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
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
ariseth the fourth rate or comparison 7. If a right line be cut into two equall parts and otherwise the oblong of the unequall segments with the quadrate of the segment betweene them is equall to the quadrate of the bisegment 5 p ij The third section doth follow from whence the fifth reason ariseth 8. If a right line be cut into equall parts and continued the oblong made of the continued and the continuation with the quadrate of the bisegment or halfe is equall to the quadrate of the line compounded of the bisegment and continuation 6 p ij From hence ariseth the Mesographus or Mesolabus of Heron the mechanicke so named of the invention of two lines continually proportionall betweene two lines given Whereupon arose the Deliacke probleme which troubled Apollo himselfe Now the Mesographus of Hero is an infinite right line which is stayed with a scrue-pinne which is to be moved up and downe in riglet And it is as Pappus saith in the beginning of his 111 booke for architects most fit and more ready than the Plato's mesographus The mechanicall handling of this mesographus is described by Eutocius at the 1 Theoreme of the 11 booke of the spheare But it is somewhat more plainely and easily thus layd downe by us 9. If the Mesographus touching the angle opposite to the angle made of the two lines given doe cut the said two lines given comprehending a right angled parallelogramme and infinitely continued equally distant from the center the intersegments shall be the meanes continually proportionally betweene and two lines given Or thus If a Mesographus touching the angle opposite to the angle made of the lines given doe cut the equall distance from the center the two right lines given conteining a right angled parallelogramme and continued out infinitely the segments shall be meane in continuall proportion with the line given H. As let the two right-lines given be a e and a i And let them comprehend the rectangled parallelogramme a o And let the said right lines given be continued infinitely a e toward u and a i toward y. Now let the Mesographus u y touch o the angle opposite to a And let it cut the sayd continued lines equally distant from the Center The center is found by the 8 e iiij to wit by the meeting of the diagonies For the equidistance from the center the Mesographus is to be moved up or downe untill by the Compasses it be found Now suppose the points of equidistancy thus found to be u and y. I say That the portions of the continued lines thus are the meane proportionalls sought And as a e is to i y so is i y to e u so is e u to a i. The fourteenth Booke of P. Ramus Geometry Of a right line proportionally cut And of other Quadrangles and Multangels THus farre of the threefold section from whence we have the five rationall rates of equality There followeth of the third section another section into two segments proportionall to the whole The section it selfe is first to be defined 1. A right line is cut according to a meane and extreame rate when as the whole shall be to the greater segment so the greater shall be unto the lesser 3. d vj. This line is cut so that the whole line it selfe with the two segments doth make the three bounds of the proportionâ— And the whole it selfe is first bound The greater segment is the middle bound The lesser the third bound 2. If a right line cut proportionally be rationall unto the measure given the segments are unto the same and betweene themselves irrationall è 6 p xiij A Triangle and all Triangulates that is figures made of triangles except a Rightangled-parallelogramme are in Geometry held to be irrationalls This is therefore the definition of a proportionall section The section it selfe followeth which is by the rate of an oblong with a quadrate 3. If a quadrate be made of a right line given the difference of the right line from the middest of the conterminall side of the said quadrate made above the same halfe shall be the greater segment of the line given proportionally cut 11 p ij Or thus If a square be made of a right line given the difference of a right line drawne from the angle of the square made unto the middest of the next side above the halfe of the side shall be the greater segment of the line given being proportionally cut H. For of y a let the quadrate a y s r be made And let s r be continued unto l. Now by the 8 e xiij the obloâ—g of o y and a y with the quadrate of u a is equall to the quadrate of u y that is by the construction of u e And therefore by the 9 e xij it is