1 Def. 6. al*: again, because B is similar to C, they are equiangular, and have their sides about the equal * 1 Def. 6. angles proportionals*: therefore

B

the figures A, B, are each of them equiangular to C, and have the sides about the equal angles of each of them and of C proportionals. Wherefore the rectilineal 1 Ax. 1. figures A and Care* equiangular, and have their sides about the equal angles* proportionals: therefore A is similar to B. Therefore rectilineal figures, &c.

11. 5. * 1 Def. 6.

11. 6.

11.5.

• 22.5.

Q. E. D.

PROP. XXII. THEOR.

If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals: and if the similar rectilineal figures similarly described upon four straight lines be proportionals, those straight lines shall be proportionals.

Let the four straight lines AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH; and upon AB, CD let the similar rectilineal figures KAB, LCD be similarly described; and upon EF, GH the similar rectilineal figures MF, NH, in like manner: the rectilineal figure KAB shall be to LCD, as MF to NH.

To AB, CD take a third proportional * X; and to EF, GH a third proportional O: and because AB is to CD as EF to GH, therefore CD is * to X as GH to O; wherefore, ex æquali *, as AB to X, so EF to 0: but as AB to X, so is the rectilineal figure KAB to the *2 Cor. 20. rectilineal figure LCD, and as EF to O, so is the rectilineal figure MF to the rectilineal figure NH: therefore, as KAB to LCD, so* is MF to ÑH.

6.

11. 5.

* 12. 6.

18.6.

And if the rectilineal figure KAB be to LCD, as MF to NH; the straight line AB shall be to CD, as EF to GH.

Make as AB to CD, so EF to PR, and upon PR describe the rectilineal figure SR similar and similarly situated to either of the figures MF, NH: then, because as AB to CD, so is EF to PR, and that upon AB, CD are described the similar and similarly situated rectilineals KAB, LCD, and upon EF, PR, in like manner, the similar rectilineals MF, SR; therefore KAB is to LCD, as MF, to SR: but by the hypothesis

KAB is to LCD, as MF to NH; and therefore the

rectilineal MF having the same ratio to each of the two NH, SR, these are equal* to one another: they are also similar, and similarly situated; therefore GH is equal to PR: and because as AB to CD, so is EF to PR, and

M

K

* 9.5.

X

BC

N

E

' Ꮐ H

that PR is equal to GH; AB is to CD +, as EF to +7.5. GH. If therefore four straight lines, &c. Q. E. D.

PROP. XXIII. THEOR.

Equiangular parallelograms have to one another the ratio See N. which is compounded of the ratios of their sides.

Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG: the ratio of the parallelogram AC to the parallelogram CF, shall be the same with the ratio which is compounded of the ratios of their sides.

DH

* 12. 6.

Let BC, CG be placed in a straight line; therefore DC and CE are also in a straight line*; and complete * 14. 1. the parallelogram DG; and taking any straight line K, make * as BC to CG, so K to L; and as DC to CE, so make* L to M: therefore, the ratios of K to L, and * 12. 6. L to M, are the same with the ratios of the sides, viz. of BC to CG, and DC to CE: but the ratio of K to M is that which is said to be compounded* of the ratios Def. A. of K to L, and L to M; therefore K has to M the ratio 5. compounded of the ratios of the sides: and because as BC to CG, so is the parallelogram AC to the parallelogram CH*; but as BC to A CG, so is K to L; therefore K is* to L, as the parallelogram AC to the parallelogram CH: again, because as DC to CE, so is the parallelogram CH to the parallelogram CF; but as DC to CE, so is L to M; wherefore L is * to M, as the parallelogram CH to the parallelogram CF: therefore since it has been proved, that as K to L, so is the parallelogram AC to the parallelogram CH; and as L to M, so the parallelogram CH to the parallelogram CF; ex æquali*, *22. 5. K is to M, as the parallelogram AC to the parallelogram CF: but K has to M the ratio which is compounded

B

1. 6. * 11.5.

G

KLM E F

* 11.5.

See N.

* 29.1.

* 34.1.

* 4.6.

* 54. 1. *7.5.

of the ratios of the sides; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides. Wherefore equiangular parallelograms, &c. Q. E. D.

PROP. XXIV. THEOR.

Parallelograms about the diameter of any parallelogram, are similar to the whole, and to one another.

Let ABCD be a parallelogram, of which the diameter is AC; and EG, HK parallelograms about the diameter: the parallelograms EG, HK shall be similar both to the whole parallelogram ABCD, and to one another.

