manner of measuring of longitudes , neither doth it here differ any whit from it , more than the yard is hang'd without the heighth given . The figure is thus : And the Demonstration is by the 12. e vij . For as a e , the segment of the yard , is unto e i , the segment of the transome : so is the assigned altitude a o , unto the length o u.

The geodesy or measuring of altitude is thus , where either the length , or some part of the length is given , as in the first and second way : Or where the distance is double , as in the third .

17 If the sight be from the beginning of the yard being right or perpendicular , by the vanes of the transome , unto the ends of the breadth ; as in the yard the difference of the segment is unto the difference of the distance , so is the distance of the vanes unto the breadth .

The measuring of breadth , that is , of a thwart or crosse line , remaineth . The Figure and Demonstration is thus : The first ayming , let it be a e i , by o , and u , the vanes of the transome o u. The second , let it be y e i , by s , and r , the vanes of the transome s r. Then by the point s , let the parallell l s m , be drawne against a o e. Here first , the triangles o u a , and s i l , are equilaters , by the 2 e vij . Because the angles at n and j , are right angles : And u a o , and j l s , the outter and inner , are equall in their bases o u , and s j , by the grant : Because here the segment of the transome remaineth the same : Therefore u a , is equall to j l. These grounds thus laid , the demonstration of the third altitude here taken place . For as y l , is unto y a : so is s j , unto e r : And , because parts are proportionall unto their multiplicants , so is s r , unto e i : for the rest doe agree .

The same shall be the geodesy or manner of measuring , if thou wouldest from some higher place , measure the breadth that is beneath thee , as in the last example . But from the distance of two places , that is , from latitude or breadth , as of Trees , Mountaines , Cities , Geographers and Chorographers do gaine great advantages and helpes .

The tenth Booke of Geometry , of a Triangulate and Parallelogramme .

ANd thus much of the geodesy of right lines , by the meanes of rectangled triangles : It followeth now of the triangulate .

1. A triangulate is a rectilineall figure compounded of triangles .

As before ( for the dichotomies sake ) of a line was made a Lineate , to signifie the genus of a surface and a Body : so now is for the same cause of a triangle made a Triangulate , to declare and expresse the genus of a Quadrilater and Multilater , and indeed more justly , then before in a Lineate . For triangles doe compound and make the triangulate , but lines doe not make the lineate .

Therefore

2. The sides of a triangulate are two more than are the triangles of which it is made .

And

3. Homogeneall Triangulates are cut into an equall number of triangles , è 20 p vj.

For if they be Quadrangles , they be cut into two triangles : If Quinquangles , into 3. If Hexangles , into 4 , and so forth .

4. Like triangulates are cut into triangles alike one to another and homologall to the whole è 20 p vj.

Or thus : Like Triangulates are divided into triangles like one unto another , and in porportion correspondent unto the whole : H.

As in these two quinqualges . First the particular triangles are like betweene themselves . For the shankes of a e u and y s m , equall angles are proportionall , by the grant . Therefore the triangles themselves are equiangles , by 14 e vij . And therefore alike , by the 12 e vii . and so forth of the rest .

The middle triangles , the equall angles being substracted shall have their other angles equall : And therefore they also shall be equiangles and alike , by the same .

Secondarily , the triangles a e u , and y s m : e i o and s r l ; e o u , and s l m , to wit , alike betweene themselves , are by the 1 e vj , in a double reason of their homologall sides e u , s m , e o , s l , which reason is the same , by meanes of the common sides . Therefore three triangles are in the same reason : And therefore they are proportionall : And , by the third composition , as one of the antecedents is , unto one of the consequents● so is the whole quinquangle to the whole .

5. A triangulate is a Quadrangle or a Multangle .

The parts of this partition are in Euclide , and yet without any shew of a division . And here also , as before , the species or severall kinds have their denomination their angles , although it had beene better and truer to have beene taken from their sides ; as to have beene called a Quadrilater , or a Multilater . But in words use must bee followed as a master .

6. A Quadrangle is that which is comprehended of foure right lines . 22 d j.

7. A quadrangle is a a Parallelogramme , or a Trapezium .

This division also in his parts is in the Elements of Euclide , but without any forme or shew of a division . But the difference of the parts shall more fitly be distinguished thus : Because in generall there are many common parallels .

8. A Parallelogramme is a quadrangle whose opposite sides are parallell .

Therefore

9. If right lines on one and the same side , doe joyntly bound equall and parallall lines , they shall make a parallelogramme .

The reason is , because they shall be equall and parallell betweene themselves , by the 26. e v.

And

10 A parallelogramme is equall both in his opposite sides , and angles , and segments cut by the diameter .

Or thus ; The opposite , both sides , and angles , and segments cut by the diameter are equall . Three things are here concluded ; The first is , that the opposite sides are equall : This manifest by the 26 e v. Because two right lines doe jointly bound equall parallells .

And

11 The Diameter of a parallelogramme is cut into two by equall raies .

As in the three figures a e i , next before : This a parallelogramme hath common with a circle , as was manifest at the 28. e iiij .

And

12 A parallelogramme is the double of a triangle of a trinangle of equall base and heighth , 41. p j.

And

13 A parallelogramme is equall to a triangle of equall heighth and double base unto it : è 42. p j.

From whence one may

14 To a triangle given , in a rectilineall angle given , make an equall parallelogramme .

15 A parallelogramme doth consist both of two diago●als , and complements , and gnomons .

For these three parts of a parallelogramme are much used in Geometricall workes and businesses , and therefore they are to be defined .

16 The Diagonall is a particular parallelogramme having both an angle and diagonall diameter common with the whole parallelogramme .

17 The Diagonall is like , and alike situate to the whole parallelogramme : è 24. p vj.

There is not any , either rate or proportion of the diagonall propounded , onely similitude is attributed to it , as in the same figure , the Diagonall a u y s , is like unto the whole parallelogramme a e i o. For first it is equianglar to it . For the angle at a , is common to them both : And that is equall to that which is at y , ( by the 10. e x : ) And therefore also it is equall to that at i , by the 10. e x. Then the angles a u y , and a s y , are equall , by the 21. e v. to the opposite inner angles at e , and o. Therefore it is equiangular unto it .

Againe , it is proportionall to it in the shankes of the equall angles . For the triangles a u y , and a e i , are alike , by the 12 e vij , because u y is parallell to the base . Therefore as a u is to u y ; so is a i to e i : Then as u y is to y a ; so is e i to i a. Againe by the 21 e v , because s y is parallell to the base i o , as a y is to y s : so is a i , to i o : Therefore equiordinately , as u y is to y s : so is e i to i o : Item as s y is to y a , so is i o to i a : And as y a is to a s : so is i a to a o. Therefore equiordinately , as y s is to s a : so is i o to o a. Lastly as s a is unto a y ; so is o a unto a i : And as a y is to a u ; so is a i unto a e. Therefore equiordinately , as s a is to a u : so is a o , to a e. Wherefore the Diagonall s u is proportionall in the shankes of equall angles to the parallelogramme o e.

The demonstration shall be the same of the Diagonall r l. The like situation is manifest , by the 21 e iiij . And from hence also is manifest , That the diagonall of a Quadrate , is a Quadrate : Of an Oblong , an Oblong : Of a Rhombe , a Rhombe : Of a Rhomboides , a Rhomboides : because it is like unto the whole , and a like situate .

Now the Diagonalls seeing they are like unto the whole and a like situate , they shall also be like betweene themselves and alike situate one to another , by the 21 and 22 e iiij .

Therefore

18. If the particular parallelogramme have one and the same angle with the whole , be like and alike situate unto it , it is the Diagonall . 26 p vj.

As for example , Let the particular parallelogramme a u y s , be coangular to the whole parallelogramme a e i o : And let it have the same angle with it at a ; like unto the whole and alike situate unto it ; I say it is the Diagonall .

Otherwise , let the diverse Diagony be a r o : And let l r be parallell against a e : Therefore a l r s , shall bee the Diagonall , by the 6 e [ 15. ] Now therefore it shall be , by 8 e [ 16 e , ] as e a is to a i : so is s a unto a l : Againe , by the grant , as e a is unto a i : so is s a to a u : Therefore the same s a is proportionall to a l , and to a u : And a l is equall to a u , the part to the whole , which is impossible .

19. The Complement is a particular parallelogramme , comprehended of the conterminall sides of the diagonals .

Or thus : It is a particular parallelogramme conteined under the next adjoyning sides of the diagonals .

20. The complements are equall . 43 p j.

Therefore

21. If one of the Complements be made equall to a triangle given , in a right-lined angle given , the other made upon a right line given shall be in like manner equall to the same triangle . 44 p j.

As if thou shouldest desire to have a parallelogramme upon a right line given , and in a right lined angle given , to be made equall to a triangle given , this proposition shall give satisfaction .

And

22 If parallelogrammes be continually made equall to all the triangles of an assigned triangulate , in a right lined angle given , the whole parallelogramme shall in like manner be equall to the whole triangulate . 45 p j.

This is a corollary of the former , of the Reason or rate of a Parallelogramme with a Triangulate ; and it needeth no father demonstration ; but a ready and steddy hand in describing and working of it .

Here thou hast 3 complements continued , and contin●ing the Parallelogramme : But it is best in making and working of them , to put out the former , and one of the sides of the inferiour or latter Diagonall , lea●t the confusion of lines doe hinder or trouble thee .

Therefore

23. A Parallelogramme is equall to his diagonals and complements .

For a Parallelogramme doth consist of two diagonals , and as many complements : Wherefore a Parallelogramme is equall to his parts : And againe the parts are equall to their whole .

24. The Gnomon is any one of the Diagonall with the two complements .

In the Elements of Geometry there is no other use , as it seemeth of the gnomons than that in one word three parts of a parallelogramme might be signified and called by three letters a e i. Otherwise gnomon is a perpendicular .

25. Parallelogrames of equall height are one to another as their bases are . 1 p vj.

Therefore

26 Parallelogrammes of equall height upon equall bases are equall . 35. 36 p j.

As is manifest in the same example .

27 If equiangle parallelogrammes be reciprocall in the shankes of the equall angle , they are equall : And contrariwise . 15 p vj.

Therefore

28 If foure right lines be proportionall , the parallelogramme made of the two middle ones , is equall to the equiangled parallelogramme made of the first and last : And contrariwise , e 16 p vj.

For they shall be equiangled parallelogrammes reciprocall in the shankes of the equall angle .

And

29 If three right lines be proportionall , the parallelogramme of the middle one is equall to the equiangled parallelogramme of the extremes : And contrariwise .

It is a consectary drawne out of the former .

Of Geometry , the eleventh Booke , of a Right angle .

1. A Parallelogramme is a Right angle or an Obliquangle .

HItherto we have spoken of certaine common and generall matters belonging unto parallelogrammes● specials doe follow in Rectangles and Obliquangles , which difference , as is aforesaid , is common to triangles and triangulates . But at this time we finde no fitter words whereby to distinguish the generals .

2. A Right angle is a parallelogramme that hath all his angles right angles .

As in a e i o. And here hence you must understand by one right angle that all are right angles . For the right angle at a , is equall to the opposite angle at i , by the 10 e x.

Therefore

3 A rightangle is comprehended of two right lines comprehending the right angle 1. d ij .

Comprehension , in this place doth signifie a certaine kind of Geometricall multiplication . For as of two numbers multiplied betweene themselves there is made a number : so of two sides ( ductis ) driven together , a right angle is made : And yet every right angle is not rationall , as before was manifest , at the 12. e iiij . and shall after appeare at the 8 e.

And

4 Foure right angles doe fill a place .

Neither is it any matter at all whether the foure rectangles be equall , or unequall ; equilaters , or unequilaters ; homogeneals , or heterogenealls . For which way so ever they be turned , the angles shall be right angles : And therefore they shall fill a place .

5 If the diameter doe cut the side of a right angle into two aquall parts , it doth cut it perpendicularly : And contrariwise .

Therefore

6 If an inscribed right line doe perpendicularly cut the side of the right angle into two equall parts , it is the diameter .

The reason is , because it doth cut the parallelogramme into two equall portions .

7 A right angle is equall to the rightangles made of one of his sides and the segments of the other .

As here the foure particular right angles are equall to the whole , which are made of a e , one of his sides , and of e i , i o , o u , u y , the segments of the other .

Lastly , every arithmeticall multiplication of the whole numbers doth make the same product , that the multiplication of the one of the whole numbers given , by the parts of the other shall make ; yea , that the multiplication of the parts by the parts shall make . This proposition is cited by Ptolomey in the 9. Chapter of the 1 booke of his Almagest .

8 If foure right lines be proportionall , the rectangle of the two middle ones , is equall to the rectangle of the two extremes . 16. p vj.

9 The figurate of a rationall rectangle is called a rectinall plaine . 16. d vij .

If therefore the Base of a Rectangle be 6. And the height 4. The plot or content shall be 24. And if it be certaine that the rectangles content be 24. And the base be 6. It shall also be certaine that the heighth is 4. The example is thus .

This manner of multiplication , say 1 , is Geometricall : Neither are there here , of lines made lines , as there of vnities were made vnities ; but a magnitude one degree higher , to wit , a surface , is here made .

Here hence is the Geodesy or manner of measuring of a rectangled triangle made knowne unto us . For when thou shalt multiply the shankes of a right angle , the one by the other , thou dost make the whole rectangled parallelogramme , whose halfe is a triangle , by the 12. e x.

Of Geometry the twelfth Booke , Of a Quadrate .

1 A Rectangle is a Quadrate or an Oblong .

THis division is made in proper termes : but the thing it selfe and the subject difference is common out of the angles and sides .

2 A Quadrate is a rectangle equilater 30. dj .

Plaines are with us , according to their diverse natures and qualities , measured with divers and sundry kindes of measures . Boord , Glasse , and Paving-stone are measured by the foote : Cloth , Wainscote , Painting , Paving , and such like , by the yard : Land , and Wood , by the Perch or Rodde ,

Of Measures● and the sundry sorts thereof commonly used and mentioned in histories we have in the former spoken at large : Yet for the farther confirmation of some thing then spoken , and here againe now upon this particular occasion repeated , it shall not be amisse to heare what our Statutes speake of these three sorts here mentioned .

