Book VI. KAB to the rectilineal LCD, and as EF to O, so is d the rectili neal MF to the rectilineal NH: therefore, as KAB to LCD, so b is MF to NH.

d 2 Cor. 20. 6.

b 11. 5.

e 12. 6.

f 18. 6.

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

Make e as AB to CD, so EF to PR, and upon PR describe f the rectilineal figure SR similar and similarly situated to either

g 9. 5.

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

PROP. XXIII. THEOR.

See N.

EQUIANGULAR parallelograms have to one another the ratio 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 is the same with the ratio which is compounded of the ratios of their sides.

Let BC, CG be placed in a straight line; therefore DC and Book VI. CE are also in a straight line; and complete the parallelogram DG; and, taking any straight line K, makeb as BC to CG, a 14. 1. so K to L, and as DC to CE, so make b L to M: therefore b 12. 6, the ratios of K to L, and 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 c A.def.5: ratios of K to L, and L to M: wherefore also K has to M the

A

B

D H

G d 1. 6.

C

e 11. 5.

ratio compounded of the ratios of the sides; and because as BC to CG, so is the parallelogram AC to the parallelogram CHd; but as BC to CG, so is K to L; therefore K ise 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 e 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 f, K is to M, as the parallelogram AC to the f 22. 5. parallelogram CF: but K has to M the ratio which is compounded 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.

KLM E F

PROP. XXIV. THEOR.

THE parallelograms about the diameter of any See N. parallelogram, are similar to the whole, and to one another.

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

Because DC, GF are parallels, the angle ADC is equal a to a 29. 1. the angle AGF: for the same reason, because BC, EF are pa

b 34. 1.

c 4. 6.

A

Book VI. rallels, 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 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 b, AB is d 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, AEFG about the equal angles are proportion- D als; and they are therefore similar to

d 7.5.

K

E

B

F

H

C

e1. def. 6. one another e: 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 f; therefore the parallelogram GE is similar to KH. Wherefore, the parallelograms, &c. Q. E. D.

f 21. 6.

PROP. XXV. PROB.

See N.

a Cor. 45.

b

TO describe a rectilineal figure which shall be similar to 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.

a

Upon the straight line BC describe a the parallelogram BE equal to the figure ABC; also upon CE describe the parallelogram CM equal to D, and having the angle FCE equal 29. 1 to the angle CBL therefore BC and CF are in a straight 14. 1. line b, as also LE and EM: between BC and CF find a mean c 13. 6. proportional GH, and upon GH described the rectilineal fi d 18. 6. gure KGH similar and similarly situated to the figure ABC: e 2. Cor. and because BC is GH as GH to CF, and if three straight 20. 6. lines be proportionals, as the first is to the third, so is the

figure upon the first to the similar and similarly described figure Book VI. upon the second; therefore as BC to CF, so is the rectilineal figure ABC to KGH: but as BC to CF, so is f the parallelogram f 1. 6. 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 parallelo- g 11. 5.

gram BE; therefore the rectilineal figure KGH is equal to the h 14. 5. parallelogram EF: but EF is equal to the figure D; wherefore also KGH is equal to D; and it is similar to ABC. Therefore the rectilineal figure KGH has been described similar to the Which was to be done. figure ABC, and equal to D.

PROP. XXVI. THEOR.

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

A

G

H

ABCD and

D

Let the parallelograms ABCD, AEFG be similar and similarly situated and have the angle DAB common. AEFG are about the same diameter. For, if not, let, if possible, the parallelogram BD have its diameter AHC in a different straight K line from AF, the diameter of the parallelogram EG, and let GF E meet AHC in H; and through H draw HK parallel to AD or BC: therefore the parallelograms ABCD, AKHG being about the

A

B

F

C

same diameter, they are similar to one another, a: wherefore, as a 24. 6. DA to AB, so is b GA to AK: but because ABCD and AEFG

are similar parallelograms, as DA is to AB, so is GA to AE; b 1.def.6.

Book VI. therefore as GA to AE, so GA to AK; wherefore GA has the same ratio to each of the straight lines AE, AK; and consec 11. 5. quently 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.

d 9. 5.

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.

E C

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 parallelogram DC; and DC is therefore called the de'fect of AE.

Α

D B

F

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 there'fore AG exceeds AC the parallelogram described upon AB in the same angle, and between the same parallels, by the paral'lelograin BG.'

PROP. XXVII. THEOR.

See N.

OF all parallelograms applied to the same straight line, and deficient by parallelograms, similar and similarly situated to that which is described upon the half of the line; that which is applied to the half, and is similar to its defect, is the greatest.

Let AB be a straight line divided into two equal parts in C; and let the parallelogram AD be applied to the half AC; which is therefore deficient from the parallelogram upon the whole line AB by the parallelogram CE upon the other half CB: of all the parallelograms applied to any other parts of