PROPOSITION XXII.

THEOREM XVI.

IF four ftraight lines (AB, CD, EF, GH) be proportionals, the fimilar

rectilineal figures & fimilarly defcribed upon them (M, N, & P, Q) fhall alfo be proportionals. And if the fimilar rectilineal figures (M, N, & P, Q) fimilarly defcribed upon four straight lines be proportionals, thofe ftraight lines fhall be proportional.

AB: Z =EF: X.

ECAUSE ABCDEF: GH.S(Hyp. 1).

(Hyp.1.Prep.&P.11.B.5).

P.22. B.5.

But the figures M,N, & P,Q_being_fimilar & fimilarly described upon the ftraight lines A B, CD, & E F, GH (Hyp. 2).

2.

AB: Z

=M: N

SP.20. B.6.
Cor. 1.
P.11. B.5.

Book VI.

Preparation.

Thefis. ABCDEF: GH.

1. To AB, CD, EF take a IVth proportional KL.

2. Upon K L defcribe the rectil. figure R, fimilar to the
rectil. figures P or Q, &fimilarly fituated."

BECAUSE

DEMONSTRATION.

ECAUSE AB: CDEF: KL (Prep. 1), & upon thofe ftraight lines have been fimilarly described the figures M, N, & P, R, fimilar each to each (Hyp. 2. & Prep. 2).

1.

But

M: NPR (ft. part of this propofition.)

M: NP: Q (Hyp. 1).

2. Confequently, P: RP: Q

3.

Wherefore,

R = Q

Moreover, thofe figures being fimilar & fimilarly defcribed upon the ftraight lines G H, KL (Prep. 2).

Q: RO of GH: O of K L

Q being R (Arg. 3).

4.

And

The of GH is

4.

to the

5. Confequently, GH = KL.

6.

of K L.

Since then ABCDEF: KL(Prep.1), & GH =KL (Arg·5).
AB: CDEF: GH.

Which was to be demonstrated.

Ff

PROPOSITION XXIII. THEOREM XVII.

EQUIANGULAR parallelograms (M & N) have to one another the

ratio which is compounded of the ratios of their fides (AC, CD & E C, CG) about the equal angles.

DEMONSTRATION.

BECAUSE the pgrs. M, P, N form a feries of three magnitudes

.L

2. And alternando

M: MPN: N.P.

P.14. B.1.
Pof.1. B.1.

D. 5. B.5.
P.16. B.5.

D. 5. B.6.

P. 1. B.6.

pgrs.

3. Confequently the ratio of the firft M to the laft N,is compounded of the ratios M: P& P: N.

:

: N.

4. The ratio of the fides AC CG is the fame as that of the
M: P; & the ratio of the fides DC: CE, the fame as that of the
pgrs. P: N.

Since then the ratio of M: N is compounded of the ratios M: P,
& P: N (Arg. 1)..

5. This fame ratio is compounded of their equals; the ratios AC: CG & CD EC, of the fides about the equal

:

6. Confequently, M: NAC.CD: EC.CG.

ACD, ECG.

D. 5. B.6.

Which was to be demonftrated.

Cor. The fame truth is applicable to the triangles (ACD, ECG) having an angle (ACD) equal to an angle (ECG): for the diagonals (A D, EG) divide the pgrs. into two equal parts (P. 34. B. 1),

PROPOSITION XXIV. THEOREM XVIII.

THE parallelograms (FH, IG) about the diagonal (AC) of any parallelo

gram (BD), are fimilar to the whole, and to one another.

Hypothefis.

1. BD is a pgr.

II. FH,IG are pgrs about the

diagonal AC

BECAUS

DEMONSTRATION,

Thefis.

1. The pgrs. AFEH, EICG are fimilar to the pgr. ABCD.

II. And fimilar to one another.

ECAUSE FE is plle. to BC (Hyp. 1. & 2. & P. 30. B. 1).

1. The AAFE is equiang. to the AABC in the order of the letters.
In like manner, becaufe HE is plle. to D C.

2. The AAHE is equiang. to the AADC, in the order of the letters.
3. Therefore the pgr. AFEH is alfo equiangular to the pgr. ABCD, in
the order of the letters.

4.

5.

And because in the AAHE,ADC, the VAHE & D are equal (Arg.2), as also in the ▲ AFE, ABC, the AFE & B (Arg. 1).

AH: HEAD: DC & AF: EFAB : CB. Moreover, because the VAEH, ACD; alfo FEA,BCA are equal (Arg. HE AEDC: AC & AE: EFAC: CB.

6. Therefore, ex æquo, HE: EF = DC: CB.

7.

P.29. B.1.

P. 4. B.6.
1.2).
P. 4. B.6.
P.22. B.5.

And because the EAH, EFA are common to the two
AAHE, ADC & AFE, A B C.

HA: EA DA: CA & EA: AFCA : AB.

8. Therefore, ex æquo, HA: AFDA: A B.

9. Wherefore the pgrs. AFEH, ABCD have their angles equal, each to each in the order of the letters (Arg. 3); & the fides about the equal angles, proportionals (Arg. 4. 6.8.).

10.Confequently, thofe pgrs. are fimilar.

11.It may be demonftrated after the fame manner that the pgrs. EICG, ABCD are fimilar.

Which was to be demonstrated. 1.

P. 4. B.6.
P.22. B.5.

D. 1. B.6.

12.Confequently, the pgrs. AFEH, EICG are alfo fimilar to one another. P.21. B.1.

Which was to be demonftrated. 11.

To defcribe a rectilineal figure (N), which shall be similar to a given reĉi

lineal figure (L), and equal to another (M).

Given.

I. The reЯilineal figure L.

II. The rectilineal figure M.

Sought.

The reail. figure N, fimilar to the recti!. figure L, &

Refolution.

1. Upon the ftraight line AC, describe the pgr.AH
given rectilineal figure L.

to the ređil. figure M

2. And on the ftraight line CH a pgr. CK

to the given recti

lineal figure M, having an

m to the Vn.

P-45. B.1.

3. Confequently, the fides AC, CI, & GH, HK will be in P.14. 29.

a ftraight line.

&34. B.í. P.13. B.6.

P.18. B.6.

4. Between A C, & CI find a mean proportional DF.
5. Upon this ftraight line D'F, defcribe the rectil. figure N,
fimilarly & fimilar to the rectilineal figure L.

ECAUSE the pgrs. AH, CK have the fame altitude (Ref.z.&3).

But the pgr. AH rectil. L, & the pgr.CK=rectil.M (Ref.1.2). 2. Confequently, L: MAC: CI.

But

AC: DF = DF CI (Ref. 4.), & upon the ftraight lines A C, DF have been fimilarly defcribed the fimilar figures L & N, (Ref. 5).

P. 1. B.6.

P.11. B.5.

6. Therefore, there has been defcribed a rectilineal figure N, fimilar to the rectilineal figure L (Ref. 5), & equal to the rectilineal figure

M (Arg. 5).

Which was to be done.

P.20. B.6.

Cor.

P.11. B.5.
P.14. B.5.