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
e vii and so forth of the rest The middle triangles the equall angles being substracted shall have their other angles equall And therefore they also shall be equiangles and alike by the same Secondarily the triangles a e u and y s m e i o and s r l e o u and s l m to wit alike betweene themselves are by the 1 e vj in a double reason of their homologall sides e u s m e o s l which reason is the same by meanes of the common sides Therefore three triangles are in the same reason And therefore they are proportionall And by the third composition as one of the antecedents is unto one of the consequentsâ— so is the whole quinquangle to the whole 5. A triangulate is a Quadrangle or a Multangle The parts of this partition are in Euclide and yet without any shew of a division And here also as before the species or severall kinds have their denomination their angles although it had beene better and truer to have beene taken from their sides as to have beene called a Quadrilater or a Multilater But in words use must bee followed as a master 6. A Quadrangle is that which is comprehended of foure right lines 22 d j. 7. A quadrangle is a a Parallelogramme or a Trapezium This division also in his parts is in the Elements of Euclide but without any forme or shew of a division But the difference of the parts shall more fitly be distinguished thus Because in generall there are many common parallels 8. A Parallelogramme is a quadrangle whose opposite sides are parallell Therefore 9. If right lines on one and the same side doe joyntly bound equall and parallall lines they shall make a parallelogramme The reason is because they shall be equall and parallell betweene themselves by the 26. e v. And 10 A parallelogramme is equall both in his opposite sides and angles and segments cut by the diameter Or thus The opposite both sides and angles and segments cut by the diameter are equall Three things are here concluded The first is that the opposite sides are equall This manifest by the 26 e v. Because two right lines doe jointly bound equall parallells And 11 The Diameter of a parallelogramme is cut into two by equall raies As in the three figures a e i next before This a parallelogramme hath common with a circle as was manifest at the 28. e iiij And 12 A parallelogramme is the double of a triangle of a trinangle of equall base and heighth 41. p j. And 13 A parallelogramme is equall to a triangle of equall heighth and double base unto it è 42. p j. From whence one may 14 To a triangle given in a rectilineall angle given make an equall parallelogramme 15 A parallelogramme doth consist both of two diagoâ—als and complements and gnomons For these three parts of a parallelogramme are much used in Geometricall workes and businesses and therefore they are to be defined 16 The Diagonall is a particular parallelogramme having both an angle and diagonall diameter common with the whole parallelogramme 17 The Diagonall is like and alike situate to the whole parallelogramme è 24. p vj. There is not any either rate or proportion of the diagonall propounded onely similitude is attributed to it as in the same figure the Diagonall a u y s is like unto the whole parallelogramme a e i o. For first it is equianglar to it For the angle at a is common to them both And that is equall to that which is at y by the 10. e x And therefore also it is equall to that at i by the 10. e x. Then the angles a u y and a s y are equall by the 21. e v. to the opposite inner angles at e and o. Therefore it is equiangular unto it Againe it is proportionall to it in the shankes of the equall angles For the triangles a u y and a e i are alike by the 12 e vij because u y is parallell to the base Therefore as a u is to u y so is a i to e i Then as u y is to y a so is e i to i a. Againe by the 21 e v because s y is parallell to the base i o as a y is to y s so is a i to i o Therefore equiordinately as u y is to y s so is e i to i o Item as s y is to y a so is i o to i a And as y a is to a s so is i a to a o. Therefore equiordinately as y s is to s a so is i o to o a. Lastly as s a is unto a y so is o a unto a i And as a y is to a u so is a i unto a e. Therefore equiordinately as s a is to a u so is a o to a e. Wherefore the Diagonall s u is proportionall in the shankes of equall angles to the parallelogramme o e. The demonstration shall be the same of the Diagonall r l. The like situation is manifest by the 21 e iiij And from hence also is manifest That the diagonall of a Quadrate is a Quadrate