right line But many doe fall out to be in a crooked line And in a Spheare a Cone Cylinderâ— 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 Euâ—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 plainâ— 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 duceâ—et orbem Therfore 10. The raiâ—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 â—eriphery must bee understood the Ray the figure contained within the periphery 11. If two equall peripâ—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 â—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 â—â—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 infinâ—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 lâ—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 cuâ— 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 obliquaâ—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 wiâ— 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 anglesâ— 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 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 liâ—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 e i then y u 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 â…” 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 vâ— 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 themselvesâ— 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 baâ—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 Mâ— 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 â—holeâ— 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 alâ—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 shankeâ— 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 halâ—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 â—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
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 continâ—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 leaâ—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 parallelogrammesâ— 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 Measuresâ— 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
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 Gâ—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 Schonerâ— 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 râ—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 sâ—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 â—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 â—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 coÌ„prehended 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
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 nâ— 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 thâ— 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 3â— or 15 â…— almost twise taken that is 31 â…• doth make 247 â…• for the summe of the whole surface But the plaine of the same base 15 â…– and the heighth 12. is 187 â…• 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 pentaâ—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
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 basesâ— 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 kindâ— 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 figuâ—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 portionsâ— And so Bisegmentum to be one such portionâ— And Bisectio such a like cutting or division 30. Rounds of equall diameters are equall Out of the 1. d. iâ—â— 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 beiâ—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 diâ—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 suâ—face with foure liâ—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 conversioÌ„ 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 Lineâ— in straightnesse and obliquity or crookednesse Obliquum oblique there signified crooked Not righâ— or straight Here uneven or bowed either upward or downeward Sn. 5. A plaine surface is a surface which lyeth â—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
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 â—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 â— 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 â— 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 linesâ— the Arabians have much under the name of The rule of sixe quantities And the Theoremes of Althinâ—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 homâ—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
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 proportionâ— 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 obloâ—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 y t that
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 â—â— 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 â—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 parallellâ— 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 otherâ— 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
an inscribed quinquangle The Diagony of an â—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 uâ—â— 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
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 aud 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
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 3â— 2â—14406 3861225 is for the altitude or heighth whose â…“ is 5 3. The quinquangled base is almost 38. Which multiplied by 5 3 doth make 63 â…“ 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 comprâ—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 â…“ in the least boundes of the same reason as 21 is unto 11. Thus much therefore of the Geodeâ—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