Unless the four quantities are of the same kind the alternation, as before observed, cannot take place, because the operation supposes the first to be some multiple, parts or part, of the third. DEFINITION XVI. Dividendo, by division, when there are four proportionals, and it is inferred that the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth. Let a b c d ; by "dividendo" it is inferred a b: b d: d. According to the above, a is supposed to be greater than b, and c greater than d; if this be not the case, but to have b greater than a, and d greater than c, b and d can be made to stand as antecedents, and a and c as consequents, by "invertion" bad: c; then, by "dividendo," we infer b — a : a :: d PROP. XVII. THEO. If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately: that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. DEFINITION XV. The term componendo, by composition, is used when there are four proportionals; and it is inferred that the first together with the second is to the second as the third together with the fourth is to the fourth. Let A B C D; then, by the term "componendo," it is inferred that By "invertion" B and D may become the first and third, A and C the second and fourth, as B: A: D: C, then, by "componendo," we infer that BAA :: D+ C: C. |