endless variety of systems of logarithms, to the same common numbers, by only changing the second term, 2, 3, or 10, &c. of the geometrical series of whole numbers; and by interpolation the whole system of numbers may be made to enter the geometric series, and receive their proportional logarithms, whether integers or decimals.

It is also apparent, from the nature of these series, that if any two indices be added together, their sum will be the index of that number which is equal to the product of the two terms, in the geometric progression, to which those indices belong. Thus, the indices 2 and 3, being added together, make 5; and the numbers 4 and 8, or the terms corresponding to those indices, being multiplied together, make 32, which is the number answering to the index 5.

In like manner, if any one index be subtracted from another, the difference will be the index of that number which is equal to the quotient of the two terms to which those indices belong. Thus, the index 6, minus the index 4, is = 2; and the terms corresponding to those indices are 64 and 16, whose quotient is 4, which is the number answering to the index 2.

For the same reason, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power. Thus, the index or logarithm of 4, in the above series, is 2; and if this number be multiplied by 3, the product will be = 6; which is the logarithm of 64, or the third power of 4.

And, if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithmi of that root. Thus, the index or logarithm of 64 is 6; and if this number be divided by 2, the quotient will be = 3; which is the logarithm of 8, or the square root of 64.

The logarithms most convenient for practice, are such as are adapted to a geometric series increasing in a tenfold pro portion, as in the last of the above forms; and are those which are to be found, at present, in most of the common tables on this subject. The distinguishing mark of this system of logarithms is, that the index or logarithm of 10 is 1; that of 100 is 2; that of 1000 is 3; &c. And, in decimals,

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 answer.

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.

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

Suppose the two numbers 18 and 30. Then,

First Position.

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.

A dd

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:

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 38 per day, for every day he worked; but with this condition, that for every day he played, he should forfeit 1s. Now it so hapHow pened, that upon the whole he had 2/ 4s to receive. Ans. 16. many of the days did he work?

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. What sum did each begin with? Ans. 100 guineas.

5. Two persons, A and B, have both the same income, A saves of his; but B, by spending 50% per annum more than A, at the end of 4 years finds himself 100/ in debt. What does each receive and spend per annum?

Ans. They receive 125/ per annum; also ▲ spends 100, and в spends 150/ per annum.

-b, therefore For since, by the supposition, r: s : : x — a : x by division, r- 8 : ba: x- - b; which is the 2d Rule.

PERMUTATIONS

PERMUTATIONS AND COMBINATIONS.

PERMUTATION is the altering, changing or varying the position or order of things; or the showing how many different ways they may be placed.—This is otherwise called Alternation, Changes, or Variation; 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: as, how many changes may be rung on a number of bells, or how many different ways any number of persons may be placed, or how many several variations may be made of any number of letters, or any other things proposed to be varied.

COMBINATION is the showing 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.

Combinations of the same Form, are those in which there are the same number of quantities, and the same repetitions : thus, aabc, bbcd, ccde, are of the same form; aabc, abbb, aabb, are of different forms.

Composition of Quantities, is the taking a given number of quantities out of as many equal rows of different quantities, one out of every row, and combining them together.

Some illustrations of these definitions are in the following Problems:

PROBLEM I.

To assign 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 scries 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 shown 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 qb, ba; whose number is expressed by 1 × 2.

EXAMPLES.

1. How many changes may be rung on 6 bells.

1

2

23

6

4

24

5

120

6

720 the Answer.

Or 1 x 2 x 3 X 4 X 5 X 6720 the Answer,

2. How many days can 7 persons be placed in a different position at dinner? Ans. 5040 days.

3. How many changes may be rung on 12 bells, and what time would it require, supposing 10 changes to be rung in 1 minute, and the year to consist of 365 days, 5 hours, and 49 minutes ?

Ans. 479001600 changes, and 91 years, 26 days, 22 hours, 41 minutes.

4. How many changes may be made of the words in the following verse: Tot tibi sunt dotes, virgo, quot sidera calo? Ans. 40320 changes.

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

In the same manner, when there are 4 things, every three, leaving out the 4th, will have 1 X 2 X 3 variations; consequently by taking in successively the 4 left out, there will be 1 X 2 X 3 x 4 variations. And so on as far as we please.

PROB