equall to the quadrates e a and a u Take away from each side the common oblong a l and the quadrate y r shall be equall to the oblong r i. Therefore the three right lines e a a r and r e by the 8 e xij are continuall proportionall And the right line a e is cut proportionally Therefore 4 If a right line cut proportionally be continued with the greater segment the whole shall be cut proportionally and the greater segment shall be the line given 5 p xiij As in the same example the right line o y is continued with the greater segment and the oblong of the whole and the lesser segment is equall to the quadrate of the greater And thus one may by infinitely proportionally cutting increase a right line and againe decrease it The lesser segment of a right line proportionally cut is the greater segment of the greater proportionally cut And from hence a decreasing may be made infinitely 5 The greater segment continued to the halfe of the whole is of power quintuple unto the said halfe that is five times so great as it is and if the power of a right line be quintuple to his segment the remainder made the double of the former is cut proportionally and the greater segment is the same remainder 1. and 2. p x iij. This is the fabricke or manner of making a proportionall section A threefold rate followeth The first is of the greater segment The converse is apparent in the same example For seeing that i o is of power five times so much as is a o the gnomon l m n shall be foure times so much as is u a Whose quadruple also by the 14. e xij is a v. Therefore it is equall to the gnomon Now a j is equall to a e Therefore it is the double also of a o that is of a y And therefore by the 24. e x. it is the double of a t And therefore it is equall to the complements i y and y s Therefore the other diagonall y r is equall to the other rectangle i v. Wherefore by the 8 e xij as e v that is a e is to y t that
is a i so is a i unto i e Wherefore by the â— e a e is proportionall cut And the greater segment is a i the same remaine The other propriety of the quintuple doth follow 6 The lesser segment continued to the halfe of the greater is of power quintuple to the same halfe è 3 p x iij. The rate of the triple followeth 7 The whole line and the lesser segment are in power treble unto the greater è 4 p xiij 8 An obliquangled parallelogramme is either a Rhombus or a Rhomboides 9 A Rhombus is an obliquangled equilater parallelogramme 32 dj It is otherwise of some called a Diamond 10 A Rhomboides is an obliquangled parallelogramâ—e not equilater 33. dj And a Rhomboides is so opposed to an oblong as a Rhombus is to a quadrate And the Rhomboides is so called as one would say Rhombuslike although beside the inequality of the angles it hath nothing like to a Rhombus An example of measuring of a Rhombus is thus 11 A Trapezium is a quadrangle not parallelogramme 34. dj The examples both of the figure and of the measure of the same let these be Therefore triangulate quadrangles are of this sort 12 A multangle is a figure that is comprehended of more than foure right lines 23. dj By this generall name all other sorts of right lined figures hereafter following are by Euclide comprehended as are the quinquangle sexangle septangle and such like inumerable taking their names of the number of their angles In every kinde of multangle there is one ordinate as we have in the former signified of which in this place we will say nothing but this one thing of the quinquangle The rest shall be reserved untill we come to Adscription 13 Multangled triangulates doe take their measure also from their triangles 14 If an equilater quinquangle have three sides equall it is equiangled 7 p 13. This of some from the Greeke is called a Pentagon of others a Pentangle by a name partly Greeke partly Latine The fifteenth Booke of Geometry Of the Lines in a Circle AS yet we have had the Geometry of rectilineals The Geometry of Curvilineals of which the Circle is the chiefe doth follow 1. A Circle is a round plaine â— 15 dj The meanes to describe a Circle is the same which was to make a Periphery But with some difference For there was considered no more but the motion the point in the end of the ray describing the periphery Here is considered the motion of the whole ray making the whole plot conteined within the periphery A Circle of all plaines is the most ordinate figure as was before taught at the 10 e iiij 2 Cirâ—les are as the quadrates or squares made of their diameters 2 p. x ij Therefore 3. The Diameters are as their peripheries Pappus 5 l x j and 26 th 18. As here thou seest in a e and i o. 