right line But many doe fall out to be in a crooked line And in a Spheare a Cone Cylinderâ— a Ruler may be applyed but it must be a sphearicall Conicall or Cylindraceall But by the example of a right line doth Vitellio 2 p j. demaund that betweene two lines a surface may be extended And so may it seeme in the Elements of many figures both plaine and solids by Euclide to be demanded That a figure may be described at the 7. and 8. e ij Item that a figure may be made vp at the 8. 14. 16. 23.28 p. vj which are of Plaines Item at the 25. 31. 33. 34. 36. 49. p.xj. which are of Solids Yet notwithstanding a plaine surface and a plaine body doe measure their rectitude by a right line so that jus postulandi this right of begging to have a thing granted may seeme primarily to bee in a right plaine line Now the Continuation of a right line is nothing else but the drawing out farther of a line now drawne and that from a point unto a point as we may continue the right line a e. unto i. wherefore the first and second Petitions of Euâ—lde do agree in one And 7. To set at a point assigned a Right line equall to another right line given And from a greater to cut off a part equall to a lesser 2. and 3. pj. Therefore 8. One right line or two cutting one another are in the same plaine out of the 1. and 2. p xj One Right line may bee the common section of two plaines yet all or the whole in the same plaine is one And all the whole is in the same other And so the whole is the same plaine Two Right lines cutting one another may bee in two plaines cutting one of another But then a plainâ— may be drawne by them Therefore both of them shall be in the same plaine And this plaine is geometrically to be conceived Because the same plaine is not alwaies made the ground whereupon one oblique line or two cutting one another are drawne when a periphery is in a sphearicall Neither may all peripheries cutting one another be possibly in one plaine And 9. With a right line given to describe a peripherie Talus the nephew of Daedalus by his sister is said in the viij booke of Ovids Metamorphosis to have beene the inventour of this instrument For there he thus writeth of him and this matter Et ex uno duo ferrea brachia nodo Iunxit ut aequali spatio distantibus ipsis Altera pars staret pars altera duceâ—et orbem Therfore 10. The raiâ—s of the same or of an equall periphery are equall The reason is because the same right line is every where converted or turned about But here by the Ray of the â—eriphery must bee understood the Ray the figure contained within the periphery 11. If two equall peripâ—eries from the ends of equall shankes of an assigned rectilineall angle doe meete before it a right line drawne from the meeting of them unto the toppe or point of the angle shall cut it into two equall parts 9. pj. Hitherto we have spoken of plaine lines Their affection followeth and first in the Bisection or dividing of an Angle into two equall parts 12. If two equall peripheries from the ends of a right line given doe meete on each side of the same a right line drawne from those meetings shall divide the right line given into two equall parts 10. pj. 13. If a right line doe stand perpendicular upon another right line it maketh on each side right angles And contrary wise The Rular for the making of straight lines on a plaine was the first Geometricall instrument The Compasses for the describing of a Circle was the second The Norma or Square for the true â—recting of a right line in the same plaine upon another right line and then of a surface and body upon a surface or body is the third The figure therefore is thus Therefore 14. If a right line do stand upon a right line it maketh the angles on each side equall to two right angles and contrariwise out of the 13. and 14. pj. And 15. If two right lines doe cut one another they doe make the angles at the top equall and all equall to foure right angles 15. pj. And 16. If two right lines cut with one right line doe make the inner angles on the same side greater then two right angles those on the other side against them shall be lesser then two right angles 17. If from â—â—oint assigned of an infinite right line given two equall parts be on each side cut off and then from the points of those sections two equall circles doe meete a right line drawne from their meeting unto the point assigned shall bee perpendicular unto the line given 11. pj. 18. If a part of an infinâ—te right line bee by a periphery from a point given without cut off a right line from the said point cutting in two the said part shall bee perpendicular upon the line given 12. pj. 19. If two right lines drawne at lâ—ngth in the same plaine doe never meete they are parallell◠è 35. dj Therefore 20. If an infinite right line doe cut one of the infinite right parallell lines it shall also cut the other As in the same example u y. cutting a e. it shall also cuâ— i o. Otherwise if it should not cut it it should be parallell unto it by the 18 e. And that against the grant 21. If right lines cut with a right line be pararellells they doe make the inner angles on the same side equall to two right angles And also the alterne angles equall betweene themselves And the outter to the inner opposite to it And contrariwise 29,28,27 p 1. The cause of this threefold propriety is from the perpendicular or plumb-line which falling upon the parallells breedeth and discovereth all this variety As here they are right