It is ordained , saith the Statute , That three Barley-cornes dry and round , doe make an Ynch : twelve ynches doe make a Foote : three foote doe make a Yard : Five yards and an halfe doe make a Perch : Fortie perches in length , and foure in breadth doe make an Aker . 33. Edwardi 1. De Terris mensurandis . Item , De compositione Vlnarum & Perticarum .

Moreover observe , that all those measures there spoken of were onely lengths : These here now last repeated , are such as the magnitudes by the measured are , in Planimetry , I meane , they are Plaines : In Stereometry they are solids , as hereafter we shall make manifest . Therefore in that which followeth , An ynch is not onely a length three barley-cornes long : but a plaine three barley-cornes long , and three broad . A Foote is not onely a length of 12. ynches : But a plaine also of 12. ynches square , or containing 144. square ynches● A yard is not onely the length of three foote : But it is also a plaine 3. foote square every way . A Perch is not onely a length of 5½ . yards : But it is a plot of ground 5½ . yards square every way .

A Quadrate therefore or square , seeing that it is equilater that is of equall sides : And equiangle by meanes of the equall right angles , of quandrangles that onely is ordinate .

Therefore

3 The sides of equall quadrates , are equall .

And

The sides of equall quadrates are equally compared : If therefore two or more quadrates be equall , it must needs follow that their sides are equall one to another .

And

4 The power of a right line is a quadrate .

Or thus : The possibility of a right line is a square H. A right line is said posse quadratum , to be in power a square ; because being multiplied in it selfe , it doth make a square .

5 If two conterminall perpendicular equall right lines be closed with parallells , they shall make a quadrate . 46. p.j.

6 The plaine of a quadrate is an equilater plaine .

Or thus : The plaine number of a square , is a plaine number of equall sides , H.

A quadrate or square number , is that which is equally equall : Or that which is comprehended of two equall numbers , A quadrate of all plaines is especially rationall ; and yet not alwayes : But that onely is rationall whose number is a quadrate . Therefore the quadrates of numbers not quadrates , are not rationalls .

Therefore

7 A quadrate is made of a number multiplied by it selfe .

Such quadrates are the first nine . 1,4,9 , 16,25,36,49,64 , 81 , made of once one , twice two , thrise three , foure times foure , five times five , sixe times sixe , seven times seven , eight times eight , and nine times nine . And this is the summe of the making and invention of a quadrate number of multiplication of the side given by it selfe .

Hereafter diverse comparisons of a quadrate or square , with a rectangle , with a quadrate , and with a rectangle and a quadrate iointly . The comparison or rate of a quadrate with a rectangle is first .

8 If three right lines be proportionall , the quadrate of the middle one , shall be equall to the rectangle of the extremes : And contrariwise : 17. p v j. and 20. p vij .

It is a corallary out of the 28. e x. As in a e , e i , i o.

9 If the base of a triangle doe subtend a right angle , the powre of it is as much as of both the shankes : And contrariwise 47,48 . p j.

10 If the quadrate of an odde number , given for the first shanke , be made lesse by an vnity ; the halfe of the remainder shall be the other shanke ; increased by an unity it shall be the base .

Or thus : If the square of an odde number given for the first foote , have an unity taken from it , the halfe of the remainder shall be the other foote , and the same halfe increased by an unitie , shall be the base : H.

Againe , the quadrate or square of 3. the first shanke is 9. and 9 — 1. is 8 , whose halfe 4 , is the other shanke . And 9 — 1 , is 10. whose halfe 5. is the base . Plato's way is thus by an even number .

11 If the halfe of an even number given for the first shanke be squared , the square number diminished by an vnity shall be the other shanke , and increased by an vnitie it shall be the base .

Againe , the quadrate or square of 3. the halfe of 6 , the first shanke , is 9. and 9 — 1 , is 8 , for the second shanke . And out of this rate of rationall powers ( as Vitruvius , in the 2. Chapter of his IX . booke ) saith Pythagoras taught how to make a most exact and true squire , by joyning of three rulers together in the forme of a triangle , which are one unto another as 3,4 . and 5. are one to another .

From hence Architecture learned an Arithmeticall proportion in the parts of ladders and stayres . For that rate or proportion , as in many businesses and measures is very commodious ; so also in buildings , and making of ladders or staires , that they may have moderate rises of the steps , it is very speedy . For 9 — 1. is , 10 , base .

12. The power of the diagony is twise asmuch , as is the power of the side , and it is unto it also incommensurable .

Or thus : The diagonall line is in power double to the side , and is incommensurable unto it , H.

This is the way of doubling of a square taught by Plato , as Vitruvius telleth us : Which notwithstanding may be also doubled , trebled , or according to any reason assigned increased , by the 25 e iiij , as there was foretold .

But that the Diagony is incommensurable unto the side it is the 116 p x. The reason is , because otherwise there might be given one quadrate number , double to another quadrate number : Which as Theon and Campanus teach us , is impossible to be found . But that reason which Aristotle bringeth is more cleare which is this ; Because otherwise an even number should be odde . For if the Diagony be 4 , and the side 3 : The square of the Diagony 16 , shall be double to the square of the side : And so the square of the side shall be 8. and the same square shall be 9 , to wit , the square of 3. And so even shall be odde , which is most absurd .

Hither may be added that at the 42 p x. That the segments of a right line diversly cut ; the more unequall they are the greater is their power .

13 If the base of a right angled triangle be cut by a perpendicular from the right angle in a doubled reason , the power of it shall be halfe as much more , as is the power of the greater shanke : But thrise so much as is the power of the lesser . If in a quadrupled reason , it shall be foure times and one fourth so much as is the greater : But five times so much as is the lesser , At the 13 , 15 , 16 p x iij.

And by the same argument it shall be treble unto the quadrate or square of e i.

The other , of the fourefold or quadruple section , are manifest in the figure following , by the like argument .

14 If a right line be cut into how many parts so ever , the power of it is manifold unto the power of segment , denominated of the square of the number of the section .

Or thus : if a right be cut into how many parts so ever it is in power the multiplex of the segment , the square of the number of the section , being denominated thereof : H.

15. If a right line be cut into two segments , the quadrate of the whole is equall to the quadrats of the segments , and a double rectanguled figure , made of them both . 4 p ij .

The third rate of a quadrate is hereafter with two rectangles , and two quadrates , and first of equality .

Therefore

16. The side of the first diagonall , is the side of one of the complements ; And being doubled , it is the side of them both together : Now the other side of the same complements both together , is the side of the other diagonall .

The side of a quadrate given is many times in numbers sought . Therefore by the former element and his consectaries the resolution of a quadrates side is framed and performed .

This is the rate of a quadrate with a rectangle & a quadrate from whence is had the analisis or resolution of the side of a quadrate expressable by a number . For it is the same way from Cambridge to London , that is from London to Cambridge . And this use of geometricall analysis remaineth , as afterward in a Cube , when as otherwise through the whole booke of Euclides Elements there is no other use at all of that .

Then beginning at the left hand , as in Division ,

Or thus :

that is where we left in multiplication , and I seeke amongst the squares the greatest conteined in the first periode , which here is 1● And the side of it , which is also 1 , I place with my quotient : Then I square this quotient , that is I multiply it by it selfe , and the product 1 , I sect under the same first periode : Lastly , I subtract it from the same periode , and there remaineth not any thing . Then as in division I set up the figures of the next periode one degree higher .

Secondly double the side now found , and it shall be 2 , which I place in manner of a Divisor , on the left hand , within the semicircle : By this I divide the 40 , the two complements or Plaines , and I finde the quotient or second side 2 ; which I place in the quotient by 1 , This side I multiply first quadrate like , that is by it selfe ; and I make 4 , the lesser Diagonall : And therefore I place under the last 4 : Then I multiply the said Divisor 2 , by the same 2 the quotient ,

Or thus :

and I make in like manner , 4 which I place under the dividend , or the first 4. Lastly I subtract these products from the numbers above them , and remaineth nothing . Therefore I say first , That 144 , the number given is a quadrate : And more-over , That 12 is the true side of it .

Againe , let the side of 15129 be sought . First divide it into imperfect periods as before was taught ; in this manner : 15129. Then I seeke amongst the former quadrates , for the side of 1 , the quadrate of the first periode ; and I finde it to be 1 : This side I place within the quotient or lunular on the right side : Lastly I subtract 1 from ● , and nothing remaineth . Then I double the said side found ; and I make 2 : This 2 , I place for my divisor within the lunular or semicircle on the left hand : By which I divide 5 ; And I finde the quotient 2 , which I place by the former quotient : Then I multiply the same 2 , first quadratelike by it selfe , and I make 4. Then I multiply the sayd divisour by 2 , the quotient , and I make likewise 4 : which I place underneath 51. Lastly , I subtract the same 44 , from 51 , and there remaine 7 , over the head of 1 ; By which I place 29 , the last periode remaining .

Againe I double 12 , my whole quotient , and I make 24. By this double I divide 72 , the double Complement remaining , and I finde 3 for the side or quotient : First this side I multiply quadratelike by it selfe , and I make 9 , which I place underneath 9 , the last figure of my dividende . Then againe , by the same quotient , or side 3 , I multiply 2● : my divisour , and I make 72 ; which I place under 72 , the two figures of my Dividende : Lastly I subtract the under figures , from the upper , and there is likewise nothing remaining : Wherefore I say , as afore ; that the figurate 15129 given , is a square : And the side thereof is 123.

Sometime after the quadrate now found , in the next places , there is neither any plaine nor square to bee found : Therefore the single side thereof shall be O. As in the quadrate 366025 , the whole side is 605 , consisting of three severall sides , of which the middle one is o.

Sometime also the middle plaine doth containe a part of the quadrate next following : Therfore if the other side remaining be greater than the side of the quadrate following , it is to be made equall unto it : As for example , Let the side of the quadrate 784 , be sought ;

Or thus :

The side of the first quadrate shall be 2 , and there shall remaine 3 , thus : Then the same side doubled is 4 for the quotient ; Which is found in 38 , the double plaine remaining 9 times , for the other side : But this side is greater than the side of the next following quadrate : Take therefore 1 out of it : And for nine take 8 , and place it in your quotient ; Which 8 multiplyed by it selfe maketh 64 , for the Lesser quadrate : And againe the same multiplyed by 4 the divisour maketh 32 ; The summe of which two products 384 , subtracted from the remaine 384 , leave nothing : Therefore 784 is a Quadrate : And the side is 28.

And from hence the invention of a meane proportionll , betweene two numbers given , ( if there be any such to be found ) is manifest . For it the product of two numbers given be a quadrate , the side of the quadrate shall be the meane proportionall , betweene the numbers given ; as is apparent by the golden rule : As for example , Betweene 4. and 9. two numbers given , I desire to know what is the meane proportion . I multiply therefore 4 and 9. betweene themselves , and the product is 36 : which is a quadrate number ; as you see in the former ; And the side is 6. Therefore I say , the meane proportionall betweene 4. and 9. is 6 , that is , As 4. is to 6. so is 6. to 9.

If the number given be not a quadrate , there shall no arithmeticall side , and to be expressed by a number be found : And this figurate number is but the shadow of a Geometricall figure , and doth not indeede expresse it fully , neither is such a quadrate rationall : Yet notwithstanding the numerall side of the greatest square in such like number may be found : As in 148. The greatest quadrate continued is 144 and the side is 12. And there doe remaine 4. Therefore of such kinde of number , which is not a quadrate , there is no true or exact side : Neither shall there ever be found any so neare unto the true one ; but there may still be one found more neare the truth . Therefore the side is not to be expressed by a number .

Of the invention of this there are two wayes : The one is by the Addition of the gnomon ; The other is by the Reduction of the number assigned unto parts of some greater denomination . The first is thus :

17 If the side found be doubled , and to the double a unity be added , the whole shall be the gnomon of the next greater quadrate .

For the sides is one of the complements , and being doubled it is the side of both together . And an unity is the latter diagonall . So the side of 148 is 12 4/25.

The reason of this dependeth on the same proposition , from whence also the whole side , is found . For seeing that the side of every quadrate lesser than the next follower differeth onely from the side of the quadrate next above greater than it but by an 1. the same unity , both twice multiplied by the side of the former quadrate , and also once by it selfe , doth make the Gnomon of the greater to be added to the quadrate . For it doth make the quadrate 169. Whereby is understood , that looke how much the numerator 4. is short of the denominatour 25. so much is the quadrate 148. short of the next greater quadrate . For it thou doe adde 21. which is the difference whereby 4 is short of 25. thou shalt make the quadrate 169. whose side is 13. The second is by the reduction , as I said , of the number given unto parts assigned of some great denomination , as 100. or 1000. or some smaller than those , and those quadrates , that their true and certaine may be knowne : Now looke how much the smaller they are , so much nearer to the truth shall the side found be .

Moreover in lesser parts , the second way beside the other , doth shew the side to be somewhat greater than the side , by the first way found : as in 7. the side by the first way is 3/25. But by the second way the side of 7. reduced unto thousands quadrates , that is unto 7000000/1000000 , that is , 2645/1000 , and beside there doe remaine 3975. But 645/1000. are greater than 3/5. For ⅗ . reduced unto 1000. are but 600/1000. Therefore the second way , in this example , doth exceed the first by 45/1000. those remaines 3975. being also neglected .

Therefore this is the Analysis or manner of finding the side of a quadrate , by the first rate of a quadrate , equall to a double rectangle and quadrate .

The Geodesy or measuring of a Triangle .

There is one generall Geodesy or way of measuring any manner of triangle whatsoever in Hero , by addition of the sides , halving of the summe , subduction , multiplication , and invention of the quadrates side , after this manner .

18 If from the halfe of the summe of the sides , the sides be severally subducted , the side of the quadrate continually made of the halfe , and the remaines shall be the content of the triangle .

This generall way of measuring a triangle is most easie and speedy , where the sides are expressed by whole numbers .

The speciall geodesy of rectangle triangle was before taught ( at the 9 e x j. ) But of an oblique angle it shall hereafter be spoken . But the generall way is farre more excellent than the speciall● For by the reduction of an obliquangle many fraudes and errours doe fall out , which caused the learned Cardine merrily to wish , that hee had but as much land as was lost by that false kinde of measuring .

