Euclid's Elements of geometry, the first three books (the fourth, fifth, and sixth books) tr. from the Lat. To which is added, A compendium of algebra (A compendium of trigonometry).

therefore, since the angles GIF and EDA are equal, but GIK and EDC are equal to the whole (by Hypoth. and Def. 1, B. 6), the remainders FIK and ADC will be equal; and since FI is to IG, as AD to DE, and IG to IK, as DE to DC (by Hypoth. and Def. 1, B. 6), ex æquali, FI will be to IK, as AD to DC (by Prop. 33, B. 5); and therefore, because the angles contained by them are equal, the triangle FIK shall be similar to the triangle ADC (by Prop. 6, B. 6), and that the remaining triangles are also equal, can be similarly proved.

PART 2.-Because the triangle FGI is similar to the triangle AED, FGI will be to AED in the duplicate ratio of the sides FI to AD (by Prop. 19, B. 6); also FIK is similarly to ADC in the duplicate ratio of FI to AD; therefore FGI is to AED, as FIK is to ADC (by Prop. 18, B. 5); and thus it can be also shown that FIK is to ADC, as FKL to ACB; therefore as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents (by Prop. 22, B. 5), or the polygon EGIKL, to the polygon AEDCB.

PART 3.-Because the polygon FGIKL is to the polygon AEDCB, as the triangle FGI is to the triangle AED, and FGI is to AED, in the duplicate ratio of the side FG to the side AE (by Prop. 19, B. 5), FGIKL will be also to AEDCB, in the duplicate ratio of FG to AE (by Prop. 18, B. 5).

COR. 1.-Hence if three right lines be proportional, a rectilineal figure upon the first, will be to a similar, and similarly situated figure upon the second, as the first to the third.

COR. 2.-Hence a rectilineal figure can be described, which is to a given one in any given ratio; for find a mean proportional between the side of the figure and the right line which is to it, in the given ratio, and upon it describe a figure similar to the given one, and similarly situated.

Rectilineal figures (A and B) which are similar to the same, (C) are also similar to one another.

For since the rectilineal figure A is similar to the rectilineal figure C, it is equiangular to it, and has the sides about the equal angles proportional (by Def. 1, B. 6), and since the rectilineal figure B, is similar to C, it is equiangular to it, and has the sides about the equal angles proportional (by Def. 1, B. 6); therefore A and B are also equiangular to each other (by Ax. 1, B. 1), and have the sides about the equal angles proportional (by Prop. 18, B. 5), therefore A and B are similar.

PROPOSITION XXII. THEOREM.

If four right lines be proportional (AB to CD, as EF to GH), the similar rectilineal figures, and similarly described on them shall be also proportional.

And if similar rectilineal figures, similarly described on four right lines, are proportional, those right lines shall be propor tional.

PART 1.-Assume a third proportional X to AB and CD, and a third proportional O, to EF and GH; because AB is to CD, as EF to GH (by Hypoth.), CD will be to X, as GH to O (by Prop. 18, B. 5); and therefore ex æquali, AB will be to X, as EF to O (by Prop. 33, B. 5); but AKB is to CLD, as AB to X (by Cor. 1, Prop. 20, B. 6); and EM to GN, as EF to O; therefore, AKB is to CLD, as EM to GN.

E F

G H

PART 2.-Letting the same construction remain; since AKB is to CLD, as EM to GN (by Hypoth.),

AB will be to X, as EF to O (by Prop. 18, B. 5) ; therefore AB is to CD as EF to GH (by Prop. 39, B.5).

Equiangular parallelograms (BD and CG) have to each other the ratio compounded of the ratios of their sides.

Place the two sides BC and CF about A the equal angles in directum; and since the angles BCD and DCF are equal to two right angles (by Prop. 13, B. 1), B but HCF is equal to BCD (by Hypoth.), DCF and FCH will be also equal to two right angles, and therefore DC and CH are in directum (by Prop. 14, B. 1), therefore complete the parallelogram CE.