Because DC, GF are parallels, the angle ADC is equal to the angle AGF: for the same reason, because BC, EF are parallels, the angle ABC is equal to the angle AEF: and each of the angles BCD, EFG is equal to the opposite angle DAB *, and therefore they are equal to one another: wherefore the parallelograms ABCD, AEFG, are equiangular; and because the angle ABC is equal to the angle AEF, and the angle BAC common to the two triangles BAC, EAF, they are equiangular to one another; therefore* as AB to BC, so is AE to EF: and because the opposite sides of parallelograms are equal to one another*, AB* is to AD, as AE to AG; and DC to CB, as GF to FE; and also CD to DA, as FG to GA: therefore the sides of the parallelograms ABCD, D K. AEFG about the equal angles are pro

G F

H

* 1 Def. 6. portionals; and they are therefore similar* to one another: for the same reason the parallelogram ABCD is similar to the parallelogram FHCK: wherefore each of the parallelograms GE, KH is similar to DB: but rectilineal figures which are similar to the same rectilineal figure are also similar to one another; therefore the parallelogram GE is similar to KH. Wherefore parallelograms, &c. Q. E. D.

* 21. 6.

*

PROP. XXV. PROB.

To describe a rectilineal figure which shall be similar to See N. one, and equal to another given rectilineal figure.

Let ABC be the given rectilineal figure to which the figure to be described is required to be similar, and D that to which it must be equal. It is required to describe a rectilineal figure similar to ABC, and equal

to D.

Upon the straight line BC describe the parallelo- Cor. 45.1, gram BE equal to the figure ABC; also upon CE de-X scribe* the parallelogram CM equal to D, and having *Cor.45.1. the angle FCE equal to the angle CBL: therefore BC and CF are in a straight line*, as also LE and EM: between BC and CF find a mean proportional GH, and upon GH describe the rectilineal figure KGH similar and similarly situated to the figure ABC.

*

*

*

* 29. 1. & 14.1.

13.6.

*18. 6.

6.

*

1.6.

11. 5.

Because BC is to GH as GH to CF, and that if three straight lines be proportionals, as the first is to the third, so is the figure upon the first to the similar and 2 Cor. 20. similarly described figure upon the second; therefore as BC to CF, so is the rectilineal figure ABC to KGH: but as BC to CF, so is the parallelogram BE to the parallelogram EF; therefore* as the rectilineal figure ABC is to KGH, so is the parallelogram BE to the parallelogram EF: and the rectilineal figure ABC is equal to the parallelogram BE; therefore the rectilineal figure KGH is equal to the parallelogram EF: but EF is equal to the figure D; where- B fore also KGH is equal to D: and it is similar to ABC. There

L

+ Constr.

* 14. 5.

D K

+ Constr.

E M

fore the rectilineal figure KGH has been described similar to the figure ABC, and equal to D. Which was to be done.

PROP. XXVI. THEOR.

If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter.

* 24.6.

* 1 Def. 6. † Hyp.

11.5.

* 9. 5.

Let the parallelograms ABCD, AEFG be similar and similarly situated, and have the angle DAB common: ABCD and AEFG shall be about the same diameter.

G D

K H

B

For, if not, let, if possible, the parallelogram BD have its diameter AHC in a different straight line from AF, the diameter of the parallelogram EG, and let GF meet AHC in H; and through H draw HK parallel to AD or BC; therefore the parallelograms ABCD, AKHG being about the same diameter, they are similar to one another: wherefore as DA to AB, so is * GA to AK: but because ABCD and AEFG are similar+ parallelograms, as DA is to AB, so is GA to AE; therefore* as GA to AE, so GA to AK; that is GA has the same ratio to each of the straight lines AE, AK; and consequently AK is equal to AE, the less to the greater, which is impossible: therefore ABCD and AKHG are not about the same diameter: wherefore ABCD and AEFG must be about the same diameter. Therefore, if two similar, &c. Q. E. D.

*

To understand the three following propositions more ' easily, it is to be observed,

1. That a parallelogram is said to be applied to a straight line, when it is described upon it as one of its 'sides. Ex. gr. the parallelogram AC is said to be ⚫ applied to the straight line AB.

EC G

2. But a parallelogram AE is said to be applied to a straight line AB, deficient by a parallelogram, when AD the base of AE is less than AB, ' and therefore AE is less than the 'parallelogram AC described upon AB in the same angle, and between 'the same parallels, by the parallelo"gram DC; and DC is therefore called the defect of AE.

D B

3. And a parallelogram AG is said to be applied to a straight line AB, exceeding by a parallelogram, when AF the base of AG is greater than AB, and therefore AG exceeds AC the parallelogram de6 soribed upon AB in the same angle, and between the same parallels, by the parallelogram BG.'