Of Geometry, the thirteenth Booke, Of an Oblong.
1 An Oblong is a rectangle of inequall sides, 31. d j.
Or thus: An Oblong is a rectangled parallelogramme, being not equilater: H. As here is ae, io.
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.
As let the right line ae, be 6. And let it be cut into two parts ai, 2. and ie, 4. The rectangle 12. made of ae, 6. the whole, and of ai, 2. the one segment, shall be equall to iu, 8. the rectangle made
of the same ai, 2. and of ie, 4. And also to ao, 4. the quadrate of the said segment ai, 2.
Now a rectangle is here therefore proposed, because it may be also a quadrate, to wit, if the line be cut into to 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 [7. e xj].
As let the line ae, 6. be cut into ai, 2. io. 2. and oe, 2. The Oblongs as, 12. ir, 12. and oy, 12. made of the whole ae, and of those segments, are equall to ay, the quadrates of the whole.
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.
As for example, let the right line ae, 8. be cut into ai, 6. and ie, 2. The oblongs ao, and iu, of the whole, and 2. the segments, are 32. The quadrate of 6. the other segment 36. And the whole 68. Now the quadrate, of the whole ae. 8. is 64. And the quadrate of the said segment 2, is 4. And the summe of these is 68.
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.
As in the triangle aei, let the angle at i, be taken for an acute angle. Here by the [4. e], two obongs of ei, and oi, with the quadrate of eo, are equall to the quadrates of ei, and oi. Let the quadrate of ao, be added to both in common. Here the quadrate of ei, with the quadrates of io, and oa, that is the [9 e xij], with the quadrate of ia, is equall to two oblongs of ei, and oi, with two quadrates of eo & oa, that is by the [9 e xij], with the quadrate of ea. Therefore two oblongs with the quadrate of the base, are equall to the quadrates of the shankes: And the base is exceeded of the shankes by two oblongs.
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.
As in the acute angled triangle aei, let the sides be 13, 20, 21. And let ae be the base of the acute angle. Now the quadrate or square of 13 the said base is 169: And the quadrate of 20, or ai, is 400: And of 21, or ei, is 441. The summe of which is 841. And 841, 169, are 672:
Whose halfe is 336. And the quotient of 336, divided by 21, is 16, the segment of the dividing shanke ei, from the angle aei, unto ao, the perpendicular of the toppe. Now 21, 16, are 5. Therefore the other segment or portion of the said ei, is 5.
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 ao, by the [9 e xij].
Here the perpendicular now found, and the sides cut, are the sides of the rectangle, whose halfe shall be the content of the Triangle: As here the Rectangle of 21 and 12 is 252; whose halfe 126, is the content of the triangle.
The second section followeth from whence ariseth the fourth rate or comparison.
7. If a right line be cut into two equall parts, and otherwise; the oblong of the unequall segments, with the quadrate of the segment betweene them, is equall to the quadrate of the bisegment. 5 p ij.
As for example, Let the right line ae 8, be cut into two equall portions, ai 4, and ie 4. And otherwise that is into two unequall portions, ao 7, and oe 1: The oblong of 7 and 1, with 9, the quadrate of 3, the intersegment, (or portion cut betweene them) that is 16; shall bee equall to the quadrate of ie 4, which is also 16. Which is also manifest by making up the diagramme as here thou seest. For as the parallelogramme as is by the [26 e x], equall to the
parallelogramme iu; And therefore by the [19 e x], it is equall to oy. For ou, is common to both the equall complements, Therefore if so be added in common to both; the ar, shall be equall to the gnomon mni: Now the quadrate of the segment betweene them is sl. Wherefore ar, the oblong of the unequall segments, with s the quadrate of the intersegment, is equall to iy the quadrate of the said bisegment.
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.
As for example, let the line ae 6, be cut into two equall portions, ai 3, and ie 3: And let it be continued unto eo 2: The oblong 16, made of 8 the continued line, and of 2, the continuation; with 9 the quadrate of 3, the halfe, (that is 25.) shall be equall to 25, the quadrate of 3, the halfe and 2, the continuation, that is 5. This as the former, may geometrically, with the helpe of numbers be expressed. For by the [26 e x], as is equall to iy: And by the [19 e x], it is equall to yr, the complement. To these equalls adde so. Now the oblong au, shall be equall to the gnomon nju. Lastly, to the equalls adde the quadrate of the bisegment or halfe. The Oblong of the continued line and of the
continuation, with the quadrate of the bisegment, shall be equall to the quadrate of the line compounded of the bisegment and continuation. These were the rates of an oblong with a rectangle.
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 III 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 II 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 ae, and ai: And let them comprehend the rectangled parallelogramme ao: And let the said right lines given be continued infinitely, ae toward u; and ai toward y. Now let the Mesographus uy, 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 ae is to iy: so is iy to eu, so is eu, to ai.
First let from s, the center, sr be perpendicular to the side ae: It shall therefore cut the said ae, into two parts, by the [5 e xj]: And therefore againe, by the [7 e], the oblong made of au, and ue, with the quadrate of re, is equall to the quadrate of ru: And taking to them in common rs, the oblong with two quadrates er, and rs, that is, by the [9 e xij], with the quadrate se is equall to the quadrates ru and rs, that is by the [9 e xij], to the quadrate su. The like is to be said of the oblong of ay, and yi, by drawing the perpendicular sl, as afore. For this oblong with the quadrates li, and sl, that is, by the [9 e xij], with the quadrate is, is equall to the quadrates yl, and ls, that is, by the [9 e 12], to ys. Therefore the oblongs equall to equalls, are equall betweene themselves: And taking from each side of equall rayes, by the [11 e x], equall quadrates se and si, there shall remaine equalls. Wherefore by the [27 e x], the sides of equall rectangles are reciprocall: And as au is to ay: so by the [13 e vij], oi, that is, by the [8 e x], ea, to iy: And so therefore by the concluded, yi is to ue; And so by the [13 e vij], is ue to eo, that is, by the [8 e x], unto ai. Therefore as ea is to yi: so is yi to ue; and so is ue, to ai. Wherefore eu, iy, the intersegments or portions cut, are the two meane proportionals betweene the two lines given.
The fourteenth Booke, of P. Ramus Geometry: Of a right line proportionally cut: And of other Quadrangles, and Multangels.
Thus farre of the threefold section, from whence we have the five rationall rates of equality: There followeth of the third section another section, into two segments proportionall to the whole. The section it selfe is first to be defined.
1. A right line is cut according to a meane and extreame rate, when as the whole shall be to the greater segment; so the greater shall be unto the lesser. 3 d vj.
This line is cut so, that the whole line it selfe, with the two segments, doth make the three bounds of the proportion: And the whole it selfe is first bound: The greater segment is the middle bound: The lesser the third bound.
2. If a right line cut proportionally be rationall unto the measure given, the segments are unto the same, and betweene themselves irrationall è 6 p xiij.
Euclide calleth each of these segments Ἀποτομὴ that is, Residuum, a Residuall or remaine: And surely these cannot otherwise be expressed, then by the name Residuum; As if a line of 7 foote should thus be given or put downe: The greater segment shall be called a line of 7 foote, from whence the lesser is substracted: Neither may the lesser otherwise be expressed, but by saying, It is the part residuall or remnant of the line of 7 foote, from which the greater segment was subtracted or taken.
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.
Let the right line gived be ae. The quadrate of the same let it be aeio: And from the angle e, unto u, the middest of the conterminal side, let the right line eu, be drawne; Then compare or lay it to the halfe ua; The difference of it above the said halfe shall be ay, This ay, say 1, is the greater segment of ae, the line given, proportionally cut.
For of ya, let the quadrate aysr, be made: And let sr, be continued unto l. Now by the [8 e, xiij]. the oblong of oy, and ay, with the quadrate of ua, is equall to the quadrate of uy, that is by the construction of ue: And therefore, by the [9 e xij]. it is equall to the quadrates ea, and au: Take away from each side the common oblong al, and the quadrate yr, shall be equall to the oblong ri. Therefore the three right lines, ea, ar, and re, by the [8 e xij]. are continuall proportionall. And the right line ae, 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 oy, 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 xiij.
This is the fabricke or manner of making a proportionall section. A threefold rate followeth: The first is of the greater segment.
Let therefore the right line ae, be cut proportionally in i: And let the greater segment be ia: and let the line cut be continued unto io, so that oa, be the halfe of the line cut. I say, the quadrate of io, is in power five times so great, as ys, the power of the quadrate of ao. Let therefore of ao, be made the quadrate iosr: We doe see the quadrate ua, to be once contained in the quadrate si. Let us now
teach that it is moreover foure times comprehended in lmn, the gnomon remaining: Let therefore the quadrate aeiu, be made of the line given: And let ri, be continued unto f. Here the quadrate ae, is ([14. e xij].) foure times so much as is that au, made of the halfe: and it is also equall to the gnomon lmn: For the part iu, is equall to ry; first by the grant, seeing that ai, is the greater segment, from whence ry, is made the quadrate, because the other Diagonall is also a quadrate: Secondarily the complements sy, and yi, by the [19. e x], are equall: And to them is equall af. For by the [23. e x]. and by the grant, it is the double of the complement yi. Therefore it is equall to them both. Wherefore the gnomon lmn, is equall to the quadruple quadrate of the said little quadrate: And the greater segment continued to the halfe of the right line given is of power five fold to the power of ao.
The converse is apparent in the same example: For seeing that io, is of power five times so much as is ao; the gnomon lmn, shall be foure times so much as is ua: Whose quadruple also, by the [14. e xij], is av. Therefore it is equall to the gnomon. Now aj, is equall to ae: Therefore it is the double also of ao, that is of ay: And therefore by the [24. e x]. it is the double of at: And therefore it is equall to the complements iy, and ys: Therefore the other diagonall yr, is equall to the other rectangle iv. Wherefore, by the [8 e xij]. as ev, that is, ae, is to yt, that is ai: so is ai, unto ie; Wherefore by the [1 e], ae, is proportionall cut: And the greater segment is ai, 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 xiij.
As here, the right line ae, let it be cut proportionally in i: And the lesser ie, let it be continued even unto o, the halfe of the greater ai. I say, that the power of oe, shall be five times as much as is the power of io. Let a quadrate
therefore be made of ae: And let the figure be made up (as you see:) And let the quadrate of the halfe be noted with su: And the gnomon rlm. Here the first quadrate oy, is five times as great, as the second su. For it doe containe it once: And the gnomon rlm, remaining containeth it foure times. For it is equall to the Oblong in; because os, the complement is equall to sy, by the [19 e x]; And therefore also it is equall to in; seeing the whole complement as, is equall to the whole complement sn: And av, is equall to os, by the construction, and [23. e x]: And adding to both the common oblong iy, the whole gnomon is equal to the whole oblong. But the oblong in, is equall to the quadrate ai, by the grant, & [8 e xij]. which by the [14. e xij]. is foure times as great, as the quadrate su. Wherefore the lesser segment ie, continued to io, the halfe of the greater segment, is of power five times as much as is the halfe of the same.
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.
Let the right line ae be proportionally cut in i, and let the figure be made up: The oblong ay, and io, with the quadrate su, by the [4 e xiij], are equall to the quadrates of ae, and ie, whose power is treble to that of ai. For they doe once containe the quadrate su; And each of the oblongs is equall to the same quadrate su, by the grant, and [8 e xij]. Therefore they doe containe it thrise.
8 An obliquangled parallelogramme is either a Rhombus, or a Rhomboides.
9 A Rhombus is an obliquangled equilater parallelogramme 32 d j.
Whereupon it is apparant that a Rhombus is a square having the angles as it were pressed, or thrust nearer together, by which name, both the Byrt or Turbot, a Fish; and a Wheele or Reele, which Spinners doe use; and the quarrels in glasse windowes, because they are cut commonly of this forme, are by the Greekes and Latines so called.
It is otherwise of some called a Diamond.
10 A Rhomboides is an obliquangled parallelogramme not equilater 33. d j.
And a Rhomboides is so opposed to an oblong, as a Rhombus is to a quadrate.
So also looke how much the straightening or pressing
together is greater, so much is the inequality of the obtuse and acute angles the greater. As here.
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 Rhomboides is thus.
11 A Trapezium is a quadrangle not parallelogramme. 34. d j.
Of the quadrangles the Trapezium remaineth for the last place: Euclide intreateth this fabricke to be granted him, that a Trapezium may be called as it were a little table: And surely Geometry can yeeld no reason of that name.
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. d j.
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.
As here, this quinquangle is measured by his three triangles. The first triangle, whose sides are 9. 10. and 17. by the [18. e xij]. is 36. The second, whose sides are 6, 17, and 17. by the same e, is 50.20/101. The third, whose sides are 17, 15. and 8. by the same, is 60. And the summe of 36. 50.20/101. and 60. is 146.20/101, for the whole content of the Quinquangle given.
14 If an equilater quinquangle have three sides equall, it is equiangled. 7 p 13.
This of some, from the Greeke is called Pentagon; of others a Pentangle, by a name partly Greeke partly Latine.
As in the Quinquangle aeiou, the three angles at a, e, and i, are equall: Therefore the other two are equall: And they are equall unto these. For let eu, ai, ia, be knit together with right lines. Here the triangles aei, and eau. by the grant, and by the [2] and [1 e vij]. are equilaters and equiangles: And the Bases ai, and eu, are
equall: And the Angles, eai, and aue, are equall: Item aeu, and eia. Therefore ay, and ye, are equall, by the [17 e vj]. Item the remainder uy, is equall to the remainder yi, when from equals equals be subtracted. Moreover by the grant, and by the [17 e vj], oui, and oiu, are equall. Wherefore three are equall; And therefore the whole angle is equall at u, to the whole angle at i. And therefore it is equall to those which are equall to it.
