Thus, if there be two ranks of magnitudes, we infer by the term “ex æquali” that DEFINITION XX. "Ex æquali in proportione perturbatâ seu inordinatâ,” from equality in perturbate, or disorderly proportion. This term is used when the first magnitude is to the second of the first rank as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank; and so on in a cross order: and the inference is in the 18th definition. It is demonstrated in 23 Prop., Book v. Thus, if there be two ranks of magnitudes, A, B, C, D, E, F, the first rank, and L, M, N, O, P, Q, the second, such that A: B :: P: Q, B: C:: 0 : P, C: DN: 0, D: E :: M: N, E: F :: L: M; the term "ex æquali in proportione perturbatâ seu inordinatâ ” infers that PROP. XX. THEO. If there be three magnitudes, and other three, which, taken two and two, have the same ratio; then, if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less. PROP. XXI. THEO. If there be three magnitudes, and other three which have the same ratio, taken two and two, but in a cross order; then if the first magnitude be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less. PROP. XXII. THEO. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last of the same. N.B. This is usually cited by the words "ex æquali," or "ex æquo." Let these magnitudes, as well as any equimultiples whatever of the antecedents and consequents of the ratios, stand as follows:— |