POSITION, in arithmetic, called also False Position, or Supposition, or Rule of False, is a rule so called, because it consists in calculating by false numbers supposed or taken at random, according to the process described in any question or problem proposed, as if they were the true numbers, and then from the results, compared with that given in the question, the true numbers are found. Thus, take or assume any number at pleasure for the number sought, and proceed with it as if it were the true number, that is, perform the same operations with it as, in the question, are described to be performed with the number required; then if the result of those operations be the same with that mentioned or given in the question, the supposed number is the same as the true one that was required; but if it be not, make this proportion, viz. as your result is to that in the question, so is your supposed false number to the true one required. Example. What number is that, to which if we add \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, and \frac{1}{5} of itself, the sum will be 240? Suppose 99 \begin{array}{r} 49.5 = \frac{1}{2} \\ 33. = \frac{1}{3} \\ 24.75 = \frac{1}{4} \\ 16.5 = \frac{1}{5} \\ \hline 222.75 = \text{result} \end{array} Then, as 222.75: 240:: 99: 106.6 = \text{Answer.} \begin{array}{r} 53.3 = \frac{1}{2} \\ 35.5 = \frac{1}{3} \\ 26.6 = \frac{1}{4} \\ 17.7 = \frac{1}{5} \\ \hline 240. = \text{proof.} \end{array} This is single position. Sometimes it is necessary to make two different suppositions or assumptions, when the same operations must be performed with each as in the single rule. If neither of the supposed numbers solve the question, find the differences between the results and the given number; multiply each of these differences into the other's position; and if the errors in both suppositions be of the same kind, i. e. if both suppositions be either less or greater than the given number, divide the differences of the products by the differences of the errors. If the errors be not of the same kind, i. e. if the one be greater and the other less than the given number, divide the sum of the products by the sum of the errors. The quotient, in either case, will be the answer. Example. Three partners, A, B, and C, bought a sugar-work which cost them L. 2000; of which A paid a certain sum unknown; B paid as much as A, and L. 50 over; C paid as much as them both, and L. 25 over: What sum did each pay? (1.) Suppose A paid L. 500 \begin{array}{r} B \text{ — } 550 \\ C \text{ — } 1075 \\ \hline 2125 \\ 2000 \\ \hline \end{array} 125 = error of excess. Pottery. (2.) Suppose A paid L. 400 B — 450 125 = excess. C — 875 1725 400 = 2d position. 2000 50000 275 = error of defect. 500 = 1st position. 1st error, 125 137500 2d — 275 50000 400 187500 Answers. 468.75 = sum paid by A. 518.75 = — — — — — B. 1012.5 = — — — — — C. 2000... = proof. This is called double position.