19 If the base of a triangle doe subtend an obtuse angle , the power of it is more than the power of the shankes , by a double right angle of the one , and of the continuation from the said obtusangle unto the perpendicular of the toppe . 12. p ij .

Or thus : If the base of a triangle doe subtend an obtuse angle , it is in power more than the feete , by the right angled figure twise taken , which is contained under one of the feete and the line continued from the said foote unto the perpendicular drawne from the toppe of the triangle . H.

There is a comparison of a quadrate with two in like manner triangles , and as many quadrates , but of unequality .

For by 9. e , the quadrate of a i , is equall to the quadrates of a o , and o i , that is , to three quadrates of i o , o e , e a , and the double rectangle aforesaid . But the quadrates of the shankes a e , e i , are equall to those three quadrates , to wit , of a i , his owne quadrate , and of e i , two , the first i o , the second o e , by the 9. e. Therefore the excesse remaineth of a double rectangle .

Of Geometry , the thirteenth Booke , Of an Oblong .

1 An Oblong is a rectangle of inequall sides , 31. d j.

This second kinde of rectangle is of Euclide in his elements properly named for a definitions sake onely .

The rate of Oblongs is very copious , out of a threefold section of a right line given , sometime rationall and expresable by a number : The first section is as you please , that is , into two segments , equall or unequall : From whence a five-fold rate ariseth .

2 An Oblong made of an whole line given , and of one segment of the same , is equall to a rectangle made of both the segments , and the square of the said segment . 3. p ij .

It is a consectary out of the 7 e xj . For the rectangle of the segments , and the quadrate , are made of one side , and of the segments of the other .

Now a rectangle is here therefore proposed , because it may be also a quadrate , to wit , if the line be cut into two equall parts .

Secondarily ,

3 Oblongs made of the whole line given , and of the segments , are equall to the quadrate of the whole 2 p ij .

This is also a Consectary out of the 4. e xj .

Here the segments are more than two , and yet notwithstanding from the first the rest may be taken for one , seeing that the particular rectangle in like manner is equall to them . This proposition is used in the demonstration of the 9. e xviij .

Thirdly ,

4 Two Oblongs made of the whole line given , and of the one segment , with the third quadrate of the other segment , are equall to the quadrates of the whole , and of the said segment . 7 p ij .

5 The base of an acute triangle is of lesse power than the shankes are , by a double oblong made of one of the shankes , and the one segment of the same , from the said angle , unto the perpendicular of the toppe . 13. p.ij.

And from hence is had the segment of the shanke toward the angle , and by that the perpendicular in a triangle .

Therefore

6. If the square of the base of an acute angle be taken out of the squares of the shankes , the quotient of the halfe of the remaine , divided by the shanke , shall be the segment of the dividing shanke from the said angle unto the perpendicular of the toppe .

Now againe from 169 , the quadrate of the base 13 , take 25 , the quadrate of 5 , the said segment : And the remaine shall be 144 , for the quadrate of the perpendicular a o , by the 9 e x ij .

Here the perpendicular now found , and the sides cut , are the sides of the rectangle , whose halfe shall be the content of the Triangle : As here the Rectangle of 21 and 12 is 252 ; whose halfe 126 , is the content of the triangle .

The second section followeth from whence ariseth the fourth rate or comparison .

7. If a right line be cut into two equall parts , and otherwise ; the oblong of the unequall segments , with the quadrate of the segment betweene them , is equall to the quadrate of the bisegment . 5 p ij .

The third section doth follow , from whence the fifth reason ariseth .

8. If a right line be cut into equall parts ; and continued ; the oblong made of the continued and the continuation , with the quadrate of the bisegment or halfe , is equall to the quadrate of the line compounded of the bisegment and continuation . 6 p ij .

From hence ariseth the Mesographus or Mesolabus of Heron the mechanicke ; so named of the invention of two lines continually proportionall betweene two lines given . Whereupon arose the Deliacke probleme , which troubled Apollo himselfe . Now the Mesographus of Hero is an infinite right line , which is stayed with a scrue-pinne , which is to be moved up and downe in riglet . And it is as Pappus saith , in the beginning of his 111 booke , for architects most fit , and more ready than the Plato's mesographus . The mechanicall handling of this mesographus , is described by Eutocius at the 1 Theoreme of the 11 booke of the spheare ; But it is somewhat more plainely and easily thus layd downe by us .

9. If the Mesographus , touching the angle opposite to the angle made of the two lines given , doe cut the said two lines given , comprehending a right angled parallelogramme , and infinitely continued , equally distant from the center , the intersegments shall be the meanes continually proportionally , betweene and two lines given .

Or thus : If a Mesographus , touching the angle opposite to the angle made of the lines given , doe cut the equall distance from the center , the two right lines given , conteining a right angled parallelogramme , and continued out infinitely , the segments shall be meane in continuall proportion with the line given : H.

As let the two right-lines given be a e , and a i : And let them comprehend the rectangled parallelogramme a o : And let the said right lines given be continued infinitely , a e toward u ; and a i toward y. Now let the Mesographus u y , touch o , the angle opposite to a : And let it cut the sayd continued lines equally distant from the Center .

( The center is found by the 8 e iiij , to wit , by the meeting of the diagonies : For the equidistance from the center the Mesographus is to be moved up or downe , untill by the Compasses , it be found . )

Now suppose the points of equidistancy thus found to be u , and y. I say , That the portions of the continued lines thus are the meane proportionalls sought : And as a e is to i y : so is i y , to e u , so is e u , to a i.

The fourteenth Booke , of P. Ramus Geometry : Of a right line proportionally cut : And of other Quadrangles , and Multangels .

THus farre of the threefold section , from whence we have the five rationall rates of equality : There followeth of the third section another section , into two segments proportionall to the whole . The section it selfe is first to be defined .

1. A right line is cut according to a meane and extreame rate , when as the whole shall be to the greater segment ; so the greater shall be unto the lesser . 3. d vj.

This line is cut so , that the whole line it selfe , with the two segments , doth make the three bounds of the proportion● And the whole it selfe is first bound : The greater segment is the middle bound : The lesser the third bound .

2. If a right line cut proportionally be rationall unto the measure given , the segments are unto the same , and betweene themselves irrationall è 6 p xiij .

A Triangle , and all Triangulates , that is figures made of triangles , except a Rightangled-parallelogramme , are in Geometry held to be irrationalls . This is therefore the definition of a proportionall section : The section it selfe followeth , which is by the rate of an oblong with a quadrate .

3. If a quadrate be made of a right line given , the difference of the right line from the middest of the conterminall side of the said quadrate made , above the same halfe , shall be the greater segment of the line given proportionally cut : 11 p ij .

Or thus : If a square be made of a right line given , the difference of a right line drawne from the angle of the square made unto the middest of the next side , above the halfe of the side , shall be the greater segment of the line given , being proportionally cut : H.

For of y a , let the quadrate a y s r , be made : And let s r , be continued unto l. Now by the 8 e , xiij . the oblo●g of o y , and a y , with the quadrate of u a , is equall to the quadrate of u y , that is by the construction of u e : And therefore , by the 9 e xij . it is equall to the quadrates e a , and a u : Take away from each side the common oblong a l , and the quadrate y r , shall be equall to the oblong r i. Therefore the three right lines , e a , a r , and r e , by the 8 e xij . are continuall proportionall . And the right line a e , is cut proportionally .

Therefore

4 If a right line cut proportionally , be continued with the greater segment , the whole shall be cut proportionally , and the greater segment shall be the line given . 5 p xiij .

As in the same example , the right line o y , is continued with the greater segment , and the oblong of the whole and the lesser segment is equall to the quadrate of the greater . And thus one may by infinitely proportionally cutting increase a right line ; and againe decrease it . The lesser segment of a right line proportionally cut , is the greater segment , of the greater proportionally cut . And from hence a decreasing may be made infinitely .

5 The greater segment continued to the halfe of the whole , is of power quintuple unto the said halfe , that is , five times so great as it is : and if the power of a right line be quintuple to his segment , the remainder made the double of the former is cut proportionally , and the greater segment , is the same remainder . 1. and 2. p x iij.

This is the fabricke or manner of making a proportionall section . A threefold rate followeth : The first is of the greater segment .

The converse is apparent in the same example : For seeing that i o , is of power five times so much as is a o ; the gnomon l m n , shall be foure times so much as is u a : Whose quadruple also , by the 14. e xij , is a v. Therefore it is equall to the gnomon . Now a j , is equall to a e : Therefore it is the double also of a o , that is of a y : And therefore by the 24. e x. it is the double of a t : And therefore it is equall to the complements i y , and y s : Therefore the other diagonall y r , is equall to the other rectangle i v. Wherefore , by the 8 e xij . as e v , that is , a e , is to that t , that is a i : so is a i , unto i e ; Wherefore by the ● e , a e , is proportionall cut : And the greater segment is a i , the same remaine .

The other propriety of the quintuple doth follow .

6 The lesser segment continued to the halfe of the greater , is of power quintuple to the same halfe è 3 p x iij.

The rate of the triple followeth .

7 The whole line and the lesser segment are in power treble unto the greater . è 4 p xiij .

8 An obliquangled parallelogramme is either a Rhombus , or a Rhomboides .

9 A Rhombus is an obliquangled equilater parallelogramme 32 dj .

It is otherwise of some called a Diamond .

10 A Rhomboides is an obliquangled parallelogram●e not equilater 33. dj .

And a Rhomboides is so opposed to an oblong , as a Rhombus is to a quadrate .

And the Rhomboides is so called as one would say Rhombuslike , although beside the inequality of the angles it hath nothing like to a Rhombus . An example of measuring of a Rhombus is thus .

11 A Trapezium is a quadrangle not parallelogramme . 34. dj .

The examples both of the figure and of the measure of the same let these be .

Therefore triangulate quadrangles are of this sort .

12 A multangle is a figure that is comprehended of more than foure right lines . 23. dj .

By this generall name , all other sorts of right lined figures hereafter following , are by Euclide comprehended , as are the quinquangle , sexangle , septangle , and such like inumerable taking their names of the number of their angles .

In every kinde of multangle , there is one ordinate , as we have in the former signified , of which in this place we will say nothing , but this one thing of the quinquangle . The rest shall be reserved untill we come to Adscription .

13 Multangled triangulates doe take their measure also from their triangles .

14 If an equilater quinquangle have three sides equall , it is equiangled . 7 p 13.

This of some , from the Greeke is called a Pentagon ; of others a Pentangle , by a name partly Greeke partly Latine .

The fifteenth Booke of Geometry , Of the Lines in a Circle .

AS yet we have had the Geometry of rectilineals : The Geometry of Curvilineals , of which the Circle is the chiefe , doth follow .

1. A Circle is a round plaine . ● 15 dj .

The meanes to describe a Circle , is the same , which was to make a Periphery : But with some difference : For there was considered no more but the motion , the point in the end of the ray describing the periphery : Here is considered the motion of the whole ray , making the whole plot conteined within the periphery .

A Circle of all plaines is the most ordinate figure , as was before taught at the 10 e iiij .

2 Cir●les are as the quadrates or squares made of their diameters 2 p. x ij .

Therefore

3. The Diameters are , as their peripheries Pappus , 5 l x j , and 26 th . 18.

As here thou seest in a e , and i o.

4. Circular Geometry is either in Lines , or in the segments of a Circle .

This partition of the subject matters howsoever is taken for the distinguishing and severing with some light a matter somewhat confused ; And indeed concerning lines , the consideration of secants is here the foremost , and first of Inscripts .

5. If a right line be bounded by two points in the periphery , it shall fall within the Circle . 2 p iij.

From hence doth follow the Infinite section , of which we spake at the 6 e j.

This proposition teacheth how a Rightline is to be inscribed in a circle , to wit , by taking of two points in the periphery .

6. If from the end of the diameter , and with a ray of it equal to the right line given , a periphery be described , a right line drawne from the said end , unto the meeting of the peripheries , shall be inscribed into the circle , equall to the right line given . 1 p iiij .

And this proposition teacheth , How a right line given is to be inscribed into a Circle , equall to a line given .

Moreover of all inscripts the diameter is the chiefe : For it sheweth the center , and also the reason or proportion of all other inscripts . Therefore the invention and making of the diameter of a Circle is first to be taught .

7. If an inscript do cut into two equall parts , another inscript perpendicularly , it is the diamiter of the Circle , and the middest of it is the center . 1 p iij.

The cause is the same , which was of the 5 e x j. Because the inscript cut into halfes if for the side of the inscribed rectangle , and it doth subtend the periphery cut also into two parts ; By the which both the Inscript and Periphery also were in like manner cut into two equall parts : Therefore the right line thus halfing in the diameter of the rectangle : But that the middle of the circle is the center , is m●nifest out of the 7 e v , and 29 e iiij .

Euclide , thought better of Impossibile , than he did of the cause : And thus he forceth it . For if y be not the Center , but s , the part must be equall to the whole : For the Triangle a o s , shall be equilater to the triangle e o s. For a o , oe , are equall by the grant : Item s a , and s e , are the rayes of the circle : And s o , is common to both the triangles . Therefore by the 1 e vij , the angles no each side at o are equall ; And by the 13 e v , they are both right angles . Therefore s o e is a right angle ; It is therefore equall by the grant , to the right angle y o e , that is , the part is equall to the whole , which is impossible . Wherefore y is not the Center . The same will fall out of any other points whatsoever ●ut of y.

Therefore

8. If two r●ght lines doe perpendicularly halfe two inscripts , the meeting of these two bisecants shall be the Center of the circle è 25 p iij.

And one may

9. Draw a periphery by three points , which doe not fall in a right line .

10. If a diameter doe halfe an inscript , that is , n●t a diameter , it doth cut it perpendicularly : And contrariwise : 3 p iij.

11. If inscripts which are not diameters doe cut one another , the segments shall be unequall . 4 p iij.

But rate hath beene hitherto in the parts of inscripts : Proportion in the same parts followeth .