Therefore, because BD is to CE, as BC to CF (by Prop. 1, B. 6), and CE to CG, as DC to CH (by same), DB has to CG a ratio compounded of the ratios of BC to CF, and DC to CH (by Def. 12, B. 5).

COR. 1.-Two right lines can be found, which are to one another in the ratio of the parallelograms BD and CG; by assuming any right line I, and by finding a fourth proportional K, to BC, CF, and I; and a fourth proportional L, to DC, CH, and K, and BD will be to CE, as I to K; and CE to CG, as K to L; and therefore ex æquali, BD will be to CG, as I to L (by Prop. 33, B. 5).

COR. 2.-Triangles which have one angle of the one equal to one angle of the other, are to each other in a ratio compounded of the ratios of the sides about the equal angles.

COR. 3.-Any parallelograms or triangles are to one another in a ratio compounded of the ratios of their bases and altitudes; for they are equal to rectangles, or to right-angled triangles, upon equal bases, and of the same altitude.

In every parallelogram (AC) the parallelograms which are about the diagonal (AF and FC) are both similar to the whole and to each other.

B K

For since the parallelograms AC and AF have a common angle A, they are equiangular (by Cor. 2, Prop. 34, B. 1); but on account of the parallels EF and E BC, the triangles AEF and ABC are similar (by Cor. 1, Prop. 4, B. 6), and A therefore AE is to EF, as AB to BC (by Def. 1, B. 6); but the remaining sides are equal to AE, EF, AB, and BC (by Prop. 34, B. 1); therefore the parallelograms AF and AC have the sides about the equal angles proportional, and therefore are similar (by Def. 1, B. 6). It can be similarly demonstrated that AC and FC are similar. Therefore, since each of the figures AF and FC is similar to the same AC, they will be similar to each other (by Prop. 21, B. 6).

To construct a rectilineal figure, equal to a given one (A), and similar to another (B).

Upon any side EF of the given figure B construct a rectangle EL, equal to B (by Cor. 1, Prop. 45, B. 1), and on its side FL construct a rectangle FD, equal to

the given one A (by Cor. 1, Prop. 45, B. 1); find a mean proportional CK between the other sides EF and FG (by Prop. 13, B. 6), then the figure described upon it, similar to the given one B, and similarly situated, will be equal to the given figure A. For the rectangle EL is to the rectangle FD, as EF to FG (by Prop. 1, B. 6);

or in the duplicate ratio of EF to CK (by Const.); and therefore, as the rectilineal figure B to the similar and similarly described one upon CK (by Prop. 20, B. 6); but EL is equal to B (by Const.), therefore the rectilineal figure described upon CK, similar to B, and similarly situated, will be equal to FD, and therefore to the given figure A (by Prop. 23, B. 5).

COR.-Hence appears a method of constructing a rectilineal figure, similar to a given one, and equal to the sum or difference of two others; if we first construct a parallelogram equal to that sum or difference (by Cors. 2 and 3, Prop. 45, B. 1).

If similar and similarly situated parallelograms (AC and AF), have a common angle, they are about the same diagonal.

For if not, but if possible, let AIF be the diagonal of the parallelogram AF, and through I draw IL parallel to AE.

Therefore since the parallelograms AI and AF, are about the same diagonal AIF, and have the angle A common to both, AI and AF shall be similar (by Prop. 24, B. 6), therefore BA is to AL, as EA to AG; but BA is to AD, as EA to AG (by Hypoth.); therefore BA is to AL, as BA to AD (by Prop. 18, B. 5); and therefore AL is equal to AD (by Prop. 15, B. 5) which is absurd. Therefore AIF is not the diagonal of AF, and it can be similarly shown that no other line is a diagonal except ACF.

If any right line (AB) be cut equally (in C) and unequally (in D), the parallelogram (FC) which is applied to the half deficient by a figure (GB) similar to itself, will be greater than the parallelogram (ED), applied to either of the other parts, whose deficiency (KB) is similar to the first (GB).

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Euclid's Elements of geometry, the first three books (the fourth, fifth, and ...