-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; or, which is the same thing, that A' B' A B then the four quantities A', A, B, B, are equal to rA+r'A, A, B, 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. But by construction mrtmrzn mrA+mr'AnA: By construction and therefore therefore by hypothesis, also mB'\nB ; so that mB' is both greater and less than nB, which is impossible. A' A It is manifest therefore that cannot be greater than; and in like manner it is shown that B' B 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. ERRATA. Page 6, line 21, for letter, read letters. I p. 19, l. 1, 3, for 8xy and 55, read 3xy and 55x. p. 44, l. 20, for 5a, read 5a3. p. 83, l. 22, 23, for (m −n) and (m±n), read (m— do. 1. 26, for members, read numbers. p. 88, l. 34, for continued, read contained. p. 104, l. 11, 12, for +63 and -b3, read +62 and p. 116, l. 12, for 3x, read 3ax. p. 125, l. 3, dele 2,. p. 132, l. 2, 9, for 4x2 and +2ub, read Cx and p. 135, l. 7, for a3, read a2. p. 137, l. 13, for = a2 read a2 1 -α p. 138, 1. 22, for law that, read law, that is to say, p. 156, 1. 7, for plynomials, read polynomials. p. 161, l. 12, for unknown, read known. p. 168, l. 7, for conditions, read condition. 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. p. 181, l. 5, 16, for formulæ, read formula. du. 1. 21, dele 199. p. 188, I. 17, for 9, read 13. p. 192, 1, 5, 11, for 2x and +-61, read 3x and G.. p. 195, l. 21, for quantities, read equations. p. 201, l. 13, for furnish, read furnishes. p. 215, 1. 5, for Analysts employ, read Analysis p. 216, 1. 2, for formule, read formula. p. 224, 1. 2, for depend, read depends. p. 248, l. 15, 18, for y= and y=3, read y=3 and p. 254, l. 19, for 8, read x. read_8y+5 18 p. 256, 1. 2, 21, for 9 and +100, read 4 and y4 100. p. 257, 1. 23, for other, read others. p. 279, 1. 7, for form, read from. p. 284, l. 12, 27, for plus and 720, read minus and 752. p. 289, l. 13, for 4, read 5. p. 301, l. 11, for 3s. and 2s. read 2s. and 3s. p. 314, l. 5, 9, for an-3b6 and is, read an-363 and - p. 317, 1. 14, for 3/2 ̄x3a=x3, &c., read - p. 323, l. 8, for 16x2y1 and 6y5, read 15x2y1 and p. 325, l. 25, 26, for square of b2 and root c2, read do. 1. 31, for +20x3-15x2+5x-1, read - 3 do. l. 33, 34, for a 4a3x, read a-4a3x, p. 329, l. 3, for square of, read square root of. a+a3, a+ p. 349, l. 5, 6, for a* and a3 +a3, read a3 and a 3 do. 1. 18, 30, for 6/243 and 10/9, read 243 |