12 If two inscripts doe cut one another , the rectangle of the segments of the one is equall to the rectangle of the segments of the other . 35 p iij.

And this is the comparison of the parts inscripts . The rate of whole inscripts doth follow , the which whole one diameter doth make :

13 Inscripts are equall distant from the center , unto which the perpendiculars from the center are equall 4 d iij.

14. If inscripts be equall , they be equally distant from the center : And contrariwise . 13 p iij.

The diameters in the same circle , by the 28 e iiij● are equall : And they are equally distant from the center , seeing they are by the center , or rather are no whit at all distant from it : Other inscripts are judged to be equall , greater , or lesser one than another , by the diameter , or by the diameters center .

Euclide doth demonstrate this proposition thus : Let first a e and i o be equall ; I say they are equidistant from the center . For let u y , and u y , be perpendiculars : They shall cut the assigned a e , & i o , into halfes , by the 5 e xj : And y a and s i a●e equall , because they are the halfes of equals . Now let the raies of the circle be u a , aund u i : Their quadrates by the 9 e xij , are equall to the paire of quadrates of the shankes , which paires are therefore equall betweene themselves . Take from equalls the quadrates y a , and s i , there shall remaine y u and u s , equalls : and therefore the sides are equall , by the 4 e 12.

The converse likewise is manifest : For the perpendiculars given do halfe them : And the halfes as before are equall .

15 Of unequall inscripts the diameter is the greatest : And that which is next to the diameter , is greater than that which is farther off from it : That which is farthest off from it , is the least : And that which is next to the least , is lesser than that which is farther off : And those two onely which are on each side of the diameter are equall è 15 e iij.

This proposition consisteth of five members : The first is , The diameter is the greatest iuscript : The second , That which is next to the diameter is greater than that which is farther off : The third , That which is farthest off from the diameter is the least : The fourth , That next to the least is lesser , than that farther off : The fifth , That two onely on each side of the diameter are equall betweene themselves . All which are manifest out of that same argument of equalitie , that is the center the beginning of decreasing , and the end of increasing . For looke how much farther off you goe from the center , or how much nearer you come unto it , so much les●er or greater doe you make the inscript .

But Euclides conclusion is by triangles of two sides greater than the other ; and of the greater angle .

The first part is plaine thus : Because the diameter a e , is equall to i l , and l o , viz. to the raies : And to those which are greater than i o , the base by the 9. e v j &c.

The second part of the nearer , is manifest by the 5 e vij . because of the triangle i l o , equicrurall to the triangle u l y , is greater in angle : And therefore it is also greater in base .

The third and fourth are consectaries of the first .

The fifth part is manifest by the second : For if beside i o , and s r , there be supposed a third equall , the same also shall be unequall , because it shall be both nearer and farther off from the diameter .

16 Of right lines drawne from a point in the diameter which is not the center unto the periphery , that which passeth by the center is the greatest : And that which is nearer to the greatest , is greater than that which is farther off : The other part of the greatest is the lest . And that which is nearest to the least , is lesser than that which is farther off : And two on each side of the greater or least are only equall . 7 p iij.

The third , that a y , is lesser than a u , because s y , which is equall to s u , is lesser than the right lines s a , and a u , by the 9 e v j : And the common s a , being taken away , a y shall be left , lesser than a u.

The fourth part followeth of the third .

The fifth let it be thus : s r , making the angle a s r , equall to the angle a s u , the bases a u , and a r , shall be equall by the 2 e v ij . To these if the third be supposed to be equall , as a l , it would follow by the 1 e v ij . that the whole angle s a , should be equall to r s a , the particular angle , which is impossible . And out of this fifth part issueth this Consectary .

Therefore

17 If a point in a circle be the bound of three equall right lines determined in the periphery , it is the center of the circle . 9 p iij.

Let the point a , in a circle be the common bound of three right lines , ending in the periphery and equall betweene themselves , be a e , a i , a u. I say this point is the center of the Circle .

18 Of right lines drawne from a point assigned without the periphery , unto the concavity or hollow of the same , that which is by the center is the greatest ; And that next to the greatest , is greater than that which is farther off : But of those which fall upon the convexiti● of the circumference , the segment of the greatest is least● And that which is next unto the least is lesser than that is farther off : And two on each side of the greatest or least are onely equall . 8 piij.

19 If a right line be perpendicular unto the end of the diameter , it doth touch the periphery : And contrariwise è 16 p iij.

As for example , Let the circle given a e , be perpendicular to the end of the diameter , or the end of the ray , in the end a , as suppose the ray be i a : I say , that e a , doth touch , not cut the periphery in the common bound a.

Therefore

20 If a right line doe passe by the center and touch-point , it is perpendicular to the tangent or touch-line . 18 p iij.

And

Or thus , as Schoner amendeth it : If a right line be the diameter by the touch point , it is perpendicular to the tangent .

21 If a right line be perpendicular unto the tangent , it doth passe by the center and touch-point . 19. piij.

Or thus : if it be perpendicular to the tangent , it is a diameter by the touch point : Schoner .

For a right line either from the center unto the touch-point ; or from the touch point unto the center is radius or semidiameter .

And

22 The touch-point is that , into which the perpendicular from the center doth fall upon the touch line .

23 A tangent on the same side is onely one .

Or touch line is but one upon one , and the same side : H. Or. A tangent is but one onely in that point of the periphery Schoner .

Euclide propoundeth this more specially thus ; that no other right line may possibly fall betweene the periphery and the tangent .

And

24 A touch-angle is lesser than any rectilineall a●ute angle , è 16 p ij .

Angulus contractus , A touch angle is an angle of a straight touch-line and a periphery . It is commonly called Angulus contingentiae : Of Proclus it is named Cornicularis , an horne-like corner● because it is made of a right line and periphery like unto a horne . It is lesse therefore than any acute or sharpe right-lined angle : Because if it were not lesser , a right line might fall between the periphery and the perpendicular .

And

25 All touch-angles in equall peripheries are equall .

But in unequall peripheries , the cornicular angle of a lesser periphery , is greater than the Cornicular of a greater .

26 If from a ray out of the center of a periphery given , a periphery be described unto a point assigned without , and from the meeting of the assigned and the ray , a perpendicular falling upon the said ray unto the now described periphery , be tied by a right line with the said center , a right line drawne from the point given unto the meeting of the periphery given , and the knitting line shall touch the assigned periphery 17 p iij.

Thus much of the Secants and Tangents severally : It followeth of both kindes joyntly together .

27 If of two right lines , from an assigned point without , the first doe cut a periphery unto the concave , the other do touch the same ; the oblong of the secant , and of the outter segment of the secant , is equall to the quadrate of the tangent : and if such a like oblong be equall to the quadrate of the other , that same other doth touch the periphery : 36 , and 37 . p iij.

Therefore

28. All tangents falling from the same point are equall .

Or , Touch lines drawne from one and the same point are equall : H.

Because their quadrates are equall to the same oblong .

And

29. The oblongs made of any secant from the same point , and of the outter segment of the secant are equall betweene themselves . Camp . 36 p iij.

The reason is because to the same thing .

And

30. To two right lines given one may so continue or joyne the third , that the oblong of the continued and the continuation may be equall to the quadrate remaining . Vitellio 127 p j.

As in the first figure , if the first of the lines given be e o , the second i a , the third o a.

Now are we come to Circular G●ometry , that is to the Geometry of Circles or Peripheries cut and touching one another : And of Right lines and Peripheries .

31. If peripheries doe either cut or touch one another , they are eccentrickes : And they doe cut one another in two points onely , and these by the touch point doe continue their diameters , 5. 6. 10,11 , 12 p iij.

All these might well have beene asked : But they have also their demonstrations , ex impossibili , not very dissicult .

Of right lines and Peripheries joyntly the rate is but one .

32. If inscripts be equall , they doe cut equall peripheries : And contrariwise , 28,29 p iij.

Or thus : If the inscripts of the same circle or of equall circles be equall , they doe cut equall peripheries : And contrariwise B.

Or thus ; If lines inscribed into equall circles or to the same be equall , they cut equall peripheries : And contrariwise , if they doe cut equall peripheries , they shall themselves be equall : Schoner●

Except with the learned Rodulphus Snellius , you doe understand aswell two equall peripheries to be given , as two equall right lines , you shall not conclude two equall sections , and therefore we have justly inserted of the same , or of equall Circles ; which we doe now see was in like manner by Lazarus Schonerus .

The sixteenth Booke of Geometry , Of the Segments of a Circle .

1. A Segment of a Circle is that which is comprehended outterly of a periphery , sand innerly of a r●ght line .

THe Geometry of Segments is common also to the spheare : But now this same generall is hard to be declared and taught : And the segment may be comprehended within of an oblique line either single or manifold . But here we follow those things that are usuall and commonly received . First therefore the generall definition is set formost , for the more easie distinguishing of the species and severall kindes .

2. A segment of a Circle is either a sectour , or a s●ction .

Segmentum a segment , and Sectio a section , and Sector a sectour , are almost the same in common acceptation , but they shall be distinguished by their definitions .

3. A Sectour is a segment innerly comprehended of two right lines , making an angle in the center ; which is called an angle in the center : As the periphery is , the base of the sectour , 9 d iij.

4. An angle in the Periphery is an angle comprehended of two right lines inscribed ,

and joyntly bounded or meeting in the periphery . 8 d iij.

This might have beene called The Sectour in the ●eriphery , to wit , comprehended innerly of two right lines joyntly bounded in the periphery ; as here a e i.

5. The angle in the center , is double to the angle of the periphery standing upon the same base , 20 p iij.

Therefore

6. If the angle in the periphery be ●quall to the angle in the center , it is double to it in base . And contrariwise .

This followeth out of the former element : For the angle in the center is double to the angle in the periphery standing upon the same base : Wherefore if the angle in the periphery be to be made equall to the angle in the center , his base is to be doubled , and thence shall follow the equality of them both : S.

7. The angles in the center or periphery of equall circles , are as the Peripheries are upon which they doe insist : And contrariwise . è 33 p vj , and 26 , 27 p iij.

Here is a double proportion with the periphery underneath , of the angles in the center : And of angles in the periphery . But it shall suffice to declare it in the angles in the center .

First therefore let the Angles in the center a e i , and o u y , be equall : The bases a i , and o y , shall be equall , by the 11 e vij : And the peripheries a i , and o y , by the 32 e x v , shall likewise be equall . Therefore if the angles be unequall , the peripheries likewise shall be equall .

The same shall also be true of the Angles in the Periphery . The Converse in like manner is true : From whence followeth this consectary :

Therefore

8. As the sectour is unto the sectour , so is the angle unto the angle : And Contrariwise .

And thus much of the Sectour .

9. A section is a segment of a circle within comprehended of one right line , which is termed the base of the section .

As here , a e i , and o u y , and s r l , are sections .

10. A section is made up by finding of the center .

11 The periphery of a section is divided into two equall parts by a perpendicular dividing the base into two equall parts . 20. p iij.

Here Euclide doth by congruency comprehende two peripheries in one , and so doe we comprehend them .

12 An angle in a section is an angle comprehended of two right lines joyntly bounded in the base and in the periphery joyntly bounded 7 d iij.

Or thus : An angle in the section , is an angle comprehended under two right lines , having the same tearmes with the bases , and the termes with the circumference : H. As a o e , in the former example .

13 The angles in the same section are equall . 21. p iij.

Here it is certaine that angles in a section are indeed angles in a periphery , and doe differ onely in base .

14 The angles in opposite sections are equall to two right angles . 22. p iij.

The reason or rate of a section is thus : The similitude doth follow .

15 If sections doe receive [ or containe ] equall angles , they are alike e 10. d iij.

16 If like sections be upon an equall base , they are equall : and contrariwise . 23,24 . p iij.

In the first figure , let the base be the same . And if they shall be said to unequall sections ; and one of them greater than another , the angle in that a o e , shall be lesse than the angle a i e , in the lesser section , by the 16 e vj. which notwithstanding , by the grant , is equall .

In the second figure , if one section be put upon another , it will agree with it : Otherwise against the first part , like sections upon the same base , should not be equall . But congruency is here sufficient .

By the former two propositions , and by the 9 e x v. one may finde a section like unto another assigned , or else from a circle given to cut off one like unto it .

17 An angle of a section is that which is comprehended of the bounds of a section .

18 A section is either a semicircle : or that which is unequall to a semicircle .

A section is two fold , a semicircle , to wit , when it is cut by the diameter : or unequall to a semicircle , when it is cut by a line lesser than the diameter .

19 A semicircle is the halfe section of a circle .

Or it is that which is made the diameter .

Therefore

20 A semicircle is comprehended of a periphery and the diameter 18 dj .

21 The angle in a semicircle is a right angle : The angle of a semicircle is lesser than a rectilineall right angle : But greater than any acute angle : The angle in a greater section is lesser than a right angle : Of a greater , it is a greater . In a lesser it is greater : Of a lesser , it is lesser , ê 31 . and 16. p iij.

Or thus : The angle in a semicircle is a right angle , the angle of a semicircle is lesse than a right rightlined angle , but greater than any acute angle : The angle in the greater section is lesse than a right angle : the angle of the greater section is greater than a right angle : the angle in the lesser section is greater than a right angle , the angle of the lesser section , is lesser than a right angle : H.

The second part , That the angle of a

semicircle is lesser than a right angle ; is manifest out of that , because it is the part of a right angle . For the angle of the semicircle a i e , is a part of the rectilineall right angle a i u.

The third part , That it is greater than any acute angle ; is manifest out of the 23. e x v. For otherwise a tangent were not on the same part one onely and no more .

The fourth part is thus made manifest : The angle at i , in the greater section a e i , is lesser than a right angle ; because it is in the same triangle a e i , which at a , is right angle . And if neither of the shankes be by the center , notwithstanding an angle may be made equall to the assigned in the same section .

The fifth is thus : The angle of the greater section e a i , is greater than a right angle ; because it containeth a right-angle .

The sixth is thus , the angle a o e , in a lesser section , is greater than a right angle , by the 14 e x v j. Because that which is in the opposite section , is lesser than a right angle .

