THEOREM.-If there be any number of ratios, and any number of other ratios such that the ratio compounded of ratios which are the same with the first ratios, each to each, is the same with the ratio compounded of ratios which are the same, each to each, with the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the same with several of the first ratios, each to each, be the same with one of the last ratios, or with the ratio compounded of ratios which are the same, each to each, with several of the last ratios: then the ratio compounded of ratios which are the same with the remaining ratios of the first, each to each, or the remaining ratio of the first, if but one remain; is the same with the ratio compounded of ratios which are the same with those remaining of the last, each to each, or with the remaining ratio of the last.

DEMONSTRATION. Let the ratios of A to B, C to D, E to F be the first ratios; and the ratios of G to H, K to L, M to N, O to P, Q to R, be the other ratios: and let A be to B, as S is to T; and C to D, as T is to V; and E to F, as V is to X; therefore, by the definition of compound ratio, the ratio S to X is compounded of

A, B; G, H; K, L; e, f, g.

h, k, l. C, D; E, F;

S, T, V, X.
M, N, O, P, Q, R. Y, Z, a, b, c, d,
m, n, o, p.

the ratios of S to T, T to V, and V to X, which are the same with the ratios of A to B, C to D, E to F, each to each; also, let G be to H, as Y is to Z; and K to L, as Z is to a; M to N, as a is to b; O to P, as b is to c; and Q to R, as c is to d: therefore, by the same definition, the ratio of Y to d is compounded of the ratios of Y to Z, Z to a, a to b, b to c, and c to d, which are the same,

each to each, with the ratios of G to H, K to L, M to N, O to P, and Q to R: therefore, by the hypothesis, S is to X, as Y is to D: also, let the ratio of A to B, that is, the ratio of S to T, which is one of the first ratios, be the same with the ratio of e to g, which is compounded of the ratios of e to f, and f to g, which, by the hypothesis, are the same with the ratios of G to H, and K to L, two of the other ratios; and let the other ratio of h to 1 be that which is compounded of the ratios of h to k, and k to 1, which are the same with the remaining first ratios, viz. of C to D, and E to F; also, let the ratio of m to p be that which is compounded of the ratios of m to n, n to o, and o to p, which are the same, each to each, with the remaining other ratios, viz. of M to N, O to P, and Q to R: then the ratio of h to 1 is the same with the ratio of m to p; that is, h is to l, as m is to p.

Because e is to f, as (G is to H, that is, as) Y is to Z; and f is to g, as (K is to L, that is, as) Z is to a; therefore, ex æquali, e is to g, as Y is to a: and by the hypothesis, A is to B, that is, S is to I, as e is to g; wherefore S is to T, as Y is to a; and, by inversion, T is to S, as a is to Y; and S is to X, as Y is to D'; therefore, ex æquali, T is to X, as a is to d: also, because h is to k, as (C is to D, that is, as) T is to V; and k is to 1, as (E is to F, is, as) V is to X; therefore, ex æquali, h is to 1, as T is to X: in like manner it may be demonstrated, that m is to p, as a is to d: and it has been shown, that T is to X, as a is to d; therefore his to 1, as m is to p (a).

that

The propositions G and K are usually, for the sake of brevity, expressed in the same terms with propositions F and H; and therefore it was proper to show the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers.

SCHOLIUM. This proposition may be algebraically expressed:

THEOREM.-If there be a number of ratios A: B, C : D, E : F, and if

ABST

C: DT: V::h: k
E:FV:X::k: l

and also a number of other ratios G : H, K: L, M: N, O: P, Q: R, and if

and if S: X:Y: d;

G:H::Y: Z:: ef
K: L::Za: f:g
M:N: a b ::m: n
0: P: b c ::n: 0
QR c

d :: 0 :p

and A Be: g; then shall
hlm: p.

A

e

THE

ELEMENTS OF EUCLID.

BOOK VI.

DEFINITIONS.

1. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.

SCHOLIUM. In the case of triangles it would have been sufficient to state that similar triangles are those which have two of their angles equal,' because it is

Да

evident from I, 32 B, the third sides must also be equal, and it is shown in the fourth proposition of this book that the sides about the equal angles of equiangular triangles are proportionals. But in the case of rectilineal figures having more than three sides both the conditions expressed above are necessary, because, as in the case of a square and rectangle, the angles are equal, each to each, but the sides about the equal angles are not proportional.

2. Two magnitudes are said to be reciprocally proportional to two others, when one of the first pair is to one of the second, as the remaining one of the second is to the remaining one of the first.

3. A straight line is said to be cut in extreme and mean ratio, when the whole is to one of the segments, as that segment is to the other.

SCHOLIUM. A straight line is said to be divided harmonically, when it is divided into three parts, such that the whole line is to one of the extreme segments, as the other extreme segment is to the middle part. Three lines

are said to be in harmonical proportion, when the first (AB) is to the third (CD), as the difference between the first (AB) and second (BC), is to the difference between the second (BC) and third (CD); and the second (BC) is called a harmonic mean between the first (AB) and third (CD) Four divergent lines (EA, EB, EC, ED) which cut a line (AD) in harmonical proportion, are called harmonicals; and this mode of dividing a line is termed harmonical section, while that described in the third definition is termed medial section.

4. The altitude of any figure is the straight line drawn from its vertex perpendicular to its base.

SCHOLIUM. Any side of a figure may be assumed as its base, and its altitude is the perpendicular distance from such side to the inost remote point in the figure.

A B C

PROPOSITION I.

THEOREM.-Triangles (ABC, ACD) and parallelograms

(EC, CF) which have the same altitude, are to one another as

their bases.

CONSTRUCTION. Produce BD

both ways to the points H, L, and take any number of straight lines BG, GH, cach equal to the base BC; and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, AL.

F

HGB C

D

K

(a) I. 38.

DEMONSTRATION. Then because CB, BG, GH are all equal, the triangles ABC, AGB, AHG are all equal (a); therefore whatever multiple the base HC is of the base BC, the same multiple is the triangle AHC of the triangle ABC: for the same reason, whatever multiple the base CL is of the base CD, the same multiple is the triangle ALC of the triangle ADC: and if the base HC be equal to the base CL, the triangle AHC is also equal to the triangle ALC (a);