POSITION.

POSITION is a method of performing certain questions, which cannot be resolved by the common direct rules. It is Sometimes called False Position, or False Supposition, because it makes a supposition of false numbers, to work with the same as if they were the true ones, and by their means discovers the true numbers sought. It is sometimes also called Trial-and-Error, because it proceeds by trials of false numbers, and thence finds out the true ones by a comparison of the errors.-Position is either Single or Double.

SINGLE POSITION.

SINGLE POSITION is that by which a question is resolved by means of one supposition only. Questions which have their result proportional to their suppositions belong to Single Position such as those which require the multiplication or division of the number sought by any proposed num ber; or when it is to be increased or diniinished by itself, or any parts of itself, a certain proposed number of times. The rule is as follows :

TAKE or assume any number for that which is required, and perform the same operations with it, as are described or performed in the question. Then say, As the result of the said operation, is to the position, or number assumed; so is the result in the question, to a fourth term, which will be the number sought*.

The reason of this Rule is evident, because it is supposed that the results are proportional to the suppositions.

EXAMPLES.

1. A person after spending and of his money, has yet remaining 601; what had he at first?

per question.

Then, 50: 120 :: 60: 144, the Answer.

2. What number is that, which being multiplied by 7, and

the product divided by 6, the quotient may be 21?

Ans. 18.

3. What number is that, which being increased by 1, 1, and of itself, the sum shall be 75?

Ans. 36.

4. A general, after sending out a foraging and of his men, had yet remaining 1000; what number had he in command? Ans. 6000.

5. A gentleman distributed 52 pence among a number of poor people, consisting of men, women, and children; to each man he gave 6d, to each woman, 4d, and to each child 2d: : moreover there were twice as many women as men, and thrice as many children as women. How many were there of each? Ans. 2 men, 4 women, and 12 children?

DOUBLE POSITION,

DOUBLE POSITION is the method of resolving certain questions by means of two suppositions of false numbers.

To the Double Rule of Position belong such questions as have their results not proportional to their positions: such are those, in which the numbers sought, or their parts, or their multiples, are increased or diminished by some given absolute number, which is no known part of the number sought.

RULE I.

TARE or assume any two convenient numbers, and proceed with each of them separately, according to the conditions of the question, as in Single Position; and find how much each result is different from the result mentioned in the question, calling these differences the errors, noting also whether the results are too great or too little.

* Demonstr. The Rule is founded on this supposition, namely, that the first error is to the second, as the difference between the true and first supposed number, is to the difference between the true and second supposed number; when that is not the case, the exact answer to the question cannot be found by this Rule.-That the Rule is true, according to that supposition, may be thus proved.

Let a and b be the two suppositions, and A and B their results, produced by similar operation; alsor and their errors, or the differences between the results A and B from the true result N; and let x denote the number sought, answering to the true result N of the question.

Then is N- Ar, and N - B 8. And, according to the supposition on which the Rule is founded, r :s :: x-ɑ : x-b: hence, by multiplying extremes and means, rx then, by transposition, rx 8x =rb

x=

rb. -87

sa;

rb 8x - sa; and, by division,

the number sought, which is the rule when the

results are both too little.

If the results be both too great, so that A and B are both greater than N; then N-A- r, and NB = -8, or rands are both negative; hence -r: −s:¦+r :+s, therefore rs':: x-ɑ: x- b; and the rest will be exactly

- 7:

as in the former case,

b, but

But if one result A only be too little, and the other B too great, or one error positive, and the other's negative, then the theorem be

rb + sa

comes x ——, which is the Rule in this case, or when the errors

are unlike. VOL. I.

Then multiply each of the said errors by the contrary supposition, namely, the first position by the second error, and the second position by the first error. Then,

If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the

answer.

But if the errors are unlike, divide the sum of the products by the sum of the errors, for the answers.

Note, The errors are said to be alike, when they are either both too great or both too little; and unlike, when one is too great and the other too little.

EXAMPLES.

the

1. What number is that, which being multiplied by 6, the product increased by 18, and the sum divided by 9, quotient shall be 20 ?

Suppose the two numbers 18 and 30. Then,

First Position.

Second Position.

Proof.

FIND, by trial, two numbers, as near the true number as convenient, and work with them as in the question; marking the errors which arise from each of them.

Multiply the difference of the two numbers assumed, or found by trial, by one of the errors, and divide the product by the difference of the errors, when they are alike, but by their sum when they are unlike.

Add

Add the quotient, last found, to the number belonging to the said error, when that number is too little, but subtract it when too great, and the result will give the true quantity sought*.

EXAMPLES.

1. So, the foregoing example, worked by this 2d rule will

be as follows:

30 positions 18;

-2 errors + 6;

their dif. 12

least error 2

sum of errors 8) 24 (3 subtr.
from the position 30

leaves the answer 27

Ex. 2. A son asking his father how old he was, received this answer: Your age is now one-third of mine; but 5 years ago, your age was only one-fourth of mine. What then are their two ages? Ans. 15 and 45.

3. A workman was hired for 20 days, at 3s per day, for every day he worked; but with this condition, that for every day he played, he should forfeit 1s. Now it so happened, that upon the whole he had 21 4s to receive. How many days did he work?

Ans. 16

4. A and B began to play together with equal sums of money: A first won 20 guineas, but afterwards lost back of what he then had; after which, в had 4 times as much as A. sum did each begin with?

What Ans. 100 guineas.

5. Two persons, A and B, have both the same income, a saves of his; but в, by spending 50l per annum more than a, at the end of 4 years finds himself 100 in debt.

each receive and spend per annum?

What does

Ans. They receive 125 per annum ; also ▲ spends 1007, and в spends 150 per annum.

For since, by the supposition, :s: : x — a: x-b, therefore by division, r

-a: x-b, which is the 2d Rule.

PERMUTATIONS