The seventh is thus . The angle e a o , is lesser than a right-angle : Because it is part of a right angle , to wit of the outter angle , if i a , be drawne out at length

And thus much of the angles of a circle , of all which the most effectuall and of greater power and use is the angle in a semicircle , and therefore it is not without cause so often mentioned of Aristotle . This Geometry therefore of Aristotle , let us somewhat more fully open and declare . For from hence doe arise many things .

Therefore

22 If two right lines jointly bounded with the diameter of a circle , be jointly bounded in the periphery , they doe make a right angle .

Or thus : If two right lines , having the same termes with the diameter , be joyned together in one point , of the circomference , they make a right angle . H.

This corollary is drawne out of the first part of the former Element , where it was said , that an angle in a semicircle is a right angle .

And

23 If an infinite right line be cut of a periphery of an externall center , in a point assigned and contingent , and the diameter be drawne from the contingent point , a right line from the point assigned knitting it with the diameter , shall be perpendicular unto the infinite line given .

Let the infinite right line be a e , from whose point a , a perpendicular is to be raised .

And

24 If a right line from a point given , making an acute angle with an infinite line , be made the diameter of a periphery cutting the infinite , a right line from the point assigned knitting the segment , shall be perpendicular upon the infinite line .

As in the same example , having an externall point given , let a perpendicular unto the infinite right line a e , be sought : Let the right line i o e , be made the diameter of the peripherie ; and withall let it make with the infinite right line giyen an acute angle in e , from whose bisection for the center , let a periphery cut the infinite , &c.

And

25 If of two right lines , the greater be made the diameter of a circle , and the lesser jointly bounded with the greater and inscribed , be knit together , the power of the greater shall be more than the power of the lesser by the quadrate of that which knitteth them both together . ad 13 p. x.

26 If a right line continued or continually made of two right lines given , be made the diameter of a circle , the perpendicular from the point of their continuation unto the periphery , shall be the meane proportionall betweene the two lines given . 13 p vj.

So if the side of a quadrate of 10. foote content , were sought ; let the sides 1 , foote and 10 , foote an oblong equall to that same quadrate , be continued ; the meane proportionall shall be the side of the quadrate , that is , the power of it shall be 10. foote . The reason of the angles in opposite sections doth follow .

27 The angles in opposite sections are equall in the alterne angles made of the secant and touch line . 32. p iij.

As let the unequall sections be e i o , and e a o : the tangent let it be u e y : And the angles in the opposite sections , e a o , and e i o. I say they are equall in the alterne angles of the secant and touch line o e y , and o e u. First that which is at a , is equall to the alterne o e y : Because also three angles o e y , o e a , and a e u , are equall to two right angles , by the 14 e v. Vnto which also are equall the three angles in the triangle a e o , by the 13 e vj. From three equals take away the two right angles a u e , and a o e : ( For a o e , is a right angle , by the 21 e ; because it is in a semicircle : ) Take away also the common angle a e o : And the remainders e a o , and o e y , alterne angles , shall be equall .

Therefore

28 If at the end of a right line given a right lined angle be made equall to an angle given , and from the toppe of the angle now made , a perpendicular unto the other side do meete with a perpendicular drawn from the middest of the line given , the meeting shall be the center of the circle described by the equalled angle , in whose opposite section the angle upon the line given shall be made equall to the assigned è 33 p iij.

And

29 If the angle of the secant and touch line be equall to an assigned rectilineall angle , the angle in the opposite section shall likewise be equall to the same . 34. piij.

Of Geometry the seventeenth Booke , Of the Adscription of a Circle and Triangle .

HItherto we have spoken of the Geometry of Rectilineall plaines , and of a circle : Now followeth the Adscription of both : This was generally defined in the first book 12 e. Now the periphery of a circle is the bound therof . Therefore a rectilineall is inscribed into a circle , when the periphery doth touch the angles of it 3 d iiij . It is circumscribed when it is touched of every side by the periphery ; 4 d iij.

1. If a rectilineall ascribed unto a circle be an equilater , it is equiangle .

Of the circumscript it is likewise true , if the circumscript be understood to be a circle . For the perpendiculars from the center a , unto the sides of the circumscript , by the 9e xij , shal make triangles on each side equilaters , & equiangls , by drawing the semidiameters unto the corners , as in the same example .

2. It is equall to a triangle of equall base to the perimeter , but of heighth to the perpendicular from the center to the side .

As here is manifest , by the 8 e vij . For there are in one triangle , three triangles of equall heighth .

The same will fall out in a Triangulate , as here in a quadrate : For here shal be made foure triangles of equall height .

Lastly every equilater rectilineall ascribed to a circle , shall be equall to a triangle , of base equall to the perimeter of the adscript . Because the perimeter conteineth the bases of the triangles , into the which the rectilineall is resolved .

3. Like rectilinealls inscribed into circles , are one to another as the quadrates of their diameters , 1 p. x i j.

In like Triangulates , seeing by the 4 e x , they may be resolved into like triangles , the same will fall out .

Therefore

4. If it be as the diameter of the circle is unto the side of rectilineall inscribed , so the diameter of the second circle be unto the side of the second rectilineall inscribed , and the severall triangles of the inscripts be alike and likely situate , the rectilinealls inscribed shall be alike and likely situate .

This Euclide did thus assume at the 2 p xij , and indeed as it seemeth out of the 18 p vj. Both which are conteined in the 23 e iiij . And therefore we also have assumed it .

Adscription of a Circle is with any triangle : But with a triangulate it is with that onely which is ordinate : And indeed adscription of a Circle is common to all .

5. If two right lines doe cut into two equall parts two angles of an assigned rectilineall , the circle of the ray from their meeting perpendicular unto the side , shall be inscribed unto the assigned rectilineall . 4 and 8. p. iiij .

The same argument shall serve in a Triangulate .

6. If two right lines do right anglewise cut into two equall parts two sides of an assigned rectilineall , the circle of the ray from their meeting unto the angle , shall be circumscribed unto the assigned rectilineall . 5 p iiij .

As in the former figures . The demonstration is the same with the former . For the three rayes , by the 2 e vij , are equall : And the meeting of them , by the 17 ex , is the center .

And thus is the common adscription of a circle : The adscription of a rectilineall followeth , and first of a Triangle .

7. If two inscripts , from the touch point of a right line and a periphery , doe make two angles on each side equall to two angles of the triangle assigned be knit together , they shall inscribe a triangle into the circle given , equiangular to the triangle given è 2 p iiij .

The circumscription here is also speciall .

8 If two angles in the center of a circle given , be equall at a common ray to the outter angles of a triangle given , right lines touching a periphery in the shankes of the angles , shall circumscribe a triangle about the circle given like to the triangle given . 3 p iiij .

Therefore

9. If a triangle be a rectangle , an obtusangle , an acute angle , the center of the circumscribed triangle is in the side , out of the sides , and within the sides : And contrariwise . 5 e iiij .

As , thou seest in these three figures underneath , the center a.

Of Geometry , the eighteenth Booke , Of the adscription of a Triangulate .

SVch is the Adscription of a triangle : The adscription of an ordinate triangulate is now to be taught . And first the common adscription , and yet out of the former adscription , after this manner .

1. If right lines doe touch a periphery in the angles of the inscript ordinate triangulate , they shall unto a circle circumscribe a triangulate homogeneall to the inscribed triangulate .

The examples shall be laid downe according as the species or severall kindes doe come in order . The speciall inscription therefore shall first be taught , and that by one side , which reiterated , as oft as need shall require , may fill up the whole periphery . For that Euclide did in the quindecangle one of the kindes , we will doe it in all the rest .

2. If the diameters doe cut one another right-anglewise , a right line subtended or drawne against the right angle , shall be the side of the quadrate . è 6 p iiij .

Therefore

3. A quadrate inscribed is the halfe of that which is circumscribed .

Because the side of the circumscribed ( which here is equall to the diameter of the circle ) is of power double , to the side of the inscript , by the 9 e x i j.

An●

4. It is greater than the halfe of the circumscribed Circle .

Because the circumscribed quadrate , which is his double , is greater than the whole circle .

For the inscribing of other multangled odde-sided figures we must needes use the helpe of a triangle , each of whose angles at the base is manifold to the other : In a Quinguangle first , that which is double unto the remainder , which is thus found .

5. If a right line be cut proportionally , the base of that triangle whose shankes shall be equall to the whole line cut , and the base to the greater segment of the same , shall have each of the angles at the base double to the remainder : And the base shall be the side of the quinquangle inscribed with the triangle into a circle . 10 , and 11. p i i i j.

6 If two right lines doe subtend on each side two angles of an inscript quinquangle , they are cut proportionally , and the greater segments are the sides of the said inscript è 8 , p x iij.

And from hence the fabricke or construction of an ordinate quinquangle upon a right line given , is manifest .

Therefore

7 If a right line given , cut proportionall , be continued at each end with the greater segment , and sixe peripheries at the distance of the line given shall meete , two on each side from the ends of the line given and the continued , two others from their meetings , right lines drawne from their meetings , & the ends of the assigned shall make an ordinate quinquangle upon the assigned .

8 If the diameter of a circle circumscribed about a quinquangle be rationall , it is irrationall unto the side of the inscribed quinquangle . è 11. p xiij .

So before the segments of a right line proportionally cut were irrationall .

The other triangulates hereafter multiplied from the ternary , quaternary , or quinary of the sides , may be inscribed into a circle by an inscript triangle , quadrate , or quinquangle . Therefore by a triangle there may be inscribed a triangulate of 6. 12,24,46 , angles : By a quadrate , a triangulate of 8. 16,32,64 . angles . By a quinquangle , a triangulate of 10 , 20. 40,80 . angles &c.

9 The ray of a circle is the side of the inscript sexangle è 15 p iiij .

Therefore

10 Three ordinate sexangles doe fill up a place .

Furthermore also no one figure amongst the plaines doth fill up a place . A Quinquangle doth not : For three angles a quinquangle may make only 3 ●/5 angles which is too little . And foure would make 4 ●/5. which is as much too great . The angles of a septangle would make onely two rightangles , and 6/7 of one : Three would make 3 , and 9/7 , that is in the whole 4. 2/7 , which is too much , &c. to him that by induction shall thus make triall , it will appeare , That a plaine place may be filled up by three sorts of ordinate plaines onely .

And

11 If right lines from one angle of an inscript sexangle unto the third angle on each side be knit together , they shall inscribe an equilater triangle into the circle given .

12 The side of an inscribed equilater triangle hath a treble power , unto the ray of the circle 12. p xiij .

13 If the side of a sexangle be cut proportionally , the greater segment shall be the side of the decangle .

Therefore

14 If a decangle and a sexangle be inscribed in the same circle , a right line continued and made of both sides , shall be cut proportionally , and the greater segment shall be the side of a sexangle ; and if the greater segment of a right line cut proportionally be the side of an hexagon , the rest shall be the side of a decagon . 9. p xiij .

The comparison of the decangle and sexangle with the quinangle followeth .

15 If a decangle , a sexangle , and a pentangle be inscribed into the same circle the side of the pentangle shall in power countervaile the sides of the others . And if a right line inscribed do countervaile the sides of the sexangle and decangle , it is the side of the pentangle . 10. p xiiij .

Let the proportion of this syllogisme be demonstrated : For this part onely remaineth doubtfull . Therefore two triangles a e i , and the e i , are equiangles , having one common angle at e : And also two equall ones a e i , and e i y , the halfes , to wit , of the same e i s : Because that is , by the 17 e , vj : one of the two equalls , unto the which e i s , the out angle , is equall , by the 15 e. vj. And this doth insist upon a halfe periphery . For the halfe periphery a l s , is equall to the halfe periphery a r s : and also a l , is equall to a r. Therefore the remnant l s , is equall to the remnant r s : And the whole r l , is the double of the same r s : And therefore e r , is the double of e o : And r s , the double of o u. For the bisegments are manifest by the 10 e , xv . and the 11 e , xvj . Therefore the periphery e r s , is the double of the periphery e o u : And therefore the angle e i u , is the halfe of the angle e i s , by the 7 e , xvj . Therefore two angles of two triangles are equall : Wherefore the remainder , by the 4 e vij , is equall to the remainder . Wherefore by the 12 e , vij , as the side a e , is to e i : so is e i , to e y. Therefore by the 8 e xij , the oblong of the extreames is equall to the quadrate of the meane .

Now let o y , be knit together with a straight : Here againe the two triangles a o e , and a o y , are equiangles , having one common angle at a : And a o y , and o e a , therefore also equall : Because both are equall to the angle at a : That by the 17 e , vj : This by the 2 e , vij : Because the perpendicular halfing the side of the decangle , doth make two triangles , equicrurall , and equall by the right angle of their shankes : And therefore they are equiangles . Therefore as e a , is to a o : so is e a , to a y. Wherefore by the 8 e , xij . the oblong of the two extremes is equall to the quadrate of the meane : And the proposition of the syllogisme , which was to be demonstrated . The converse from hence as manifest Euclide doth use at the 16 p xiij .

16. If a triangle and a quinquangle be inscribed into the same Circle at the same point , the right line inscribed betweene the bases of the both opposite to the sayd point , shall be the side of the inscribed quindecangle . 16. p. iiij .

Therefore

17. If a quinquangle and a sexangle be inscribed into the same circle at the same point , the periphery intercepted betweene both their sides , shall be the thirtieth part of the whole periphery .

Of Geometry the ninteenth Booke ; Of the Measuring of ordinate Multangle and of a Circle .

OVt of the Adscription of a Circle and a Rectilineall is drawne the Geodesy of ordinate Multangles , and first of the Circle it selfe . For the meeting of two right lines equally , dividing two angles is the center of the circumscribed Circle : From the center unto the angle is the ray : And then if the quadrate of halfe the side be taken out of the quadrate of the ray , the side of the remainder shall be the perpendicular , by the 9 e xij . Therefore a speciall theoreme is here thus make :

1. A plaine made of the perpendicular from the center unto the side , and of halfe the perimeter , is the content of an ordinate multangle .