Of an Oblong an Oblong Of a Rhombe a Rhombe Of a Rhomboides a Rhomboides because it is like unto the whole and a like situate Now the Diagonalls seeing they are like unto the whole and a like situate they shall also be like betweene themselves and alike situate one to another by the 21 and 22 e iiij Therefore 18. If the particular parallelogramme have one and the same angle with the whole be like and alike situate unto it it is the Diagonall 26 p vj. As for example Let the particular parallelogramme a u y s be coangular to the whole parallelogramme a e i o And let it have the same angle with it at a like unto the whole and alike situate unto it I say it is the Diagonall Otherwise let the diverse Diagony be a r o And let l r be parallell against a e Therefore a l r s shall bee the Diagonall by the 6 e 15. Now therefore it shall be by 8 e 16 e as e a is to a i so is s a unto a l Againe by the grant as e a is unto a i so is s a to a u Therefore the same s a is proportionall to a l and to a u And a l is equall to a u the part to the whole which is impossible 19. The Complement is a particular parallelogramme comprehended of the conterminall sides of the diagonals Or thus It is a particular parallelogramme conteined under the next adjoyning sides of the diagonals 20. The complements are equall 43 p j. Therefore 21. If one of the Complements be made equall to a triangle given in a right-lined angle given the other made upon a right line given shall be in like manner equall to the same triangle 44 p j. As if thou shouldest desire to have a parallelogramme upon a
right line given and in a right lined angle given to be made equall to a triangle given this proposition shall give satisfaction And 22 If parallelogrammes be continually made equall to all the triangles of an assigned triangulate in a right lined angle given the whole parallelogramme shall in like manner be equall to the whole triangulate 45 p j. This is a corollary of the former of the Reason or rate of a Parallelogramme with a Triangulate and it needeth no father demonstration but a ready and steddy hand in describing and working of it Here thou hast 3 complements continued and 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
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
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
it cannot be divided into a more simple solid figure although it may be divided into an infinite sort of other figures Of the Triangle all plaines are made as of a Pyramis all bodies or solids are compoundedâ— such are a e i. and a e i o. 12. A rationall figure is that which is comprehended of a base and height rationall betweene themselves So Euclide at the 1. d. ij saith that a rightangled parallelogramme is comprehended of two right lines perpendicular one to another videlicet one multiplied by the other For Geometricall comprehension is sometimes as it were in numbers a multiplication Therefore if yee shall grant the base and height to bee rationalls betweene themselves that their reason I meane may be expressed by a number of the assigned measure then the numbers of their â—ides being multiplyed one by another the bignesse of the figure shall be expressed Therefore a Rationall figure is made by the multiplying of two rationall sides betweene themselves Therefore 13. The number of a rationall figure is called a Figurate number And the numbers of which it is made the Sides of the figurate As if a Right angled parallelogramme be comprehended of the base foure and the height three the Rationall made shall be 12. which wee here call the figurate and 4. and 3. of which it was made we name sides 14. Isoperimetrall figures are figures of equall perimeter This is nothing else but an interpretation of the Greeke word So a triangle of 16. foote about is a isoperimeter to a triangle 16. foote about to a quadrate 16. foote about and to a circle 16. foote about 15. Of isoperimetralls homogenealls that which is most ordinate is greatest Of ordinate isoperimetralls heterogenealls that is greatest which hath most bounds So an equilater triangle shall bee greater then an isoperimeter inequilater triangle and an equicrurall greater then an unequicrurall so in quadrangles the quadrate is greater then that which is not a quadrate so an oblong more ordinate is greater then an oblong lesse ordinate So of those figures which are heterogeneall ordinates the quadrate is greater then the Triangle And the Circle then the Quadrate 16. If prime figures be of equall height they are in reason one unto another as