: = 3. 2 ACCB, or 89 = 52 + 32 +2 X 5 X 3 = 25 + 9 + 30 = 64, and 82 = 64. The propositions of the fifth book may also be illustrated by numbers. If A = 12, B = 4, C=18, D= 6, then A:B=C:D A -C or 12:45 18:6. Also A +B:B=C+D:D or 12 + 4:4=18+6:6 or 16:4= 24:6. A C 12 18 D 4 6 II. The following are examples of products and quotients, and simple cases of fractions :3, and n = 5, then, 1. mn = 3 X 5 = 15. 2. mn2 =3 X 52 = 3 X 25 = = 75. mn2 3. =mn, for mn multiplied by n, and di 3 X 52 3x5x5 vided by n, is just mn; or — =3X5=15 5 5 If m = mnn n 12 = mn. mn mn n2 nn n 5. mnpq 3X5 3X5 3 4. or n 52 5 X 5 5 If also p = :2, and 9 = 4, then, = 3X 5 X 2 X 4= 120; or mpnq=3X 2X 5 X 4 = 120. mnpa 3X5 X 2X4 6. = mnp, or =3X5 X 2 = 30 = 2 4 mnp. тпpg mping 3X 2X5 X4 7. = mp, or =3 X 2 = ng ng 5X4 52 X 42 ng' 6 20 mnpq naqa mp ng The rules for numerical vulgar fractions apply to the simple algebraical fractions treated of here. The three following articles are propositions respecting common fractions : b III. Let m = m; a, b, and m, being integers, then , 1 a 三4m m a та a ma ma IV. If a, b, and m, be integers, then mbi 6 Let a = 6,6 = 2, and m = 3, then ū= = 3; and 3X6 18 mb = 3X2 6 = 3; or ī mbi V. Let a, b, c, and d, be integers, then 7 a=id , , 6 8 If a= 6,6 = 3,c=8, and d=5, then a ūd=3*5 X 3 1 6x8 48 1 = 2x1 3 and 6 d a с ac . a ac a id VI. The propositions in the last three articles are also true when a, b, c, d, and m, are fractional terminate numbers. а, g k For example, let a= b = and m = 7 the numh T. ma bers e, f, g, h, k, and l, being integers ; then b mbi k 9 kg therefore ī Ih' e f ma a M = ma a 721 Therefore a ke lh h eh mb If kg fg And = ; therefore f g fg ma mbi 2 7 9 As a numerical illustration, let a = 6 3 5 II 9 2 6 9 7 63 ; mb= Х therefore 55 ; 6 55 10 2 5 10 Also mb =ī X 63 =21: 6 3 X7-21 ma 6 mb In a similar manner, the propositions in Art. III. and V. may be proved. The same propositions are true when a, b, c, d, and m, are interminate numbers. Their demonstrations are given in the four following articles. The accented letter in every case denotes an interminate number greater than that denoted by the same letter unaccented, and the same letter doubly accented denotes a terminate number intermediate between the other two. Thus a' z a, a" <a', and a" za; where a' is the intermediate terminate number. When a contradictory conclusion is arrived at on the hypothesis that a' 7 a, it is of course false, and it may in each case be similarly proved that the hypothesis of a' za is also false ; and hence if the first hypothesis be proved false in any case, it be concluded that a' must = a. These remarks will prevent unnecessary repetitions. VII. Let a, b, and m, be one or all of them interminate, 6 1 6 1 The values of and each supposed to be expressed by a decimal fraction, and the proposition asserts the equality of these decimals. 1. When a, and therefore m, is interminate. 3 1 6 1 1 1 Let m (VI.); but m7 for and is = m, then are m a m a 1 =m" a 7> 6 b m" <m'; therefore or a" za. But a" a" (VI.), and m" 7 m, therefore or a' > a, and it 7 6 1 was also shown to be less, therefore 2. When 6, and therefore m, is interminate. 3 1 Then (1st case) ; a m a m a n a q=n; but a 1 1 > ", and therefore or n 7 m, and was also m 6 shown to be less ; hence a m a c a Letī= 7 d COR.-If -If==m, then 7 VIII. When a, b, and m, are, one or all, interminate, mat mbi 1. When a is interminate. ma' a'' ma' ma' ma Then mb' 7 mb (VI.); but mb mb' a" o <t za; therea ma fore 7 mb ma" mb mb a therefore , and hence a" <a; but a” a ma' a' therefore o <ão or a' <a; but a" z a; hence mo' a a ūmo ma (VI.); and m'a < m'a, 7 'm" a m'a and m'b 7 mb, therefore < but these are.equal, ml Let y = mb Then 7 = m'a mbi a a = ĝ, therefore ma for they are each = ū mbi a" ma' Let Then mb mb ma a" therefore < ū and a" Za; but a' Za; therefore a a IN N 7 mb 6. When 6 and m are interminate. ma" (6th case); but 6 mbo mb mb ma' a' and hence a' ca; but a' a; there õi a ma fore 7 mbo IX. Let a, b, c, and d, be one or all interminate, d ac =id a'c a" ac d=bd ī'ā= od . |