I say moreover that the angle at o, is likewise equall, if ao, and oe, be knit together with a right line, as here: For three in like manner do come to be equall.
But if the three angles non deinceps not successively following be equall, as aio, the businesse will yet be more easie, as here: For the angles eua, and eoi, are equall by the grant: And the inner also eou, and euo. Therefore the wholes of two are equall. Of the other at e, the same will fall out, if iu, be knit together with a right line iu, as here: For the wholes of two shall be equall.
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. è 15 d j.
As here thou seest. A Rectilineall plaine was at the [3 e vj], defined to be a plaine comprehended of right lines. And so also might a circle have beene defined to be a plaine comprehended of a periphery or bought-line, but this is better.
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 Circles are as the quadrates or squares made of their diameters 2 p. xij.
For Circles are like plaines. And their homologall sides are their diameters, as was foretold at the [24 e iiij]. And therefore by the [1 e vj], they are one to another, as the quadrates of their diameters are one to another, which indeed is the double reason of their homologall sides. As here the Circle aei, is unto the Circle ouy as 25, is unto 16, which are
the quadrates of their Dieameters, 5 and 4.
Therefore
3. The Diameters are, as their peripheries Pappus, 5 l. xj, and 26th. 18.
As here thou seest in ae, and io.
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.
As here ae, because the right within the same points is shorter, than the periphery is, by the [5 e ij].
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.
As let the right line given be a: And from e, the end of the diameter ei: And with eo, a part of it equall to a, the line given, describe the circle eu: A right line eu, drawne from the end e, unto u, the meeting of the two peripheries, shall be inscribed in the circle given, by the [5 e], equall to the line given; because it is equall to eo, by the [10 e v], seeing it is a ray of the same Circle.
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.
As let the Inscript ae, cut the inscript iu perpendicularly: dividing it into two equall parts in o. I say that the one inscript thus halfing the other, is the diameter of the Circle: And that the middest of it is the center thereof: As in the circle, let the Inscript is, cut the inscript ae, and that perpendicularly dividing into two equall parts in o. I say that iu, thus dividing ae, is the Diameter of the Circle: And y, the middest of the said iu, is the Center of the same.
The cause is the same, which was of the [5 e xj]. Because the inscript cut into halfes is 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 manifest 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 aos, shall be equilater to the triangle eos. For ao, oe, are equall by the grant: Item sa, and se, are the rayes of the circle: And so, is common to both the triangles. Therefore by the [1 e vij], the angles on each side at o are equall; And by the [13 e v], they are both right angles. Therefore soe is a right angle; It is therefore equall by the grant, to the right angle yoe, that is, the part is equall to the whole, which is impossible. Wherefore s is not the Center. The same will fall out of any other points whatsoever out of y.
Therefore
8. If two right lines doe perpendicularly halfe two
inscripts, the meeting of these two bisecants shall be the Center of the circle è 25 p iij.
As here ae, and io, let them cut into halfes the right lines uy, and ys. And let them meete, that they cut one another in r. I say r is the center of the circle ayoseiu. For before, at the [6], and [7 e], it was manifest that the Center was in the Diameter. And in the meeting of the diameters. [Therefore two manner of wayes is the Center found; First by the middle of the diameter: And then againe by the concourse, or meeting of the diameters, in the middest of the lines halfed or cut into two equall portions.] Here is no neede of the meeting of many diameters, one will serve well enough.
And one may
9. Draw a periphery by three points, which doe not fall in a right line.
As here, by aei, (First from a, to e, let a right line be drawne; And likewise from e to i. Then, by the [12 e v], let both these lines be cut into equall parts, by two infinite right lines: These halfing lines also shall meete: And in their meeting shall be the Center, by the [8 e]. And therefore from that meeting unto any of the sayd points given is the ray of the periphery desired.)
10. If a diameter doe halfe an inscript, that is, not a diameter, it doth cut it perpendicularly: And contrariwise: 3 p iij.
As let the diameter ae, halfe the inscript io, which is not a diameter: And let the raies of the circle bee ui, and uo. The cause in all is the same, which was of the [5 e xj].
11. If inscripts which are not diameters doe cut one another, the segments shall be unequall. 4 p iij.
This is a consectary drawne out of the [28 e iiij]. For if the inscripts were halfed, they should be diameters, against the grant.
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.
If the inscripts thus cut be diameters, the proportion is manifest, as in the first figure. For the Rectangle of the segments, of the one is equall to the rectangle of the segments of the other, seeing they be both quadrates of equall sides. If they be not diameters let them otherwise as ae, and io: I say the Oblong of au, and ue, is equall to the Oblong of ou, and ui. For let the raies from the Center y, be ye, and yi. To the quadrate of each of these both the rectangles of the segments shall be equall. For by the [7 e], let the diameter yu, fall upon the point of the common section u; And let ys, and sr, be perpendiculars. Here by the [5 e xj]. the inscripts are cut equally in the points r and s: And unequally in the point u: Therefore by the [7 e xiij], the
oblong, of ou, and ui, with the quadrate su, is equall to the quadrates si; And adding ys, the same oblong, with the quadrates us and sy, that is, by the [9 e xij], with the quadrate yu, is equall to the quadrates is and sy, that is, by the [9 e xij], to the quadrate iy, that is, by the [5 e xij], to ye, to the which by the same cause it is manifest the other oblong with the quadrate yu is equall. Let the quadrate yu, bee taken from each of them: And then the oblongs shall be equall to the same: And therefore betweene themselves.
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.
As it appeareth in the next figure, of the lines ae and io, unto which the perpendiculars uy and us, from the Center u, are equall.
14. If inscripts be equall, they be equally distant from the center: And contrariwise. 13 p iij.
The diameters in the same circle, by the [28 e iiij], are equall: And they are equally distant from the center, seeing they are by the center, or rather are no whit at all distant from it: Other inscripts are judged to be equall, greater, or lesser one than another, by the diameter, or by the diameters center.
Euclide doth demonstrate this proposition thus: Let first ae and io be equall; I say they are equidistant from the center. For let uy, and us, be perpendiculars: They shall cut the assigned ae, & io, into halfes, by the [5 e xj]: And ya and si are equall, because they are the halfes of equals. Now let the raies of the circle be ua, and ui: 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 ya, and si, there shall remaine yu, and us, 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 inscript: 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 lesser or greater doe you make the inscript.
Let there be in a circle; many inscripts, of which one, to wit, ae, let it be the diameter: I say, that it is of them all the greatest or longest. But let io, be nearer to the diameter, (or as in the former Elements was said) nearer to the center, than uy. I say that io, is longer than uy. Moreover, let uy, be the farthest off from the same diameter or center; I say the same uy, is the shortest of them all. Now to this shortest uy, let io, be nearer than ae; I say therefore that io, also is lesser than ae. Let at length io, be not the diameter: I say that beyond the diameter ae, there may onely a line be inscribed equall unto it, such as is sr. And those equal betweene themselves on each side of the diametry may only be given, not three, nor more. And after the same manner also, onely one beyond the diameter, may possibly be equall to uy, to wit, that which is as farre off from the diameter as it is; and so in others.
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 ae, is equall to il, and lo, viz. to the raies; And to those which are greater than io, the base by the [9. e vj] &c.
The second part of the nearer, is manifest by the [5 e vij]. because of the triangle ilo, equicrurall to the triangle uly, 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 io, and sr, 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 left. 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 first part of ae, and ai, is manifest, as before, by the [9 e vj]. The second of ai, and ao; Item of ao, and au, is plaine by the [5 e vij].
The third, that ay, is lesser than au, because sy, which is equall to su, is lesser than the right lines sa, and au, by the [9 e vj]: And the common sa, being taken away, ay shall be left, lesser than au.
The fourth part followeth of the third.
The fifth let it be thus: sr, making the angle asr, equall to the angle asu, the bases au, and ar, shall be equall by the [2 e vij]. To these if the third be supposed to be equall, as al, it would follow by the [1 e vij]. that the whole angle sa, should be equall to rsa, 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 ae, ai, au. I say this point is the center of the Circle.
Otherwise from a point of the diameter which is not the center, not onely two right lines on each side should be equall. For by any point whatsoever the diameter may be drawne. Such was before observed in a quinquangle; If three angles be equall, all are equall; so in a Circle: If three right lines falling from the same point unto the perephery be equall, all are equall.
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 convexitie of the circumference, the segment of the greatest is least. And that which is next unto the least is lesser than that is farther off: And two on each side of the greatest or least are onely equall. 8 p iij.
The demonstration of this is very like unto the above mentioned, of five parts. And thus much of the secants, the Tangents doe follow.
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 ae, be perpendicular to the end of the diameter, or the end of the ray, in the end a, as suppose the ray be ia: I say, that ea, doth touch, not cut the periphery in the common bound a.
This was to have beene made a postulatum out of the definition of a perpendicle: Because if this should leane never so little, it should cut the periphery, and should not be perpendicular: Notwithstanding Euclide doth force it thus: Otherwise let the right line ae, be perpendicular to the diameter ai. And a right line from o, with the center i, let it fall within the circle at o, and let oi, joyned together. Here in the triangle aoi, two angles, contrary to the [13 e vj], should be right angles at a, by the grant: And at o, by the [17 e vj].
The demonstration of the converse is like unto the former. For if the tangent, or touch-line ae, be not perpendicular to the diameter iou, let oe, from the center o, be drawne perpendicular; Then shall the angle oei, be right angle: And oie an acutangle: And therefore by the [22 e vj], oi, that is oy, shall be greater then oye, that is the part, then the whole.
Therefore
20 If a right line doe passe by the center and touch-point, it is perpendicular to the tangent or touch-line. 18 p iij.
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. p iij.
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.
It is a consectary drawne out of the [xiij. e ij]. Because a tangent is a very perpendicular.
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 acute 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 contingentiæ: Of Proclus it is named Cornicularis, an horne-like corner; because it is made of a right line and periphery like unto a horne. It is lesse therefore than any acute or sharpe right-lined angle: Because if it were not lesser, a
right line might fall between the periphery and the perpendicular.
And
25 All touch-angles in equall peripheries are equall.
But in unequall peripheries, the cornicular angle of a lesser periphery, is greater than the Cornicular of a greater.
26 If from a ray out of the center of a periphery given, a periphery be described unto a point assigned without, and from the meeting of the assigned and the ray, a perpendicular falling upon the said ray unto the now described periphery, be tied by a right line with the said center, a right line drawne from the point given unto the meeting of the periphery given, and the knitting line shall touch the assigned periphery 17 p iij.
As with the ray ae, from the center a, of the periphery assigned, unto the point assigned e, let the periphery eo, be described: And let io, be perpendicular to the ray unto the described periphery. This knit by a right line unto the center a, let eu, be drawne. I say, that eu, doth touch the periphery iu, assigned: Because it shall be perpendicular unto the end of the diameter. For the triangles eau, and oai, by the [2 e vij], seeing they are equicrurall; And equall in shankes of the common angle; they are equall in the angles at the base. But the angle aio, is a right angle: Therefore the angle eua, shall be a right angle. And therefore the right line eu, by the [13 e ij], is perpendicular to ao.
Thus much of the Secants and Tangents severally: It followeth of both kindes joyntly together.
27 If of two right lines, from an assigned point without, the first doe cut a periphery unto the concave,
the other do touch the same; the oblong of the secant, and of the outter segment of the secant, is equall to the quadrate of the tangent: and if such a like oblong be equall to the quadrate of the other, that same other doth touch the periphery: 36, and 37. p iij.
If the secant or cutting line do passe by the center, the matter is more easie and as here, Let ae, cut; And ai, touch: The outter segment is ao, and the center u, Now ui, shall be perpendicular to the tangent ai, by the 20. e: Then by [8 e xiij], the oblong of ea, and ao, with the quadrate of au, that is, of iu, is equall to the quadrate of au, that is, by the [9 e xij]. to the quadrates of ai, and iu. Take iu, the common quadrate: The Rectangle shall be equall to the quadrate of the tangent.
If the secant doe not passe by the center, as in this figure, the center u, found by the [7 e], iu, shall be by the [20 e] perpendicular unto the tangent ai; then draw ua, and uo, and the perpendicular halving oe, by the [10 e]. Here by the [8 e xiij], the oblong of ae, and ao, with the quadrate oy, is equal to the quadrate ay: Therefore yu, the common quadrate added, the same oblong, with the quadrates oy, and yu, that is by the [9 e xij]. with the quadrate ou, is equall to the quadrates ay, and uy, that is, by the [9 e xij], to au, that is, againe, to ai, and iu. Lastly, let ur, and iu, two equall quadrates be taken from each, and there wil remaine the oblong equall to the quadrate of the tangent.
The converse is likewise demonstrated in this figure. Let the Rectangle of ae, and ay, be equall to the quadrate of ai.
I say, that ai doth touch the circle. For let, by the [26 e], ao the tangent be drawne: Item let au, ui, and uo bee drawne. Here the oblong of ea, and ay, is equall to the quadrate of ao, by the [27 e]: And to the quadrate of ai, by the grant. Therefore ai, and ao, are equall. Then is uo, by the [20 e], perpendicular to the tangent. Here the triangles auo, and aui, are equilaters: And by the [1 e vij], equiangles. But the angle at o is a right angle: Therefore also a right angle and equall to it is that at i, by the [13 e iij], wherefore ai is perpendicular to the end of the diameter: And, by the [19 e], it toucheth the periphery.
Therefore
28. All tangents falling from the same point are equall.
Or, Touch lines drawne from one and the same point are equall: H.
Because their quadrates are equall to the same oblong.