their bases are And contrariwise Therefore 17. If prime figures of equall heighth have also equall bases they are equall The reason is because then those two figures compared have equall sides which doe make them equall betweene themselves For the parts of the one applyed or laid unto the parts of the other doe fill an equall place as was taught at the 10. e. j. Sn. So Triangles so Parallelogrammes and so other figures proposed are equalled upon an equall base 18. If prime figures be reciprocall in base and height they are equall And contrariwise 19. Like figures are equiangled figures and proportionall in the shankes of the equall angles First like figures are defined then are they compared one with another similitude of figures is not onely of prime figures and of such as are compounded of prime figures but generally of all other whatsoever This similitude consisteth in two things to witt in the equality of their angles and proportion of their shankesâ— Therefore 20. Like figures have answerable bounds subtended against their equall angles and equall if they themselves be equall Or thus They have their termes subtended to the equall angles correspondently proportionall And equall if the figures themselves be equall H. This is a consectary out of the former definition And 21. Like figures are situate alike when the proportionall bounds doe answer one another in like situation The second consectary is of situation and place And this like situation is then said to be when the upper parts of the one figure doe agree with the upper parts of the other the lower with the lower and so the other differences of places Sn. And 22. Those figures that are like unto the same are like betweene themselves This third consectary is manifest out of the definition of like figures For the similitude of two figures doth conclude both the same equality in angles and proportion of sides betweene themselves And 23. If unto the parts of a figure given like parts and alike situate be placed upon a bound given a like figure and likely situate unto the figure given shall bee made accordingly This fourth consectary teacheth out of the said definition the fabricke and manner of making of a figure alike and likely situate unto a figure given Sn. 24. Like figures have a reason of their homologallor correspondent sides equally manifold unto their dimensions and a meane proportionall lesse by one Thus farre of the first part of this element The second that like figurs have a meane proportional lesse by one then are their dimensions shall be declared by few words For plaines having but two dimensions have but one meane proportionall solids having three dimensions have two meane proportionalls The caâ—se is onely Arithmeticall as afore For where the bounds are but 4. as they are in two plaines there can be found no more but one meane proportionall as in the former example of 8. and 18. where the homologall or correspondent sides are 2. 3. and 4. 6. Therefore Againe by the same ruâ—e where the bounds are 6. as they are in two solids there may bee found no more but two meane proportionalls as in the former solids 30. and 240. where the homologall or correspondent sides are 2. 4. 3. 6. 5. 10. Therefore Therefore 25. If right lines be continually proportionall more by one then are the dâ—mensions of like figures likelily situate unto the first and second it shall be as the first right line is unto the last so the first figure shall be unto the second And contrariwise Out of the similitude of figures two consectaries doe arise in part only as is their axiome rationall and expressable by numbers If three right lines be continually proportionall it shall be as the first is unto the third So the rectineall figure made upon the first shall be unto the rectilineall figure made upon the second alike and likelily situate This may in some part be conceived and understood by numbers As for example Let the lines given be 2. foot 4. foote and 8 foote And upon the first and second let there be made like figures of 6. foote and 24. foote So I meane that 2. and 4. be the bases of them Here as 2. the first line is unto 8. the third line So is 6. the first figure unto 24. the second figure as here thou seest Againe let foure lines continually proportionall be 1. 2. 4. 8. And let there bee two like solids made upon the first and second vpon the first of the sides 1. 3. and 2. lee it be 6. Vpon the second of the sides 2. 6. and 4. let it be 48. As the first right line 1. is unto the fourth 8. So is the figure 6. unto the second 48.
