OPERATIONS.

LOGARITHMIC OPERATIONS.

373

For the reasons already explained, no addition or subtraction of numbers can be expressed by logarithms; and the sum or difference of a number and a logarithm does not express any thing which has a meaning or could exist. In stating this, we are to understand that the word number is used as a short expression for all common numbers, and for all quantities that are capable of being, in whole or in part, expressed by common numbers, without in any way altering their nature.

Multiplication is performed by adding the logarithm of the factors; and Division by subtracting the logarithm of the divisor from that of the dividend. If a number of successive multiplications and divisions, without any intervening additions or subtractions, have to be performed, all the logarithms of the factors may be collected into one sum, and the logarithms of all the divisors into another; and when the last of these is subtracted from the first, the remainder is the logarithm of the ultimate quotient, or of the value of the expression in a single number.

In performing the subtractions, it must not be lost sight of that they are really divisions; and that though the logarithm which we have to subtract be really greater, both in the index and in the decimal part, than the logarithm from which we have to subtract it, there will still be a real and positive remainder; whereas if, in common arithmetical subtraction, we try to take a greater number from a less, we should find ourselves minus the difference; that is to say, the numerical difference would be wholly affected by the sign, and really be a quantity less than nothing. But in division we never can have a negative quotient, if the divisor and dividend have the

same sign; and as the decimal part of every logarithm is. + or positive, there is always something positive to divide, as well as something positive to divide it by, whatsoever may be the signs of the indices in the case of logarithms.

Involution is performed logarithmically, by multiplying the logarithm of the root by the exponent of the power; and the natural number answering to the product is the power required, as a natural number: and Evolution is performed by dividing the logarithm of the power by the exponent of the root; and the natural number answering to the quotient is the root itself.

In multiplying a logarithm with a negative index, the product which arises from multiplying the left-hand figure of the decimal part, if it contains any thing which has to be carried to the product of the negative index, must, though positive itself, send the quantity carried to the index as negative. This may at first sight seem a little singular; but it is nevertheless strictly true. The index affects the whole value of the number for which the logarithm stands, in as far as it determines its place in the scale of numbers; and thus, though it makes no alteration in the figures of which the logarithm is composed, it affects the whole value of that logarithm; for if it did not, the logarithm would not be a faithful representative of the number. Therefore, when the decimal or positive part of the logarithm, having a negative index, is so multiplied as that there is something to carry from the left-hand figure to the index, the number which is thus carried becomes of itself an index by the transfer, and acquires the negative sign by passing the decimal point, just as a number which is passed the decimal point into the decimals requires - before its exponent.

On the other hand, in dividing logarithms with negative indices, the quotient, if it amount to 1 or more, must have the

sign—; but whatever remainder may be over on dividing the negative index, will go over to the first figure of the decimals as a positive quantity, every 1 in it counting as 10, as in every other case in division.

Such are the leading principles in the arithmetic of exponents, and the nature and use of common logarithms; and as there is no contrivance in the whole compass of the mathematical sciences so well calculated for abridging labour, or enabling us to perform operations which we could not perform by common arithmetic, this is a portion of the subject which ought to be fully understood by every one who wishes to be even a tolerable accountant; while those who aim at any of the practical connexions between arithmetic and geometry, will find most untoward work of it if they do not learn the ready use of this powerful instrument. It is true that the tables are easily understood, and that the operations are very simple; and that these may be committed to memory, and put in practice without any understanding, just as a parrot learns to repeat a sentence, or whistle a tune; but those who are thus unfortunate, never know when they are right; and therefore when the slightest trifle arises which is not found in the formula, they are completely at a loss, and sure to be wrong.

SECTION XV.

INTERSECTIONS OF LINES AND CIRCLES.

THE circle is the foundation of all our purely geometrical notions of the equality of lines and angles; and these are the elements of which all our other and more complex notions of

geometrical equality are formed; therefore, after we have once fully understood the nature of lines and angles, and the general doctrine of relations, of which an outline has been attempted in the former sections of this volume, our next object should be to make ourselves well acquainted with the uses of the circle, in enabling us to compare the lengths of lines, and the magnitudes of angles. This we shall attempt in the present section.

It will be borne in mind that all radii of the same circle, or of equal circles, are necessarily equal; which follows from the way in which a circle is described, and is our primary notion of it, or that upon which the definition is founded. This, therefore, is a simple cause of the equality of lines, which does not need or admit of any proof.

It will also be borne in mind, that the circumference of a circle is the measure of all the angular space round a point; that is, it is equal to four right angles; and that any portion of the circumference of any circle, and the same portion of four right angles, are naturally and reciprocally the measures of each other; consequently the larger the portion of the circumference, the larger is the angle; and conversely, the smaller the portion of the circle, the smaller is the angle. It must be understood, however, that it is not the absolute length of the arc in any known measure, but its length as compared with that of the whole circumference of which it is a part, which determines the value of the angle.

When it is not otherwise mentioned, the centre of the circle is the point to which the angle is referred, because it is the only point within the circle which stands in the same relation to every point of the circumference. In the language of

geometry, the arc is said to suttend the angle, that is, to "hold under it ;" so as to tie it to one definite value; and

of which c is the centre; and the portions of the circumference from A to B, and from в to c, any two arcs.

Draw the radii

▲ C, B C, and D c, from the extremities of the arcs, to c the centre, and the angle A CB, and arc a в are reciprocally the the measures of each other, and so are the angle B C D and the arc B D. Also the angle A c D, which is the sum of the two angles A C B and B C D, and the arc A D, which is the sum of the two arcs A B and B D, are reciprocally the measures of each other. If the arc A B is greater than the arc B D, the angle a C B is greater than the angle B C D ; if equal, equal; and if less, less. Further, if the one arc is any multiple or part of the other arc, the angle standing on or subtended by the first arc, is the same multiple or part of the angle standing on or subtended by the second arc. Arcs, and the angles which stand on them, are therefore proportional quantities.

But if we, as in the following figure,