angles which are the inner on the same part or side Item the alterne angles Item the inner and the outter And therefore they are equall both I meane the two inner to two right angles and the alterne angles between themselvs And the outter to the inner opposite to it If so be that the cutting line be oblique that is fall not upon them plumbe or perpendicularly the same shall on the contrary befall the parallels For by that same obliquaâ—ion or slanting the right lines remaining and the angles unaltered in like manner both one of the inner to wit e u y is made obtuse the other to wiâ— u y o is made acute And the alterne angles are made acute and obtuse As also the outter and inner opposite are likewise made acute and obtuse The same impossibility shall be concluded if they shall be sayd to be lesser than two right anglesâ— The second and third parts may be concluded out of the first The second is thus Twise two angles are equall to two right
alterne o e y Because also three angles o e y o e a and a e u are equall to two right angles by the 14 e v. Vnto which also are equall the three angles in the triangle a e o by the 13 e vj. From three equals take away the two right angles a u e and a o e For a o e is a right angle by the 21 e because it is in a semicircle Take away also the common angle a e o And the remainders e a o and o e y alterne angles shall be equall Therefore 28 If at the end of a right line given a right lined angle be made equall to an angle given and from the toppe of the angle now made a perpendicular unto the other side do meete with a perpendicular drawn from the middest of the line given the meeting shall be the center of the circle described by the equalled angle in whose opposite section the angle upon the line given shall be made equall to the assigned è 33 p iij. And 29 If the angle of the secant and touch line be equall to an assigned rectilineall angle the angle in the opposite section shall likewise be equall to the same 34. piij. Of Geometry the seventeenth Booke Of the Adscription of a Circle and Triangle HItherto we have spoken of the Geometry of Rectilineall plaines and of a circle Now followeth the Adscription of both This was generally defined in the first book 12 e. Now the periphery of a circle is the bound therof Therefore a rectilineall is inscribed into a circle when the periphery doth touch the angles of it 3 d iiij It is circumscribed when it is touched of every side by the periphery 4 d iij. 1. If a rectilineall ascribed unto a circle be an equilater it is equiangle Of the circumscript it is likewise true if the circumscript be understood to be a circle For the perpendiculars from the center a unto the sides of the circumscript by the 9e xij shal make triangles on each side equilaters equiangls by drawing the semidiameters unto the corners as in the same exāple 2. It is equall to a triangle of equall base to the perimeter but of heighth to the perpendicular from the center to the side As here is manifest by the 8 e vij For there are in one triangle three triangles of equall heighth The same will fall out in a Triangulate as here in a quadrate For here shal be made foure triangles of equall height Lastly every equilater rectilineall ascribed to a circle shall be equall to a triangle of base equall to the perimeter of the adscript Because the perimeter conteineth the bases of the triangles into the which the rectilineall is resolved 3. Like rectilinealls inscribed into circles are one to another as the quadrates of their diameters 1 p. x i j. In like Triangulates seeing by the 4 e x they may be resolved into like triangles the same will fall out Therefore 4. If it be as the diameter of the circle is unto the side of rectilineall inscribed so the diameter of the second circle be unto the side of the second rectilineall inscribed and the severall triangles of the inscripts be alike and likely situate the rectilinealls inscribed shall be alike and likely situate This Euclide did thus assume at the 2 p xij and indeed as it seemeth out of the 18 p vj. Both which are conteined in the 23 e iiij And therefore we also have assumed it Adscription of a Circle is with any triangle But with a triangulate it is with that onely which is ordinate And indeed adscription of a Circle is common to all 5. If two right lines doe cut into two equall parts two angles of an assigned rectilineall the circle of the ray from their meeting perpendicular unto the side shall be inscribed unto the assigned rectilineall 4 and 8. p. iiij The same argument shall serve in a Triangulate 6. If two right lines do right anglewise cut into two equall parts two sides of an assigned rectilineall the circle of the ray from their meeting unto the angle shall be circumscribed unto the assigned rectilineall 5 p iiij As in the former figures The demonstration is the same with the former For the three rayes by the 2 e vij are equall And the meeting of them by the 17 ex is the center And thus is the common adscription of a circle The adscription of a rectilineall followeth and first of a Triangle 7. If two inscripts from the touch point of a right line and a periphery doe make two angles on each side equall to two angles of the triangle assigned be knit together they shall inscribe a triangle into the circle given equiangular to the triangle given è 2 p iiij The circumscription here is also speciall 8 If two angles in the center of a circle given be equall at a common ray to