2 AC CB, or 8252+32+2 × 5 × 3 = 25 +9 +30 =64, and 82 = = 64. The propositions of the fifth book may also be illustrated by numbers. If A 12, B4, C 18, D6, then A:B-C: D or 12:418:6. Also A + B:B=C+D: D or 12+ 4:4186: 6 or 16:4=24: 6. II. The following are examples of products and quotients, and simple cases of fractions : If m 3, and n = : 5, then, 1. mn = 3 × 5 = 15. 2. mn2 = 3 × 52 = 3 × 25 = 75. =mn, for mn multiplied by n, and di p = If also 5. mnpq 5 3X5 3 m = 52 5X5 5 n' 2, and q = 4, then, = 3× 5 × 2 × 4 = 120; or mpnq = 3×2× 5 X 4120. The rules for numerical vulgar fractions apply to the simple algebraical fractions treated of here. The three following articles are propositions respecting common fractions : = a III. Let =m; a, b, and m, being integers, then 1 m a 8 b a If a = 8, and b = 2, then m =7=3=4; and2 = 2 a ma a ac V. Let a, b, c, and d, be integers, then=" If a 6, 63, c = 8, and d = 5, then = 3 ; or b d VI. The propositions in the last three articles are also true when a, b, c, d, and m, are fractional terminate numbers. bers e, f, g, h, k, and l, being integers; then therefore In a similar manner, the propositions in Art. III. and V. may be proved. every 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 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'a, a" <a', and a" >a; where a" is the intermediate terminate number. When a contradictory conclusion is arrived at on the hypothesis that a'a, it is of course false, and it may in each case be similarly proved that the hypothesis of a'<a is also false; and hence if the first hypothesis be proved false in any case, it may be concluded that a' musta. These remarks will prevent unnecessary repetitions. VII. Let a, b, and m, be one or all of them interminate, a and % b 1 =m, then 7= 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. 2. When b, and therefore m, is interminate. The proof is similar to the last. 3. When a, b, and m, are interminate. -= ; b" 1 and also let =- Then (1st case) b 1 Let a a b" =n; but α n therefore nm, for b” — b. But b′ 1 1 b", and therefore 7 or nm, and it was also m n VIII. When a, b, and m, are, one or all, interminate, mb a" b mb (VI.); but mb a and hence a" <a; but a" a ; there therefore |