5. A person bought a chaise, horse and harness, for 6ol.; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness : what did he give for each?

Ans. 131. 6s. 8d.. for the horse, 61. 13s. 4d. for the har ness, and 40l. for the chaise.

6. A vessel has three cocks, A, B and C; A can fill it in 1 hour, B in 2, and C in 3: in what time will they all fill it together? Ans. hour.

DOUBLE POSITION.

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Double Position teaches to resolve questions by making two suppositions of false numbers.

RULE.*

1. Take any two convenient numbers, and proceed with each according to the conditions of the question.

2. Find

*The rule is founded on this supposition, that the first error ip 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 the supposition, may be thus demonstrated.

Let A and B be any two numbers, produced from a and b by similar operations; it is required to find the number, from which N is produced by a like operation.

Put x number required, and let N-A-r, and N—B=s. Then, according to the supposition, on which the rule is found:: x-a: x-b, whence, by multiplying means and

ed, r

་ ་་

extremes,

2. Find how much the results are different from the result in the question.

3. Multiply each of the errors by the contrary supposi tion, and find the sum and difference of the products.

4. If the errors be alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer.

5. If the errors be unlike, divide the sum of the prod ucts by the sum of the errors, and the quotient will be the answer.

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

EXAMPLES.

1. A lady bought tabby at 4s. a yard, and Persian at 25. a yard; the whole number of yards she bought was 8, and the whole price 20s. : how many yards had she of cach sort?

Suppose

extremes, rx―rb=sx-sa; and, by transposition, rx-sx=rb—sa ;

Again, if rands be both negative, we shall have

:: x-a ; x—b, and therefore -rx+rb=—sx+sa; and rx

rb-sa

$x=rb-sa; whence x =

as before.

r-s

In like manner, if r or s be negative, we shali have x=

by working as before, which is the rule.

rb+sa

NOTE. It will be often advantageous to make 1 and o the suppositions.

Suppose 4 yards of tabby, value 16s.

Then she must have 4 yards of Persian, value 8

Sum of their values 24

So that the first error is + 4

Again, suppose she had 3 yards of tabby at 12s.
Then she must have 5 yards of Persian at 10

Then 4-22

Also 4×28

second error.

Sum of their values 22

So that the second error is +2

difference of the errors.

product of the first supposition and

And 3X412 product of the second supposition by

the first error.

And 12-8=4= their difference.

Whence 422

yards of tabby, the answer.

And 8-26 yards of Persian,

2. Two persons, A and B, have both the same income ; A saves of his yearly; but B, by spending 50l. per annum more than A, at the end of 4 years finds himself Iool. in debt what is their income, and what do they spend per annum?

:

Ans. Their income is 1251. per annum; A spends rool. and B 150l. per annum.

3. Two persons, A and B, lay out equal sums of money in trade; A gains 261. and B loses 871. and A's money is now double that of B: what did each lay out ?

Ans. 300l.

4. A labourer was hired for 40 days, upon this condition, that he should receive 20d. for every day he wrought, and forfeit 10d. for every day he was idle ; now he receiv

ed

ed at last 21. 1s. 8d. : how many days did he work, and how many was he idle?

Ans. He wrought 30 days, and was idle 10.

5. A gentleman has two horses of considerable value, and a saddle worth 50l. ; now, if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first: what is the value of each horse?

Ans. One 30l. and the other 401.

6. There is a fish, whose head is 9 inches long, and his tail is as long as his head and half as long as his body, and his body is as long as his tail and his head

whole length of the fish?

what is the

Ans. 3 feet.

PERMUTATION AND COMBINATION.

THE Permutation of Quantities is the shewing how many different ways the order or position of any given number of things may be changed.

This is also called Variation, Alternation, or Changes; and the only thing to be regarded here is the order they stand in; for no two parcels are to have all their quantities placed in the same situation.

The Combination of Quantities is the shewing how often a less number of things can be taken out of a greater, and combined together, without considering their places, or the order they stand in.

This is sometimes called Election, or Choice; and here every parcel must be different from all the rest, and no two are to have precisely the same quantities, or things.

The Composition of Quantities is the taking a given number of quantities out of as many equal rows of different quantities,

quantities, one out of each row, and combining them together.

Here no regard is had to their places; and it differs from combination only, as that admits of but one row, or set, of things.

Combinations of the same form are those, in which there is the same number of quantities, and the same repetitions thus, abcc, bbad, deef, &c. are of the same form; but abbc, abbb, aacc, &c. are of different forms.

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PROBLEM Ì.

To find the number of permutations, or changes, that can be made of any given number of things, all different from each other.

RULE.*

Multiply all the terms of the natural series of numbers, from 1 up to the given number, continually together, and the last product will be the answer required.

EXAMPLES.

* The reason of the rule may be shewn thus: any one thing a is capable only of one position, as a.

Any two things, a and b, are only capable of two variations; as ab, ba; whose number is expressed by IX 2.

If there be 3 things, a, b and c, then any two of them, leaving out the third, will have 1×2 variations; and consequently, when the third is taken in, there will be 1 X2 X3 variations.

In the same manner, when there are 4 things, every 3, leaving out the fourth, will have 1X2 X 3 variations. Then, the fourth being taken in, there will be 1 X2 X3 X4 variations. And so on, as far as you please.