the outter angles of a triangle given right lines touching a periphery in the shankes of the angles shall circumscribe a triangle about the circle given like to the triangle given 3 p iiij Therefore 9. If a triangle be a rectangle an obtusangle an acute angle the center of the circumscribed triangle is in the side out of the sides and within the sides And contrariwise 5 e iiij As thou seest in these three figures underneath the center a. Of Geometry the eighteenth Booke Of the adscription of a Triangulate SVch is the Adscription of a triangle The adscription of an ordinate triangulate is now to be taught And first the common adscription and yet out of the former adscription after this manner 1. If right lines doe touch a periphery in the angles of the inscript ordinate triangulate they shall unto a circle circumscribe a triangulate homogeneall to the inscribed triangulate The examples shall be laid downe according as the species or severall kindes doe come in order The speciall inscription therefore shall first be taught and that by one side which reiterated as oft as need shall require may fill up the whole periphery For that Euclide did in the quindecangle one of the kindes we will doe it in all the rest 2. If the diameters doe cut one another right-anglewise a right line subtended or drawne against the right angle shall be the side of the quadrate è 6 p iiij Therefore 3. A quadrate inscribed is the halfe of that which is circumscribed Because the side of the circumscribed which here is equall to the diameter of the circle is of power double to the side of the inscript by the 9 e x i j. An◠4. It is greater than the halfe of the circumscribed Circle Because the circumscribed quadrate which is his double is greater than the whole circle For the inscribing of other multangled odde-sided figures we must needes use the helpe of a triangle each of whose angles at the base is manifold to the other In a Quinguangle first that which is double
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
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
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
these latter daies the Germaines especially as Regiomontanus Werner Schoner and Appian have grac'd it But above all other the learned Gemma Phrisius in a severall worke of that argument onely hath illustrated and taught the use of it plainely and fully The Iacobs staffe therefore according to his owne and those Geometricall parts shall here be described The astronomicall distribution wee reserve to his time and place And that done the use of it shall be shewed in the measuring of lines 2 The shankes of the staffe are the Index and the Transome 3 The Index is the double and one tenth part of the transome Or thus The Index is to the transversary double and 1 10 part thereof H. As here thou seest 4 The Transome is that which rideth upon the Index and is to be slid higher or lower at pleasure Or The transversary is to be moved upon the Index sometimes higher sometimes lower H. This proportion in defining and making of the shankes of the instrument is perpetually to be observed as if the transome be 10. parts the Index must be 21. If that be 189. this shall be 90. or if it be 2000. this shall be 4200. Neither doth it skill what the numbers be so this be their proportion More than this That the greater the numbers be that is the lesser that the divisions be the better will it be in the use And because the Index must beare and the transome is to be borne let the index be thicker and the transome the thinner But of what matter each part of the staffe be made whether of brasse or wood it skilleth not so it be firme and will not cast or warpe Notwithstanding the transome will more conveniently be moved up and downe by brasen pipes both by it selfe and upon the Index higher or lower right angle wise so touching one another that the alterne mouth of the one may touch the side of the other The thrid pipe is to be moved or slid up and downe from one end of the transome to the other and therefore it may be called the Cursor The fourth and fifth pipes fixed and immoveable are set upon the ends of the transome are unto the third and second of equall height with â—innes to restraine when neede is the opticke line and as it were with certaine points to define it in the transome The three first pipes may as occasion shall require be fastened or staied with brasen scrues With these pipes therefore the transome may be made as great as need shall require as here thou seest The fabricke or manner of making the instrument hath hitherto beene taught the use thereof followeth unto which in generall is required First a just distance For the sight is not infinite Secondly that one eye be closed For the optick faculty conveighed from both the eyes into one doth aime more certainely and the instrument is more fitly applied and set to the cheeke bone then to any other place For here the eye is as it were the center of the circle into which the transome is inscribed Thirdly the hands must be steady for if they shake the proportion of the Geodesy must needes be troubled and uncertaine Lastly the place of the station is from the midst of the foote 5 If the sight doe passe from the beginning of one shanke it passeth by the end of the other And the one shanke is perpendicular unto the magnitude to be measured the other parallell These common and generall things are premised That the sight is from the beginning