In an ordinate Sexangle also the ray , by the 9 e xviij , is knowne by the side of the sexangle● As here , the quadrate of 6 , the ray is 36. The quadrate of 3 , the halfe of the side , is 9● And 36 — 9. are 27 , for the quadrate of the Perpendicular , whose side 5 12/11 is the perpendicular it selfe . Now the whole perimeter , as you see , is 36. Therefore the halfe is 18. And the product of 18 by 5 2/11 is 93 3/11 for the content of the sexangle given .

Lastly in all ordinate Multangles this theoreme shall satisfie thee .

2 The periphery is the triple of the diameter and almost one seav●nth part of it .

Therefore

3. The plaine of the ray , and of halfe the periphery is the content of the circle .

And

4. As 14 is unto 11 , so is the quadrate of the diameter unto the Circle .

For here 3 bounds of the proportion are given in potentia : The fourth is found by the multiplication of the third by the second , and by the Division of the product by the first : As here the Quadrate of the diameter 14 , is 196. The product of 196 by 11 is 2156. Lastly 2156 divided by 14 , the first bound , giveth in the Quotient 154 , for the content of the circle sought . This ariseth by an analysis out of the quadrate and Circle measured . For the reason of 196 , unto a 154 ; is the reason of 14 unto 11 , as will appeare by the reduction of the bounds .

This is the second manner of squaring of a circle taught by Euclide as Hero telleth us , but otherwise layd downe , namely after this manner . If from the quadrate of the diameter you shall take away 3/14 parts of the same , the remainder shall be the content of the Circle . As if 196 , the quadrate be divided by 14 , the quotient likewise shall be 14. Now thrise 14 , are 42 : And 196 — 42 , are 154 , the quadrate equall to the circle .

Out of that same reason or rate of the pheriphery and diameter ariseth the manner of measuring of the Parts of a circle , as of a Semicircle , a Sector , a Section , both greater and lesser .

And

5. The plaine of the ray and one quarter of the periphery , is the content of the semicircle .

This may also be done by taking of the halfe of the circle now measured .

And

6. The plaine made of the ray and halfe the base , is the content of the Sector .

And

7. If a triangle , made of two raies and the base of the greater section , be added unto the two sectors in it , the whole shall be the content of the greater section : If the same be taken from his owne sector , the remainder shall be the content of the lesser .

In the former figure the greater section is a e i : The lesser is a i. The base of them both is as you see , 6. The perpendicular from the toppe of the triangle , or his heighth is 4. Therefore the content of the triangle is 12. Wherefore 30 — 30 — 12 , that is 72 , is the content of the greater section a e i. And the lesser sectour , as in the former was taught , is 184/7. Therefore 184/7 — 12 , that is , 6 4/7 , is the content of a i , the lesser section .

And

8. A circle of unequall isoperimetrall plaines is the greatest .

Of Geometry the twentieth Booke , Of a Bossed surface .

1. A bossed surface is a surface which lyeth unequally betweene his bounds .

IT is contrary unto a Plaine surface , as wee heard at the 4 e v.

2. A bossed surface is either a sphericall , or varium .

Therefore

4 It is made by

the turning about of an halfe circumference the diameter standeth still . è 14 d x j.

As here if thou shalt conceive the space betweene the periphery and the diameter to be empty .

5. The greatest periphery in a sphericall surface is that which cutteth it into two equall parts .

Those things which were before spoken of a circle , the same almost are hither to bee referred . The greatest periphery of a sphericall doth answere unto the Diameter of a Circle .

Therefore

6. That periphery that is neerer to the greatest , is greater than that which is farther off : And on each side those two which are equally distant from the greatest , are equall .

7 The plaine made of the greatest periphery and his diameter is the sphericall .

Therefore

8 A plaine of the greatest circle and 4 , is the sphericall .

This consectarium is manifest out of the former element .

And

9 As 7 is to 22. so is the quadrate of the diameter unto the sphericall .

For 7 , and 22 , are the two least bounds in the reason of the diameter unto the periphery : But in a circle , as 14 , is to 11. so is the quadrate of the diameter unto the circle . The analogie doth answer fitly : Because here thou multipliest by the double , and dividest by the halfe : There contrariwise thou multipliest by the halfe , and dividest by the double . Therefore there one single circle is made , here the quadruple of that . This is , therefore the analogy of a circle and sphericall ; from whence ariseth the hemispherical , the greater and the lesser section .

And

10 The plaine of the greatest periphery and the ray , is the hemisphericall .

As here , the greatest periphery is 44. the ray 7. The product therefore of 44. by 7. that is , 308. is the hemisphericall .

11 If looke what the part be of the ray perpendicular from the center unto the base of the greater section , so much the hemisphericall be increased , the whole shall be the greater section of the sphericall : But if it be so much decreased , the remainder shall be the lesser .

12 The varium is a bossed surface , whose base is a periphery , the side a right line from the bound of the toppe , unto the bound of the base .

13 A varium is a conicall or a cylinder like forme .

Therefore

15 It is made by turning about of the side about the periphery beneath .

16 The plaine of the side and halfe the base is the conicall surface .

As in the example next afore going , the side is 13. The halfe periphery is 15 5/7 : And the product of 15. 5/7 . by 13. is 204. 2/7 . for the conicall surface . To which if you shall adde the circle underneath , you shall have the whole surface .

Therefore

18 It is made by the turning of the side about two equall and parallell peripheries .

19 The plaine of his side and heighth is the cylinder like surface .

As here the periphery is 22. as is gathered by the Diameter , which is 7. The heighth is 12. The base therefore is 38. ½ . And 38 . ½ by 12. are 462. for the cylinderlike surface . To which if you shall adde both the bases on each side , to wit , 38. ½ . twise , or 77. once , the whole surface shall be 539.

Geometry , the one and twentieth Book . Of Lines and Surfaces in solids●

1 A body or solid is a lineate broad and high 1 d xj .

2 The bound of a solid is a surface 2 d xj .

The bound of a line is a point : and yet neither is a point a line , or any part of a line . The bound of a surface is a line : And yet a line is not a surface , or any part of a surface . So now the bound of a body is a surface : And yet a surface is not a body , or any part of a body . A magnitude is one thing ; a bound of a magnitude is another thing , as appeared at the 5 e j.

As they were called plaine lines , which are conceived to be ●● a plaine , so those are named solid both lines and surfaces which are considered in a solid ; And their perpendicle and parallelisme are hither to be recalled from simple lines .

3 If a right line be unto right lines cut in a plaine underneath , perpendicular in the common intersection , it is perependicular to the plaine beneath : And if it be perpendicular , it is unto right lines , cut in the same plaine , perpendicular in the common intersection è 3 d and 4 pxj.

If thou shalt conceive the right lines , a e , i o , u y , to cut one another in the plaine beneath , in the common intersections : And the line r s , falling from above , to be to every one of them perpendicular in the common point s , thou hast an example of this consectary .

4 If three right lines cutting one another , be unto the same right line perpendicular in the common section , they are in the same plaine 5. p x j.

For by the perpendicle and common section is understood an equall state on all parts , and therefore the same plaine : as in the former example , a s , y s , o s , suppose them to be to s r , the same loftie line , perpendicular , they shall be in the same nearer plaine a i u e o y.

5 If two right lines be perpendicular to the underplaine , they are parallells : And if the one of two parallells be perpendicular to the under plaine , the other is also perpendicular to the same . 6.8 p xj .

6 If right lines in diverse plaines be unto the same right line parallell , they are also parallell betweene themselves . 9 p xj .

7 If two right lines be perpendiculars , the first from a point above , unto a right line underneath , the second from the common section in the plaine ●nderneath , a third , from the sayd point perpendicular to the second , shall be perpendicular to the plaine beneath . è 11 p xj .

If the right line i o , doe with equall angles agree to r , the third element .

8. If a right line from a point assigned of a plaine underneath , be parallell to a right line perpendicular to the same plaine , it shall also be perpendicular to the plaine underneath . e x 12 p xj .

9. If a right line in one of the plaines cut , perpendicular to the common section , be perpendicular to the other , the plaines are perpendicular : And if the plaines be perpendicular , a right line in the one perpendicular to the common section is perpendicular to the other è 4 d , and 38 p xj .

10. If a right line be perpendicular to a plaine , all plaines by it , are perpendicular to the same : And if two plaines be unto any other plaine perpendiculars , the common section is perpendicular to the same . e 15 , and 19 p. xj .

11. Plaines are parallell which doe leane no way . 8 d x j.

And

12. Those which divided by a common perpendicle . 14 p xj .

It is also out of the definition of parallels , at the 17 e i j.

And

13. If two paires of right in them be joyntly bounded , they are parallell . 15 p xj .

The same will fall out if thou shalt imagine the joyntly bounded to infinitely drawn out ; for the plaines also infinitely extended shall be parallell●

14. If two parallell plaines are cut with another plaine , the common sections are parallels , 16 p xj .

The twenty second Booke , of P. Ramus Geometry , Of a Pyramis .

1. The axis of a solid is the diameter about which it is turned , e 15,19,22 d x j.

2. A right solid is that whose axis is perpendicular to the center of the base .

Thus Serenus and Apollonius doe define a Cone and a Cylinder : And these onely Euclide considered : Yea and indeed stereometry entertaineth no other kinde of solid but that which is right or perpendicular .

3. If solids be comprehended of homogeneall surfaces , equall in multitude and magnitude , they are equall . 10 d x j.

Equality of lines and surfaces was not informed by any peculiar rule ; farther than out of reason and common sense , and in most places congruency and application was enough and did satisfie to the full : But here the congruency of Bodies is judged by their surfaces . Two cubes are equall , whose sixe sides or plaine surfaces , are equall , &c.

4. If solids be comprehended of surfaces in multitude equall and like , they are equall , 9 d x j.

This is a consectary drawne out of the generall difinition of like figures , at the 19 e. iiij . For there like figures were defined to be equiangled and proportionall in the shankes of the equall angles : But in like plaine solids the angles are esteemed to be equall out of the similitude of their like plaines : And the equall shankes are the same plaine surfaces , and therefore they are proportionall , equall and alike .

5 Like solids have a treble reason of their homologall sides , and two meane proportionalls . 33. p xj . 8 p xij .

It is a consectary drawne out of the 24 e. iiij . as the example from thence repeated shall make manifest .

6 A solid is plaine or embosed .

7 A plaine solid is that which is comprehended of plaine surfaces .

8 The plaine angles comprehending a solid angle , are lesse than foure right angles . 21. p x j .

For if they should be equall to foure right angles , they would fill up a place , by the 22 e , vj. neither would they at all make an angle , much lesse therefore would they doe it if they were greater .

9 If three plaine angles lesse than foure right angles , do comprehend a solid angle , any two of them are greater than the other● And if any two of them be greater than the other , then may comprehend a solid angle , 21. and 23. p xj .

The converse from hence also is manifest .

Euclide doth thus demonstrate it : First if three angles are equall , then by and by two are conceived to be greater than the remainder . But if they be unequall , let the angle a e i , be greater than the angle a e o : And let a e u , equall to a e o , be cut off from the greater a e i : And let e u , be equall to e o. Now by the 2 e , vij . two triangles a e u , and a e o , are equall in their bases a u , and a o. Item a o , and e i , are greater than a i , and a o : And a o , is equall to a u. Therefore o i , is greater than i u. Here two triangles , u e i , and i e o , equall in two shankes ; and the base o i , greater than the base i u. Therefore , by the 5 e vij . the angle o e i , is greater than the angle i e u. Therefore two angles a e o , and o e i , are greater than a e i.

10 A plaine solid is a Pyramis or a Pyramidate .

11 A Pyramis is a plaine solid from a rectilineall base equally decreasing .

As here thou conceivest from the triangular base a e i , unto the toppe o , the triangles a o e , a o i , and e o i , to be reared up .

Therefore

12 The sides of a pyramis are one more than are the base .

The sides are here named Hedrae .

And

13 A pyramis is the first figure of solids .

For a pyramis in solids , is as a triangle is in plaines . For a pyramis may be resolved into other solid figures , but it cannot be resolved into any one more simple than it selfe , and which consists of fewer sides than it doth .

Therefore

14 Pyramides of equall heighth , are as their bases are 5 e , and 6. p xij .

And

15 Those which are reciprocall in base and heighth are equall 9 p xij .

These consectaries are drawne out of the 16 , 18 e. iiij .

16 A tetraedrum is an ordinate pyramis comprehended of foure triangles 26. d xj .

Therefore

17 The edges of a tetraedrum are sixe , the plaine angles twelve , the solide angles foure .

For a Tetraedrum is comprehended of foure triangles , each of them having three sides , and three corners a peece : And every side is twise taken : Therefore the number of edges is but halfe so many .

And

18 Twelve tetraedra's doe fill up a solid place .

Because 8. solid right angles filling a place , and 12. angles of the tetraedrum are equall betweene themselves , seeing that both of them are comprehended of 24. plaine right-angles . For a solid right angle is comprehended of three plaine right angles : And therefore 8. are comprehended of 24. In like manner the angle of a Tetraedrum is comprehended of three plaine equilaters , that is of sixe third of one right angle : and therefore of two right angles : Therefore 12 are comprehended of 24.

And

19 If foure ordinate and equall triangles be joyned together in solid angles , they shall comprehend a tetraedrum .

20. If a right line whose power is sesquialter unto the side of an equilater triangle , be cut after a double reason , the double segment perpendicular to the center of the triangle , knit together with the angles thereof shall comprehend a tetraedrum . 13 p xiij .

For a solid to be comprehended of right lines understand plaines comprehended of right lines , as in other places following .

The twenty third Booke of Geometry , of a Prisma ,

1 A Pyramidate is a plaine solid comprehended of pyramides .

2. A pyramidate is a Prisma , or a mingled polyedrum .

3. A prisma is a pyramidate whose opposite plaines are equall , alike , and parallell , the rest parallelogramme . 13 dxj.

Therefore

4. The flattes of a prisma are two more than are the angles in the base .

And indeed as the augmentation of a Pyramis from a quaternary is infinite : so is it of a Prisma from a quinary : As if it be from a triangular , quadrangular , or quinquangular base ; you shal have a Pentaedrum , Hexaedrum , Heptaedrum , and so in infinite .