And
29. The oblongs made of any secant from the same point, and of the outter segment of the secant are equall betweene themselves. Camp. 36 p iij.
The reason is because to the same thing.
And
30. To two right lines given one may so continue or joyne the third, that the oblong of the continued and the continuation may be equall to the quadrate remaining. Vitellio 127 p j.
As in the first figure, if the first of the lines given be eo, the second ia, the third oa.
Now are we come to Circular Geometry, that is to the Geometry of Circles or Peripheries cut and touching one another: And of Right lines and Peripheries.
31. If peripheries doe either cut or touch one another, they are eccentrickes: And they doe cut one another in two points onely, and these by the touch point doe continue their diameters, 5. 6. 10, 11, 12 p iij.
All these might well have beene asked: But they have also their demonstrations, ex impossibili, not very difficult.
The first part is manifest, because the part should be equall to the whole, if the Center were the same to both, as a. For two raies are equall to the common raie ao: And therefore ae and ai, that is, the part and the whole, are equall one to another.
The second part is demonstrated as the first: For otherwise the part must be equall to the whole, as here ae and ai, the raies of the lesser periphery; And ae, and ao, the raies of the greater are equall. Wherefore ai, should be equall to ao the Part to the whole.
If the Peripheries be outwardly contiguall, the matter is more easie, and by the judgement of Euclide it deserved not a demonstration, as here.
The third part is apparent out of the first: Otherwise those which cut one another should be concentrickes. For, by the [7 e], the center being found: And by the [9 e], three right lines being drawne from the center unto three points of
the sections, the three raies must be equall, as here.
The fourth part is demonstrated after the same manner: Because otherwise the Part must be greater then the whole. For let the right line aeio, be drawne by the centers a and e: And let the particular raies be eu, and au. Here two sides ue, and ea, of the triangle uea, by the [9 e vj], are greater than ua: And therefore also then ao; Take away ae, the remainder ue, shall be greater than eo. But ei is equall to eu. Wherefore ei is greater than eo, the part, than the whole.
The same will fall out, if the touch be without, as here: For, by the [9 e vj], ea and ia, are greater than ie. But eo and iu, are equall to ea, and ia. Wherefore eo, and iu, are greater than ie, the parts than the whole.
Of right lines and Peripheries joyntly the rate is but one.
32. If inscripts be equall, they doe cut equall peripheries: And contrariwise, 28, 29 p iij.
Or thus: If the inscripts of the same circle or of equall circles be equall, they doe cut equall peripheries: And contrariwise B.
Or thus: If lines inscribed into equall circles or to the same be equall, they cut equall peripheries: And contrariwise, if they doe cut equall peripheries, they shall themselves be equall: Schoner.
The matter is apparent by congruency or application: as here in this example. For let the circles agree, and then shall equall inscripts and peripheries agree.
Except with the learned Rodulphus Snellius, you doe understand aswell two equall peripheries to be given, as two equall right lines, you shall not conclude two equall sections, and therefore we have justly inserted of the same, or of equall Circles; which we doe now see was in like manner by Lazarus Schonerus.
The sixteenth Booke of Geometry, Of the Segments of a Circle.
1. A Segment of a Circle is that which is comprehended outterly of a periphery, and innerly of a right line.
The Geometry of Segments is common also to the spheare: But now this same generall is hard to be declared and taught: And the segment may be comprehended within of an oblique line either single or manifold. But here we follow those things that are usuall and commonly received. First therefore the generall definition is set formost,
for the more easie distinguishing of the species and severall kindes.
2. A segment of a Circle is either a sectour, or a section.
Segmentum a segment, and Sectio a section, and Sector a sectour, are almost the same in common acceptation, but they shall be distinguished by their definitions.
3. A Sectour is a segment innerly comprehended of two right lines, making an angle in the center; which is called an angle in the center: As the periphery is, the base of the sectour, 9 d iij.
As aei is a sectour. Here a sectour is defined, and his right lined angle, is absolutely called The greater Sectour which notwithstanding may be cut into two sectours by drawing of a semidiameter, as after shall be seene in the measuring of a section.
4. An angle in the Periphery is an angle comprehended of two right lines inscribed, and jointly bounded or meeting in the periphery. 8 d iij.
This might have beene called The Sectour in the Periphery, to wit, comprehended innerly of two right lines joyntly bounded in the periphery; as here aei.
5. The angle in the center, is double to the angle of the periphery standing upon the same base, 20 p iij.
The variety or the example in Euclide is threefold, and yet
the demonstration is but one and the same: As here eai, the angle in the center, shall be prooved to be double to eoi, the angle in the periphery, the right line ou cutting it into two triangles on each side equicrurall; And, by the [17 e vj], at the base equiangles: Whose doubles severally are the angles, eau, of eoa: And iau, of ioa, For seeing it is equall to the two inner equall betweene themselves by the [15 e vj]; it shall be the double of one of them. Therefore the whole eai, is the double of the whole eoi.
The second example is thus of the angle in the center aei: And in the periphery aoi. Here the shankes eo, and ei, by the [28 e iiij], are equall: And by the [17 e vj], the angles at o and i are equall: To both which the angle in the center is equall, by the [15 e vj]. Therefore it is double of the one.
The third example is of the angle in the center, aei, And in the periphery aoi, Let the diameter be oeu. Here the whole angle ieu, by the [15 e vj], is equall to the two inner angles eoi, and eio, which are equall one to another, by the [17 e vj]: And therefore it is double of the one. Item the particular angle aeu, is equall by the [15 e vj], to the angles eoa, and eao, equall also one to another, by the [17 e vj]. Therefore the remainder aei, is the double of the other aoi, in the periphery.
Therefore
6. If the angle in the periphery be equall to the
angle in the center, it is double to it in base. And contrariwise.
This followeth out of the former element: For the angle in the center is double to the angle in the periphery standing upon the same base: Wherefore if the angle in the periphery be to be made equall to the angle in the center, his base is to be doubled, and thence shall follow the equality of them both: S.
7. The angles in the center or periphery of equall circles, are as the Peripheries are upon which they doe insist: And contrariwise. è 33 p vj, and 26, 27 p iij.
Here is a double proportion with the periphery underneath, of the angles in the center: And of angles in the periphery. But it shall suffice to declare it in the angles in the center.
First therefore let the Angles in the center aei, and ouy be equall: The bases ai, and oy, shall be equall, by the [11 e vij]: And the peripheries, ai, and oy, by the [32 e xv], shall likewise be equall. Therefore if the angles be unequall, the peripheries likewise shall be unequall.
The same shall also be true of the Angles in the Periphery. The Converse in like manner is true: From whence followeth this consectary:
Therefore
8. As the sectour is unto the sectour, so is the angle unto the angle: And Contrariwise.
And thus much of the Sectour.
9. A section is a segment of a circle within cōprehended of one right line, which is termed the base of the section.
As here, aei, and ouy, and srl, are sections.
10. A section is made up by finding of the center.
The Invention of the center was manifest at the [7 e xv]: And so here thou seest a way to make up a Circle, by the [8 e xv].
11 The periphery of a section is divided into two equall parts by a perpendicular dividing the base into two equall parts. 20. p iij.
Let the periphery of the section aoe, to be halfed or cut into two equall parts. Let the base ae, be cut into two equall parts by the pendicular io, which shall cut the periphery in o, I say, that ao, and oe, are bisegments. For draw two right lines ao, and oe, and thou shalt have two triangles aio, and eio, equilaters by the [2 e vij]. Therefore the bases ao, and oe, are
equall: And by the [32. e xv]. equall peripheries to the subtenses.
Here Euclide doth by congruency comprehende two peripheries in one, and so doe we comprehend them.
12 An angle in a section is an angle comprehended of two right lines joyntly bounded in the base and in the periphery joyntly bounded 7 d iij.
Or thus: An angle in the section, is an angle comprehended under two right lines, having the same tearmes with the bases, and the termes with the circumference: H. As aoe, in the former example.
13 The angles in the same section are equall. 21. p iij.
Let the section be eauo, And in it the angles at a, & u: These are equall, because, by the [5 e], they are the halfes of the angle eyo, in the center: Or else they are equall, by the [7 e], because they insist upon the same periphery.
Here it is certaine that angles in a section are indeed angles in a periphery, and doe differ onely in base.
14 The angles in opposite sections are equall to two right angles. 22. p iij.
For here the opposite angles at a, and i, are equall to the three angles of the triangle eoi, which are equall to two right angles, by the [13 e vj]. For first i, is equall to it selfe: Then a, by parts is equall to the two other. For eai, is equall to eoi, and iao, to oei, by the [13 e]. Therefore the opposite angles are equall to two right angles.
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 è 10. d iij.
As here aei, and ouy. The triangle here inscribed, seeing they are equiangles, by the grant; they shall also be alike, by the [12 e vij].
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 aoe, shall be lesse than the angle aie, 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 xv]. one may finde a section like unto another assigned, or else from a circle given to cut off one like unto it.
17 Angle of a section is that which is comprehended of the bounds of a section.
As here eai: And eia.
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 half 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 d j.
As aei, is a semicircle: The other sections, as oyu, and oeu, are unequall sections: that greater; this lesser.
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, è 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.
There are seven parts of this Element: The first is that The angle in a semicircle is a right angle: as in aei: For if the ray oe, be drawne, the angle aei, shall be divided into two angles aeo, and oei, equall to the angles eao, and eio, by the [17 e vj]. Therefore seeing that one angle is equall to the other two, it is a right angle, by the [6 e viij]. Aristotle saith that the angle in a semicircle is a right angle, because it is the halfe of two right angles, which is all one in effect.
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 aie, is part of the rectilineall right angle aiu.
The third part, That it is greater than any acute angle; is manifest out of the [23. e xv]. 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 aei, is lesser than a right angle; because it is in the same triangle aei, which at a, is a right angle. And if neither of the shankes be by the center, not withstanding an angle may be made equall to the assigned in the same section.
The fifth is thus: The angle of the greater section eai, is greater than a right angle: because it containeth a right-angle.
The sixth is thus, the angle aoe, in a lesser section, is greater than a right angle, by the [14 e xvj]. Because that which is in the opposite section, is lesser than a right angle.
The seventh is thus. The angle eao, is lesser than a right-angle: Because it is part of a right angle, to wit of the outter angle, if ia, 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 ae, from whose point a, a perpendicular is to be raised.
The right line ae, let it be cut by the periphery aei, (whose center o, is out of the assigned ae,) and that in the point a, and a contingent point, as in e: And from e, let the
diamiter be eoi: The right line ai, from a, the point given, knitting it with the diameter ioe, shall be perpendicular upon the infinite line ae; Because with the said infinite, it maketh an angle in a semicircle.
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 ae be sought: Let the right line ioe, be made the diameter of the peripherie; and withall let it make with the infinite right line given 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.
As in this example; The power of the diameter ae, is greater than the power of ei, by the quadrate of ai. For the triangle aei, shall be a rectangle; And by the [9 e xij.] ae, the greater shall be of
power equall to the shankes. Out of an angle in a semicircle Euclide raiseth two notable fabrickes; to wit, the invention of a meane proportionall betweene two lines given: And the Reason or rate in opposite sections. The genesis or invention of the meane proportionall, of which we heard at the [9 e viij]. is thus:
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.
As for example, let the assigned right lines be ae, and ei, of the which aei, is continued. And let eo, be perpendicular from the periphery aoi, unto e, the point of continuation or joyning together of the lines given. This eo, say I, shall be the meane proportionall: Because drawing the right lines ao, and io, you shall make a rectangled triangle, seeing that aoi, is an angle in a semicircle: And, by the [9 e viij]. oe, shall be proportionall betweene ae, and ei.
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.
If the sections be equall or alike, then are they the sections of a semicircle, and the matter is plaine by the [21 e]. But if they be unequall or unlike the argument of demonstration
is indeed fetch'd from the angle in a semicircle, but by the equall or like angle of the tangent and end of the diameter.
As let the unequall sections be eio, and eao: the tangent let it be uey: And the angles in the opposite sections, eao, and eio. I say they are equall in the alterne angles of the secant and touch line oey, and oeu. First that which is at a, is equall to the alterne oey: Because also three angles oey, oea, and aeu, are equall to two right angles, by the [14 e v]. Unto which also are equall the three angles in the triangle aeo, by the [13 e vj]. From three equals take away the two right angles aue, and aoe: (For aoe, is a right angle, by the [21 e]; because it is in a semicircle:) Take away also the common angle aeo: And the remainders eao, and oey, alterne angles, shall be equall.
Secondarily, the angles at a, and i, are equall to two right angles, by the [14, e]: To these are equall both oey, and oeu. But eao, is equall to the alterne oey. Therefore that which is at i, is equall to, the other alterne oeu. Neither is it any matter, whether the angle at a, be at the diameter or not: For that is onely assumed for demonstrations sake: For wheresoever it is, it is equall, to wit, in the same section. And from hence is the making of a like section, by giving a right line to be subtended.
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.
This you may make triall of in the three kindes of angles, all wayes by the same argument: as here the angle given is a: The right line given ei: at the end e, the equalled angle, ieo: The perpendicular to the side eo, let it be eu: But from the middest of the line given let it be yu. Here u, shall be the center desired. And from hence one may make a section upon a right line given, which shall receive a rectilineall angle equall to an angle assigned.
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. p iij.
As in this figure underneath. And from hence one may from a circle given cut off a section, in which there is an
angle equall to the assigned. As let the angle given be a: And the circle eio. Thou must make at the point e, of the secant eo, and the tangent yu, an angle equall to the assigned, by the [11 e iij]. such as here is oeu: Then the section oei, shall containe an angle equall to the assigned.
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 rectilineall ascribed unto a circle be an equilater, it is equiangle.