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
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
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
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
quem Agricola alijex antiquis monumentis tradi derunt Now by any one of these knowne and compared with ours to all English men well knowne the rest may easily be proportioned out 2. The thing proposed to bee measured is a Magnitude Magnitudo a Magnitude or Bignesse is the subject about which Geometry is busied For every Art hath a proper subject about which it doth employ al his rules and precepts And by this especially they doe differ one from another So the subject of Grammar was speech of Logicke reason of Arithmeticke numbers and so now of Geometry it is a magnitude all whose kindes differences and affections are hereafter to be declared 3. A Magnitude is a continuall quantity A Magnitude is quantitas continua a continued or continuall quantity A number is quantitas discreta a disjoined quantity As one two three foure doe consist of one two three foure unities which are disjoyned and severed parts whereas the parts of a Line Surface and Body are contained and continued without any manner of disjunction separation or distinction at all as by and by shall better and more plainely appeare Therefore a Magnitude is here understood to be that whereby every thing to be measured is said to bee great As a Line from hence is said to be long a Surface broade a Body solid Wherefore Length Breadth and solidity are Magnitudes 4. That is continuum continuall whose parts are contained or held together by some common bound This definition of it selfe is somewhat obscure and to be understand onely in a geometricall sense And it dependeth especially of the common bounde For the parts which here are so called are nothing in the whole but in a potentia or powre Neither indeede may the whole magnitude bee conceived but as it is compact of his parts which notwithstanding wee may in all places assume or take as conteined and continued with a common bound which Aristotle nameth a Common limit but Euclide a Common section as in a line is a Point in a surface a Line in a body a Surface 5. A bound is the outmost of a Magnitude Terminus a Terme or Bound is here understood to bee that which doth either bound limite or end actu in deede as in the beginning and end of a magnitude Or potentia in powre or ability as when it is the common bound of the continuall magnitude Neither is the Bound a parte of the bounded magnitude For the thing bounding is one thing and the thing bounded is another For the Bound is one distance dimension or degree inferiour to the thing bounded A Point is the bound of a line and it is lesse then a line by one degree because it cannot bee divided which a line may A Line is the bound of a surface and it is also lesse then a surface by one distance or dimension because it is only length wheras a surface hath both length and breadth A Surface is the bound of a body and it is lesse likewise then it is by one dimension because it is onely length and breadth whereas as a body hath both length breadth and thickenesse Now every Magnitude actu in deede is terminate bounded and finite yet the geometer doth desire some time to have an infinite line granted him but no otherwise infinite or farther to bee drawane out then may serve his turne 6. A Magnitude is both infinitely made and continued and cut or divided by those things wherewith it is bounded A line a surface and a body are made gemetrically by the motion of a point line and surface Item they are conteined continued and cut or divided by a point line and surface But a Line is bounded by a point a surface by a line And a Body by a surface as afterward by their severall kindes shall be understood Now that all magnitudes are cut or divided by the same wherewith they are bounded is conceived out of the definition of Continuum e. 4. For if the common band to containe and couple together the parts of a Line surface Body be a Point Line and Surface it must needes bee that a section or division shall be made by those common bandes And that to bee dissolved which they did containe and knitt together 7. A point is an undivisible signe in a magnitude A Point as here it is defined is not naturall and to bee perceived by sense Because sense onely perceiveth that which is a body And if there be any thing lesse then other to be perceived by sense that is called a Point Wherefore a Point is no Magnitude But it is onely that which in a Magnitude is conceived and imagined to bee undivisible And although it be voide of all bignesse or Magnitude yet is it the beginning of all magnitudes the beginning I meane potentiâ in powre 8. Magnitudes commensurable are those which one and the same measure doth measure contrariwise Magnitudes incommensurable are those which the same measure cannot measure 1 2. d. X. Magnitudes compared betweene themselves in respect of numbers have Symmetry or commensurability and Reason or rationality Of themselves Congruity and Adscription But the measure of a magnitude is onely by supposition and at the discretion of the Geometer to take as pleaseth him whether an