of the Index by the end of the transome Or contrariwise From the beginning of the transome unto the end of the Index And that the Index is right that is perpendicular to the line to be measured the transome parallell Or contrariwise Now the perpendicularity of the Index in measurings of lengthts may be tried by a plummet of lead appendentâ— But in heights and breadths the eye must be trusted although a little varying of the plummet can make no sensible errour By the end of the transome understand that which is made by the line visuall whether it be the outmost finne or the Cursour in any other place whatsoever 6 Length and Altitude have a threefold measure The first and second kinde of measure require but one distance and that by granting a dimension of one of them for the third proportionall The third two distances and such onely is the dimension of Latitude Geodesy of right lines is two fold of one distance or of two Geodesy of one distance is when the measurer for the finding of the desired dimension doth not change his place or standing Geodesy of two distances is when the measurer by reason of some impediment lying in the way betweene him and the magnitude to be measured is constrained to change his place and make a double standing Here observe That length and heighth may be joyntly measured both with one and with a double station But breadth may not be measured otherwise than with two 7 If the sight be from the beginning of the Index râ—ght or plumbe unto the length and unto the father end of the same as the segment of the Index is unto the segment of the transome so is the heighth of the measurer unto the length The same manner of measuring shall be used form an higher place as out of y the segment of the Index is 5. parts the segment of the transome 6 and then the height be 10 foote the same Length shall be found to bee 12 foote Neither is it any matter at all whether the length in a plaine or levell underneath Or in an ascent or descent of a mountaine as in the figure under written Thus mayest thou measure the breadths of Rivers Valleys and Ditches For the Length is alwayes after this manner so that one may measure the distance of shippes on the Sea as also Thales Milesius in Proclus at the 26 pj did measure them An example thou hast here Hereafter in the measuring of Longitude and Altitude fight is unto the toppe of the heighth Which here I doe now forewarne thee of least afterward it should in vaine be reitered often The second manner of measuring a Length is thus 8. If the sight be from the beginning of the index parallell to the length to be measured as the segment of the transome is unto the segment of the index so shall the heighth given be to the length As if the segment of the Transome be 120 parts the height given 400-foote The segment of the Index 210 parts The length by the golden rule shall be 700 foote The figure is thus And the demonstration is like unto the former or indeed more easier For the triangles are equiangles as afore Therefore as o u is to u a so is e i to i a. This is the first and second kinde of measuring of a Longitude by one single distance or station The third which is by a double distance doth now
follow Here the transome if there be roome enough for the measurer to goe farre enough backe must be put lower in the second distance 9. If the sight be from the beginning of the transverie parallell to the length to be measured as in the index the difference of the greater segment is unto the lesser so is the difference of the second station unto the lenth This kinde of Geodaesy is somewhat more subtile than the former were The figure is thus in which let the first ayming be from a the beginning of the transome and out of a i the length sought by o the end of the Index unto e the toppe of the heighth And let the segment of the Index be o u The second ayming let it be from y the beginning of the transome out of a greater distance by s the end of the Index unto e the same note of the heighth And let the segment of the Index be s r. Here the measuring performed is the taking of the difference betweene o u and s r. The rest are faigned onely for demonstrations sake Therefore in the first station let a m l be from the beginning of the transome be parallell to y e. Here first m u is equall to s r. For the triangleâ— m u a and s r y are equall in their shankes u a and r y by the grant Because the transome standeth still in his owne place And the angles at m u a u a m are equall to the angles And all right angles are equall by the 14 e iij. These are the outter and inner opposite one to another And such are equall by the 1. e. v. Therefore they are equilaters by the 2 e vij And o m is the difference of the segments of the Index Then as o m is to m u so is e l to l i as the equation of three degrees doth shew For by the 12 e vij as o m is to m a so is e l to l a And as m a is to m u so is l a to l i. Therefore by right as o m is to m u so is e l to l i And by the 12 e v j so is y a to a i As if the difference of the first segment be 36 parts The second segment be 72 parts The difference of the second sâ—ation 40 foote The length sought shall be 80 foote And here indeed is no heighth definitely given that may make any bound of the principall proportion Notwithstanding the Heighth although it be of an unknowne measure is the bound of the length sought And therefore it is an helpe and meanes to argue the question Because it is