Of the inscript it is manifest; And that of a Triangle by it selfe: Because if it be equilater, it is equiangle, by the [19 e vj]. But in a Triangulate the matter is to be prooved by demonstration. As here, if the inscripts ou, and sy, be equall, then doe they subtend equall peripheries, by the 32
e xv. Then if you doe omit the periphery in the middest betweene them both, as here uy, and shalt adde oies the remainder to each of them, the whole oiesy, subtended to the angle at u: And uoies, subtended to the angle at y, shall be equall. Therefore the angles in the periphery, insisting upon equall peripheries are equall.
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 [9 e 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. xij.
Because by the [1 e vj], like plains have a doubled reasó of their homologall sides. But in rectilineals inscribed the diameters are the homologall sides, or they are proportionall to their homologall sides. As let the like rectangled triangles be aei, and ouy; Here because ae and ou, are the diameters, the matter appeareth to be plaine at the first sight. But in the Obliquangled triangles, sei, and ruy, alike also, the diameters are proportionall to their homologall sides, to wit, ei and uy. For by the grant, as se is to ru: so is ei to uy, And therefore, by the former, as the diameter ea and uo.
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.
As in the Triangle aei, let the right lines ao, and eu, halfe the angles a and e: And from y, their meeting, let the perpendiculars unto the sides be yo, yu, ys; I say that the center y, with the ray yo, or ya, or ys, is the circle inscribed, by the [17 e xv]. Because the halfing lines with the perpendiculars shall make equilater triangles, by the [2 e vij]. And therefore the three perpendiculars, which are the bases of the equilaters, shall be equall.
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 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 e x], 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.
Let the Triangle aei be given: And the circle, o, into which a Triangle equiangular to the triangle given, is to be inscribed. Therefore let the right line uys, touch the periphery yrl: And from the touch y, let the inscripts yr, and yl, make with the tangent two angles uyr, and syl, equall to the assigned angles aei, and aie: And let them be knit together with the right line rl: They shall by the [27 e xvj], make the angle of the alterne segments equall to the angles uyr, and syl. Therefore by the [4 e vij] seeing that two are equall, the other must needs be equall to the remainder.
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.
Let there be a Triangle, and in it the outter angles aei, and aou: The Circle let it be ysr; And in the center l, let the angles ylr, and slr; at the common side lr, bee made equall to the said outter angles aei, and aou. I say the angles of the circumscribed triangle, are equall to the angles of the triangle given. For the foure inner angles of the quadrangle ylrm, are equall to the foure right angles, by the [6 e x]: And two of them, to wit, at y and r, are right angles, by the construction: For they are made by the secant and touch line, from the touch point by the center, by the [20 e xv]. Therefore the remainders at l and m, are equall to two right angles: To which two aei and aeo are equall. But the angle at l, is equall to the outter: Therefore the remainder m, is equall to aeo. The same shall be sayd of the angles aoe, and aou. Therefore two being equall, the rest at a and i, shall be equall.
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.
Such 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 onto a circle cirumscribe 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-angle-wise, a right line subtended or drawne against the right angle, shall be the side of the quadrate. è 6 p iiij.
As here. For the shankes of the angle are the raies whose diameters knit together shall make foure rectangled triangles, equall in shankes: And by the [2 e vij], equall in bases. Therefore they they shall inscribe a quadrate.
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 xij].
And
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 or 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 Quinquangle first, that which is double unto the remainder, which is thus found.
5. If a right line be cut proportionally, the base of that triangle whose shankes shall be equall to the whole line cut, and the base to the greater segment of the same, shall have each of the angles at 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 iiij.
Here first thou shalt take for the fabricke or making of the Triangle, for the ray the right line ae by the [3 e xiiij], cut proportionally in o: A circle also shalt thou make upon the center a, with the ray ae: And then shalt thou by the [6 e xv], inscribe a right line equall to the greater segment: And shalt knit the same inscript with the whole line cut with another right line. This triangle shall be your desire. For by the [17 e vj], the angles at the base ei are equall, so that looke whatsoever is prooved of the one, is by and by also prooved of the other. Then let oi be drawne; And a Circle, by the [8 e xvij], circumscribed about the triangle aoi. This circle the right line ei, shall touch, by the [27 e xv]. Because, by the grant, the right line ae, is cut proportionally, therefore the Oblong of the secant and outter segment, is equall to the quadrate of the greater segment, to which by the grant, the base ei, is equall. Here therefore the angle aie is the double of the angle at a: because it is equall to the angles aio, and oai, which are equall betweene themselves. For by the [27 e xvj] it is equall to the angle oai in the alterne segment. And the remainder aio, is equall to it selfe. Therefore also the angle aei, is equall to the same two angles, because it is equall to the angle aie. But the outter angle eoi, is equall to the same two, by the [15 e vj]. Therefore the angles ioe and oei (because they are equall to the same) they are equall betweene themselves. Wherefore by the [17 e vj], the sides oi and ei are equall. And there also ao and oi: And the angles oai & oia are equall by the [17 e vj]. Wherefore seeing
that to both the angle aie is equall, it shall be the double of either of the equalls.
But the base ei, is the side of the equilater quinquangle. For if two right lines halfing both the angles of a triangle which is the double of the remainder, be knit together with a right line, both one to another, and with the angles, shall inscribe unto a circle an equilater triangle, whose one side shall be the base it selfe: As here seeing the angles eoa, eoi, uio, uia, iao, are equal in the periphery, the peripheries, by the [7 e, xvj]. subtending them are equall: And therefore, by the [32 e, xv]. the subtenses ae, ei, io, ou, ua, are also equall. Now of those five, one is ae. Therefore a right line proportionall cut, doth thus make the adscription of a quinquangle: And from thence againe is afforded a line proportionally cut.
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 xiij.
As here, Let ai, and eu, subtending the angles on each side aei, and eau: I say, That they are proportionally cut in the point s: And the greater segments si, and su, are equall to ae, the side of the quinquangle. For here two triangles are equiangles: First aei, and uae, are equall by the grant, and by the [2 e, vij]. Therefore the angles aie, and aes, are equall. Then aei, and ase, are equall: Because the
angle at a, is common to both: Therefore the other is equall to the remainder, by the [4 e, 7]. Now, by the [12. e, vij.] as ia, is unto ae, that is, as by and by shall appeare, unto is: so is ea, unto as: Therefore, by the [1 e, xiiij]. ia, is cut proportionally in s. But the side ea, is equall to is: Because both of them is equall to the side ei, that by the grant, this by the [17. e, vj]. For the angles at the base, ise, and ies, are equall, as being indeed the doubles of the same. For ise, by the [16. e vj]. is equall to the two inner, which are equall to the angle at u, by the [17 e vj]. and by the former conclusion. Therefore it is the double of the angles aes: Whose double also is the angle uei, by the [7 e. xvj]. insisting indeede upon a double periphery.
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.
The example is thus.
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, 48, 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.
A sexangle is inscribed by an inscript equilaterall triangle, by halfing of the three angles of the said triangle: But it is done more speedily by the ray or semidiameter of the circle, sixe times continually inscribed. As in the circle given, let the diameter be ae; And upon the center o, with the ray ie, let the periphery uio, be described: And from the points o and u, let the diameters be oy, and us; These knit both one with another, and also with the diameter ae shall inscribe an equilaterall sexangle into the circle given, whose side shal be equal to the ray of the same circle. As eu, is equal to ui, because they both equall to the same ie, by the [29 e, iiij]. There fore eiu, is an equilater triangle: And likewise eio, is an equilater. The angles also in the center are ⅔ of one rightangle: And therefore they are equall. And by the [14. e v], the angle sio, is ⅓. of two rightangles: And by the [15. e v]. the angles at the toppe are also equall. Wherefore sixe are equall: And therefore, by the [7 e xvj]. and [32. e, xv], all the bases are equall, both betweene themselves, and as was even now made manifest, to the ray of the circle given. Therefore the sexangle inscript by the ray of a circle is an
equilater; And by the [1 e xvij]. equiangled.
Therefore
10 Three ordinate sexangles doe fill up a place.
As here. For they are sixe equilater triangles, if you shal resolve the sexangles into sixe triangls: Or els because the angle of an ordinate sexangle is as much as one right angle and ⅓. of a right angle.
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 3.3/5 angles which is too little. And foure would make 4.4/5 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.
As here; Because the sides shall be subtended to equall peripheries: Therefore by the [32 e xv]. they shall be equall betweene themselves: And againe, on the contrary, by such a like triangle, by halfing the angles, a sexangle is inscribed.
12 The side of an inscribed equilater triangle hath a
treble power, unto the ray of the circle 12. p xiij.
As here, with ae, one side of the triangle aei, two third parts of the halfe periphery are imployed: For with one side one third of the whole eu, is imployed: Therefore eu, is the other third part, that is, the sixth part of the whole periphery. Therefore the inscript eu, is the ray of the circle, by the [9 e]. Now the power of the diameter aou, by the [14 e xij]. is foure times so much as is the power of the ray, that is, of eu: And by [21. e xvj], and [9 e xij], ae, and eu, are of the same power; take away eu, and the side ae, shall be of treble power unto the ray.
13 If the side of a sexangle be cut proportionally, the greater segment shall be the side of the decangle.
Pappus lib. 5. ca. 24. & Campanus ad 3 p xiiij. Let the ray ao, or side of the sexangle be cut proportionally, by the [3 e xiiij]: And let ae, be equall to the greater segment. I say that ae, is the side of the decangle. For if it be moreover continued with the whole ray unto i, the whole aei, shall by the [4 e xiiij]. be cut proportionally: and the greater segment ei, shal be the same ray. For the if the right line iea, be cut proportionally, it shall be as ia, is unto ie, that is to oa, to wit, unto the ray: so ao, shal be unto ae. Therefore, by the [15. e vij]. the triangles iao, and oae, are equiangles: And the angle aoe, is equall to the angle oia. But the angle uoe, is foure times as great as the angle aoe: for it is equall to the two inner at a, and e, by the [15 e vj]: which are equall between themselves, by the [10 e v]. and by the [17 e vj]. And therefore it is the double of
aeo, which is the double, for the same cause, of aio, equall to the same aoe. Therefore uoe, is the quadruple of the said aoe. Therefore ue, is the quadruple of the periphery ea. Therefore the whole uea, is the quintuple of the same ea: And the whole periphery is decuple unto it. And the subtense ae, is 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 the 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 side of the inscribed quinquangle be ae: of the sexangle, ei: Of the decangle ao. I say, the side ae, doth in power countervaile the rest. For let there be two perpēdiculars: The first io, the second iu, cutting the sides of the quinquangle and decangle into halves: And the meeting of the second perpendicular with the side of the quinquangle let it be y. The syllogisme of the demonstration is this: The oblongs of the side of the quinquangle, and the segments of the same, are equall to the quadrates of the other sides. But the quadrate of the same whole side, is equall to the oblongs of the whole, and the segments, by the [3 e, xiij]. Therefore it is equall to the quadrates of the other sides.
Let the proportion of this syllogisme be demonstrated: For this part onely remaineth doubtfull. Therefore two triangles, aei, and yei, are equiangles, having one common angle at e: And also two equall ones aei, and eiy, the halfes, to wit, of the same eis: Because that is, by the [17 e, vj]: one of the two equalls, unto the which eis, the out angle, is equall, by the [15 e. vj]. And this doth insist upon a halfe periphery. For the halfe periphery als, is equall to the halfe periphery ars: and also al, is equall to ar. Therefore the remnant ls, is equall to the remnant rs: And the whole rl, is the double of the same rs: And therefore er, is the double of eo: And rs, the double of ou. For the bisegments are manifest by the [10 e, xv]. and the [11 e, xvj]. Therefore the periphery ers, is the double of the periphery eou: And therefore the angle eiu, is the halfe of the angle eis, 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 ae, is to ei: so is ei, to ey. Therefore by the [8 e xij], the oblong of the extreames is equall to the quadrate of the meane.
Now let oy, be knit together with a straight: Here againe the two triangles aoe, and aoy, are equiangles, having one common angle at a: And aoy, and oea, 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 ea, is to ao: so is ea, to ay. 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 said
point, shall be the side of the inscribed quindecangle. 16. p. iiij.
For the side of the equilaterall triangle doth subtend 1/3 of the whole pheriphery. And two sides of the ordinate quinquangle doe subtend 2/5 of the same. Now 2/5 - 1/3 is 1/15: Therefore the space betweene the triangle, and the quinquangle shall be the 1/15 of the whole periphery.
Therefore
17. If a quinquangle and a sexangle be inscribed into the same circle at the same point, the periphery intercepted beweene both their sides, shall be the thirtieth part of the whole periphery.
As here. Therefore the inscription of ordinate triangulates, of a Quadrate, Quinquangle, Sexangle, Decangle, Quindecangle is easie to bee performed by one side given or found, which reiterated as oft as need shall require, shal subtend the whole periphery. Jun. 4. A. C.
Of Geometry the ninteenth Booke; Of the Measuring of ordinate Multangle and of a Circle.
Out 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 made:
1. A plaine made of the perpendicular from the center unto the side, and of halfe the perimeter, is the content of an ordinate multangle.
As here; The quadrate of 10, the ray is 100. The quadrate of 6, the halfe of the side 12, is 36: And 100. 36 is 64, the quadrate of the Perpendicular, whose side 8, is the Perpendicular it selfe. Now the whole periphery of the Quinquangle, is 60. The halfe thereof therefore is 30. And the product of 30, by 8, is 240, for the content of the sayd quinquangle.
The Demonstration here also is of the certaine antecedent cause thereof. For of five triangles in a quinquangle, the plaine of the perpendicular, and of halfe the base is one of them, as in the former hath beene taught: Therefore five
such doe make the whole quinquangle. But that multiplication, is a multiplication of the Perpendicular by the Perimeter or bout-line.
