Page images
PDF
EPUB

Corollary. 2. And if a third proportional x be found to AB, FG: [by 10. def. 5.] A B will be to x, in the duplicate ratio of that of AB to FG: But the ratio of one polygon to the other, and of one quadrilateral figure to the other, is the duplicate of the ratio that one homologous fide has to another; that is, of AB to FG: This is alfo demonftrated of triangles: therefore univerfally it is manifeft, that if three right lines be proportional, as the first is to the third, fo is any right lined figure defcribed upon the first, to a fimilar right lined figure alike defcribed upon the fecond.

Otherwife.

We shall demonftrate otherwise more expeditiously that the triangles are homologous.

For again let ABCDE, FGHKL be the fimilar polygons; and join BE, EC, GL, LH: I say as the triangle ABE is to the triangle FGL, fo is the triangle EBC to the triangle LGH, and the triangle CDE to the triangle

HK L.

[blocks in formation]

to G L. By the fame reafon the triangle BEC to the triangle GLH is in the duplicate ratio of BE to GL: then [by 11. 5.] as the triangle ABE is to the triangle FGL, fo is the triangle EBC to the triangle LGH. Again, because the triangle EBC is fimilar to the triangle LGH; [by 19. 6.] the triangle EBC has a ratio to the triangle LGH, the duplicate of the ratio of the right line CE to the right line HL. By the fame reafon alfo the triangle ECD has a ratio to the triangle L HK the duplicate of that of CE to HL: Therefore [by 11. 5.] as the triangle EBC is to the triangle LGH, fo is the triangle ECD to the triangle LHK: But it has been proved that as the triangle EBC is to the triangle LG H, fo is the triangle ABE to the triangle FGL: Therefore as the triangle ABE is to the triangle FG L, fo is the triangle BEC to the tri

4

angle

angle GLH, and the triangle E CD to the triangle LHK ; and therefore [by 12. 5.] as one of the antecedents is to one of the confequents, fo are all the antecedents to all the confequents, and the reft, as in the former demonstration. Which was to be demonstrated.

PROP. XXI. THEOR.

Right lined figures which are fimilar to the fame right lined figure, are fimilar to one another.

Let the right lined figures A, B be each of them fimilar to the right lined figure c: I fay the right lined figure A is alfo fimilar to the right lined figure B.

For because the right lined figure A is fimilar to the right lined figure c; [by def. 1. 6.] it will be equiangular to it, and have the fides about the equal angles proportional. Again, because the right lined figure B is fimilar to the right lined figure c, it will be equiangular to it, and have the fides about the equal angles

AA

B

proportional; therefore each of the figures A, B is equiangular to the right lined figure c; and has the fides about the equal angles proportional: Wherefore the right lined figure A [by 1. ax.] is equiangular to B, and [by 11. 5.] has the fides about the equal angles proportional; and accordingly [by 1. def. 6.] A is fimilar to B. Which was to be demonftrated.

THEOR.

PROP. XXII. If four right lines be proportional, the right lined figures that are defcribed upon them fimilar and alike fituate will be alfo proportional: and if right lined figures defcribed upon four right lines, Similar and alike fituate be proportional; thofe right lines will be proportionali.

Let the four right lines A B, CD, EF, GH be propor tional, viz. as AB is to CD, fo is EF to GH; and upon AB, CD, let the right lined figures KAB, LCD fimilar

and

Book VI. and alike fituate be described; but upon EF, GH, let the right lined figures MF, NH fimilar and alike fituate, be defcribed: 1 fay as the fight lined figure KAB is to the right lined figure LCD, fo is the right lined figure MF to the right lined figure NH.

For [by 11. 6.] find a third proportional x tỏ Â B, C D ; and a third proportional o to £ F, GH. And because A B is to CD, ás E F is

K

X

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

to GH.

And as 'CD is to x fo is GH to o it will be, by equality [by 22. 5.] as A B is to x, fo is EE to o. But

[bý 2. cor. 20. 6.1 as A B is to X, fo is the right lined fgure K A B to the right lined figurë LCD. But as EF is to o, fo is the right lined figure

MF, to the right lined figure NH: Therefore [by 11. 5. as K A B is to LCD, fo is MF to N H.

But now let the right lined figure K A B be to the right lined figure LCD, as the right lined figure MF is to the right lined figure NH: 1 fay as A B is to CD, fo will EF

be to GH.

PR.

For [by 12. 6.] make as AB is to CD, fo is EF tổ And [by 18. 6.] upon PR, defcribe the right lined figure SR fimilar and alike fituate to either of the right lined figures MF or NH.

