UPON

PON a given straight line (AD) to describe a rectilineal figure (M) fimilar, and fimilarly fituated to a given rectilineal figure (N).

Given.

1. The ftraight line A D.

II. The rectilineal figure N,

1. Join HF.

Pof.1. B.1.

to

P. 23. 32.

2. At the points A & D in AD, make VAVYE & Ym="
Vn, wherefore the remaining

the remaining EFH.

ABD will be

موع

3. At the points D & B in D B make Yo=p&q=
Vr, confequently the remaining VC will be to the Lem. B..
remaining G.

BECAUSE

DEMONSTRATION,

ECAUSE the A ABD is equiangular to the AEF H, & the AD B C equiangular to the AHFG (Ref. 2. & 3).

I.

&

BD FH BA: FEAD: E H.
BD: FH DC: HG=CB: G F.

} P. 4. B.6.

P.11. B.5.

Ax.2. B.1.

2. Confequently, BA: FEAD: EH DC: HG = CB: GF.

But m being Vn (Ref. 2), & V o = V p (Ref. 3). 3. The whole Vm+ is to the whole

Moreover, VAVE (Ref. 2), & VC=VG (Ref. 3).

4. Wherefore, the rectilineal figure M is equiangular to the rectilineal
figure N, & their fides about the equal are proportionals.

6. Therefore, the rectilineal figure M defcribed upon the given line AD
is fimilar to the rectilineal figure EG, & is fimilarly fituated.
Which was to be done.

D. 1. B.6.

B

E

G

A

PROPOSITION XIX.

THEOREM XIII.

SIMILAR triangles (ABC, DEF) are to one another in the dupli

cate ratio of their homologous fides (CB, F E or AC, DF, &c).

Hypothefis.

Thefis.

The AABC is to the ▲ DEF in the duplicate ratio of CB to FE that is as CB2 : FE2.*

Take CG a third proportional to CB, F E, & draw A G.

DEMONSTRATION.

ECAUSE AC: CBDF: FE (Hyp. & D. 1. B. 6).

BECAUSE

CB: FEFE CG (Prep.).

I. Alternando AC: DF CB: FE.

But

2. Confequently, AC: DF FE: CG.

3. Therefore, the fides of the AAGC, DEF about the equal VC & F

6. Confequently, the ▲ ABC: ADEF=CB: CG.

7. CB: CG in the duplicate ratio of CB to FE, or as CB2: FE2* 8. Wherefore, the AABC: ADEF in the duplicate ratio of C Boto FE, or as CB2: F E2*.

FROM

Which was to be demonstrated.
COROLLARY.

SP.11. B.6.

Pof.1. B.1.

P.16. B.5

P.11. B.5.

P.15. B.6.

P. 1. B.6.

P. 7. B.5.

D.10. B.5.

P.11. B.5.

ROM this it is manifeft, that if three lines (CB, F E, CG) be proportionals, as the firft is to the third, jo is any ▲ upon the first to a fimilar, Ɛ$ fimilarly described A upon the fecond.

• See Cor. 2. of the following propofition.

PROPOSITION XX.

THEOREM XIV.

SIMILAR polygons (M & N) may be divided by the diagonals (A C, AD; F H, FI) into the fame number of fimilar triangles (ABC, ACD, ADE, & FGH, FHI, FIK) having the fame ratio to one another, that the polygons (M&N) have; and the polygons (M & N) have to one another the duplicate ratio of that which their homologous fides (AB, FG; or BC, GH &c.)

have.

BEECAUS

DEMONSTRATION.

EECAUSE VB=VG & AB : BC=FG: GH (Hyp. & D.1. B.6).

1. The AABD is equiangular to the AFGH. 2. Wherefore thofe A are fimilar, & \m=V a.

But the whole Vm+n is to the whole Va+c (Hyp).

3. Confequently, Vn is to Vc.

Since then by the fimil. of the ▲ ABC & FGH (Arg.2),
AC: BC FH: GH,

& by the fimil. of the polyg. M & N, BC: CD=GH: HI.
AC: CD

4. It follows, Ex quo, that

Pof.1. B..

P. 6. B.6.
P. 4. B.6.
Cor.
Ax.3. B.1.

D. 1.B6.

FH: HI

P.22. B.5.

P. 6. B.6.

SP. 4. B.6.
Cor.

That is, the fides about the equal Vn & c are proportionals.
5. Therefore the AA CD is equiangular to the AFHI.
And confequently is fimilar to it.

6. For the fame reason, all the other ▲ ADE, FIK, &c. are similar.
7. Therefore, fimilar polygons may be divided into the same number of
fimilar A.
Which was to be demonftrated. 1.

See Cor. 2. of this propofition.

Likewife, because the ▲ A B C, F G H are fimilar (Arg. 2).

6. The

And the

AABC:AFGH=AC : FH2 *
AACD: AFHIA C2: F H2.*

7. Therefore, the AABC: AFGHA ACD: AFHI.
be demonftrated after the fame manner, that

It may

8. The

9. Wherefore, ABC: AFGH≈^ACD: AFHI=^ADE: AFIK. P.11. B.5. 10. Therefore, comparing the fum of the anteced. to that of the confeq. AABC+AACD, &c. : AFGH+AFHI,&c=^ABC: AFGH,&c. P.12. B.5. That is, the polyg. M: polyg. NAAB-C: AFGH= AACD: AFHI, &c.

Which was to be demonstrated. 11. Since then the AABC: AFGHA B2: F G2* (P.19. B. 6). 11. The polyg. M: polyg. NA B2 F G2*.

As

P. 7. B.5.

P.11. B.S

Which was to be demonftrated. 111.

COROLLARY I

S this Demonftration may be applied to quadrilateral figures, the fame truth has already been proved in triangles (P.19), it is evident univerfally, that fimilar rectilineal figures are to one another in the duplicate ratio of their homologous fides. Wherefore, if to A B, FG two of the homologous fides a third proportional X be taken; becaufe A B is to X in the duplicate ratio of A B: FG ; & that a rectilineal figure M is to another fimilar rectilineal figure N, in the duplicate ratio of the fame fides AB FG; it follows, that if three ftraight lines be proportionals, as the firft is to the third, fo is any rectilineal figure described upon the first to a fimilar & fimilarly defcribed rectilineal figure upon the fecond. (P.11. B.5).

ALL

LL fquares being fimilar figures (D. 30. B. 1. & D. 1. B. 6), fimilar rectilineal figures M&N, are to one another as the fquares of their homologous fides AB, CD (expressed thus A B C D2) for those figures are in the duplicate ratio of thefe fame fides.

AAA

PROPOSITION XXI. THEOREM XV.

RECTILINEAL figures (A, C) which are fimilar to the fame

rectilineal figure (B), are also similar to one another.

BECAUS

Thefis.

The rectilineal figure A is fimilar to the rectilineal figure C.

DEMONSTRATION.

ECAUSE each of the figures A & C is fimilar to the figure B (Hyp.).

1. Each of thofe figures will be alfo equiangular to the figure B, & will have the fides about the equal V, proportional to the fides of the figure B.

2. Confequently, thofe figures A & C will be alfo equiangular to one another, and their fides about the equal V, will be proportional. 3. Confequently, the figures A & C are fimilar.

Which was to be demonftrated.

D. 1. B.6. S Ax.1. B.1. P.11. B.5. D. 1. B.6.