5. The plaine of the base and heighth is the solidity of a right prisma .

6. A prisma is the triple of a pyramis of equall base and heighth . è 7 p. x i j.

If the base be triangular , the Prisma may be resolved into prisma's of triangular bases , and the theoreme shall be concluded as afore .

Therefore

7. The plaine made of the base and the third part of the heighth is the solidity of a pyramis of equall base and heighth .

So in the example following , Let 36 , the quadrate of 6 the ray , be taken out of 292 9/1156 the quadrate of the side 17 3/34 the side 16 3/34 of 256 9/1156 the remainder shall be the height , whose third part is 5 37/102 ; the plaine of which by the base 72 ¼ shall be 387 11/24 for the solidity of the pyramis given .

After this manner you may measure an imperfect Prisma :

8. Homogeneall Prisma's of equall heighth are one to another as their bases are one to another , 29 , 30,31 , 32 p xj .

This element is a consectary out of the 16 e iiij .

And

9. If they be reciprocall in base and heighth , they are equall .

This is a Consectary out the 18 e iiij .

And

10. If a Prisma be cut by a plaine parallell to his opposite flattes , the segments are as the bases are . 25 p. xj .

11. A Prisma is either a Pentaedrum , or Compounded of pentaedra's .

Here the resolution sheweth the composition .

12 If of two pentaedra's , the one of a triangular base , the other of a parallelogramme base , double unto the triangular , be of equall heighth , they are equall 40. p xj .

The canse is manifest and briefe : Because they be the halfes of the same prisma : As here thou maist perccive in a prisma cut into two halfes by the diagoni's of the opposite sides .

Euclide doth demonstrate it thus : Let the Pentaedra's a e i o u , and y s r l m , be of equall heighth : the first of a triangular base e i o : The second of a parallelogramme base s l , double unto the triangular . Now let both of them be double and made up , so that first be n● The second y s r l v f. Now againe , by the grant , the base s l , is the double of the base e i o , : whose double is th● base e o , by the 12 e x. Therefore the bases s l , and e o , are equall : And therefore seeing the prisma's , by the grant , here are of equall heighth , as the bases by , the conclusion are equall , the prisma's are equall ; And therefore also their halfes a e i o u , and y s n l r , are equall .

The measuring of a pentaedrall prisma was even now generally taught : The matter in speciall may be conceived in these two examples following .

The plaine of 18. the perimeter of the triangular base , and 12 , the heighth is 216. This added to the triangular base , 15 18/3● . or 15 , ⅗ , almost twise taken , that is , 31 ⅕ , doth make 247 ⅕ . for the summe of the whole surface . But the plaine of the same base 15 ⅖ , and the heighth 12. is 187 ⅕ , for the whole solidity .

So in the pentaedrum , the second prisma , which is called Cuneus , ( a wedge ) of the sharpnesse , and which also more properly of cutting is called a prisma , the whole surface is 150 , and the solidity 90.

13 A prisma compounded of penta●dra's , is either an Hexaedrum or Polyedrum : And the Hexaedrum is either a Parallelepipedum or a Trapezium .

14 A parallelepipedum is that whose opposite plaines are parallelogrammes ê 24. p xj .

Therefore a Parallelepipedum in solids , answereth to a Parallelogramme in plaines . For here the opposite Hedrae or flattes are parallell : There the opposite sides are parallell .

Therefore

15 It is cut into two halfes with a plaine by the diagonies of the opposite sides . 28 p xj . It answereth to the 34. pj.

And

16 If it be halfed by two plaines halfing the opposite sides , the common bisection and diagony doe halfe one another 39. p xj .

17 If three lines be proportionall , the parallelepipedum of meane shall be equall to the equiangled p●rallelepipedum of all them . è 36. p x j.

It is a consectary out of the 8 e.

18 Eight rectangled parallelepiped's doe fill a solid place .

19 The Figurate of a rectangled parallelepipedum is called a solid , made of three numbers 17. d vij .

As if thou shalt multiply 1,2,3 . continually , thou shalt make the solid 6. Item if thou shalt in like manner multiply 2,3,4 . thou shalt make the solid 24. And the sides of that solid 6 solid shall be 1,2,3 . Of 24 , they shall be 2,3,4 .

Therefore

20 If two solids be alike , they have their sides proportionalls , and two meane proportionalls 21 d vij , 19. 21. p viij .

It is a consectary out of the 5 e xxij . But the meane proportionalls are made of the sides of the like solids , to wit , of the second , third , and fourth : Item of the third , fourth , and fifth , as here tho● seest .

Of Geometry the twentie fourth Book . Of a Cube .

1 A Rightangled parallelepipedum is either a Cube , or an Oblong .

2 A Cube is a right angled parallelepipedum of equall flattes , 25. d xj .

As here thou seest in these two figures .

Therefore

3 The sides of a cube are 12. the plaine angles 24. the solid 8.

Therefore

4 If sixe equall quadrates be joyned with solid angles , they shall comprehend a cube .

As here in these two examples .

Therefore

5 If from the angles of a quadrate , perpend●culars equall to the sides be tied together aloft , they shall comprehend a Cube . è 15 p xj .

It is a consectary following upon the former consectary : For then shall sixe equall quadrates be knit together :

6 The diagony of a Cube is of treble power unto the side .

For the Diagony of a quadrate is of double power to the side , by the 12 e , xij . And the Diagony of a Cube is of as much power as the side the diagony of the quadrate , by the same e. Therefore it is of treble power to the side .

7 If of foure right lines continually proportionally the first be the halfe of the fourth , the cube of the first shall be the halfe of the Cube of the second è 33 p xj .

It is a consectary out of the 25 e , iiij . From hence Hippocrates first found how to answer Apollo's Probleme .

8 The solid plaine of a cube is called a Cube , to wit , a solid of equall sides . 19 , d vij .

Therefore

9 It is made of a number multiplied into his owne quadrate .

So is a Cube made by multiplying a number by it selfe , and the product againe by the first . Such are these nine first cubes made of the nine first Arithmeticall figures .

This is the generall invention of a Cube , both Geometricall and Arithmeticall .

10 If a right line be cut into two segments , the Cube of the whole shall be equall to the Cubes of the segments , and a double solid thrice comprehended of the quadrate of his owne segment and the other segment .

As for example , the side 12 , let it be cut into two segments 10 and 2. The cube of 12. the whole , which is 1728 , shall be equall to two cubes 1000 , and 8 made of the segments 10. and 2. And a double solid ; of which the first 600. is thrise comprehended of 100. the quadrate of his segment 10. and of 2. the other segment : The second 120. is thrice comprehended of 4 , the quadrate of his owne segment , and of 10. the other segment . Now 1000 — 600 — 120. — 8 , is equall to 1728 : And therefore a right &c.

Therefore

11. The side of the first severall cube is the other side of the second solide : And the quadrate of the same side is the other side of the first solide , whose other side is the side of the second cube ; and the quadrate of the same other side is the other side of the second solid .

In that equation therefore of foure solids with one solid , thou shalt consider a peculiar making and composition : First that the last cube be made of the last segment 2● Then that the second solid of 4 , the quadrate of his owne segment , and of 10 , the other segment be thrise comprehended : Lastly that the first solid of 100 , the square of his owne segment 10 and the other segment 2 , be also thrice comprehended : Lastly , that the Cube 1000 , be made of the greater segment 10. Out of this making &c.

The plaine of the perimeter of the base 20 , and the altitude 5 is 100. This added to 25 and 25 , both the bases that is to 50 , maketh 150 , for the whole surface . Now the plaine of 25 the base , and the heighth 5 is 125 , for the whole solidity .

So in the Oblong , the plaine of the base's perimeter 20 , and the heighth 11 , is 220 , which added to the bases 24 and 24 , that is 48 , maketh 268 , for the whole surface . But the plaine of the base 24 , and the height 11 , is 264 , for the solidity .

Thus are such kinde of walls whether of mudde , bricke , or stone , of most large houses to bee measured . The same manner of Geodesy is also to be used in the measuring of a Rhombe , Rhomboides , Trapezium or mensall , and any kinde of multangled body . The base is first to be measured , as in the former : Then out of that and the heighth the solidity shall be manifested : As in the Rhombe the base is 24 , the heighth 4. Therefore the solidity is 96.

In the Rhomboides , the base is 64 3●/12● : The heigh 11. Therefore the solidity is 1028 44/1●9 .

The same is the geodesy of a trapezium , as in these examples : The surface of the first is 198 : The solidity 192 ½ .

The surface of the second is 158 3/49 ; The solidity is 91 29/4● .

And from hence also may the capacity or content of vessells or measures , made after any manner of plaine solid bee esteemed and judged of as here thou seest . For here the plaine of the sexangular base is 41 1/7 ; ( For the ray , by the 9 e xviij , is the fide : ) and the heighth 5 , shall be 205 5/7. Therefore if a cubicall foote doe conteine 4 quarters , as we commonly call them , then shall the vessell conteine 822 6/7 quartes , that is almost 823 quartes .

Of Geometry the twenty fifth Booke ; Of mingled ordinate Polyedra's .

1. A mingled ordinate polyedrum is a pyramidate , compounded of pyramides with their toppes meeting in the center , and their bases onely outwardly appearing .

SEeing therefore a Mingled ordinate pyramidate is thus made or compounded of pyramides the geodesy of it shall be had from the Geodesy of the pyramides compounding it : And one Base multiplyed by the number of all the bases shall make the surface of the body . And one Pyramis by the number of all the pyramides ; shall make the solidity .

2 The heighth of the compounding pyramis is found by the ray of the circle circumscribed about the base , and by the semidiagony of the polyedrum .

The base of the pyramis appeareth to the eye : The heighth lieth hidde within , but it is discovered by a right angle triangle , whose base is the semidiagony or halfe diagony , the shankes the ray of the circle , and the perpendicular of the heighth . Therefore subtracting the quadrate of the ray , from the quadrate of the half● diagony the side of the remainder , by the 9 e x ij . shall be the heighth . But the ray of the circle shall have a speciall invention , according to the kindes of the base , first of a triangular , and then next of a quinquangular .

3 A mingled ordinate polyedrum hath either a triangular , or a quinquangular base .

The division of a Polyhedron ariseth from the bases upon which it standeth .

4 If a quadrate of a triangular base be divided into three parts , the side of the third part shall be the ray of the circle circumscribed about the base .

As is manifest by the 12 e. xviij . And this is the invention or way to finde out the circular ray for an octoedrum , and an icosoedrum .

5 A mingled ordinate polyedrum of a triangular base , is either an Octoedrum , or an Icosoedrum .

This division also ariseth from the bases of the figures .

6 An octoedrum is a mingled ordinate polyedrum , which is comprehended of eight triangles . 27 d xj .

As here thou seest , in this Monogrammum and solidum , that is lined and solid octahedrum .

Therefore

7 The sides of an octoedrum are 12. the plaine angles 24 , and the solid 6.

And

8 Nine octoedra's doe fill a solid place .

For foure angles of a Tetraedrum are equall to three angles of the Octoedrum : And therefore 12. are equall to nine . Therefore nine angles of an octaedrum doe countervaile eight solid right angles .

And

9 If eight triangles , equilaters and equall be joyned together by their edges ; they shall comprehend an octaedrum .

10 If a right line on each side perpendicular to the center of a quadrate and equall to the halfe diagony be tied together with the angles , it shall comprehend an octaedrum , 14. d xiij .

Therefore

11 The Diagony of an octaedrum is of double power to the side .

As is manifest by the 9 e xij .

And

12 If the quadrate of the side of an octaedrum , be doubled , the side of the double shall be the diagony .

As in the figure following , the side is 6. The quadrate 36. the double is 72. whose side 8 8/17. is the diagony .

13 An Icosaedrum is an ordinate polyedrum comprehended of 20 triangles 29 d xj .

Therefore

14 The sides of an Icosaedrum are 30. the plaine angles 60. the solid 12.

15 If twentie ordinate and equall triangles be joyned with solid angles , they shall comprehend an Icosaedrum .

16 If ordinate figures , to wit , a double quinquangle , and one decangle be so inscribed into the same circle , that the side of both the quinquangle doe subtend two sides of the decangle , sixe right lines perpendicular to the circle and equall to his ray , five from the angles of one of the quinquangles , knit together both betweene themselves , and with the angles of the other quinquangle ; the sixth from the center on each side continued with the side of the decangle , and knit therewith the five perpendiculars , here with the angles of the second quinquangle , they shall comprehend an icosaedrum . è 15 p xiij .

In like manner also shall it be proved of the five upper triangles , by drawing the right lines d y , and c n , which as afore ( because they knit together equal parallells , to wit , d c , and y n ) they shall be equall . But d y , is the side of a sexangle : Therefore c n , shall be also the side of a sexangle : And c g , is the side of a decangle : Therefore a n , whose power is equall to both theirs , by the 9 e xij . shall by the converse of the 15 e xviij , be the side of a quinquangle : And in like manner g t , shall be concluded to be the side of a quinquangle . Wherefore n g t , is an equilater : And the foure other shall likewise be equilaters .

The other five triangles beneath shall after the like manner be concluded to be equilaters . Therefore one shall be for all , to wit , i b e , by drawing the raies d i , and d e. For i b , whose power , as afore , is asmuch as the sides of the sexangle , and decangle , shall be the side of the quinquangle : And in like sort b e , being of equall power with d e , and d o , the sides of the sexangle and decangle , shall be the side of the quinquangle . Wherefore the triangle e b i , is an equilater : And the foure other in like manner may be shewed to be equilaters . Therefore all the side of the twenty triangles , seeing they are equall , they shall be equilater triangles : And by the 8 e , vij . equall .

17 The diagony of an icosaedrum is irrational unto the side .

This is the fourth example of irrationality , or incommensurability . The first was of the Diagony and side of a square or quadrate . The second was of the segments of a line proportionally cut . The third of the Diameter of a circle and the side of a quinquangle .

And

18 The power of the diagony of an icosaedrum is five times as much as the ray of the circle .

19. A mingled ordinate polyedrum of a quinquangular base is that which is comprehended of 12 quinquangles , and it is called a Dodecaedrum .