conceived to stand plumbe upon the outmost end of the length Therefore that third kinde of measuring of length is oftentimes necessary when by neither of the former wayeâ— the length may possibly be taken by reason of some impediment in the way to wit of a wall or tree or house or mountaine whereby the end of the length may not be seene which was the first way Nor an height next adjoyning to the end of the length is given which is the second way Hitherto we have spoken of the threefold measure of longitude the first and second out of an heighth given the third out of a double distance The measuring of heighth followeth next and that is also threefold Now heighth is a perpendicular line falling from the toppe of the magnitude unto the ground or plaine whereon the measurer doth stand after which manner Altitude on heighth was defined at the 9 e iiij The first geodesy or manner of measuring of heighths is thus 10. If the sight be from the beginning of the transome perpendicular unto the height to be measured as the segment of the transome is unto the segment of the Index so shall the length given be to the height Let the segment of the transome be 60 parts the segment of the Index 36 the Length given 120 foote the height sought shall be by the golden rule 72 foote The Figure is thus And the demonstration is by the 12 e vij as afore but here is to be added the height of the measurer which if it be 4 foot the whole height shall be 76 foote Therefore in an eversed altitude 11. If the sight be from the beginning of the Index parallell to the height as the segment of the transome is unto the segment of the index so shall the length given be unto the height sought Eversa altitudo An eversed altitude Reversed H is that which we call depth which indeed is nothing else in the Geometers sense but heighth turned topsie turvie as we say or with the heeles upward For out of the heighth concluded by subducting that which is above ground the heighth or depth of a Well shall remaine 12. If the sight be from the beginning of the Index perpendicular to the heighth to be measured as the segment of the Index is unto the segment of the Transome so shall the length given be to the heighth Therefore 13. If the sight be from the beginning of the Index perpendicular to the magnitude to be measured by the names of the transome unto the ends of some known part of the height as the distance of the Names is unto the rest of the transome above them so shall the known part be unto the part sought Or thus If the sight passe from the beginning of the Index being right by the vanes of the transversary to the tearmes of some parts as the distance of the vanes is unto the rest of the transversary above the index so is the part knowne unto the remainder H. This is a consectary of a knowne part of an height from whence the rest may be knowne as in the figure The first and second kinde of measuring of heights is thus The third followeth 14 If the sight be from the beginning of the Index perpendicular to the heighth as in the Index the difference of the segmeut is unto the difference of the distance or station so is the segment of the transome unto the heighth Hitherto you must recall that subtilty which was used in the third manner of measuring of lengths Let the first aime be taken from a the beginning of the Index perpendicular unto the height to be measured And from an unknowne length a i by o the end of the transome unto e the toppe of the height e i And let the segment of the Index be u a. The second ayme let it be taken from y the beginning of the same Index and out of a greater distance by s the end of the transome unto the same toppe e. And the segment of the Index let it be r l. Here as afore the measuring is performed and done by the taking of the difference of the said y r above a u Now the demonstration is concluded as in the former was taught Let the parallel l s m be erected against a o e. Here
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
distant from it Other inscripts are judged to be equall greater or lesser one than another by the diameter or by the diameters center Euclide doth demonstrate this proposition thus Let first a e and i o be equall I say they are equidistant from the center For let u y and u y be perpendiculars They shall cut the assigned a e i o into halfes by the 5 e xj And y a and s i aâ—e equall because they are the halfes of equals Now let the raies of the circle be u a aund u i Their quadrates by the 9 e xij are equall to the paire of quadrates of the shankes which paires are therefore equall betweene themselves Take from equalls the quadrates y a and s i there shall remaine y u and u s equalls and therefore the sides are equall by the 4 e 12. The converse likewise is manifest For the perpendiculars given do halfe them And the halfes as before are equall 15 Of unequall inscripts the diameter is the greatest And that which is next to the diameter is greater than that which is farther off from it That which is farthest off from it is the least And that which is next to the least is lesser than that which is farther off And those two onely which are on each side of the diameter are equall è 15 e iij. This proposition consisteth of five members The first is The diameter is the greatest iuscript The second That which is next to the diameter is greater than that which is farther off The third That which is farthest off from the diameter is the least The fourth That