4. Circular Geometry is either in Lines or in the segments of a Circle This partition of the subject matters howsoever is taken for the distinguishing and severing with some light a matter somewhat confused And indeed concerning lines the consideration of secants is here the foremost and first of Inscripts 5. If a right line be bounded by two points in the periphery it shall fall within the Circle 2 p iij. From hence doth follow the Infinite section of which we spake at the 6 e j. This proposition teacheth how a Rightline is to be inscribed in a circle to wit by taking of two points in the periphery 6. If from the end of the diameter and with a ray of it equal to the right line given a periphery be described a right line drawne from the said end unto the meeting of the peripheries shall be inscribed into the circle equall to the right line given 1 p iiij And this proposition teacheth How a right line given is to be inscribed into a Circle equall to a line given Moreover of all inscripts the diameter is the chiefe For it sheweth the center and also the reason or proportion of all other inscripts Therefore the invention and making of the diameter of a Circle is first to be taught 7. If an inscript do cut into two equall parts another inscript perpendicularly it is the diamiter of the Circle and the middest of it is the center 1 p iij. The cause is the same which was of the 5 e x j. Because the inscript cut into halfes if for the side of the inscribed rectangle and it doth subtend the periphery cut also into two parts By the which both the Inscript and Periphery also were in like manner cut into two equall parts Therefore the right line thus halfing in the diameter of the rectangle But that the middle of the circle is the center is mâ—nifest out of the 7 e v and 29 e iiij Euclide thought better of Impossibile than he did of the cause And thus he forceth it For if y be not the Center but s the part must be equall to the whole For the Triangle a o s shall be equilater to the triangle e o s. For a o oe are equall by the grant Item s a and s e are the rayes of the circle And s o is common to both the triangles Therefore by the 1 e vij the angles no each side at o are equall And by the 13 e v they are both right angles Therefore s o e is a right angle It is therefore equall by the grant to the right angle y o e that is the part is equall to the whole which is impossible Wherefore y is not the Center The same will fall out of any other points whatsoever â—ut of y. Therefore 8. If two râ—ght lines doe perpendicularly halfe two inscripts the meeting of these two bisecants shall be the Center of the circle è 25 p iij. And one may 9. Draw a periphery by three points which doe not fall in a right line 10. If a diameter doe halfe an inscript that is nâ—t a diameter it doth cut it perpendicularly And contrariwise 3 p iij. 11. If inscripts which are not diameters doe cut one another the segments shall be unequall 4 p iij. But rate hath beene hitherto in the parts of inscripts Proportion in the same parts followeth 12 If two inscripts doe cut one another the rectangle of the segments of the one is equall to the rectangle of the segments of the other 35 p iij. And this is the comparison of the parts inscripts The rate of whole inscripts doth follow the which whole one diameter doth make 13 Inscripts are equall distant from the center unto which the perpendiculars from the center are equall 4 d iij. 14. If inscripts be equall they be equally distant from the center And contrariwise 13 p iij. The diameters in the same circle by the 28 e iiijâ— are equall And they are equally distant from the center seeing they are by the center or rather are no whit at all
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
as is manifest by division The examples are thus And 26. If foure right lines bee proportionall betweene themselves Like figures likelily situate upon them shall be also proportionall betweene themselves And contrariwise out of the 22. pvj. and 37. pxj. The proportion may also here in part bee expressed by numbers And yet a continuall is not required as it was in the former In Plaines let the first example be as followeth The cause of proportionall figures for that twice two figures have the same reason doubled In Solids let this bee the second example And yet here the figures are not proportionall unto the right lines as before figures of equall heighth were unto their basesâ— but they themselves are proportionall one to another And yet are they not proportionall in the same kinde of proportion The cause also is here the same that was before To