In an ordinate Sexangle also the ray, by the [9 e xviij], is knowne by the side of the sexangle. As here, the quadrate of 6, the ray is 36. The quadrate of 3, the halfe of the side, is 9: And 36 - 9. are 27, for the quadrate of the Perpendicular, whose side 5.2/11 is the perpendicular it selfe. Now the whole perimeter, as you see, is 36. Therefore the halfe is 18. And the product of 18 by 5.2/11 is 93.3/11 for the content of the sexangle given.
Lastly in all ordinate Multangles this theoreme shall satisfie thee.
2 The periphery is the triple of the diameter and almost one seaventh part of it.
Or the Periphery conteineth the diameter three times and almost one seventh of the same diameter. That it is triple of it, sixe raies, (that is three diameters) about which the periphery, the [9 e xviij], is circumscribed doth plainely shew: And therefore the continent is the greater: But the excesse is not altogether so much as one seventh part. For there doth want an unity of one seventh: And yet is the same excesse farre greater than one eighth part. Therefore because the difference was neerer to one seventh, than it was to one eighth, therefore one seventh was taken, as neerest unto the truth, for the truth it selfe.
Therefore
3. The plaine of the ray, and of halfe the periphery is the content of the circle.
For here 7, the ray, of halfe the diameter 14, Multiplying 22, the halfe of the periphery 44, maketh the oblong 154, for the content of the circle. In the diameter two opposite sides, and likewise in the perimeter the two other opposite sides of the rectangle are conteined. Therefore the halfes of those two are taken, of the which the rectangle is comprehended.
And
4. As 14 is unto 11, so is the quadrate of the diameter unto the Circle.
For here 3 bounds of the proportion are given in potentia: The fourth is found by the multiplication of the third by the second, and by the Division of the product by the first: As here the Quadrate of the diameter 14, is 196. The product of 196 by 11 is 2156. Lastly 2156 divided by 14, the first bound, giveth in the Quotient 154, for the content of the circle sought. This ariseth by an analysis out of the quadrate and Circle measured. For the reason of 196, unto a 154; is the reason of 14 unto 11, as will appeare by the reduction of the bounds.
This is the second manner of squaring of a circle taught by Euclide as Hero telleth us, but otherwise layd downe, namely after this manner. If from the quadrate of the diameter you shall take away 3/14 parts of the same, the remainder shall be the content of the Circle. As if 196, the quadrate be divided by 14, the quotient likewise shall be 14. Now thrise 14, are 42: And 196 - 42, are 154, the quadrate equall to the circle.
Out of that same reason or rate of the pheriphery and
diameter ariseth the manner of measuring of the Parts of a circle, as of a Semicircle, a Sector, a Section, both greater and lesser.
And
5. The plaine of the ray and one quarter of the periphery, is the content of the semicircle.
As here thou seest: For the product of 7, the halfe of the diameter, multiplyed by 11, the quarter of the periphery, doth make 77, for the content of the semicircle.
This may also be done by taking of the halfe of the circle now measured.
And
6. The plaine made of the ray and halfe the base, is the content of the Sector.
Here are three sectours, ae the base of 12 foote: And ei in like manner of 12 foote. The other or remainder ia of 7 f. and 3/7 of one foote. The diameter is 10 foote. Multiply therefore 5, halfe of the diameter, by 6 halfe of the base, and the product 30, shall be the content of the first sector. The same shall also be for the second sectour. Againe multiply the same ray or semidiameters 5, by 3.5/7, the halfe of 7.3/7, the product of 18.4/7 shall be the content of the third sector. Lastly, 30 + 30 + 18.4/7 are 78.4/7, the content of the whole circle.
And
7. If a triangle, made of two raies and the base of the greater section, be added unto the two sectors in it, the whole shall be the content of the greater section: If the same be taken from his owne sector, the remainder shall be the content of the lesser.
In the former figure the greater section is aei: The lesser is ai. The base of them both is as you see, 6. The perpendicular from the toppe of the triangle, or his heighth is 4. Therefore the content of the triangle is 12. Wherefore 30 + 30 + 12, that is 72, is the content of the greater section aei. And the lesser sectour, as in the former was taught, is 18.4/7. Therefore 18.4/7 - 12, that is, 6.4/7, is the content of ai, the lesser section.
And
8. A circle of unequall isoperimetrall plaines is the greatest.
The reason is because it is the most ordinate, and
comprehended of most bounds; see the [7], and [15 e iiij]. As the Circle a, of 24 perimeter, is greater then any rectilineall figure, of equall perimeter to it, as the Quadrate e, or the Triangle i.
Of Geometry the twentieth Booke, Of a Bossed surface.
1. A bossed surface is a surface which lyeth unequally betweene his bounds.
It is contrary unto a Plaine surface, as wee heard at the [4 e v].
2. A bossed surface is either a sphericall, or varium.
3. A sphericall surface is a bossed surface equally distant from the center of the space inclosed.
Therefore
4. It is made by the turning about of an halfe circumference the diameter standeth still. è 14 d xj.
As here if thou shalt conceive the space betweene the periphery and the diameter to be empty.
5. The greatest periphery in a sphericall surface is that which cutteth it into two equall parts.
Those things which were before spoken of a circle, the same almost are hither to bee referred. The greatest periphery of a sphericall doth answere unto the Diameter of a Circle.
Therefore
6. That periphery that is neerer to the greatest, is greater than that which is farther off: And on each
side those two which are equally distant from the greatest, are equall.
The very like unto those which are taught at the [15], [16], [17], [18. e. xv]. may here againe be repeated: As here.
7 The plaine made of the greatest periphery and his diameter is the sphericall.
So the plaine made of the diameter 14. and of 44. the greatest periphery, which is 616. is the sphericall surface. So before the content of a circle was measured by a rectangle both of the halfe diameter, and periphery. But here, by the whole periphery and whole diameter, there is made a rectangle for the measure of the sphericall, foure times so great as was that other: Because by the [1 e vj]. like plaines (such as here are conceived to be made of both halfe the diameter, and halfe the periphery, and both of the whole diameter and whole periphery) are in a doubled reason of their homologall sides.
Therefore
8 A plaine of the greatest circle and 4, is the sphericall.
This consectarium is manifest out of the former element.
And
9 As 7 is to 22. so is the quadrate of the diameter unto the sphericall.
For 7, and 22, are the two least bounds in the reason of the diameter unto the periphery: But in a circle, as 14, is to 11, so is the quadrate of the diameter unto the circle. The analogie doth answer fitly: Because here thou multipliest by the double, and dividest by the halfe: There contrariwise thou multipliest by the halfe, and dividest by the double. Therefore there one single circle is made, here the quadruple of that. This is, therefore the analogy of a circle and sphericall; from whence ariseth the hemispherical, the greater and the lesser section.
And
10 The plaine of the greatest periphery and the ray, is the hemisphericall.
As here, the greatest periphery is 44. the ray 7. The product therefore of 44. by 7. that is, 308. is the hemisphericall.
11 If looke what the part be of the ray perpendicular from the center unto the base of the greater section, so much the hemisphericall be increased, the whole shall be the greater section of the sphericall: But if it be so much decreased, the remainder shall be the lesser.
As in the example, the part of the third ray, that is, of 3/7, is from the center: such like part of the hemispherical 308, is 132. (For the 7, part of 308. is 44. And three times 44. is 132.) Therefore 132. added to 308. do make 440. for the greater section of the sphericall. And 132. taken from 308. doe leave 176. for the lesser section of the same.
12 The varium is a bossed surface, whose base is a
periphery, the side a right line from the bound of the toppe, unto the bound of the base.
13 A varium is a conicall or a cylinderlike forme.
14 A conicall surface is that which from the periphery beneath doth equally waxe lesse and lesse unto the very toppe.
Therefore
15. It is made by turning about of the side about the periphery beneath.
16 The plaine of the side and halfe the base is the conicall surface.
As in the example next aforegoing, the side is 13. The halfe periphery is 15.5/7: And the product of 15.5/7 by 13. is 204.2/7. for the conicall surface. To which if you shall adde the circle underneath, you shall have the whole surface.
17 A cylinderlike forme is that which from the periphery underneath unto the the upper one, equall and parallell unto it, is equally raised.
Therefore
18 It is made by the turning of the side about two equall and parallell peripheries.
19 The plaine of his side and heighth is the cylinderlike surface.
As here the periphery is 22. as is gathered by the Diameter, which is 7. The heighth is 12. The base therefore is 38.1/2. And 38.1/2 by 12. are 462. for the cylinderlike surface. To which if you shall adde both the bases on each side, to wit, 38.1/2. twise, or 77. once, the whole surface shall be 539.
Geometry, the one and twentieth Book, Of Lines and Surfaces in solids.
1 A body or solid is a lineate broad and high 1 d xj.
For length onely is proper to a line: Length and breadth, to a surface: Length breath, and heighth joyntly, belong unto a body: This threefold perfection of a magnitude, is proper to a body: Whereby wee doe understand that are in a body, not onely lines of length, and surfaces of breadth, (for so a body should consist of lines and surfaces.) But we do conceive a solidity in length, breadth and heighth. For every part of a body is also a body. And therefore a solid we doe understand the body it selfe. As in the body aeio, the length is ae; the breadth, ai, And the heighth, ao.
2 The bound of a solid is a surface 2 d xj.
The bound of a line is a point: and yet neither is a point a line, or any part of a line. The bound of a surface is a line: And yet a line is not a surface, or any part of a surface. So now the bound of a body is a surface: And yet a surface is not a body, or any part of a body. A magnitude is one thing;
a bound of a magnitude is another thing, as appeared at the [5 e j].
As they were called plaine lines, which are conceived to be in a plaine, so those are named solid both lines and surfaces which are considered in a solid; And their perpendicle and parallelisme are hither to be recalled from simple lines.
3 If a right line be unto right lines cut in a plaine underneath, perpendicular in the common intersection, it is perpendicular to the plaine beneath: And if it be perpendicular, it is unto right lines, cut in the same plaine, perpendicular in the common intersection è 3 d and 4 p xj.
Perpendicularity was in the former attributed to lines considered in a surface. Therefore from thence is repeated this consectary of the perpendicle of a line with the surface it selfe.
If thou shalt conceive the right lines, ae, io, uy, to cut one another in the plaine beneath, in the common intersections: And the line rs, falling from above, to be to every one of them perpendicular in the common point s, thou hast an example of this consectary.
4 If three right lines cutting one another, be unto the same right line perpendicular in the common section, they are in the same plaine 5. p xj.
For by the perpendicle and common section is understood an equall state on all parts, and therefore the same plaine: as in the former example, as, ys, os, suppose them to be to sr, the same loftie line, perpendicular, they shall be in the same nearer plaine aiueoy.
5 If two right lines be perpendicular to the under-plaine, they are parallells: And if the one two
parallells be perpendicular to the under plaine, the other is also perpendicular to the same. 6. 8 p xj.
The cause is out of the first law or rule parallells. For if two right lines be perpendicular to the same under plaine, being joyned together by a right line, they shall make their inner corners equall to two right angles: And therefore they shall be parallells, by the [21. e v]. And if in two parallells knit together with a right line, one of the inner angles, be a right angle: the other also shall be a right angle. Because they are divided by a common perpendicular; As in the example. If the angles at a, and e, be right angles, ai, and eo, are parallells, and contrariwise, if ai, and eo be parallells, and the angle at a, be a right angle, the angle at e, also shall be a right angle.
6 If right lines in diverse plaines be unto the same right line parallel, they are also parallell betweene themselves. 9 p xj.
As here ae, and uy, right lines in diverse plaines suppose them to be parallell to io: I say, they are parallell one to another. For from the point i, let ia, and iu, be erected at right angles to io to cut the parallells, by the [17. e v]. Therefore, by the [3 e], oi, seeing that it is perpedicular to ia, and iu, two lines cutting one another, it is perpendicular to the plaine beneath. Therefore by the the [6 e], yu, and ea, are perpendicular to the same plaine: And therefore, by the same, they are parallell.
7 If two right lines be perpendiculars, the first from a point above, unto a right line underneath, the second
from the common section in the plaine underneath, a third, from the sayd point perpendicular to the second, shall be perpendicular to the plaine beneath. è 11 p xj.
It is a consectary out of the [3 e]. As for example, if from a lofty point a, ae, be by the [18 e v], perpendicular to e, a point of the right line io underneath: And from e the common section, by the [17 e v], there be eu, another perpendicular: Lastly ay, a lofty right line, be by the [18 e v], perpendicular unto eu, at the point y, ay shall be perpendicular unto the plaine underneath. For that ae is perpendicular to io, the same ae declineth neither to the right hand, nor to the left, by the [13 e ij]. And in that againe ay is perpendicular to eu, it leaneth neither forward nor backeward. Therefore it lyeth equally or indifferently, betweene the foure quarters of the world.
If the right line io, doe with equall angles agree to r, the third element.
8. If a right line from a point assigned of a plaine underneath, be parallell to a right line perpendicular to the same plaine, it shall also be perpendicular to the plaine underneath. ex 12 p xj.
As for example let the plaine be aeio: And the assigned point in it u: From this point a lofty perpendicular is to be erected. Let there be made from the point y, the perpendicular ys, unto the plaine underneath, by the [7 e]. And to it let ur, be made parallell by the [24 e v]. Now ur, seeing it is parallell to a perpendicular upon the plaine underneath, it shall be perpendicular to the same, by the [5 e].
9. If a right line in one of the plaines cut, perpendicular to the common section, be perpendicular to the other, the plaines are perpendicular: And if the plaines be perpendicular, a right line in the one perpendicular to the common section is perpendicular to the other è 4 d, and 38 p xj.