next to the least is lesser than that farther off The fifth That two onely on each side of the diameter are equall betweene themselves All which are manifest out of that same argument of equalitie that is the center the beginning of decreasing and the end of increasing For looke how much farther off you goe from the center or how much nearer you come unto it so much lesâ—er or greater doe you make the inscript But Euclides conclusion is by triangles of two sides greater than the other and of the greater angle The first part is plaine thus Because the diameter a e is equall to i l and l o viz. to the raies And to those which are greater than i o the base by the 9. e v j c. The second part of the nearer is manifest by the 5 e vij because of the triangle i l o equicrurall to the triangle u l y is greater in angle And therefore it is also greater in base The third and fourth are consectaries of the first The fifth part is manifest by the second For if beside i o and s r there be supposed a third equall the same also shall be unequall because it shall be both nearer and farther off from the diameter 16 Of right lines drawne from a point in the diameter which is not the center unto the periphery that which passeth by the center is the greatest And that which is nearer to the greatest is greater than that which is farther off The other part of the greatest is the lest And that which is nearest to the least is lesser than that which is farther off And two on each side of the greater or least are only equall 7 p iij. The third that a y is lesser than a u because s y which is equall to s u is lesser than the right lines s a and a u by the 9 e v j And the common s a being taken away a y shall be left lesser than a u. The fourth part followeth of the third The fifth let it be thus s r making the angle a s r equall to the angle a s u the bases a u and a r shall be equall by the 2 e v ij To these if the third be supposed to be equall as a l it would follow by the 1 e v ij that the whole angle s a should be equall to r s a the particular angle which is impossible And out of this fifth part issueth this Consectary Therefore 17 If a point in a circle be the bound of three equall right lines determined in the periphery it is the center of the circle 9 p iij. Let the point a in a circle be the common bound of three right lines ending in the periphery and equall betweene themselves be a e a i a u. I say this point is the center of the Circle 18 Of right lines drawne from a point assigned without the periphery unto the concavity or hollow of the same that which is by the center is the greatest And that next to the greatest is greater than that which is farther off But of those which fall upon the convexitiâ— of the circumference the segment of the greatest is leastâ— And that which is next unto the least is lesser than that is farther off And two on each side of the greatest or least are onely equall 8 piij. 19 If a right line be perpendicular unto the end of the diameter it doth touch the periphery And contrariwise è 16 p iij. As for example Let the circle given a e be perpendicular to the end of the diameter or the end of the ray in the end a as suppose the ray be i a I say that e a doth touch not cut the periphery in the common bound a. Therefore 20 If a right line doe passe by the center and touch-point it is perpendicular to the tangent or touch-touch-line 18 p iij. And Or thus as Schoner amendeth it If a right line be the diameter by the touch point it is perpendicular to the tangent 21 If a right line be perpendicular unto the tangent it doth passe by the center and touch-point 19. piij. Or thus if it be perpendicular to the tangent it is a diameter by the touch point Schoner For a right line either from the center unto the touch-point or from the touch point unto the center is radius or semidiameter And 22 The touch-point is that into which the perpendicular from the center doth fall upon the touch line 23 A tangent on the same side is onely one Or touch line is but one upon one and the same side H. Or. A tangent is but one onely in that point of the periphery Schoner Euclide propoundeth this more specially thus that no other right line may possibly fall betweene the periphery and the tangent And 24 A touch-angle is lesser than any rectilineall aâ—ute angle è 16 p ij Angulus contractus A touch angle is an angle of a straight touch-line and a periphery It is commonly called Angulus contingentiae Of Proclus it is named Cornicularis an horne-like cornerâ— because it is made of a right line and periphery like unto a horne It is lesse therefore than any acute or sharpe right-lined angle Because if it were not lesser a right line might fall between the periphery and the
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