witt because twice two figures have the same reason trebled 27. Figures filling a place are those which being any way set about the same point doe leave no voide roome This was the definition of the ancient Geometers as appeareth out of Simplicius in his commentaries upon the 8. chapter of Aristotle's iij. booke of Heaven which kinde of figures Aristotle in the same place deemeth to bee onely ordinate and yet not all of that kindâ— But only three among the Plaines to witt a Triangle a Quadrate and a Sexangle amongst Solids two the Pyramis and the Cube But if the filling of a place bee judged by right angles 4. in a Plaine and 8. in a Solid the Oblong of plaines and the Octahedrum of Solids shall as shall appeare in their places fill a place And yet is not this Geometrie of Aristotle accurate enough But right angles doe determine this sentence and so doth Euclide out of the angles demonstrate That there are onely five ordinate solids And so doth Potamon the Geometer as Simplicus testifieth demonstrate the question of figuâ—es filling a place Lastly if figures by laying of their corners together doe make in a Plaine 4. right angles or in a Solid 8. they doe fill a place Of this probleme the ancient geometers have written as we heard even now And of the latter writers Regiomontanus is said to have written accurately And of this argument Maucolycus hath promised a treatise neither of which as yet it hath beene our good hap to see Neither of these are figures of this nature as in their due places shall be proved and demonstrated 28. A round figure is that all whose raies are equall Rotundum a Roundle let it be here used for Rotunda figura a round figure And in deede Thomas Finkius or Finche as we would call him a learned Dane sequestring this argument from the rest of the body of Geometry hath intituled that his worke De Geometria rotundi Of the Geometry of the Round or roundle 29. The diameters of a roundle are cut in two by equall raies The reason is because the halfes of the diameters are the raies Or because the diameter is nothing else but a doubled ray Therefore if thou shalt cut off from the diameter so much as is the radius or ray it followeth that so much shall still remaine as thou hast cutte of to witt one ray which is the other halfe of the diameter Sn. And here observe That Bisecare doth here and in other places following signifie to cutte a thing into two equall parts or portionsâ— And so Bisegmentum to be one such portionâ— And Bisectio such a like cutting or division 30. Rounds of equall diameters are equall Out of the 1. d. iâ—â— Circles and Spheares are equall which have equall diameters For the raies which doe measure the space betweene the Center and Perimeter are equall of which beiâ—g doubled the Diameter doth consist Sn. The fifth Booke of Ramus his Geometry which is of Lines and Angles in a plaine Surface 1. A lineate is either a Surface or a Body LIneatum or Lineamentum a magnitude made of lines as was defined at 1. e. iij. is here divided into two kindes which is easily conceived out of the said definition there in which a line is excluded and a Surface a body are comprehended And from hence arose the division of the arte Metriall into Geometry of a surface and Stereometry of a body after which maner Plato in his vij booke of his Common-wealth and Aristotle in the 7. chapter of the first booke of his Posteriorums doe diâ—tinguish betweene Geometry and Stereometry And yet the name of Geometry is used to signifie the whole arte of measuring in generall 2. A Surface is a lineate only broade 5. dj Epiphania the Greeke word which importeth onely the outter appearance of a thing is here more significant because of a Magnitude there is nothing visible or to bee seene but the surface 3. The bound of a surface is a line 6. dj The matter in Plaines is manifest For a three cornered surface is bounded with 3. lines A foure cornered suâ—face with foure liâ—es and so forth A Circle is bounded with one line But in a Sphearicall surface the matter is not so plaine For it being whole seemeth not to be bounded with a line Yet if the manner of making of a Sphearicall surface by the conversioÌ„ or turning about of a semiperiphery the beginning of it as also the end shal be a line to wit a semiperiphery And as a point doth not only actu or indeede bound and end a line But is potentia or in power the middest of it So also a line boundeth a Surface actu and an innumerable company of lines may be taken or supposed to be throughout the whole surface A Surface therefore is made by the motion of a line as a Line was made by the motion