Then because A B is to CD, as E F is to PR, and upon A B, C D are described the right lined figures K ́A B, LCD fimilar and alike fituate. But upon EF, PR the right lined figures MF, SR, fimilar and alike fituate. [by the first part of this] it is as the right lined figure KAB is to the right lined figure LCD, fo is the right lined figure MF, to the right lined figure s R. But [by fuppofition] as the right lined figure K AB, is to the right lined figure LCD, fo is the right lined figure M F to the right lined figure NH: Therefore [by 11. 5.] MF has the fame ratio to each of the right lined figures NH, SR: Confe

quently

quently [by 9. 5.] the right lined figures NH, SR are equal: They are alfo fimilar and alike fituate: Therefore [by the following lemma] G H is equal to PR. And because AB is to CD, as E F is to PR, and PR is equal to GH [by 7. 5.] A B will be to CD, as EF is to GH

If therefore four right lines be proportional, the right lined figures that are described upon them fimilar and alike fituate, will be alfo proportional: and if right lined figures defcribed upon four right lines fimilar and alike fituate, be proportional; these right lines will be proportional Which was to be demonstrated.

LEMM A.

But we fhall thus demonftrate, that the homologous fides of fimilar and equal right lined figures, are equal.

Let the right lined figures NH, SR be fimilar and equal :and let HG be to GN, as R P is to PS: I fay RP is equal

[merged small][ocr errors]

For if they be unequal, one of them will be the greater. Let this be R P. Then because RP is to PS, as HG is to GN and alternately [by 16. 5.] as R P is to GH, fo is Ps to GN. But PR is greater than GH: therefore Ps will be greater than GN: Wherefore [by 20. 6.] the right lined figure R s is greater than the right lined figure HN; and alfo equal; which cannot be: Therefore PR is not unequal to GH: Wherefore it is equal to it. Which was to be demonftrated.

i This propofition is always true; if all the four figures be fimilar and alike fituate. But when only two and two are fo: it will not always be true, unless the two fimilar and alike fituate figures be both described upon thofe two right lines that are the antecedents and confequents of the equal ratios; as if the right line A be to B, as c is to D, and the figures G, н defcribed upon the first and fourth right lines A and D be fimilar and alike fi.

[blocks in formation]

third right lines B, C be both fimilar and alike fituate, thefe four figures G, I, K, L will not be proportional, as iveafily apprehended, from the bare contemplation of the figures.

Should

Should not the lemma to this propofition have been made a propofition, and fet down before this propofition?

PROP. XXIII. THEOR.

Equiangular parallelograms are to one another in a ratio compounded of the ratios of their fides *.

Let A C, CF be equiangular parallelograms, having the angle B C D equal to the angle ECG: I say the parallelogram A c is to the parallelogram CF in the ratio compounded of the ratios of the fides: that is, of the ratio of BC to CG, and of the ratio of DC to CE. For put BC, CG in the fame right line. Then [by 14. 1.] DC will be in the fame right line with CE and compleat the parallelogram DG, alfo let K be any right line, and [by 22. 6.] make as BC to CG, fo is K to L, and as DC is to CE, fo is L to M.

A

B

D

H

G

[ocr errors]

Then the ratios of K to L, and L to м are the fame as the ratios of the fides, viz. that of BC to CG, and of DC to CE: But [by 5. def. 6.] the ratio of K to M is com pounded of the ratio of K to L, and of the ratio of L to M. Wherefore the ratio of K to M is the ratio compounded of the ratios of the fides. And because [by 1. 6.] as BC is to C G, fo is the parallelogram A to the parallelogram: CH :But as BC to CG, fo is K to L" Therefore [by 11. 5.] as K is to L, fo is the parallelogram A C to the parallelogram C H. Again, because [by 1.6.] as DC is to CE, fo is the parallelogram C H, to the parallelogram CF; and as DC is to CE, fo is L to M: Therefore as L is to M, fo [by 11. 5.] will the parallelogram CH be to the parallelgram C F. And fo fince it has been proved that as K is to L, fo is the parallelogram AC to the parallelogram CH: But as L to M, fo is the parallelogram C H to the parallelogram CF it will be, by equality [by 22. 5.] as K is to M, fo is the parallelogram AC to the parallelogram CF. But the ratio of K to M is that compounded of the ratios of the fides: Therefore the ratio of the parallelogram AC

KLM

E F

to

« PreviousContinue »