POSITION, or the rule of false Post 'nos, otherwise called the rule of FALSE noon, in arithmetic, is a rule so called, because, in calculating on several false numbers taken at random, as if they were the true ones, and from the differences found therein, the number sought is de termined. This rule is either single or double. Single position is when there happens in the proposition some parti tion of numbers Into parts proportional, in which case the question may be re solved, at one operation, by this rule, Imagine a number at pleasure, and work therewith according to the tenor of the question, as if it were the true number and what proportion there is between the false conclusion and the false propor tion, such proportion the given number has to the number sought. -the number found by argumentation shall he the first term of the rule of three ; the second number supposed, the second term ; and the given number, the third, Or the result is to be regulated by this proportion, viz. As the total arising from the error to the true total, so is the sup posed' part to the true one. Example, A, B, and C, designing to buy a quantity of lead to the value of 1401. agree that 1.1 shall pay as much again as A, and C as much again as B ; what then must each PaY? Now suppose A to pay 101. then B must pay 201. and C 40/. the total of which is 701. but should be 1401. There fore, if 701. should be 1401. what should 101. be ? Answer, 201. for A's share, which dou bled makes 401. for B's share, and that again doubled gives 801. for C's share, the total of which is 1401. Double posi tion is when there can be no partition in the numbers to make a proportion. In
this case, therefore, you must make a supposition twice, proceeding therein according to the tenor of the question. If neither of the supposed numbers solve the proportion, observe the errors, and whether they be greater or less than the supposition requires, and mark the errors accordingly with the sign + and—.
Then multiply contrarywise the one position by the other error, and if the errors be both too great, or both too lit tle, subtract the one product from the other, and divide the difference of the Vitoducts by the difference of the errors. if the errors be unlike, as the one and the other —, add the products, and di vide the sum thereof' by the sum of the errors added together : for the propor tion of the errors is the same with the proportion of the excesses or defects of the numbers supposed to be the numbers sought : or, the suppositions and their errors being placed as before, work by this proportion as a general rule, viz. as the difference of the errors, if alike ; or their sum, if unlike, to the difference of the suppositions, so either error to a fourth which accordingly, added to or subtracted from the supposition against it, will answer the question.
Posmox, in geometry, is a term some times used in contradistinction to magni tude: thus a line is said to be given in position, fiositione data, when its situation, bearing, or direction, with regard to some other line, is given : on the contrary, a line is given in magnitude, when its length is given, but not its situation.