first the triangles o u a s r l are equilaters by the 2 e vij seeing that the angles at a and l the externall and internall are equall in bases o u and s r for the segment in each distance is the same still Therefore u a is equall to r l. Now the rest is concluded by a sorites of foure degrees As y r is unto y u so by the 12. e vij is s r that is o u unto e i And as o u is unto e i so is a u that is l r unto a i. Therefore the remainder y l unto the remainder y a shall be as y r is unto the whole y i and therefore from the first unto the last as s r is to e i. Therefore let the difference of the Index be 23 parts◠The difference of the distance 30. foote The segment of the transome 23. parts The height shall be 57. 9 23. or foote Therefore 15 Out of the Geodesy of heights the difference of two heights is manifest Or thus By the measure of one altitude we may know the difference of two altitudes H. For when thou hast taken or found both of them by some one of the former wayes take the lesser out of the greater and the remaine shall be the heighth desired From hence therefore by one of the towers of unequall heighth you may measure the heighth of the other First out of the lesser let the length be taken by the first way Because the height of the lesser wherein thou art is easie to be taken either by a plumbe-line let fall from the toppe to the bottom or by some one of the former waies Then measure the heighth which is above the lesser And adde that to the lesser and thou shalt have the whole heighth by the first or second way The figure is thus and the demonstration is out of the 12. e vij For as a e is to e i so is a o to o u. Contrariwise out of an higher Tower one may measure a lesser 16 If the sight be first from the toppe then againe from the base or middle place of the greater by the vanes of the transome unto the toppe of the lesser heighth as the said parts of the yards are unto the part of the first yard so the heighth betweene the stations shall be unto his excesse above the heighth desired For let the parts of the yard be 12. and 6. and the summe of them 18. Now as 18. is 12. so is the whole altitude u y 190. foote unto the excesse 126⅔ foote The remainder therefore 63⅓ foote shall be a s the lesser heighth sought The second station might have beene in o the end of the perpendicular from a. But by taking the ayme out of the toppe of the lesser altitude the demonstration shall be yet againe more easie and short by the two triangles at the yard a e i and a e f resembling the two whole triangles a o u and a o y in like situation the parts of the shanke cut are on each side the segments of the transome One may againe also out of the toppe of a Turret measure the distance of two turrets one from another For it is the first manner of measuring of longitudes neither doth it here differ any whit from it more than the yard is hang'd without the heighth given The figure is thus And the Demonstration is by the 12. e vij For as a e the segment of the yard is unto e i the segment of the transome so is the assigned altitude a o unto the length o u. The geodesy or measuring of altitude is thus where either the length or some part of the length is given as in the first and second way Or where the distance is double as in the third 17 If the sight be from the beginning of the yard being right or perpendicular by the vanes of the transome unto the ends of the breadth as in the yard the difference of the segment is unto the differēce of the distance so is the distance of the vanes unto the breadth The measuring of breadth that is of a thwart or crosse line remaineth The Figure and Demonstration is thus The first ayming let it be a e i by o and u the vanes of the transome o u. The second let it be y e i by s and r the vanes of the transome s r. Then by the point s let the parallell l s m be drawne against a o e. Here first the triangles o u a and s i l are equilaters by the 2 e vij Because the angles at n and j are right angles And u a o and j l s the outter and inner are equall in their bases o u and s j by the grant Because here the segment of the transome remaineth the same Therefore u a is equall to j l. These grounds thus laid the demonstration of the third altitude here taken place For as y l is unto y a so is s j unto e r And because parts are proportionall unto their multiplicants so is s r unto e i for the rest doe agree The same shall be the geodesy or manner of measuring if thou wouldest from some higher place measure the breadth that is beneath thee as in the last example But from the distance of two places that is from latitude or breadth as of Trees Mountaines Cities Geographers and Chorographers do gaine great advantages and helpes The tenth Booke of Geometry of a Triangulate and Parallelogramme ANd thus much of the geodesy of right lines by the meanes of rectangled triangles It followeth now of the triangulate 1. A triangulate is a rectilineall figure compounded of triangles As before for the dichotomies sake of a line was made a Lineate to signifie the genus of a surface and a Body so now is for the same cause of a triangle made a Triangulate to declare and expresse the genus of a Quadrilater and Multilater and indeed more justly then before in a Lineate For triangles doe compound and make the triangulate but lines doe not make the lineate Therefore 2. The sides of a triangulate are two more than are the triangles of which it is made And 3. Homogeneall Triangulates are cut into an equall number of triangles è 20 p vj. For if they be Quadrangles they be cut into two triangles If Quinquangles into 3. If Hexangles into 4 and so forth 4. Like triangulates are cut into triangles alike one to another and homologall to the whole è 20 p vj. Or thus Like Triangulates are divided into triangles like one unto another and in porportion correspondent unto the whole H. As in these two quinqualges First the particular triangles are like betweene themselves For the shankes of a e u and y s m equall angles are proportionall by the grant Therefore the triangles themselves are equiangles by 14 e vij And therefore alike by the 12