Let N be any number, and its logarithm; that is, let 10% N; also, let N' be any other number, and x' its logarithm; that is, let 10 N'. Then, by multiplying the two together, we have (Algebra, pa. 13),

=

10x+x'=NN';

that is (17), x+x' is the logarithm of the product NN'. Hence, in order to find the product NN', without actually multiplying, we shall merely have to take out of the table the logarithms of N and of N', viz. x and x', add them together, and then seek in the table for the number, NN', which has the sum x+x for its logarithm.

THEOREM II. The difference of the logarithms of any two numbers is equal to the logarithm of the quotient of those numbers.

As before, let 10% N and 10x = N'. Then, by dividing (Algebra, pa. 13),

that is (17), x ' is the logarithm of the quotient

Hence, in order to find the quotient

N
N"

N"

without actually

dividing, we shall merely have to take out of the table the logarithms of N and of N', subtract one from the other, and

N then seek for the number of which the difference — 1′ N

is the logarithm.

Thus, let it be required to find, without multiplication, the product of 2 and 3; these easy numbers being chosen, to come within the limits of the short specimen of a table, given above.

log 23010300

log 34771213

The sum 7781513 log 6.

In like manner, to find the quotient of 8 and 2 without division,

log 89030900
log 2 =3010300

The difference 6020600 log 4.

Instead of the short numbers here selected, if we had taken numbers ever so large, provided the tables were sufficiently extensive, the results would have been obtained with the same ease. Thus, for the product of 23·14 by 5'062, the process would be

log 23.14

1.3643634

log 5.062

•7043221

2.0686855 log 117.135

therefore the product is 117.135.

It is necessary to apprise the student that in the Tables the decimal part only of each logarithm is inserted; the integral part, called the characteristic, is omitted, because this integral part is always immediately suggested by the number itself, whose logarithm is sought. If this number consist of two figures, it must be either 10 or else lie between 10 and 100; its logarithm, therefore, must be either 1 or else a number lying between 1 and 2; that is, the integral part must be 1. Again, if the proposed number consist of three figures, the integral part of the logarithm of it must be 2; if of four figures, the integral part of the logarithm must be 3, and so on. Hence, when any number is proposed, we have only to count the number of figures in the integral part of that number, in order to ascertain the cha

racteristic of its logarithm; this characteristic will always be the number, a unit below the number counted. Thus, whatever be the decimal part of the logarithm of 785-26, we know that the integral part of it is 2; and whatever be the decimal part of log 3279-5, the integral part is 3; also, the integral part of log 27.53 is 1, and that of log 6.283 is 0.

It moreover follows, as another important consequence of 10 being chosen for the root or base of our system of logarithms, that the decimal parts of the logarithms of all numbers consisting of the same figures, wherever the decimal point may intervene, remain the same; the characteristic being the only thing that changes with a change of place in the decimal point, thus:

The mark is placed over the 1 here to intimate that this is sub

tractive, but the annexed decimal additive.

It thus appears that the logarithm of a number which is entirely decimal consists of two parts of different signthe negative characteristic, and the positive decimal. As the sines and cosines to radius unity are all such numbers, except at the extremities of the quadrant, their logarithms would always be thus made up of a negative and a positive part, a combination which, in many instances, would be attended with inconvenience. To avoid this inconvenience, therefore, the logarithmic sines, cosines, &c. are not computed to the radius unity, like the natural sines and cosines, but to the radius 1010, which is so large as to render the occurrence of a sine, a cosine, &c. below 10 in numerical value, an impossibility, except at points so near to the extremities of the quadrant that their distances from those extremities are too minute to be appreciated.* Hence the lines, of which the logarithms form the table of logarithmic sines, tangents, &c. are no other than the natural trigonometrical lines, multiplied severally by 1010; so that the ratio of any two of the natural trigonometrical lines is kept undisturbed.

:

(19.) We have but one more observation to make-it is this that though the subtraction of one row of figures from another row is a very easy matter, yet where there are several rows, of which some are to be added and others to be subtracted, the twofold operation may become a little perplexing. In order, therefore, to give to our logarithmic processes the utmost degree of simplicity, an obvious contrivance has been adopted, whereby the subtractive quantities are converted into additive, and thus the twofold operation avoided.

The artifice by which the simplification is effected is this: the subtractive number is not actually taken out of the tables; but what that number wants of 10, and this added, along with the numbers really additive, and then 10 is subducted from the result, or rather this result is written down 10 short of the whole amount. It is clear that when we have to subtract a number below 10, but that instead of

In the logarithmic tables the log sine of an arc so small as 1" is no less than 4 6855749; the sine itself being 104-6855749.

so doing we add what that number wants of 10, we commit an error in excess, to the amount of 10; so that, by rejecting 10 in the result, we remove that error, and leave the true remainder that we should have got if we had actually subtracted the proposed number. Suppose, for instance, we have to subtract 7 from 13, but instead of doing so we add 3, what 7 wants of 10, the sum will be 16, 10 too much; rejecting this, therefore, we have 6 for the true result. What a number wants of 10 is called the arithmetical complement of the number; and with a table of numbers below 10 before us, it is very nearly as easy to write down from it the arithmetical complement of a number, as to transcribe the number itself. The shortest way of doing this will be to commence with the left-hand figure of the number in the book, and proceeding towards the right, to put down what each figure wants of 9, till we arrive at the final decimal, when its deficiency from 10 must be written. It is easy to see that this is the same in effect as to commence in the usual way, with the right-hand figure, and subtract from 10; but the figures may, in the former method, be written down with greater facility.*

Plane Triangles in general.

We shall now proceed to investigate rules for the solution of the different cases of plane triangles in general. Of these cases there are three; requiring three distinct rules for the determination of the unknown parts.

In the first case, a side and the angle opposite to it are two of the three given parts.

In the second case, two sides and the included angle are the given parts.

In the third case, the three sides are the given parts.

* In the logarithmic tangents and cotangents the integral part is sometimes equal to, or even greater than 10. In this case, 10 is at once to be rejected from the number, and the complement of what is left taken; 20 being rejected from the sum.