The perpendicularity of plaines, is drawne out of the former condition of the perpendicle: And the state of plaines on each side equall betweene themselves, is fetch'd from a perpendicularity of a right line falling upon a plaine. Because from hence it is understood that the plaine it selfe doth lye indifferently betweene all parts signified by right lines: Which in a Booke with the pages each way opened, is perceived by the verses or lines of the pages, both to the section and plaine underneath, perpendicular as here thou seest.
10. If a right line be perpendicular to a plaine, all plaines by it, are perpendicular to the same: And if two plaines be unto any other plaine perpendiculars, the common section is perpendicular to the same. e 15, and 19 p. xj.
The first is a consectary drawne out of the [9 e]. And the latter is from hence manifest, because that same common section is a right line, in any manner of lofty plaines intersected, perpendicular both to the common section and plaine underneath. For if the common section, were not perpendicular to the plaine underneath, neither should the plaines
cutting one another be perpendicular to the plaine underneath, but some one should be oblique, against the grant, as here thou seest.
11. Plaines are parallell which doe leane no way. 8 d xj.
And
12. Those which divided by a common perpendicle. 14 p xj.
It is a consectary out of the [3], and [6 e]. For if the middle right line be perpendicular to both the plaines, it is also to the right lines on either side cut, perpendicular in the common intersection: And the inner angles on each side, being right angles, will evince them to be parallels.
It is also out of the definition of parallels, at the [15 e ij].
And
13. If two paires of right in them be joyntly bounded, they are parallell. 15 p xj.
Such are the opposite walls in the toppe or ridge of houses. As let aei, and uoy, be plaine which have two payres of
right lines, ea, and ia: Item uo, and yo, joyntly bounded in a, and o: And parallels, to wit ea, against uo: and ia, against yo. I say that the plaines themselves are parallels: For the right lines ue, and oa: item yi, and oa, doe knit together equall parallels, they shal by the [27 e v], be equall and parallels: And so they shall prove the equidistancie.
The same will fall out if thou shalt imagine the joyntly bounded to infinitely drawn out; for the plaines also infinitely extended shall be parallell.
14. If two parallell plaines are cut with another plaine, the common sections are parallels, 16 p xj.
As here thou seest the parallell plaines aeio, and uysr, cut by the plaine ljvf, the common sections lj, and fv, shall also be parallell: Otherwise they themselves, and therefore also the plaines in which they are, shall meete, as in the point t, which is against the grant.
The twenty second Booke, of P. Ramus Geometry, Of a Pyramis.
1. The axis of a solid is the diameter about which it is turned, e 15, 19, 22 d xj.
The Axis or Axeltree is commonly thought to be proper to the sphere or globe, as here ae: But it is attributed to other kindes of solids, as well as to that.
2. A right solid is that whose axis is perpendicular to the center of the base.
Thus Serenus and Apllonius doe define a Cone and a Cylinder: And these onely Euclide considered: Yea and indeed stereometry entertaineth no other kinde of solid but that which is right or perpendicular.
3. If solids be comprehended of homogeneall surfaces, equall in multitude and magnitude, they are equall. 10 d xj.
Equality of lines and surfaces was not informed by any peculiar rule; farther than out of reason and common sense, and in most places congruency and application was enough and did satisfie to the full: But here the congruency of Bodies is judged by their surfaces. Two cubes are equall, whose sixe sides or plaine surfaces, are equall, &c.
4. If solids be comprehended of surfaces in multitude equall and like, they are equall, 9 d xj.
This is a consectary drawne out of the general difinition of like figures, at the [19 e. iiij]. For there like figures were defined to be equiangled and proportionall in the shankes of the equall angles: But in like plaine solids the angles are esteemed to be equall out of the similitude of their like plaines: And the equall shankes are the same plaine surfaces, and therefore they are proportionall, equall and alike.
5 Like solids have a treble reason of their homologall sides, and two meane proportionalls. 33. p xj. 8 p xij.
It is a consectary drawne out of the [24 e. iiij]. as the example from thence repeated shall make manifest.
6 A solid is plaine or embosed.
7 A plaine solid is that which is comprehended of plaine surfaces.
8 The plaine angles comprehending a solid angle, are lesse than foure right angles. 21. p xj.
For if they should be equall to foure right angles, they would fill up a place by the [27 e, iiij]. neither would they at all make an angle, much lesse therefore would they doe it if they were greater.
9 If three plaine angles lesse than foure right angles, do comprehend a solid angle, any two of them are greater
than the other: And if any two of them be greater than the other, then may comprehend a solid angle, 21. and 23. p xj.
It is an analogy unto the [10 e vj]. and the cause is in a readinesse. For if two plaine angles be equall to the remainder, they shall with that third include no space betweene them: But if thou shalt conceit to fit the plaine to the shankes, with the congruity they should of two make one: but much lesse if they be lesser.
The converse from hence also is manifest.
Euclide doth thus demonstrate it: First if three angles are equall, then by and by two are conceived to be greater than the remainder. But if they be unequall, let the angle aei, be greater than the angle aeo: And let aeu, equall to aeo, be cut off from the greater aei: And let eu, be equall to eo. Now by the [2 e, vij]. two triangles aeu, and aeo, are equall in their bases au, and ao. Item ao, and ei, are greater than ai, and ao: And ao, is equall to au. Therefore oi, is greater than iu. Here two triangles, uei, and ieo, equall in two shankes; and the base oi, greater than the base iu. Therefore, by the [5 e vij]. the angle oei, is greater than the angle ieu. Therefore two angles aeo, and oei, are greater than aei.
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 aei, unto the toppe o, the triangles aoe, aoi, and eoi, to be
In the pyramis aeiou, thou seest from the quadrangular base aeio, unto the toppe u, foure triangles in like manner to be raised.
Therefore
12 The sides of a pyramis are one more than are the base.
The sides are here named Hedræ.
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
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.
As here thou seest. In rectilineall plaines we have in the former signified, in every kinde there is but one ordinate figure: Amongst the triangles the equilater: Amongst the
quadrangles, the Quadrate: so now of all kinde of Pyramides, there is one kinde ordinate onely, and that is the Tetraedrum. And yet not every Tetraedrum is such, but that only which is comprehended of triangles, not onely severally ordinate, but equall one to another altogether alike.
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.
This fabricke or construction is very easie, as you may see in these examples: For if thou shalt joyne or fold together these triangles here thus expressed, thou shalt make 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.
As here, Let first ae be the right line whose power is sesquialter unto ai the side of the equilater triangle, as in the forme was manifest at the [13 e xij]. And let it be by the [29 e v], be cut in a double reason in o: And let the double segment ao, be perpendicular to the equilater triangle uys, unto the center r, by the [7 e xxj]. And let lr be knit with the angles, by lu, ls, ly. I say that the triangles uys, usl, uyl, are equilater and equall, because all the sides are equall. First the three lower ones are equall by the grant: And the three higher ones are equall by the [9 e xij]. And every one of the higher ones are equall to the under one. For if a Circle bee supposed to bee circumscribed about the triangle, the side
shall be of treble power to the ray ur, by the [12 e xviij]. But the higher one also is of treble power to the same ray, as is manifest in the first figure of the ray oi, which is for the ray of the second figure ur. For as ao, is to oi, so by the [9 e viij], is oi, unto oe: And by the [25 e iiij], as the first rect line ao, is unto the third oe: so is the quadrate ao, unto the quadrate oi. And by compounding ao with oe; As ae is to oe; so are the quadrates ao; and oi, that is, by the [9 e xij], the quadrate ai, unto the quadrate oi, But ae is the triple of oe. Therefore the quadrate ai, is the triple of the quadrate oi. Wherefore the higher side equall to ai, is of treble power to the ray: And therefore also all the sides are equall: And therefore againe the triangles themselves are equall.
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 d xj.
As here thou seest. The base of a pyramis was but one: Of a Prisma, they are two, and they opposite one against another, First equall; Then like: Next parallell. The other are parallelogramme.
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 Pentraedrum, 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. xij.
As in the example a prisma pentaedrum is cut into three equall pyramides. For the first consisting of the plaines aei, aeo, aoi, eio; is equall to the second consisting of the plaines aoi, aou, aiu, iou, by the [10 e vij]. Because it is equall to it both in common base and heighth. Therefore the first and second are equall. And the same second is equall to it selfe, seeing the base is iou, and the toppe a. Then also it is equall to the third consisting of the plaines aiu, aiy, uiy, auy. Therefore three are equall.
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.
The heighth of a pyramis shall be found, if you shall take the square of the ray of the base out of the quadrate of the side: for the side of the remainder, by the [9 e xij], shall be the altitude or heighth, as in the example following.
Here the content of the triangle by the [18 e xij], is found to be 62.44/125 for the base of the pyramis. The altitude is 9.15/19: Because by the [12 e xviij], the side is of treble power to the ray. But if from 144, the quadrate of 12 the side, you take the subtriple i. 48, the remainder 96, by the [9 e xij], shall be the square of the heighth. And the side of the quadrate shall be 9.15/19. Now the third part of 9.15/19 is 3.5/19. And the plaine of 62.44/125 and 3.5/19, shall be 203.1103/2375 for the solidity of the pyramis.
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 72.1/4 shall be 387.11/24 for the solidity of the pyramis given.
If the pyramis be unperfit, first measure the whole, and then that part which is wanting: Lastly from the whole
subtract that which was wanting, and the remaine shall be the solidity of the unperfect pyramis given: As here, let ao, the side of the whole be 16.5/12, eo the side of the particular be 8.1/16. Therefore the perpendicular of the whole ou, shall be 15.5/32: Whose third part is 5.5/96: Of which, and the base 93.3/11 the plaine shall be 471.134/1056 for the whole pyramis. But in the lesser pyramis, 9 the square of the ray 3, taken out of 65.1/256 the quadrate of the side 8.1/16 the remaine shall be 56.1/256; whose side is almost 7-1/2 for the heighth. The third part of which is 2-1/2. The base likewise is almost 22. The plaine of which two is 55, for the solidity of the lesser pyramis: And 471 - 55 is 416, for the imperfect pyramis.
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.
The reason is, because they consist equally of like number
of pyramides. Now it is required that they be homogeneall or of like kindes; Because a Pentaedrum with an Hexaedrum will not so agree.
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.
The segments are homogeneall because the prismas. Therefore seeing they are of equall heighth (by the heighth I meane of plaine dividing them) they shall be as their bases are: And here the bases are to be taken opposite to the heighth.
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 cause is manifest and briefe: Because they be the halfes of the same prisma: As here thou maist perceive in a prisma cut into two halfes by the diagoni's of the opposite sides.
Euclide doth demonstrate it thus: Let the Pentaedra's aeiou, and ysrlm, be of equall heighth: the first of a triangular base eio: The second of a parallelogramme base sl, double unto the triangular. Now let both of them be double and made up, so that first be aeioun. The second ysrlvf. Now againe, by the grant, the base sl, is the double of the base eio,: whose double is the base eo, by the [12 e x]. Therefore the bases sl, and eo, 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 aeiou, and ysmlr, 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, 15.18/31. or 15.3/5, almost twise taken, that is, 31.1/5, doth make 247.1/5, for the summe of the whole surface. But the plaine of the same base 15.3/5, and the heighth 12. is 187.1/5, 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 pentaedra'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 Hedræ 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. p j.
Let the Prisma be of sixe bases ai, yo, ye, ui, si, au. The diagonies doe cut into halfes, by the [10. e x]. the opposite bases: And the other opposite bases or the two prisma's cut, are equall by the [3 e]. Wherefore two prisma's are comprehended of bases, equall both in multitude and magnitude: therfore they are equall.
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.
Because here the diameters (such as is that bisection) are halfed betweene themselves [or doe halfe one another.] Let the parallelepipedum aeiouy, be cut in to y the halfs by two plains, fro srlm, uivf, halfing the opposite sides: Here the common section ts, and the diagony ao, doe cut one another.
17 If three lines be proportionall, the parallelepipedum of meane shall be equall to the equiangled parallelepipedum of all them. è 36. p xj.
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 thou seest.
| 2, | 3, | 5, | 4, | 6, | 10. |
| 30, | 60, | 120, | 240. | ||
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, perpendiculars 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.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Latera. |
| 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | Quadrates. |
| 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | Cubes. |
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.
But the genesis of the whole cube will make all this whole matter more apparant, to wit, how the extreme and meane solids are made. Let therefore a cube be made of three equall sides, 12, 12, and 12: And first of all let the second side be multiplied by the first, after this manner: And not adding the severall figures of the same degree, as was taught in multiplication, but multiply againe every one of them by the other side; and lastly, add the figures of the same degrees severally, thus:
|
12 12 —— 24 12 12 —— 48 24 24 12 —— 1,6,12,8 | Or thus, |
12 12 —— 4 20 20 100 —— 12 —— 8 40 40 40 200 200 200 1000 —— 1728 |
Therefore
11. The side of the first severall cube is the other side of the second solide: And the quadrate of the same side is the other side of the first solide, whose other side is the side of the second cube; and the quadrate of the same other side is the other side of the second solid.
In that equation therefore of foure solids with one solid, thou shalt consider a peculiar making and composition: First that the last cube be made of the last segment 2: Then that the second solid of 4, the quadrate of his owne segment, and of 10, the other segment be thrise comprehended: Lastly that the first solid of 100, the square of his owne segment 10 and the other segment 2, be also thrice comprehended: Lastly, that the Cube 1000, be made of the greater segment 10. Out of this making &c.