ynch an hand breadth foote or any other thing whatsoever for a measure Therefore two magnitudes the one a foote long the other two foote long are commensurable because the magnitude of one foote doth measure them both the first once the second twice But some magnitudes there are which have no common measure as the Diagony of a quadrate and his side 116. p. X. actu in deede are Asymmetra incommensurable And yet they are potentiâ by power symmetra commensurable to witt by their quadrates For the quadrate of the diagony is double to the quadrate of the side 9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given Contrariwise they are irrationalls 5. d. X. Ratio Reason Rate or Rationality what it is our Authour and likewise Salignacus have taught us in the first Chapter of the second booke of their Arithmetickes Thither therefore I referre thee Data mensura a Measure given or assigned is of Euclide called Rhetè that is spoken or which may be uttered definite certaine to witt which may bee expressed by some number which is no other then that which as we said was called mensura famosa a knowne or famous measure Therefore Irrationall magnitudes on the contrary are understood to be such whose reason or rate may not bee expressed by a number or a measure assigned As the side of the side of a quadrate of 20. foote unto a magnitude of two foote of which kinde of magnitudes thirteene sorts are mentioned in the tenth booke of Euclides Elements such are the segments of a right line proportionally cutte unto the whole line The Diameter in a circle is rationall But it is irrationall unto the side of
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
whole shall be the gnomon of the next greater quadrate For the sides is one of the complements and being doubled it is the side of both together And an unity is the latter diagonall So the side of 148 is 12 4 25. The reason of this dependeth on the same proposition from whence also the whole side is found For seeing that the side of every quadrate lesser than the next follower differeth onely from the side of the quadrate next above greater than it but by an 1. the same unity both twice multiplied by the side of the former quadrate and also once by it selfe doth make the Gnomon of the greater to be added to the quadrate For it doth make the quadrate 169. Whereby is understood that looke how much the numerator 4. is short of the denominatour 25. so much is the quadrate 148. short of the next greater quadrate For it thou doe adde 21. which is the difference whereby 4 is short of 25. thou shalt make the quadrate 169. whose side is 13. The second is by the reduction as I said of the number given unto parts assigned of some great denomination as 100. or 1000. or some smaller than those and those quadrates that their true and certaine may be knowne Now looke how much the smaller they are so much nearer to the truth shall the side found be Moreover in lesser parts the second way beside the other doth shew the side to be somewhat greater than the side by the first way found as in 7. the side by the first way is 3 25. But by the second way the side of 7. reduced unto thousands quadrates that is unto 7000000 1000000 that is 2645 1000 and beside there doe remaine 3975. But 645 1000. are greater than 3 5. For â…— reduced unto 1000. are but 600 1000. Therefore the second way in this example doth exceed the first by 45 1000. those remaines 3975. being also neglected Therefore this is the Analysis or manner of finding the side of a quadrate by the first rate of a quadrate equall to a double rectangle and quadrate The Geodesy or measuring of a Triangle There is one generall Geodesy or way of measuring any manner of triangle whatsoever in Hero by addition of the sides halving of the summe subduction multiplication and invention of the quadrates side after this manner 18 If from the halfe of the summe of the sides the sides be severally subducted the side of the quadrate continually made of the halfe and the remaines shall be the content of the triangle This generall way of measuring a triangle is most easie and speedy where the sides are expressed by whole numbers The speciall geodesy of rectangle triangle was before taught at the 9 e x j. But of an oblique angle it shall hereafter be spoken But the generall way is farre more excellent than the speciallâ— For by the reduction of an obliquangle many fraudes and errours doe fall out which caused the learned Cardine merrily to wish that hee had but as much land as was lost by that false kinde of measuring 19 If the base of a triangle doe subtend an obtuse angle the power of it is more than the power of the shankes by a double right angle of the one and of the continuation from the said obtusangle unto the perpendicular of the toppe 12. p ij Or thus If the base of a triangle doe subtend an obtuse angle it is in power more than the feete by the right angled figure twise taken which is contained under one of the feete and the line continued from the said foote unto the perpendicular drawne from the toppe of the triangle H. There is a comparison of a quadrate with two in like manner triangles and as many quadrates but of unequality For by 9. e the quadrate of a i