of Proof of 12 are 8, and of 8 are 6 of of 12; of 12 are 10, and the half of 10 is 5= 12: but 5 + 6 = 11. Therefore, &c. PROBLEM XV. What number is that of of which of of it are equal to 19? 19x But equal 19 by the problem. Therefore, 19x= 120 19 x 120; and equal 120. Proof: of of 45; but 64 45 19. = are 64, and of of 120 is Therefore, &c. PROBLEM XVI. What number is that of which of multiplied by of of it will be equal to 6? Therefore Then by the conditions of the problem, 144; and consequently x = 6. 24 12. Proof: of but 6 x 16. of 12 are 6; and of of 12 is 1; PROBLEM XVII. What number is that of which 3 + 2 are equal to I? Proof of 43; and 2 of 4=; but + or 1. Therefore, &c. PROBLEM XVIII. What number is that the,, and of which make 12? Let a be the required number. Proof of 11, is 57; of 11 is 3; and of 11 is 213; but 57 +31 + 219 = 12. PROBLEM XIX. The triple, the half, and the fourth of a certain number, are equal to 104: What is the number? Let r be the number required. We shall then have, by the conditions of the problem: If and of the hull of a ship be immersed in the sea, and only 4 feet of it above the surface of the water: What is the depth of the vessel? And x 48 feet, the depth of the vessel. A banker at his death, being desirous to reward 10 of his clerks, gave orders in his will, that 5500 guineas should be divided among them, in such a manner, that the first 5 should have each an equal share of the whole legacy; that the next 3 men should have shared among them one-half of what was bequeathed to the first_5; and that the 2 last should have divided between them one-third of that sum: What was the share of each? Let r be the share of each of the first five clerks, and a the 5500 guineas. Then, by the conditions of the problem, the share of the first five will be 5r; that of the next three r; and that of the two last r. But as these three quantities are equal to a, or the whole, we have the following equation: 5x + 3x + 3x = a. By multip. and reduct. 55x — 6a By division ба x= =600 guineas. 55 Each of the first five then had ....600 Each of the next three And each of the two last. ...500 Proof: 5 x 600 = 3000 3 × 500 1500 2 × 500 = 1000 5500 The application which we have here made of analysis to the solution of a few problems, evidently shews that this method, by its precision, brevity, and extent, is far superior to arithmetic. The latter confines our attention to determinate quantities, and, if we may use the expression, enchains it by the slowness of its progress; while the other, more rapid, enables us to pass over the intermediate operations, and to direct our attention to the real point of difficulty. The chief advantages, therefore, derived from this science are, that it facilitates the discovery and comprehension of mathematical truths, and that it supplies us with easy methods, and general rules, for resolving all problems that may be proposed respecting quantities. When we have obtained a result by the rules of arithmetic, there is nothing indeed that exhibits to the mind the chain of operations which conducted to it. When, after a few arithmetical operations, we have obtained 12 for result, we see nothing in 12 which can indicate whether this number has arisen from the multiplication of 3 by 4, of 2 by 6, or by the addition of 5 to 7, or of 2 to 10; or, in general, from the combination of any other operations. Arithmetic gives rules for finding certain results, but these results of themselves can furnish no rules. Algebra, however, or that mode of calculation which employs indeterminate characters, preserves, as we may say, the traces of all the intermediate operations, which conduct to the last result. |