And thus much of the Cube: Of other sorts of parallelepipedes, as of the Oblong, the Rhombe, the Rhomboides, and of the Trapezium, and many flatted pentaedra's there is no
peculiar stereometry. The measuring of a Prisma hath in the former beene generally declared, and is now onely farther be made more plaine by speciall examples; as here:
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.
The same also Geodesie or manner of measuring is used in the measuring of rectangled walls or gates and doores, which have either any window, or any hollow
or voyde space cut out of them, if those voyde places be taken out of them; as here thou seest in the next following example. The thickenesse is 3 foote; the breadth 12, the heighth 11. Therefore the whole solidity is 396. Now the Gate way is of thickenesse 3 foote, of breadth 4: of heighth 6. And therefore the whole solidity of the Gate is 72 foote. But 396 - 72 are 314. Therefore the solidity of the rest of the wall remaining is 324.
In the second example, the length is 10. The breadth 8, the heighth 7. Therefore the whole body if it were found, were 560 foote. But there is an hollow in it, whose length is 6, breadth 5, heighth 7. Therefore the cavity or hollow place is 168. Now 560 - 168 is 392, for the solidity of the rest of the sound body.
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 64.35/129: The heigh 16. Therefore the solidity is 1028.44/129.
The same is the geodesy of a trapezium, as in these examples: The surface of the first is 198: The solidity 192.1/2.
The surface of the second is 158.3/49: The solidity is 91.29/49.
The same shall be also the geodesy of a many flatted Prisma: As here thou seest in an Octoedrum of a sexangular base: The surface shall bee 762.6/11: The solidity 1492.4/11.
And from hence also may the capacity or content of vessels 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 side:) and the heighth 5, shall be 205.5/7. Therefore if a cubicall foote doe conteine 4 quarters, as we commonly call them, then shall the vessell conteine 822.6/7 quartes, that is almost 823 quartes.
Of Geometry the twenty fifth Booke; Of mingled ordinate Polyedra's.
1. A mingled ordinate polyedrum is a pyramidate, compounded of pyramides with their toppes meeting in the center, and their bases onely outwardly appearing.
Seeing therefore a Mingled ordinate pyramidate is thus made or compounded of pyramides the geodesy of it shall be had from the Geodesy of the pyramides compounding it: And one Base multiplyed by the number of all the bases shall make the surface of the body. And one Pyramis by the number of all the pyramides; shall make the solidity.
2 The heighth of the compounding pyramis is found by the ray of the circle circumscribed about the base, and by the semidiagony of the polyedrum.
The base of the pyramis appeareth to the eye: The heighth lieth hidde within, but it is discovered by a right angle triangle, whose base is the semidiagony or halfe diagony, the shankes the ray of the circle, and the perpendicular of the heighth. Therefore subtracting the quadrate of the ray, from the quadrate of the halfe diagony the side of the remainder, by the [9 e xij]. shall be the heighth. But the ray of the circle shall have a speciall invention, according to the kindes of the base, first of a triangular, and then next of a quinquangular.
3 A mingled ordinate polyedrum hath either a triangular, or a quinquangular base.
The division of a Polyhedron ariseth from the bases upon which it standeth.
4 If a quadrate of a triangular base be divided into three parts, the side of the third part shall be the ray of the circle circumscribed about the base.
As is manifest by the [12 e. xviij]. And this is the invention or way to finde out the circular ray for an octoedrum, and an icosoedrum.
5 A mingled ordinate polyedrum of a triangular base, is either an Octoedrum, or an Icosoedrum.
This division also ariseth from the bases of the figures.
6 An octoedrum is a mingled ordinate polyedrum, which is comprehended of eight triangles. 27 d xj.
As here thou seest, in this Monogrammum and solidum, that is lines and solid octahedrum.
Therefore
7 The sides of an octoedrum are 12. the plaine angles 24, and the solid 6.
And
8 Nine octoedra's doe fill a solid place.
For foure angles of a Tetraedrum are equall to three angles of the Octoedrum: And therefore 12. are equall to
nine. Therefore nine angles of an octaedrum doe countervaile eight solid right angles.
And
9 If eight triangles, equilaters and equall be joyned together by their edges; they shall comprehend an octaedreum.
This construction is easie, as it is manifest in the example following: Where thou seest as it were two equilater and equall triangles of a double pentaedrum to cut one another.
10 If a right line of each side perpendicular to the center of a quadrate and equall to the halfe diagony be tied together with the angles, it shall comprehend an octaedrum, 14. d xiij.
For the perpendicular yu, and su, with the semidiagoni's, ua, uo, ui, ue, shall be made equall by the [2 e vij], the eight sides ya, ye, yo, yi, se, si, sa, so; And also eight triangles.
Therefore
11 The Diagony of an octaedrum is of double power to the side.
As is manifest by the [9 e xij].
And
12 If the quadrate of the side of an octaedrum, be
doubled, the side of the double shall be the diagony.
As in the figure following, the side is 6. The quadrate is 36. the double is 72. whose side 8.8/17, is the diagony.
And from hence doth arise the geodesy of the octaedrum. For the semidiagony is 4.4/17. whose quadrate is 17.171/289. And the quadrate of 6, the side of the equilater triangle, being of treble power to the ray, by the [12 e, xviij]. is 36. And the side of 12. the third part 3.3/7 is the ray of the circle. Wherefore 8.8/17. that is 5.21/289. is the quadrate of the perpendicular, whose side 2.1/5 is the height of the same perpendicular: whose third part againe 11/25. multiplied by 15.18/31. the triangular base doe make 11.66/155 for one of the eight pyramides: Therefore the same 11.66/155 multiplied by eight, shall make 91.63/155 for the whole octoedrum.
13 An Icosaedrum is an ordinate polyedrum comprehended of 20 triangles 29 d xj.
Therefore
14 The sides of an Icosaedrum are 30. plaine angles 60. the solid 12.
And
15 If twentie ordinate and equall triangles be joyned with solid angles, they shall comprehend an Icosaedrum.
This fabricke is ready end easie, as is to be seene in this example following.
16. If ordinate figures, to wit, a double quinquangle, and one decangle be so inscribed into the same circle, that the side of both the quinquangle doe subtend two sides of the decangle, sixe right lines perpendicular to the circle and equall to his ray, five from the angles of one of the quinquangles, knit together both betweene themselves, and with the angles of the other quinquangle; the sixth from the center on each side continued with the side of the decangle, and knit therewith the five perpendiculars, here with the angles of the second quinquangle, they shall comprehend an icosaedrum. è 15 p xiij.
For there shall be made 20 triangles, both equilaters and equall. Let there be therefore two ordinate quinquangles, the first aeiou; The second ysrlm; each of whose sides let them subtend two sides of a decangle; to wit, utym, let it subtend ya, and am. Then let there be five perpendiculars from the angles of the second quinquangle yj, sy, rv,
lf, mt. And let them be knit first one with another, by the lines nj, jv, vf, ft, tn. Secondarily, with the angles of the first quinquangle, by the lines ne, ej, ji, iv, of, fu, ut, ta, an. The sixth perpendicular from the center d, let it be bg, the ray dc, continued at each end with the side of the decangle, cg, and db, tied together about with the perpendiculars, as by the lines ng, tg: Beneath with the angles of the first quinquangle, as by the lines be, bi, and in other places in like manner, and let all the plaines be made up. This say I, is an Icosaedrum; And is comprehended of 20. triangles, both equilaters and equall. First, the tenne middle triangles, leaving out the perpendiculars, that they are equilaters and equall, one shall demonstrate, as nat. For mt and yu, because they are perpendiculars, they are also, by the [6 e xxj]. parallells: And by the grant, equall. Therefore by the [27 e, v], nt, is equall to ym, the side of the quinquangle. Item na, by the [6 e xij]. is of as great power, as both the shankes ny, and ya, that is, by the construction, as the sides of the sexangle and decangle: And, by the converse of the [15. e xviij]. it is the side of the quinquangle. The same shall fall out of ot. Wherefore nat, is an equilater triangle. The same shall fall out of the other nine middle triangles, nae, nej, eji, jiv, ivo, vof, fou, fut, uta, tan.
In like manner also shall it be proved of the five upper triangles, by drawing the right lines dy and cn which as afore (because they knit together equall parallells, to wit, dc, and yn) they shall be equall. But dy, is the side of a sexangle: Therefore cn, shall be also the side of a sexangle: And cg, is the side of a decangle: Therefore an, whose power is equall to both theirs by the [9 e xij]. shall by the converse of the [15 e xviij], be the side of a quinquangle: And in like manner gt, shall be concluded to be the side of a quinquangle. Wherefore ngt, is an equilater: And the foure other shall likewise be equilaters.
The other five triangles beneath shall after the like manner be concluded to be equilaters. Therefore one shall be for all, to wit, ibe, by drawing the raies di, and de. For ib,
whose power, as afore, is as much as the sides of the sexangle, and decangle, shall be the side of the quinquangle: And in like sort be, being of equall power with de, and do, the sides of the sexangle and decangle, shall be the side of the quinquangle. Wherefore the triangle ebi, is an equilater: And the foure other in like manner may be shewed to be equilaters. Therefore all the side of the twenty triangles, seeing they are equall, they shall be equilater triangles: And by the [8 e, vij]. equall.
17 The diagony of an icosaedrū is irrational unto the side.
This is the fourth example of irrationality, or incommensurability. The first was of the Diagony and side of a square or quadrate. The second was of the segments of a line proportionally cut. The third of the Diameter of a circle and the side of a quinquangle.
And
18 The power of the diagony of an icosaedrum is five times as much as the ray of the circle.
For by the [13 e, xviij]. the line continually made of the side of the sexangle and decangle is cut proportionally, and the greater segment is the side of the sexangle: As here. Let the perpendicular ae, be cut into two equall parts in i. Then eo, that is the lesser segment continued with the halfe of the greater, that is, with ie. it shall by the [6 e xiiij], be of power five times so great as is the power of the same halfe. Therefore seeing that io, the halfe of the diagony is of power fivefold to the halfe: the whole diagony shall be of power fivefold to the whole cut.
And from hence also shall be the geodesy of the Icosaedrum. For the finding out of the heighth of the pyramis, there is the semidiagony of the side of the decangle and the halfe ray of the circle: But the side of the decangle is a right line subtending the halfe periphery of the side of the quinquangle, or else the greater segment of the ray
proportionally cut. For so it may be taken Geometrically, and reckoned for his measure. Therefore if the quadrate of the side of the decangle, be taken out of the quadrate of the side of the quinquangle, there shall by the [15 e xviij], remaine the quadrate of the sexangle, that is of the ray. The side of the decangle (because the side of the quinquangle here is 6) shall be 3.3/35 to wit a right line subtending the halfe periphery. Now the halfe ray shall thus be had. The quadrates of the quinquangle and decangle are 36, and 9.639/1225. And this being subducted fro that, the remaine 26.386/1225 by the [15 e xviij], shall be the quadrate or square of the sexangle: And the side of it, 5, and almost 5/7 shall be the ray: The halfe ray therefore shall be 2.6/7. To the side of the decangle 3.3/35 adde 2.6/7: the whole shal be 5.33/35 for the semi-diagony of the Icosaedrum. The ray of the circle circumscribed about the triangle, is by the [12 e xviij], the same which was before 3.3/7 to wit of the quadrate 12. Therefore if the quadrate of the circular ray, be taken out of the quadrate of the halfe diagony, there shall remaine the quadrate of the heighth and perpendicular: the quadrate of the halfe-diagony is 35.389/1225: the quadrate of the circular ray is 12. This taken out of that beneath 23.639/1225: whose side is almost 5, for the perpendicular and heighth proposed: From whence now the Pyramis is esteemed. The case of a triangular pyramis is 15.18/31. The Plaine of this base and the third part of the heighth is 25.30/31 for the solidity of one Pyramis. This multiplyed by 20 maketh 519.11/31 for the summe or whole solidity of the Icosaedum. And this is the geodesy or manner of measuring of an Icosaedrum.
19. A mingled ordinate polyedrum of a
quinquangular base is that which is comprehended of 12 quinquangles, and it is called a Dodecaedrum.
Therefore
20. The sides of a Dodecaedrum are 30, the plaine angles 60. the solid 20.
And
21. If 12 ordinate equall quinquangles be joyned with solid angles, they shall comprehend a Dodecaedrum.
As here thou seest.
22. If the sides of a cube be with right lines cut into two equall parts, and three bisegments of the bisecants in the abbuting plaines, neither meeting one the other, nor parallell one unto another, two of one, the third of that next unto the remainder, be so proportionally cut that the lesser segments doe bound the bisecant: three lines without the cube perpendicular unto the sayd
plaines from the points of the proportionall sections, equall to the greater segment knit together, two of the same bisecant, betweene themselves and with the next angles of cube; the third with the same angles, they shall comprehend a dodecaedrum. 17 p xiij.
Let there be two plaines for a cube for all, that one quinquangle for twelve may be described, and they abutting one upon another, aeio, and euyi, having their sides halfed by the bisecantes, sr, lm, rn, jv: And the three bisegments or portions of the bisegments lm, and rn, neither concurring or meeting, nor parallell one to another; two of the said lm, to wit, fl, and fm: The third next unto the remainder, that is lr. And let each bisegment be cut proportionally in the points d, c, g; so that the lesser segments doe bound the bisecant, to wit, dl, cm, and gr. Lastly let there be three perpendiculars from the points db, cg, to the said d, cp, gz: And the two first knit one to another, by bp: And againe with the angles of the cube, by be, and pi: The third knit with the same angles, by ze, and zi: And let all the plaines be made up. I say first, that the five sides bp, pi, iz, ze, and eb are equall; Because, every one of them severally are the doubles of the same greater segment. For in drawing the right lines de and eg, ig, it shall be plaine of two of them; And after the same manner of the rest. First therefore cd, and bp, are equall by the [6 e xxj], and by the [27 e v]. Therefore bp, is the double of the greater segment. Then the whole fl, cut proportionally, and the lesser segment dl, they are by the [7 e xiiij], of treble power to the greater fd, that is, by the fabricke db. Therefore le wich is equall to lf, the line cut, and ld, are of treble power to the same db: But by the [9 e xij], de is of as much power as le, and ld too.