of a point 4. A surface is either Plaine or Bowed The difference of a Surface doth answer to the difference of a Lineâ— in straightnesse and obliquity or crookednesse Obliquum oblique there signified crooked Not righâ— or straight Here uneven or bowed either upward or downeward Sn. 5. A plaine surface is a surface which lyeth â—qually betweene his bounds out of the 7. dj Planum a Plaine is taken and used for a plaine surface as before Rotundum a Round was used for a round figure Therefore 6. From a point unto a point we may in a plaine surface draw a right line 1 and 2. post j. Three things are from the former ground begg'd The first is of a Right line A right line and a periphery were in the ij booke defined But the fabricke or making of them both is here said to bee properly in a plaine Now the Geometricall instrument for the drawing of a right plaine is called Amussis by Petolemey in the 2. chapter of his first booke of his Musicke Regula a Rular such as heere thou seest And from a point unto a point is this justly demanded to be done not unto points For neither doe all points fall in a
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
unto the remainder which is thus found 5. If a right line be cut proportionally the base of that triangle whose shankes shall be equall to the whole line cut and the base to the greater segment of the same shall have each of the angles at the base double to the remainder And the base shall be the side of the quinquangle inscribed with the triangle into a circle 10 and 11. p i i i j. 6 If two right lines doe subtend on each side two angles of an inscript quinquangle they are cut proportionally and the greater segments are the sides of the said inscript è 8 p x iij. And from hence the fabricke or construction of an ordinate quinquangle upon a right line given is manifest Therefore 7 If a right line given cut proportionall be continued at each end with the greater segment and sixe peripheries at the distance of the line given shall meete two on each side from the ends of the line given and the continued two others from their meetings right lines drawne from their meetings the ends of the assigned shall make an ordinate quinquangle upon the assigned 8 If the diameter of a circle circumscribed about a quinquangle be rationall it is irrationall unto the side of the inscribed quinquangle è 11. p xiij So before the segments of a right line proportionally cut were irrationall The other triangulates hereafter multiplied from the ternary quaternary or quinary of the sides may be inscribed into a circle by an inscript triangle quadrate or quinquangle Therefore by a triangle there may be inscribed a triangulate of 6. 12,24,46 angles By a quadrate a triangulate of 8. 16,32,64 angles By a quinquangle a triangulate of 10 20. 40,80 angles c. 9 The ray of a circle is the side of the inscript sexangle è 15 p iiij Therefore 10 Three ordinate sexangles doe fill up a place Furthermore also no one figure amongst the plaines doth fill up a place A Quinquangle doth not For three angles a quinquangle may make only 3 ◠5 angles which is too little And foure would make 4 ◠5. which is as much too great The angles of a septangle would make onely two rightangles and 6 7 of one Three would make 3 and 9 7 that is in the whole 4. 2 7 which is too much c. to him that by induction shall thus make triall it will appeare That a plaine place may be filled up by three sorts of ordinate plaines onely And 11 If right lines from one angle of an inscript sexangle unto the third angle on each side be knit together they shall inscribe an equilater triangle into the circle given 12 The side of an inscribed equilater triangle hath a treble power unto the ray of the circle 12. p xiij 13 If the side of a sexangle be cut proportionally the greater segment shall be the side of the decangle Therefore 14 If a decangle and a sexangle be inscribed in the same circle a right line continued and made of both sides shall be cut proportionally and the greater segment shall be the side of a sexangle and if the greater segment of a right line cut proportionally be the side of an hexagon the rest shall be the side of a decagon 9. p xiij The comparison of the decangle and sexangle with the quinangle followeth 15 If a decangle a sexangle and a pentangle be inscribed into the same circle the side of the pentangle shall in power countervaile the sides of the others And if a right line inscribed do countervaile the sides of the sexangle and decangle it is the side of the pentangle 10. p xiiij Let the proportion of this syllogisme be demonstrated For this