Therefore

20. The sides of a Dodecaedrum are 30 , the plaine angles 60. the solid 20.

And

21. If 12 ordinate equall quinquangles be joyned with solid angles , they shall comprehend a Dodecaedrum .

As here thou seest .

22. If the sides of a cube be with right lines cut into two equall parts , and three bisegments of the bisecants in the abbuting plaines , neither meeting one the other , nor parallell one unto another , two of one , the third of that next unto the remainder , be so proportionally cut that the lesser segments doe bound the bisecant : three lines without the cube perpendicular unto the sayd plaines from the points of the proportionall sections , equall to the greater segment knit together , two of the same bisecant , betweene themselves and with the next angles of cube ; the third with the same angles , they shall comprehend a dodecaedrum . 17 p xiij .

I say also that it is a Plaine quinquangle : For it may be said to be an oblique quinquangle ; and to be seated in two plaines . Let therefore f h be parallell to d b , and c p : and be equall unto them . And let h z , be drawne : This h z shall be cut one line , by the 14 e vij . For as the whole tr , that is r f , is unto the greater segment that is to f h : so f h , that is z g , is unto g r. And two paire of shankes f h , g r , f c , g z , by the 6 e xxj , are alternely or crosse-wise parallell . Therefore their bases are continuall .

Hitherto it hath beene prooved that the quinquangle made is an equilater and plaine : It remaineth that it bee prooved to be Equiangled . Let therefore the right lines e p , and e c , be drawne : I say that the angles , p b e , and e z i , are equall : Because they have by the construction , the bases of equall shankes equall , being to wit in value the quadruple of l e. For the right line l f , cut proportionally , and increased with the greater segment d f , that is f c , is cut also proportionally , by the 4 e xiiij , and by the 7 e xiiij , the whole line proportionally cut , and the lesser segment , that is c p , are of treble value to the greater f l , that is of the sayd l e. Therefore e l , and l c , that is e c , and c p , that is e p , is of quadruple power to e l : And therefore by the 14 e xij , it is the double of it : And e i , it selfe in like manner , by the fabricke or construction , is the double of the same . Therefore the bases are equall . And after the same manner , by drawing the right lines i d , and i b , the third angle b p i , shall be concluded to be equall to the angle e z i. Therefore by the 13 e xiiij , five angles are equall .

23. The Diagony is irrationall unto the side of the dodecahedrum .

This is the fifth example of irrationality and incommensurability . The first was of the diagony and side of a quadrate or square . The second was of a line proportionally cut and his segments : The third is of the diameter of a Circle and the side of an inscribed quinquangle . The fourth was of the diagony and side of an icosahedrum . The fifth now is of the diagony and side of a dodecahedrum .

24 If the side of a cube be cut proportionally , the greater segment shall be the side of a dodecahedrum .

The semidiagony and ray of the circle thus found , the altitude remaineth . Take out therefore the quadrate of the ray of the circle , 16 4/225 out of the quadrate of the semidiagony 47. 12458 / 17161. the side of the remainder 3● 2●14406/3861225 is for the altitude or heighth : whose ⅓ is 5/3. The quinquangled base is almost 38. Which multiplied by 5/3 doth make 63 ⅓ for the solidity of one Pyramis ; which multiplied by 12 , doth make 760. for the soliditie of the whole dodetacedrum .

25 There are but five ordinate solid plaines .

This appeareth plainely out of the nature of a solid angle , by the kindes of plaine figures . Of two plaine angles a solid angle cannot be comprehended . Of three angles of an ordinate triangle is the angle of a Tetrahedrum comprehended : Of foure , an Octahedrum : Of five , an Icosahedrum : Of sixe none can be compr●hended : For sixe such like plaine angles , are equall to 12 thirds of one right angle , that is to foure right angles . But plaine angles making a solid angle , are lesser than foure right angles , by the 8 e xxij . Of seven therefore , and of more it is , much lesse possible . Of three quadrate angles the angle of a cube is comprehended : Of 4. such angles none may be comprehended for the same cause . Of three angles of an ordinate quinquangle , is made the angle of a Dodecahedrum . Of 4. none may possibly be made : For every such angle : For every one of them severally doe countervaile one right angle and 1/5 of the same , Therefore they would be foure , and three fifths . Of more therefore much lesse may it be possible .

This demonstration doth indeed very accurately and manifestly appeare , Although there may be an innumerable sort of ordinate plaines , yet of the kindes of angles five onely ordinate bodies may be made ; From whence the Tetrahedrum , Octahedrum , and Icosahedrum are made upon a triangular base : the Cube upon a quadrangular : And the Dodecahedrum , upon a quinquangular .

Of Geometry the twenty sixth Booke ; Of a Spheare .

1 AN imbossed solid is that which is comprehended of an imbossed surface .

2. And it is either a spheare or a Mingled forme .

3. A spheare is a round imbossement .

Therefore

4. A Spheare is made by the conversion of a semicircle , the diameter standing still . 14 d xj .

As here thou seest :

5. The greatest circle of a spheare , is that which cutteth the spheare into two equall parts .

Therefore

6. That circle which is neerest to the greatest , is greater than that which is farther off .

And

7. Those which are equally distant from the greatest are equall .

As in the example above written .

8. The plaine of the diameter and sixth part of the sphearicall is the solidity of the spheare .

Therefore

9. As 21 is unto 11 , so is the cube of the diameter unto the spheare .

As here the Cube of 14 is 2744. For it was an easy matter for him that will compare the cube 2744 , with the spheare , to finde that 2744 to be to 1437 ⅓ in the least boundes of the same reason , as 21 is unto 11.

Thus much therefore of the Geode●y of the spheare : The geodesy of the Setour and section of the spheare shall follow in the next place .

And

10. The plaine of the ray , and of the sixth part of the sphearicall is the hemispheare .

But it is more accurate and preciser cause to take the halfe of the spheare .

11. Spheares have a trebled reason of their diameters .

So before it was told you ; That circles were one to another , as the squares of their diameters were one to another , because they were like plaines : And the diameters in circles were , as now they are in spheares , the homologall sides . Therefore seeing that spheres are figures alike , and of treble dimension , they have a trebled reason of their diameters .

12 The five ordinate bodies are inscribed into the same spheare , by the conversion of a semicircle having for the diameter , in a tetrahedrum , a right line of value sesquialter unto the side of the said tetrahedrum ; in the other foure ordinate bodies , the diagony of the same ordinate .

The adscription of ordinate plainebodies is unto a spheare , as before the Adscription plaine surfaces was into a circle ; of a triangle , I meane , and ordinate triangulate , as Quadrangle , Quinquangle , Sexangle , Decangle , and Quindecangle . But indeed the Geometer hath both inscribed and circumscribed those plaine figures within a circle . But these five ordinate bodies , and over and above the Polyhedrum the Stereometer hath onely inscribed within the spheare . The Polyhedrum we have passed over , and we purpose onely to touch the other ordinate bodies .

13 Out of the reason of the axeltree of the sphearicall the sides of the tetraedrum , cube , octahedrum and dodecahedrum are found out .

If the same axis be cut into two halfes , as in u : And the perpendicular u y , be erected : And y , and a , be knit together , the same y a , thus knitting them , shall be the side of the Octahedrum , as is manifest in like manner , by the said 10 e , viij . and 25 e iiij .

The side of the Icosahedrum is had by this meanes .

14 If a right line equall to the axis of the sphearicall , and to it from the end of the perpendicular be knit unto the center , a right line drawne from the cutting of the periphery unto the said end shall be the side of the Icosahedrum .

15 Of the five ordinate bodies inscribed into the same spheare , the tetrahedrum in respect of the greatnesse o● his side is first , the Octahedrum , the second ; the Cube , the third ; the Icosahedrum , the fourth ; and the Dodecahedrum , the fifth .

The latter , Euclide doth demonstrate with a greater circumstance . Therefore out of the former figures and demonstrations , let here be repeated , The sections of the axis first into a double reason in s : And the side of the sexangle r l : And the side of the Decangle a r , inscribed into the same circle , circumscribing the quinquangle of an icosahedrum : And the perpendiculars i s , and u l.

Here the two triangles a i e , and i e s , are by the 8 e , viij . alike ; And as s e , is unto e i : So is i e , unto e a : And by 25 e , iiij , as s e , is to e a : so is the quadrate of s e , to the quadrate of e i : And inversly or backward , as a e , is to s e : so is the quadrate of i e , to the quadrate of s e. But a e , is the triple of s e. Therefore the quadrate of i e , is the triple of s e. But the quadrate of a s , by the grant , and 14 e xij , is the quadruple of the quadrate of s e. Therefore also it is greater than the quadrate of i e : And the right line a s , is greater than i e , and a l , therefore is much greater . But a l , is by the grant compounded of the sides of the sexangle and decangle r l , and a r. Therefore by the 1 c. 5 e , 18. it is cut proportionally : And the greater segment is the side of the sexangle , to wit , r l : And the greater segment of i e , proportionally also cut , is the e. Therefore the said r l , is greeter than y e : And even now it was shewed u l , was equall to r l. Therefore u l. is greater than y e : But u e , the side of the Icosahedrum , by 22. e vj. is greater than u l. Therefore the side of the Icosahedrum is much greater , then the side of the dodecahedrum .

Of Geometry the twenty seventh Book ; Of the Cone and Cylinder .

1 A mingled solid is that which is comprehended of a variable surface and of a base .

FOr here the base is to be added to the variable surface .

2 If variable solids have their axes proportionall to their bases , they are alike . 24. d xj .

It is a Consectary out of the 19 e , iiij . For here the axes and diameters are , as it were , the shankes of equall angles , to wit , of right angles in the base , and perpendicular axis .

3 A mingled body is a Cone or a Cylinder .

The cause of this division of a varied or mingled body , is to be conceived from the division of surfaces .

4 A Cone is that which is comprehended of a conicall and a base .

Therefore

5 It is made by the turning about of a rightangled triangle , the one shanke standing still .

As it appeareth out of the definition of a variable body .

And

6 A Cone is rightangled , if the shanke standing still be equall to that turned about : It is Obtusangeld , if it be lesse : and acutangled , if it be greater . ê 18 d xj .

And

7 A Cone is the first of all variable .

For a Cone is so the first in variable solids , as a triangle is in rectilineall plaines : As a Pyramis is in solid plaines ; For neither may it indeed be divided into any other variable solids more simple .

And

8 Cones of equall heighth are as their bases are 11. p xij .

As here you see .

And

9 They which are reciprocall in base and heighth are equall , 15 p x ij .

These are consectaries drawne out of the 12 and 13 e , iiij . As here you see .

10 A Cylinder is that which is comprehended of a cyliudricall surface and the opposite bases .

Therefore

11 It is made by the turning about of a right angled parallelogramme , the one side standing still . 21. dxj.

As is apparant out the same definition of a varium .

12. A plaine made of the base and heighth is the solidity of a Cylinder .

This manner of measuring doth answeare , I say , to the manner of measuring of a prisma , and in all respects to the geodesy of a right angled parallelogramme .

If the cylinder in the opposite bases be oblique , then if what thou cuttest off from one base thou doest adde unto the other , thou shalt have the measure of the whole , as here thou seest in these cylinders , a and b.

As here the diameter of the inner Circle is 6 foote : The periphery is 18 6/7 : Therefore the plot or content of the circle is 28 2/7 Of which , and the heighth 10 , the plaine is 282 6/7 for the capacity of the vessell . Thus therefore shalt thou judge , as afore , how much liquour or any thing esle conteined , a cubicall foote may hold .

13. A Cylinder is the triple of a cone equall to it in base and heighth . 10 p xij .

The demonstration of this proposition hath much troubled the interpreters . The reason of a Cylinder unto a Cone , may more easily be assumed from the reason of a Prisme unto a Pyramis : For a Cylinder doth as much resemble a Prisme , as the Cone doth a Pyramis : Yea and within the same sides may a Prisme and a Cylinder , a Pyramis and a Cone be conteined : And if a Prisme and a Pyramis have a very multangled base , the Prisme and Clinder , as also the Pyramis and Cone , do seeme to be the same figure . Lastly within the same sides , as the Cones and Cylinders , so the Prisma and Pyramides , from their axeletrees and diameters may have the similitude of their bases . And with as great reason may the Geometer demand to have it granted him , That the Cylinder is the treble of a Cone● As it was demanded and granted him , That Cylinders and Cones are alike , whose axletees are proportionall to the diameters of their bases .

Therefore

14. A plaine made of the base and third part of the height , is the solidity of the cone of equall base & height ,

Of two cones of one common base is made Archimede's Rhombus , as here , whose geodaesy shall be cut of two cones .

And

15. Cylinder of equall heighth are as their bases are . 11 p xij .

And

16 Cylinders reciprocall in base and heighth are eequall . 15 p xij .

Both these affections are in common attributed to the equally manifold of first figures .

And

17. If a Cylinder be cut with a plaine surface parallell to his opposite bases , the segments are , as their axes are 13 p xij .

The unequall sections of a spheare we have reserved for this place : Because they are ●omprehended of a surface both sphearicall and conicall , as is the sectour . As also of a plaine and sphearicall , as is the section : And in both like as in a Circle , there is but a greater and lesser segment . And the sectour , as before , is considered in the center .

18. The sectour of a spheare is a segment of a spheare , which without is comprehended of a sphearicall● within of a conicall bounded in the center , the greater of a concave , the lesser of a convex .

Archimides , maketh mention of such kinde of Sectours , in his 1 booke of the Spheare . From hence also is the geodesy following drawne . And here also is there a certaine analogy with a circular sectour .

19. A plaine made of the diameter , and sixth part of the greater , or lesser sphearicall , is the greater or lesser sector .

And from hence lastly doth arise the solidity of the section , by addition and subduction .

20. If the greater sectour be increased with the internall cone , the whole shall be the greater section : If the lesser be diminished by it , the remaine shall be the lesser section .

As here the inner cone measured is 126 4/63. The greater sectour , by the former was 1026 ⅔ . And 126 ⅔ — 126 4/63 doe make 1152 46/63.

Againe the lesser sectour , by the next precedent , was 410 ⅔ : And here the inner cone is 126 4/63 And therefore 410 2/● — 126 4/63 that is 284 38/63 is the lesser section .

FINIS .