Therefore de is of treble power to db. Therefore both ed, and db, are of quadruple power to db. But be, by the [9 e xij], is of as much power as ed, and db. And therefore be, is of quadruple value to db: And by the [14 e xij], it is the double of the said db. Therefore the two sides eb, and bp, are equall: And by the same argument pi, iz, and ze, are equall. Therefore the quinquangle is equilater.
I say also that it is a Plaine quinquangle: For it may be said to be an oblique quinquangle; and to be seated in two plaines. Let therefore fh be parallell to db, and cp: and be equall unto them. And let hz, be drawne: This hz shall be cut one line, by the [14 e vij]. For as the whole tr, that is rf, is unto the greater segment that is to fh: so fh, that is zg, is unto gr. And two paire of shankes fh, gr, fc, gz, by the [6 e xxj], are alternely or crosse-wise parallell. Therefore their bases are continuall.
Hitherto it hath beene prooved that the quinquangle made is an equilater and plaine: It remaineth that it bee prooved to be Equiangled. Let therefore the right lines ep, and ec, be drawne: I say that the angles, pbe, and ezi, are equall: Because they have by the construction, the bases of equall shankes equall, being to wit in value the quadruple of le. For the right line lf, cut proportionally, and increased with the greater segment df, that is fc, 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 cp, are of treble value to the greater fl, that is of the sayd le. Therefore el, and lc, that is ec, and cp, that is ep, is of quadruple power to el: And therefore by the [14 e xij], it is the double of it: And ei, 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 id, and ib, the third angle bpi, shall be concluded to be equal to the angle ezi. 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.
For that hath beene told you even now.
But from hence also doth arise the geodesy or māner of measuring of a dodecahedrum. For if the quadrate of the line subtending the angle of a quinquangle be trebled, the half of the treble shall be the side of the semidiagony of the dodecahedrum: Because by the [6 e xxiiij], the diagony of the cube, that is of the dodecahedrum is of treble power to the side of the cube. But if the quadrate of the side of the decangle be taken out of the quadrate of the side of the quinquangle; The side of the remainder shall be the ray of the circle circumscribed about a quinquangle. Lastly if the quadrate of the ray, be taken of the quadrate of the half-diagony; the side of the remainder shall be the heighth of perpendicular. As if the side of the decangle be 7.3/5: The quadrate of that shall be 57.19/25: the treble of which is 173.7/25 whose side is about 13.107/131 for the side of the Dodecahedrum, therefore 6.119/131 the halfe shall be the semidiagony of the dodecahedrum. The ray of the
Circle shall now thus be found. If the quadrate of the side of the decangle be taken out of the quadrate of the side of the sexangle; the side of the remainder, shall be the Ray of the Circle, by the [15] and [9 e xviij]. As here the side of the Quinquangle is 4.2/3. The side of the Decangle 2.2/5: And the quadrates therefore are 21.7/9, and 5.19/25. This subducted from that leaveth 16.4/225 whose side is 4.2/15 for the Ray of the Circle.
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 31.2714406/3861225 is for the altitude or heighth: whose 1/3 is 5/3. The quinquangled base is almost 38. Which multiplied by 5/3 doth make 63.1/3 for the solidity of one Pyramis; which multiplied by 12, doth make 760. for the soliditie of the whole dodecaedrum.
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 comprehended: 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 speare is a round imbossement.
It may also be defined to be that which is comprehended of a sphearical surface. A sphearicall body in Greeke is called Sphæra, in Latine Globus, a Globe.
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.
As before there was an analogy betweene a Circle and a Sphericall: so now is there betweene a Cube and a spheare. A cubicall surface is comprehended of sixe quadrate or square and equall bases: And a spheare in like manner is comprehended of sixe equall sphearicall bases compassing the
cubicall bases. A cube is made by the multiplication of the sixth part of the base, by the side: And a spheare likewise is made by multiplying the sixth part of the sphearicall by the diameter, as it were by the side: so the plaine of 616/6 and 14, the diameter is 1437.1/3 for 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 1437.1/3 in the least boundes of the same reason, as 21 is unto 11.
Thus much therefore of the Geodesy of the spheare: The geodesy of the Sectour 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 semicicle 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 orginate.
The Adscription of ordinate plaine bodies 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.
The axeltree in the three first bodies is rationall unto the side, as was manifested in the former. For it is of the sesquialter valew unto the side of the tetrahedrum; of treble, to the side of the cube: Of double, to the side of the Octahedrum. Therefore if the axis ae, be cut by a double reason in i: And the perpendicular io, be knit to a, and e, shall be the side of the tetrahedrum; and oe, of the cube, as was manifest by the [10 e viij], and [25 iiij]: And the greater segment of the side of the cube proportionally cut, is by the [24 e, xxv].
If the same axis be cut into two halfes, as in u: And the perpendicular uy, be erected: And y, and a, be knit together, the same ya, thus knitting them, shall be the side of the Octahedrum, as is manifest in like manner, by the said [10 e, viij], and [25 e iiij].
The side of the Icosahedrum is had by this meanes.
14. If a right line equall to the axis of the sphearicall, and to it from the end of the perpendicular be knit unto the center, a right line drawne from the cutting of the
periphery unto the said end shall be the side of the Icosahedrum.
As here let the Axis ae; be the diameter of the circle aue, and ai, equall to the same axis, and perpendicular from the end, be knit unto the center, by the right line io: A right drawne from the section u, unto a, shall be the side of the Icosahedrum. From u, let the perpendicular uy, be drawne: Here the two triangles iao, & uyo, are equiangles by the [13 e, vij]. Therfore by the [12 e, vij]. as ia, is unto ao: so is uy, unto yo. But ia, is the double of the said ao: Therefore uy, is the double of the same yo: Therefore by the [14 e, xij], it is of quadruple power unto it: And therefore also uy, and yo, that is, by the [9 e xij], uo, that is againe by the [28 e, iiij], ao, is of quintuple power to yo. But yo, is lesser than ao, that is, than oe: Let therefore os, be cut off equall to it. Now as the halfe of ao, is of quintuple valew to the halfe of yo: so the double ae, is of quintuple power to the double ys. Therefore, by the [18 e xxv]. seeing that the diagony ae, is of quintuple power to ys; the said ys, shall be the side of the sexangle inscribed into a circle, circumscribing the quinquangle of the Icosahedrum. But the perpendicular uy, is equall to ys; because each of them is the double of yo. Wherefore uy, is the side of the sexangle. But ay, is the side of the Decangle: For it is equall to se: Because if from equall rayes ao, and oe, you take equall portions oy, and os: There shall remaine equall, ya, and se. And the Diagony of an Icosahedrum by the [16 e xxv], is compounded of the side of the sexangle, continued at each end with the side of the decangle. Wherefore ay, is the side of the decangle. Lastly, ua, whose power is as much as the sides of the
sexangle and decangle, by the [15. e, xviij], shall be the side of an Icosahedrum.
15 Of the five ordinate bodies inscribed into the same spheare, the tetrahedrum in respect of the greatnesse of his side is first, the Octahedrum, the second; the Cube, the third; the Icosahedrum, the fourth; and the Dodecahedrum, the fifth.
As it will plainely appeare, if all of them be gathered into one, thus. For ai, the side of the Tetrahedrum, subtendeth a greater periphery than ao, the side of the Octahedrum; And ao, a greater than ie, the side of the Cube; because it subtendeth but the halfe: And ie, greater than ue, the side of the Icosahedrum: And ue, greater than ye, the side of Dodecahedrum.
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 rl: And the side of the Decangle ar, inscribed into the same circle, circumscribing the quinquangle of an icosahedrum: And the perpendiculars is, and ul.
Here the two triangles aie, and ies, are by the [8 e, viij]. alike; And as se, is unto ei: So is ie, unto ea: And by [25 e, iiij], as se, is to ea: so is the quadrate of se, to the quadrate of ei: And inversly or backward, as ae, is to se: so is the quadrate of ie, to the quadrate of se. But ae, is the triple of se. Therefore the quadrate of ie, is the triple of se. But the quadrate of as, by the grant, and [14 e xij], the quadruple of the quadrate of se. Therefore also it is greater than the quadrate of ie: And the right line as, is greater than ie, and al, therefore is much greater. But al, is by the grant
compounded of the sides of the sexangle and decangle rl, and ar. Therefore by the 1 c. [5 e, 18.] it is cut proportionally: And the greater segment is the side of the sexangle, to wit, rl: And the greater segment of ie, proportionally also cut, is ye. Therefore the said rl, is greeter than ye: And even now it was shewed ul, was equall to rl. Therefore ul, is greater than ye: But ue, the side of the Icosahedrum, by [22. e vj]. is greater than ul. 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.
Here the base is a circle.
Therefore
5 It is made by the turning about of a right angled 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.
Here a threefold difference of the heighth of a Cone is professed, out of the threefold difference of the angles, whereby the toppe of the halfed cone is distinguished: Notwithstanding this consideration belongeth rather to the Optickes, than to Geometry. For a Cone a farre off seeme like triangle. Therefore according to the difference of the heighth, it
appeareth with a right angled, or obtusangled or acutangled toppe: As here the least Cone is obtusangled: the middle one rightangled: and the highest acutangled. But the cause of this threefold difference in the angles from of the difference of the shankes, is out of the consectaries of the threefold triangle of a right line cutting the base into two equall parts, as appeareth at the end of the [viij] Booke.
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 xij.
These are consectaries drawne out of the [16] and [18 e. iiij]. As here you see.
10 A Cylinder is that which is comprehended of a cylindricall surface and the opposite bases.
For here two circles, parallell one to another are the bases of a Cylinder.
Therefore
11 It is made by the turning about of a right angled parallelogramme, the one side standing still. 21. d xj.
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.
The geodesy here is fetch'd from the prisma: As if the base of the cylinder be 38.1/2: Of it and the heighth 12, the solidity of the cylinder is 462.
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.
From hence the capacity or content of cylinder-like
vessell or measure is esteemed and judged of. For the hollow or empty place is to be measured as if it were a solid body.
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 liquor or any thing else 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 Cylinder, 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 axeltrees and diameters may have the similitude of their bases. And with as great reason may the Geometer demand to have it granted him, That the Cylinder is the treble of a Cone: As it was demanded and granted him, That Cylinders and Cones are alike, whose axletees are proportionall to the diameters of their bases.
Therefore
14. A plaine made of the base and thid part of the height, is the solidity of the cone of equall base & height;
The heighth is thus had. If the square of the ray of the base, be taken out of the square of the side, the side of the remainder shall bee the heighth, as is manifest by the [9 e xij]. Here therefore the square of the ray 5, is 25. The square of 13, the side is 169. And 169 - 25, are 144; whose side is 12 for the heighth: The third part of which is 4. Now the circular base is 78.4/7: And the plaine of these is 314.2/7 for the solidity of the Cone.
But the analogie of a conicall unto a Cylinder like surface doth not answeare, that the Conicall should be the subtriple of the Cylindricall, as the Cone is the subtriple of the Cylinder.
Of two cones of one common base is made Archimede's Rhombus, as here, whose geodæsy shall be cut of two cones.
And
15. Cylinder of equall heighth are as their bases are. 11 p xij.
Sackes in which they carry corne, are for the most part of
a cylinderlike forme. If an husbandman therefore shall lend unto his neighbour a sacke full or corne, and the base of the sacke be 4 foote over. And the neighbour afterward for that one sacke, shall pay him 4 sacke fulls, every sacke being as long as that was, yet but one foote over in the diameter, he may be thought peradventure to have repayed that which he borrowed in equall measure, to wit in heighth and base. But it shall be indeed farre otherwise: For there is a great difference betweene the quadrate of the foure severall diameters, 1. 1. 1. 1. that is 4: and 16, the quadrate of 4, the diameter of that sacke by which it was lent. For Circles are one unto another as the quadrates of their diameters are one to another, by the [2 e xv]. Therefore he payd him but one fourth part of that which he borrowed of him.
And
16 Cylinders reciprocall in base and heighth are equall. 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.
As here thou seest. For the axes are the altitudes or heights. It is likwise a consectary following upon that generall theoreme of first figure, but somewhat varyed from it. It doth answere unto the [10 e 23].
The unequall sections of a spheare we have reserved for this place: Because they are comprehended of a surface both sphearicall and conicall, as is the sectour. As also of a plaine and sphearicall, as is the section: And in both like as in a Circle, there is but a greater and lesser segment. And the sectour, as before, is considered in the center.
18. The sectour of a spheare is a segment of a spheare, which without is comprehended of a sphearicall within of a conicall bounded in the center, the greater of a concave, the lesser of a convex.
Archimedes, 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.
As here of the Diameter 14, and of 73.1/3 and 4.2/3 (which is the one sixth part of the greater sphearicall) the plaine is 1026.2/3 for the solidity of the greater sectour, so of the same diameter 14, and 29.1/3 which is the 1/6 part of 176, the lesser sphæricall, the plaine is 410.2/3 for the solidity of the lesser sectour.
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 1026.2/3. And 1026.2/3 + 126.4/63 doe make 1152.46/63.
Againe the lesser sectour, by the next precedent, was 410.2/3: And here the inner cone is 126.4/63 And therefore 410.2/3 - 126.4/63 that is 284.38/63 is the lesser section.
FINIS.