part onely remaineth doubtfull Therefore two triangles a e i and y e i are equiangles having one common angle at e And also two equall ones a e i and e i y the halfes to wit of the same e i s Because that is by the 17 e vj one of the two equalls unto the which e i s the out angle is equall by the 15 e. vj. And this doth insist upon a halfe periphery For the halfe periphery a l s is equall to the halfe periphery a r s and also a l is equall to a r. Therefore the remnant l s is equall to the remnant r s And the whole r l is the double of the same r s And therefore e r is the double of e o And r s the double of o u. For the bisegments are manifest by the 10 e xv and the 11 e xvj Therefore the periphery e r s is the double of the periphery e o u And therefore the angle e i u is the halfe of the angle e i s by the 7 e xvj Therefore two angles of two triangles are equall Wherefore the remainder by the 4 e vij is equall to the remainder Wherefore by the 12 e vij as the side a e is to e i so is e i to e y. Therefore by the 8 e xij the oblong of the extreames is equall to the quadrate of the meane Now let o y be knit together with a straight Here againe the two triangles a o e and a o y are equiangles having one common angle at a And a o y and o e a therefore also equall Because both are equall to the angle at a That by the 17 e vj This by the 2 e vij Because the perpendicular halfing the side of the decangle doth make two triangles equicrurall and equall by the right angle of their shankes And therefore they are equiangles Therefore as e a is to a o so is e a to a y. Wherefore by the 8 e xij the oblong of the two extremes is equall to the quadrate of the meane And the proposition of the syllogisme which was to be demonstrated The converse from hence as manifest Euclide doth use at the 16 p xiij 16. If a triangle and a quinquangle be inscribed into the same Circle at the same point the right line inscribed betweene the bases of the both opposite to the sayd point shall be the side of the inscribed quindecangle 16. p. iiij Therefore 17. If a quinquangle and a sexangle be inscribed into the same circle at the same point the periphery intercepted betweene both their sides shall be the thirtieth part of the whole periphery Of Geometry the ninteenth Booke Of the Measuring of ordinate Multangle and of a Circle OVt of the Adscription of a Circle and a Rectilineall is drawne the Geodesy of ordinate Multangles and first of the Circle it selfe For the meeting of two right lines equally dividing two angles is the center of the circumscribed Circle From the center unto the angle is the ray And then if the quadrate of halfe the side be taken out of the quadrate of the ray the side of the remainder shall be the perpendicular by the 9 e xij Therefore a speciall theoreme is here thus make 1. A plaine made of the
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
sides 28 p xj It answereth to the 34. pj. And 16 If it be halfed by two plaines halfing the opposite sides the common bisection and diagony doe halfe one another 39. p xj 17 If three lines be proportionall the parallelepipedum of meane shall be equall to the equiangled pâ—rallelepipedum of all them è 36. p x j. It is a consectary out of the 8 e. 18 Eight rectangled parallelepiped's doe fill a solid place 19 The Figurate of a rectangled parallelepipedum is called a solid made of three numbers 17. d vij As if thou shalt multiply 1,2,3 continually thou shalt make the solid 6. Item if thou shalt in like manner multiply 2,3,4 thou shalt make the solid 24. And the sides of that solid 6 solid shall be 1,2,3 Of 24 they shall be 2,3,4 Therefore 20 If two solids be alike they have their sides proportionalls and two meane proportionalls 21 d vij 19. 21. p viij It is a consectary out of the 5 e xxij But the meane proportionalls are made of the sides of the like solids to wit of the second third and fourth Item of the third fourth and fifth as here thoâ— seest Of Geometry the twentie fourth Book Of a Cube 1 A Rightangled parallelepipedum is either a Cube or an Oblong 2 A Cube is a right angled parallelepipedum of equall flattes 25. d xj As here thou seest in these two figures Therefore 3 The sides of a cube are 12. the plaine angles 24. the solid 8. Therefore 4 If sixe equall quadrates be joyned with solid angles they shall comprehend a cube As here in these two examples Therefore 5 If from the angles of a quadrate perpendâ—culars equall to the sides be tied together aloft they shall comprehend a Cube è 15 p xj It is a consectary following upon the former consectary For then shall sixe equall quadrates be