the common algebraic definition, they will also be proportionals according to Euclid's definition. Let the four DEMONSTRATION. rA, A, rB, B, be the proportionals according to our 5th definition; that is, according to the common algebraic definition; it is to be proved that the same four rA, A, rB, B, are proportionals by Euclid's fifth def. of the fifth book. Let m and n be any two integers, each greater than unity, so that mrA, mrB, are any equimultiples whatever of the first and third; and nA, nB are any whatever of the second and fourth; and the four multiples are therefore mrA, nA, mrB, nB; Now the thing to be proved is, that according as the multiple mrA is greater than, equal to, or less than nA; the multiple mrB will also be greater than, equal to, or less than nB. then by division. First let mrA nA, and by multiplication Secondly, if then and therefore mrn, mrBnB. mrA=nA, mrang mrBnB. If four magnitudes be proportionals by Euclid's fifth definition, they will also be proportionals by the common algebraic definition. DEMONSTRATION. Let A', A, B', B, be any four magnitudes, such that m, n, being any integers greater than unity, and the equimultiples, mA', mB', being taken, and likewise the equimultiples nA, nB; making the four multiples mA', nA, mB', nB; the hypothesis is, that if mA' be greater than nA, mB is also greater than nB; if equal, equal; and if less, less and it is to be proved that A': A:: B': B; then the four quantities A', A, B', B, are equal to ́rA+r'A, A, rB, B. Now, let m be such an integer greater than unity, that mr and mr' may be each greater than 2; and take n the next integer greater than mr, of course n will be less than mr+mr' ;) and the four multiples mA', nA, mB', nB, become mrA+mr'A, nA, mrB', nB. By 'construction and therefore mrtmrong mrA+mr'AnA: But by construction and therefore mr A<nA therefore by hypothesis, also mBnB ; so that mB' is both greater and less than nB, which is impossible. A' A It is manifest therefore that cannot be greater B; and in like manner it is shown that B' than A' -i A not be greater than and therefore that is, A': A: B': B. SCHOLIUM. Thus we have shown, that if four quantities be proportionals by the common algebraic definition, they will also be proportionals according to Euclid's definition; and conversely, that if four quantities be proportionals by Euclid's definition, they will also be proportionals by the common algebraic definition; and by a similar method of reasoning we may easily show, that when four quantities are not proportionals by one of these two definitions, they cannot be proportionals by the other definition. Thus it appears, that the two definitions are altogether equivalent; each comprehending, or excluding, whatever is comprehended, or excluded, by the other. THE END. Page 6, line 21, for letter, read letters. p. 19, l. 1, 3, for 8xy and 552, read 3xy and 55x2. p. 33, l. 12, for to, read to the addition of. p. 44, l. 20, for 5a, read 5a3. p. 83, 1. 22, 23, for (m −n) and (m±n), read (m n)c and (min)c. do. 1. 26, for members, read numbers. p. 88, 1. 34, for continued, read contained. p. 96, l. 15, for ac2, read ac2. p. 104, l. 11, 12, for +63 and -b2, read +b2 and +b3 p. 107, l. 4, for -x read -x2. p. 116, l. 12, for 3x, read 3ar. p. 125, 1. 3, dele 2,. p. 132, l. 2, 9, for 4x2 and +2ab, read 6x2 and +2bx. p. 135, l. 7, for a3, read a2. p. 137, l. 13, for a2 1 -a a2 1 p. 138, l. 22, for law that, read law, that is to say. at the general formula, that. p. 156, l. 7, for plynomials, read polynomials. p. 159, l. 16, for (a+x2), read (a+x)2. p. 161, l. 12, for unknown, read known. p. 168, l. 7, for conditions, read condition. p. 173, l. 4, for x=2, read x=4. p. 176, l. 14, for 30, read 20. p. 179, l. 7, for 198, read 199. p. 180, 1. 8, for quantities, read quantities, values. do. 1. 29, for formulæ, read formula. p. 181, l. 5, 16, for formulæ, read formula. du. 1. 21, dele 199. p. 188, l. 17, for 9, read 13. p. 192, 1, 5, 11, for 2x and +6a, read 3x and 6.0. |