is equall to the quadrates of a o and o i that is to three quadrates of i o o e e a and the double rectangle aforesaid But the quadrates of the shankes a e e i are equall to those three quadrates to wit of a i his owne quadrate and of e i two the first i o the second o e by the 9. e. Therefore the excesse remaineth of a double rectangle Of Geometry the thirteenth Booke Of an Oblong 1 An Oblong is a rectangle of inequall sides 31. d j. This second kinde of rectangle is of Euclide in his elements properly named for a definitions sake onely The rate of Oblongs is very copious out of a threefold section of a right line given sometime rationall and expresable by a number The first section is as you please that is into two segments equall or unequall From whence a five-fold rate ariseth 2 An Oblong made of an whole line given and of one segment of the same is equall to a rectangle made of both the segments and the square of the said segment 3. p ij It is a consectary out of the 7 e xj For the rectangle of the segments and the quadrate are made of one side and of the segments of the other Now a rectangle is here therefore proposed because it may be also a quadrate to wit if the line be cut into two equall parts Secondarily 3 Oblongs made of the whole line given and of the segments are equall to the quadrate of the whole 2 p ij This is also a Consectary out of the 4. e xj Here the segments are more than two and yet notwithstanding from the first the rest may be taken for one seeing that the particular rectangle in like manner is equall to them This proposition is used in the demonstration of the 9. e xviij Thirdly 4 Two Oblongs made of the whole line given and of the one segment with the third quadrate of the other segment are equall to the quadrates of the whole and of the said segment 7 p ij 5 The base of an acute triangle is of lesse power than the shankes are by a double oblong made of one of the shankes and the one segment of the same from the said angle unto the perpendicular of the toppe 13. p.ij. And from hence is had the segment of the shanke toward the angle and by that the perpendicular in a triangle Therefore 6. If the square of the base of an acute angle be taken out of the squares of the shankes the quotient of the halfe of the remaine divided by the shanke shall be the segment of the dividing shanke from the said angle unto the perpendicular of the toppe Now againe from 169 the quadrate of the base 13 take 25 the quadrate of 5 the said segment And the remaine shall be 144 for the quadrate of the perpendicular a o by the 9 e x ij Here the perpendicular now found and the sides cut are the sides of the rectangle whose halfe shall be the content of the Triangle As here the Rectangle of 21 and 12 is 252 whose halfe 126 is the content of the triangle The second section followeth from whence
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
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
by this meanes 14 If a right line equall to the axis of the sphearicall and to it from the end of the perpendicular be knit unto the center a right line drawne from the cutting of the periphery unto the said end shall be the side of the Icosahedrum 15 Of the five ordinate bodies inscribed into the same spheare the tetrahedrum in respect of the greatnesse oâ— his side is first the Octahedrum the second the Cube the third the Icosahedrum the fourth and the Dodecahedrum the fifth The latter Euclide doth demonstrate with a greater circumstance Therefore out of the former figures and demonstrations let here be repeated The sections of the axis first into a double reason in s And the side of the sexangle r l And the side of the Decangle a r inscribed into the same circle circumscribing the quinquangle of an icosahedrum And the perpendiculars i s and u l. Here the two triangles a i e and i e s are by the 8 e viij alike And as s e is unto e i So is i e unto e a And by 25 e iiij as s e is to e a so is the quadrate of s e to the quadrate of e i And inversly or backward as a e is to s e so is the quadrate of i e to the quadrate of s e. But a e is the triple of s e. Therefore the quadrate of i e is the triple of s e. But the quadrate of a s by the grant and 14 e xij is the quadruple of the quadrate of s e. Therefore also it is greater than the quadrate of i e And the right line a s is greater than i e and a l therefore is much greater But a l is by the grant compounded of the sides of the sexangle and decangle r l and a r. Therefore by the 1 c. 5 e 18. it is cut proportionally And the greater segment is the side of the sexangle to wit r l And the greater segment of i e proportionally also cut is y e. Therefore the said r l is greeter than y e And even now it was shewed u l was equall to r l. Therefore u l. is greater than y e But u e the side of the Icosahedrum by 22. e vj. is greater than u l. Therefore the side of the Icosahedrum is much greater then the side of the dodecahedrum Of Geometry the twenty seventh Book Of the Cone and Cylinder 1 A mingled solid is that which is comprehended of a variable surface and of a base FOr here the base is to be added to the variable surface 2 If variable solids have their axes proportionall to their bases