knit together 6 The diagony of a Cube is of treble power unto the side For the Diagony of a quadrate is of double power to the side by the 12 e xij And the Diagony of a Cube is of as much power as the side the diagony of the quadrate by the same e. Therefore it is of treble power to the side 7 If of foure right lines continually proportionally the first be the halfe of the fourth the cube of the first shall be the halfe of the Cube of the second è 33 p xj It is a consectary out of the 25 e iiij From hence Hippocrates first found how to answer Apollo's Probleme 8 The solid plaine of a cube is called a Cube to wit a solid of equall sides 19 d vij Therefore 9 It is made of a number multiplied into his owne quadrate So is a Cube made by multiplying a number by it selfe and the product againe by the first Such are these nine first cubes made of the nine first Arithmeticall figures This is the generall invention of a Cube both Geometricall and Arithmeticall 10 If a right line be cut into two segments the Cube of the whole shall be equall to the Cubes of the segments and a double solid thrice comprehended of the quadrate of his owne segment and the other segment As for example the side 12 let it be cut into two segments 10 and 2. The cube of 12. the whole which is 1728 shall be equall to two cubes 1000 and 8 made of the segments 10. and 2. And a double solid of which the first 600. is thrise comprehended of 100. the quadrate of his segment 10. and of 2. the other segment The second 120. is thrice comprehended of 4 the quadrate of his owne segment and of 10. the other segment Now 1000 600 120. 8 is equall to 1728 And therefore a right c. Therefore 11. The side of the first severall cube is the other side of the second solide And the quadrate of the same side is the other side of the first solide whose other side is the side of the second cube and the quadrate of the same other side is the other side of the second solid In that equation therefore of foure solids with one solid thou shalt consider a peculiar making and composition First that the last cube be made of the last segment 2â— Then that the second solid of 4 the quadrate of his owne segment and of 10 the other segment be thrise comprehended Lastly that the first solid of 100 the square of his owne segment 10 and the other segment 2 be also thrice comprehended Lastly that the Cube 1000 be made of the greater segment 10. Out of this making c. The plaine of the perimeter of the base 20 and the altitude 5 is 100. This added to 25 and 25 both the bases that is to 50 maketh 150 for the whole surface Now the plaine of 25 the base and the heighth 5 is 125 for the whole solidity So in the Oblong the plaine of the base's perimeter 20 and the heighth 11 is 220 which added to the bases 24 and 24 that is 48 maketh 268 for the whole surface But the plaine of the base 24 and the height 11 is 264 for the solidity Thus are such kinde of walls whether of mudde bricke or stone of most large houses to bee measured The same manner of Geodesy is also to be used in the measuring of a Rhombe Rhomboides Trapezium or mensall and any kinde of multangled body The base is first to be measured as in the former Then out of that and the heighth the solidity shall be manifested As in the Rhombe the base is 24 the heighth 4. Therefore the solidity is 96. In the Rhomboides the base is 64 3â— 12â— The heigh 11. Therefore the solidity is 1028 44 1â—9 The same is the geodesy of a trapezium as in these examples The surface of the first is 198 The solidity 192 ½ The surface of the second is 158 3 49 The solidity is 91 29 4â— And from hence also may the capacity or content of vessells or measures made after any manner of plaine solid bee esteemed and judged of as here thou seest For here the plaine of the sexangular base is 41 1 7 For the ray by the 9 e xviij is the fide and the heighth 5 shall be 205 5 7. Therefore if a cubicall foote doe conteine 4 quarters as we commonly call them then shall the vessell conteine 822 6 7 quartes that is almost 823 quartes Of Geometry the twenty fifth Booke Of mingled ordinate Polyedra's 1. A mingled ordinate polyedrum is a pyramidate compounded of pyramides with their toppes meeting in the center and their bases onely outwardly appearing SEeing therefore a Mingled ordinate pyramidate is thus made or compounded of pyramides the geodesy of it shall be had from the Geodesy of the pyramides compounding it And one Base multiplyed by the number of all the bases shall make the surface of the body And one Pyramis by the number of all the pyramides shall make the solidity 2 The heighth of the