A SYLLABUS OF PLANE GEOMETRY. 39 DEF. 2. One magnitude is said to be a measure or part of another magnitude when the former is contained an exact number of times in the latter. The following property of multiples is axiomatic :— 1. As A> = or<B, so is mA>= or<mB (Euc. Ax. 1 & 3). 2. = As mA>= 3. mA+mB+ = m (A+B+ ...) (Euc. v. 1.) mA - mB = m (A – B) (A being greater than B) (Euc. v. 5.) (m + n) A mA+nA 4. 5. 6. 7. m.nA mΑ-nA (Euc. v. 2.) (m − n) A (m being greater than n) (Euc. v. 6.) mn.A = nm.A = n.mA (Euc. v. 3) DEF. 3. The ratio of one magnitude to another of the same kind is the relation of the former to the latter in respect of quantuplicity. The ratio of A to B is denoted thus A : B, and A is OBS. This inter-distribution of multiples is definite for two DEF. 4. The ratio of two magnitudes is said to be equal to that of two other magnitudes (whether of the same or of a different kind from the former), when any equimultiples whatever of the antecedents of the ratios being taken and likewise any equimultiples whatever of the consequents, the multiple of one 40 A SYLLABUS OF antecedent is greater than, equal to, or less than that of its consequent, according as that of the other antecedent is greater than, equal to, or less than that of its consequent. Or in other words: The ratio of A to B is equal to that of P to Q, when mA is greater than, equal to, or less than nB, according as mP is greater than, equal to or less than nQ, whatever whole numbers m and n may be. It is an immediate consequence that : The ratio of A to B is equal to that of P to Q; when, m being any number whatever, and n another number determined so that either mA is between nB and (n + 1)B or equal to nB, according as mA is between B and (n + 1)B or is equal to »B, so is mp between nQ and (n + 1)Q or equal to nQ. The definition may also be expressed thus: The ratio of A to B is equal to that of P to Q when the multiples of A are distributed among those of B in the same manner as the multiples of P are among those of Q. DEF. 5. The ratio of two magnitudes is greater than that of two other magnitudes, when equimultiples of the antecedents and equimultiples of the consequents can be found such that, while the multiple of the antecedent of the first is greater than or equal to that of its consequent, the multiple of the antecedent of the other is not greater or is less than that of its consequent. Or in other words: The ratio of A to B is greater than that of P to Q, when whole = nB, mP is DEF. 6. When the ratio of A to B is equal to that of P to Q, the four magnitudes are said to be proportionals or to form a proportion. The proportion is denoted thus: A:BP: Q which is read, "A is to B as P is to Q." A and Q are called the extremes, B and P the means, and Q is said to be the fourth proportional to A, B and P. The antecedents A, P are said to be homologous, and so are the consequents B, Q. DEF. 7. Three magnitudes (A, B, C) of the same kind are said to be proportionals, when the ratio of the first to the second is equal to that of the second to the third that is when A: B :: B : C. In this case C is said to be the third proportional to A and B, and B the mean proportional between A and C. DEF. 8. The ratio of any magnitude to an equal magnitude is said to be a ratio of equality. If A be greater than B, the ratio A: B is said to be a ratio of greater inequality, and the ratio B: A a ratio of less inequality. Also the ratios A: B and B : A are said to be reciprocal to one another. THEOR. 1. Ratios that are equal to the same ratio are equal to one another. [Let A B :: P: Q and X: Y :: P: Q, then A : B :: X: Y. For the multiples of A being distributed among those of B as the multiples of P among those of Q, and the same being true of the multiples of X and Y, the multiples of A are distributed among those of B as the multiples of X among those of Y.] THEOR. 2. If two ratios are equal, as the antecedent of the first is greater than, equal to, or less than its consequent, so is the antecedent of the other greater than, equal to, or less than its consequent. 42 A SYLLABUS OF [Let A : B :: P: Q, then as A > = or <B, so is P > = or<Q. This is contained in Def. 4, if the multiples taken be the magnitudes themselves.] THEOR. 3. If two ratios are equal, their reciprocal ratios are For, since the multiples of A are distributed among those of B as the multiples of P among those of Q, the multiples of B are distributed among those of A as the multiples of Q among those of P.] THEOR. 4. If the ratios of each of two magnitudes to a third magnitude be taken, the first ratio will be greater than, equal to, or less than the other as the first magnitude is greater than, equal to, or less than the other: and if the ratios of one magnitude to each of two others be taken, the first ratio will be greater than, equal to, or less than the other as the first of the two magnitudes is less than, equal to, or greater than the other. [Let A, B, C be three magnitudes of the same kind, then A: C> or <B : C, as A> = = or <B If A=B, it follows directly from Def. 4 that A: C :: B: C and If A>B, m can be found such that mB is less than mA by a Hence if mA be between nС and (n+1)C, or if mAnC, mB will be less than C, whence (Def. 6) A : C>B:C; Also, since nC>mB_while_nC is not>mA (Def. 6) C: B > C: A or C: A<C: B. If A <B, then B>A and therefore B : C>A : C, that is A : C <B: C, and so also C: A > C: B. Hence the proposition is proved.] COR. The converses of both parts of the proposition are true, since the "Rule of Conversion" is applicable. THEOR. 5. The ratio of equimultiples of two magnitudes is equal to that of the magnitudes themselves. [Let A, B be two magnitudes, then mA : mB :: A: B. For as pA>=or<qB, so is m.pA>=or<m.qB; but m.pA= p.mA and m.qB=q.mB, therefore as pA>=or<qB, so is p.mA > or <q.mB, whatever be the values of p and q, and hence mA: mB:: A: B.] THEOR. 6. If two magnitudes (A, B) have the same ratio as two whole numbers (m, n), then nA = mB: and conversely if nA = mB, A has to B the same ratio as m to n. [Of A and m take the equimultiples nA and n.m, and of B and ʼn take the equimultiples mB and m.n, then since n.m=m.n, it follows (Def. 4) that nAmB. Again since by Def. 4 mB : nB :: m : n we have, if nA = mB, nA: nB :: mn; whence it follows (Theor. 5) that A: B :: m: n.] COR. If A: B:: P: Q and nA = mB, then nP = mQ; whence if A be a multiple, part, or multiple of a part of B, P is the same multiple, part, or multiple of a part of Q. THEOR. 7. If four magnitudes of the same kind be proportionals, the first will be greater than, equal to, or less than the third, according as the second is greater than, equal to, or less than the fourth. Then if A=C, A: B :: C: B, and therefore C: D :: C: B, whence B D. Also if A> C, A: B>C: B, and therefore C: D>C: B, whence BD. Again if A<C, A: B<C: B, and therefore C: D<C: B, whence B<D.] THEOR. 8. If four magnitudes of the same kind be proportionals, the first will have to the third the same ratio as the second to the fourth. |