(Dem.) Since A: B:: E: F, and B: C:: D: E. and multiplying equals by equals, X B с E cancelling the common multipliers from numerator and denominator on both sides (Arith. Prin. 5), therefore A: C:: D:F. Next, let there be given four magnitudes, A, B, C, and D, and other four, E, F, G, and H, which, taken two and two in a cross order, have the same ratio; namely, A: B:: G: H, B: C::F:G, and C:D :: E:F; to prove that A:D:: E: H. (Dem.) From the above given proportions, the numerator and denominator of both sides (Arith. Prin. 5), In the same manner the demonstration may be extended to any number of such magnitudes. PROPOSITION XXIV. THEOREM. (EUCLID XXIV.) If the first has to the second the same ratio which the third has to the fourth; and the fifth to the second the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth. Let the six given magnitudes, A, B, C, D, E, and F, be such that A: B:: C: D, and E: B:: F: D; it is required to prove that A+E:B:: C+ F: D. PROPOSITION XXV. THEOREM. (EUCLID E.) Ratios which are compounded of the same or equal ratios are equal to one another. Given the ratios of A to B, and of B to C, which compound the ratio of A to C, equal, each to each, to the ratios of D to E, and E to F, which compound the ratio of D to F; to prove that A:C :: D: F. (Dem.) For, first, if the ratio of A to B be equal to that of D to E, and the ratio of B to C equal to that of E to F, then (V. 22), A: C:: D: F. and And next, if the ratio of A to B be equal to that of E to F, the ratio of B to C equal to that of D to E, then (V. 23), A:C:: D:F. In the same manner the proposition may be demonstrated whatever be the number of ratios. TI PROPOSITION XXVI. THEOREM. (Euclid g.) The terms of a proportion are also proportional by mixing. Given that A: B:: C: D; to prove that C+D:C- D. A+B:AB:: being equal to unity, then if AB, adding the second equals to (Arith. Prin. 5), and therefore A+B:AB::C+D: C-D. In the same manner, if AB it can be demonstrated that A+B:BA::C+D:D – C. If four magnitudes be proportional, the sum of the greatest and least is greater than the sum of the other two. Given that A: B:: C: D, and that A is the greatest term; to prove that A+DB + C. (Dem.) Since A is the greatest, it is greater than either B or C, but (V. 14) when AC, BD, and, by alternation (V. 16), A:C:: B:D, therefore (V. 14) since AB, CD, hence but (V. 20) A: A-B::C: CD, and since C D (V. 14), add BD to both, B+C. SIXTH BOOK. DEFINITIONS.. 1. Straight lines are said to be similarly divided when their corresponding segments are proportional. 2. A line is said to be cut in a given ratio when its segments> have that ratio. 3. Straight lines that meet in the same point are called convergent or divergent lines, according as they are considered to verge towards or from the point of concourse. 4. A straight line is said to be cut in extreme and mean ratio,, when the whole is to the greater segment, as the greater segment is to the less. 5. A straight line is said to be harmonically divided, when it is divided into three segments such, that the whole line is to one extreme segment, as the other extreme segment to the middle segment. 6. Three straight lines are said to be in harmonical progression or harmonical progressionals, when the first is to the third, as the difference between the first and second is to the difference between the second and third. 7. The second of three lines in this progression is called a harmonic mean between the other two; and the third is called a third harmonical progressional to the other two. 8. Four divergent lines that cut any line harmonically are called harmonicals. 9. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals. 10. Two sides of one figure are said to be reciprocally proportional to two sides of another, when one of the sides of the first is to one of the sides of the other, as the remaining side of the other is to the remaining side of the first. 11. A lune is a figure contained by two arcs of different circles. Triangles and parallelograms of the same altitude are one to another as their bases. Given the triangles ABC, ACD, and the parallelograms EC, CF, having the same altitude; namely, the perpendicular drawn from the point A to BD; to prove that as the base BC is to the base CD, so is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF. (Const.) Produce BD both ways to the points I and L, and take any number of straight lines BG, GH, HI, each equal to the base BC, and DK, KL, any number of them, each equal to the base CD; and join AI, AH, AG, AK, and AL. (Dem.) Then, because CB, BG, GH, and HI are all equal, the triangles AIH, AHG, AGB, ABC are all equal (I. 38); therefore, whatever multiple the base IC is of the base BC, the same multiple is the triangle AIC of the triFor the same angle ABC. reason, whatever multiple the base LC is of the base CD, ALC of the triangle ADC. the same multiple is the triangle And if the base IC be equal to the base CL, the triangle AIC is also equal to the triangle ALC; and if the base IC be greater than the base CL, the triangle AIC is likewise greater than the triangle ALC; and if less, less. Therefore, since there are four magnitudes; namely, 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; namely, the base IC and triangle AIC; and of the base CD and triangle ACD, the second and fourth, there have been taken any equimultiples whatever; namely, the base CL and triangle ALC; and since it has been shewn that if the base IC be greater than the base CL, the triangle AIC is greater than the triangle ALC; and if equal, equal; and if less, less. Therefore (V. Def. 10), as the base BC is to the base CD, so is the triangle ABC to the triangle ACD. Again, because the parallelogram CE is double of the triangle ABC (I. 41), and the parallelogram CF double of the triangle and that magnitudes have the same ratio which their ACD, |