they are alike 24. d xj It is a Consectary out of the 19 e iiij For here the axes and diameters are as it were the shankes of equall angles to wit of right angles in the base and perpendicular axis 3 A mingled body is a Cone or a Cylinder The cause of this division of a varied or mingled body is to be conceived from the division of surfaces 4 A Cone is that which is comprehended of a conicall and a base Therefore 5 It is made by the turning about of a rightangled triangle the one shanke standing still As it appeareth out of the definition of a variable body And 6 A Cone is rightangled if the shanke standing still be equall to that turned about It is Obtusangeld if it be lesse and acutangled if it be greater ê 18 d xj And 7 A Cone is the first of all variable For a Cone is so the first in variable solids as a triangle is in rectilineall plaines As a Pyramis is in solid plaines For neither may it indeed be divided into any other variable solids more simple And 8 Cones of equall heighth are as their bases are 11. p xij As here you see And 9 They which are reciprocall in base and heighth are equall 15 p x ij These are consectaries drawne out of the 12 and 13 e iiij As here you see 10 A Cylinder is that which is comprehended of a cyliudricall surface and the opposite bases Therefore 11 It is made by the turning about of a right angled parallelogramme the one side standing still 21. dxj. As is apparant out the same definition of a varium 12. A plaine made of the base and heighth is the solidity of a Cylinder This manner of measuring doth answeare I say to the manner of measuring of a prisma and in all respects to the geodesy of a right angled parallelogramme If the cylinder in the opposite bases be oblique then if what thou cuttest off from one base thou doest adde unto the other thou shalt have the measure of the whole as here thou seest in these cylinders a and b. As here the diameter of the inner Circle is 6 foote The periphery is 18 6 7 Therefore the plot or content of the circle is 28 2 7 Of which and the heighth 10 the plaine is 282 6 7 for the capacity of the vessell Thus therefore shalt thou judge as afore how much liquour or any thing esle conteined a cubicall foote may hold 13. A Cylinder is the triple of a cone equall to it in base and heighth 10 p xij The demonstration of this proposition hath much troubled the interpreters The reason of a Cylinder unto a Cone may more easily be assumed from the reason of a Prisme unto a Pyramis For a Cylinder doth as much resemble a Prisme as the Cone doth a Pyramis Yea and within the same sides may a Prisme and a Cylinder a Pyramis and a Cone be conteined And if a Prisme and a Pyramis have a very multangled base the Prisme and Clinder as also the Pyramis and Cone do seeme to be the same figure Lastly within the same sides as the Cones and Cylinders so the Prisma and Pyramides from their axeletrees and diameters may have the similitude of their bases And with as great reason may the Geometer demand to have it granted him That the Cylinder is the treble of a Coneâ— As it was demanded and granted him That Cylinders and Cones are alike whose axletees are proportionall to the diameters of their bases Therefore 14. A plaine made of the base and third part of the height is the solidity of the cone of equall base height Of two cones of one common base is made Archimede's Rhombus as here whose geodaesy shall be cut of two cones And 15. Cylinder of equall heighth are as their bases are 11 p xij And 16 Cylinders reciprocall in base and heighth are eequall 15 p xij Both these affections are in common attributed to the equally manifold of first figures And 17. If a Cylinder be cut with a plaine surface parallell to his opposite bases the segments are as their axes are 13 p xij The unequall sections of a spheare we have reserved for this place Because they are â—omprehended of a surface both sphearicall and conicall as is the sectour As also of a plaine and sphearicall as is the section And in both like as in a Circle there is but a greater and lesser segment And the sectour as before is considered in the center 18. The sectour of a spheare is a segment of a spheare which without is comprehended of a sphearicallâ— within of a conicall bounded in the center the greater of a concave the lesser of a convex Archimides maketh mention of such kinde of Sectours in his 1 booke of the Spheare From hence also is the geodesy following drawne And here also is there a certaine analogy with a circular sectour 19. A plaine made of the diameter and sixth part of the greater or lesser sphearicall is the greater or lesser sector And from hence lastly doth arise the solidity of the section by addition and subduction 20. If the greater sectour be increased with the internall cone the whole shall be the greater section If the lesser be diminished by it the remaine shall be the lesser section As here the inner cone measured is 126 4 63. The greater sectour by the former was 1026 â…” And 126 â…” 126 4 63 doe make 1152 46 63. Againe the lesser sectour by the next precedent was 410 â…” And here the inner cone is 126 4 63 And therefore 410 2 â— 126 4 63 that is 284 38 63 is the lesser section FINIS Or thus Or thus Or thus