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 l 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 T, 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, that 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). 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 ABS:T C: DT: V::h: k 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 would have been sufficient to state that similar triangles are those which have two of their angles equal,' because it is Дл 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. B 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, I, and take any number of straight lines BG, GH, each equal to the base BC; and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, AL. DEMONSTRATION. Then because CB, BG, GH are all equal, the triangles ABC, base HC is of the base BC, the same multiple is the triple the AGB, AHG are all equal (a); therefore whatever АНС 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); E A F and if the base HC be greater than the base CL, likewise the triangle AHC is greater than the triangle ALC; and if less, less therefore, since there are four magnitudes, viz. the two bases BC, CD, and the two triangles ABC, ACD; and of the base BC, and the triangle ABC, the first and third, any equimultiples whatever have been taken, viz. the base HC, and the triangle AHC; and also of the base CD and the triangle ACD, the second and fourth, any equimultiples H G C D K (b) V. Def. 5. (c) I. 41. (d) V. 15. V. 11. whatever have been taken, viz. the base CL, and the triangle ALC; and since it has been shown, that if the base HC be greater than the base CL, the triangle AHC is greater than the triangle ALC; and if equal, equal; and if less, less; therefore, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD (6). And because the parallelogram CE is double of the triangle ABC (c), and the parallelogram CF double of the triangle ACD (c), and that magnitudes have the same ratio which their equimultiples have (d); as the triangle ABC is to the triangle ACD, so is the parallelogram CE to the parallelogram CF: and because it has been shown, that, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD; and as the triangle ABC is to the triangle ACD, so is the parallelogram CE to the parallelogram CF; therefore, as the base BC is to the base CD, so is the parallelogram CE to the parallelogram CF (e). COROLLARY 1. From this it is evident, that triangles and parallelograms which have equal altitudes, are to one another as their bases. For, let the figures be so placed as to have their bases in the same straight line, and draw perpendiculars from the vertices of the triangles to the bases, then, because the perpen- (a) I. 28. diculars are both equal and parallel to one another (a), (b) 1. 33. the straight line which joins the vertices is parallel to that in which their bases are (6). Then, if the same construction be made as in the proposition, the demonstration will be identical. COROLLARY 2. THEOREM. Triangles (ABC, DBC) and parallelograms which have equal bases, are to one another as their altitudes. |