Double Position Approximate. 435. We must here inform the student that the above is no longer true when the data of the question require the involution of S, or any part of it, into itself; for, in this case, instead of finding, from the formula, the true result, we only arrive at an approximation. Put Now suppose the numbers S', S", and S to have been involved, so that instead of each number, we had its square. Then, that the proportion may remain, it will be necessary to show that the differences of the squares of the numbers are as the differences of the numbers themselves. Let S' = 8; S" 10, and S=6. Then the above proportion becomes 6:10 6: ee", or 2: 4 :: e': e". Then instead of the numbers, taking their squares, if the proportion still exists, we shall have 64 36: 100-36 :: 2:4; that is, 28: 64 :: 2 : 4, or (403) 112: 128, which is absurd. We shall, however, obtain an approximation. For, let a and aq be the suppositions, both greater than c, the true number. Also put a +q= x; then a — c: x-c::e': e", and 8 ac = ratio of differences. Then taking the square of each supposition and the square of the true number, we have a2 c2 ха C2 = ratio of the involved numbers. To show the approximation of these ratios, we divide one by the other, thus: Now, if we take each supposition as near as possible to the true number, their difference q is a very small number, and consequently, 1+ a near approximation to a unit, which a + c is the result when the ratios are equal. To solve a question of this kind, therefore, assume two numbers, each as near as possible to the true, and find the result as above, by the appropriate formula. Take the result which is an approximation, and of the two suppositions, that which is nearest to the true number, or any other which may appear still nearer, and proceed as before. The true number may thus, by repeated trials, be obtained to any degree of exactness. Examples. 1. Two merchants begin business with equal sums of money; one gains 25 per cent., the other loses 10 per cent. of his capital, and $2000 more; after which the first has $5000 more than the other. With what sum did each begin? 5714,284 diff. $5000, as required by the question. -2000 capital of second; and, by the question, as the difference is 5000, if we add this to capital of second, we have х х the equation + =x +3000; multiplying by 40, 4 we have 40x + 10x and 10 = 40x 4x+120000; subtract 40x, 10x=4x+120000; transpose 4x, 14x 120000, or 7x x = 60000 7 = = 60000, and All questions in Double Position, which, in their operation, require neither Involution nor Evolution, belong to Double Position Absolute; and may, as above, be solved, either by one operation, according to the formula, or by a simple equation. 24. 2. What number exceeds five times its square root by 13? Take 64: then 1/64 = 8, and 5 × 8=40; 64 - 40 1311, error in plus. 24; Take 49: then 1/49 = 7; 5 × 7 = 35; 49 — 35 = 14; 1, error in plus. 14- 13 Mult. in a cross order 64 X 1= = 64 and 49 X 11 = 539. Then, as the errors have like signs 539-64475, difference of products. 11—1= 10 difference of errors, and 475 10 approximate number. 1/47,5=6,892; 5 × 6,892 = 34,46 47,534,4613,04; 13,04 13,04, error in plus. Assuming 49 and 47,5, and multiplying as before, we have 49,041,96; 47,5 × 147,5; 47,5-1,9645,54, difference of products. 1—,04,96, difference of errors; then 45,54,9647,4375, a greater approximation, which exceeds 5 times its square root 6,887488 by 13,00006. Now take 47,4375 error in plus, 00006 47,5 error in plus, 04. 47,4375 X,04 = 1,8975 47,5 X,00006,00285 1,89465 difference of products. ,04,00006 =,03994 difference of errors. 1,89465,03994 47,4374 required number. 5 × √/47,4374 = 5 × 6,88748 = 34,4374 13 required difference. By a Quadratic Equation. The square of a+b is a2+2ab+ b2, or a2 + 2ba + b2. Take away b ; = 21, and let b 2: then = a3+4a=21. This equation, in which we find the square of the unknown quantity in the first term, and the quantity itself in the second, is properly called quadratic from the Latin quadrans, a square. In the equation a2 + 2ba 21, the coefficient of a in the second term is 26. Now b is half this coefficient; and, as the square of this half is the part which is wanting to make the first member a complete square, we infer that in every equation of this kind, if we add the square of half the coefficient of the second term to each member, that member which contains the unknown quantity becomes a complete square; and consequently, by extracting the square root of each side, the index is expunged, and the equation becomes simple. In the equation a2 + 4a =21, the coefficient of a being 4, we add the square of its half to each side, and have a2 + 4a + 4 25. Now ab, which is the root of a2 + 2ab+ b3, is merely the sum of the roots of the first and last terms of the ex pression. Also, as the square of a b is a3-2ab+ b2, we may infer that the root of the complete square is the root of the first term, and the root of the last connected by the sign preceding the second term. Hence, by extracting the square root of each side of the equation a2+4a +4=25, we have a + 2 = 5, or a = 5 -23; and substituting 3 for a in the equation a2+4a =21, we have 9+12: - 21. To solve example 2, let x number required. Then sides, 135/x; divide by 5, 25 =x; mult. by 25, x2 - 26x+169 =25x; transpose 25x and complete the square, we add and have 169, x2. 51x 2601. 2601 676 - - 169; to 1925 ; extracting roots, From the above it is evident that questions in this rule, in 51 /1925 51 +/1925 + or 47,4374 required number. which the required number depends on the increase or diminu tion of the given number by quantities obtained by its involution or evolution, can only be solved by Approximate Double Position or by a Quadratic Equation. 3. A father gives, in unequal portions, $10000 to his two sons. If $1000 be taken from the portion of the younger, the remainder will be to the portion of the elder as 3 to 5. What is the cube root of the ratio of the whole portion of the younger to that of the elder? Ans. ,9196413 +. 4. Having diminished a number by four times its square root, the remainder, diminished by 4, is of the number. What is the ratio of the last remainder to of the square root of the number? Ans. 1. 5. Required a number, from which, if you take 5 times. its square root, the remainder will be 3. Ans. 30,7069+ approx. number. Alligation. 436. Alligation, which means tying together, is the rule which determines the value of compounds formed of ingredients of different values, and is of two kinds, Medial and Alternate. Alligation Medial. This rule is so called, because when several quantities, and the respective value, rate, or price of each is given, it finds the medium, or mean value, rate, or price of the mixture compounded of those quantities. The rule is as follows: Multiply each quantity by the value of its principal unit, and add the products together: the sum is the value of the whole composition. Next find the sum of the given quantities; then as the mixture, when compounded, is a uniform quantity, it is plain that the values of any two quantities of the mixture must be to each other as the quantities themselves; we therefore say As the sum of the given quantities To any proposed part of the mixture, To the value of the part proposed. The price or value of a number of units is evidently the result of the price of one unit, added to itself as often as the unit is added in forming the number; that is, the price of one unit multiplied by the number of units: hence, inversely, |