progression, the series of natural numbers; that is, the two following progressions are selected. 1: 10: 100: 1000 10000 100000: 1000000. 4. 5. 6 0 1 2 3 219. The logarithm of an unit followed by any given number of ciphers, may always be easily determined; for it will consist of as many units as there are ciphers following that unit. We shall not explain the method for finding the logarithms of the intermediate terms in the decuple progression. It depends upon principles, with which the learner is not yet supposed to be acquainted, But we will explain the formation of logarithms by a method, which is not the most expeditious for calculating them, but suffices, as well to give an idea of their formation, as to afford a reason for the uses, to which these artificial numbers are applied. 220. After the definition which has been given of logarithms, it is plain, that, to have the logarithm of any number, (of 3, for example,) this number must form a part of the fundamental geometrical progression. And although 3 does not appear to form a part of this geometrical progression 1: 10 100 1000: &c.; yet it may be seen in article (214,) that between 1 and 10 a great number of geometrical means may be inserted. Then, as we ascend from 1 to 10 by gradations the more closely united, that is, by numbers, that differ from each other the less, in proportion as the number of such means shall be greater; either some one of these means will be precisely the number 3, or two consecutive numbers will be found, between which the number 3 is contained, and of which each shall differ as little from 3, as the number of means inserted shall be greater. Then, in like manner, if we insert between 0 and i as many arithmetical means as there were geometrical means inserted between 1 and 10, each term of the geometrical progression will have for its logarithm, the term, which corresponds to it in the arithmetical progression. That is, for the logarithm of 3, we take the number, that is found in the same place, in which 3 is found in the geometrical progression. But if no one of these numbers be exactly the number 3, we take, in the arithmetical progression, the term, that corresponds to that term of the geometrical progression, which approaches nearest to the number 3. On this principle is founded the calculation of logarithms; and although more expeditious methods may be followed, the results will be the same. 221. Insert, or suppose to be inserted, 10000000 geometrical means between 1 and 10, the same between 10 and 100, the same number between 100 and 1000, &c.; and then insert the same number of arithmetical means between O and 1, the same number between 1 and 2, between 2 and 3, &c.; then, having placed all the first in one line, and all the second underneath them, seek among the first, the number 2, or number nearest to it, and take the corresponding number in the lower series. In like manner, seck among the first series the number 3, or number nearest to it, and take the corresponding number in the lower series. Do the same successively, for the numbers 4, 5, 6, &c. and writing the numbers 1, 2, 3, 4, 5, &c. in the same vertical column, (as may be seen in the annexed Table,) place in a column by the side of them, the terms of the arithmetical progression, which have been found to correspond to these numbers, or to the numbers, which approach nearest to them. From this operation an idea may be obtained of the formation of Logarithms, and of their arrangement in common Ta bles. 222. It may be remarked concerning logarithmic tables, that the first figure to the left of each logarithm is called the characteristic; because it is by this figure, that we know in what decade the number is contained, to which the logarithm belongs. For example, if a number have for its characteristic, the number 3, we know this number must be a certain number of thousands; because, the logarithm of 1000 being 3, and that of 10000 being 4, every number between 1000 and 10000 must have for its logarithm, 3 and a fraction. It will then have 3 for its characteristic; and the other figures express the fraction reduced to decimals. PROPERTIES OF LOGARITHMS. 223. As we treat of only such logarithms as are found in ordinary Tables, our explanations will be confined to such properties as belong to geometrical progressions, that have unity for their first term, and arithmetical pro gressions, that have cipher for their first term. Let us compare term for term, any geometrical progression whose first term is unity, with any arithmetical progression whose first term is cipher. For example, the two following progressions. 1:39: 27: 81 : 243: 729 : 2187: 6561: &c. It is inferred from the nature and perfect correspondence of these two progressions, that as often as the ratio of the first progression is factor in any one of its terms, so often will the ratio of the second progression be contained in the term corresponding to the term considered in the first progression. For example, in the term 2187, the ratio 3 is seven times factor; and in 28 the term corresponding to it, the ratio 4 is contained seven times. And, according to what has been shown in articles (206 and 212), the ratio is factor in any term of the first progression, as often as there are terms preceding it; and in the second progression, any term is composed of as many times the ratio, as there are terms preceding it. But the number of terms is the same in both progressions. Hence, any term of a geometrical progression will always have for its corresponding term in the arithmetical progression, a term which shall contain the ratio of this progression as often as the ratio of the other progression is factor in the first. 224. Then, if we multiply together any two terms of a geometrical progression, and at the same time add together the two corresponding terms of the arithmetical progression, the product and sum will be two corresponding terms in these progressions. For it is plain, that the ratio will be factor in the product, whatever it is, as often as in both the terms that are multiplied together; and the ratio of the arithmeti cal progression will be contained in the sum, whatever it is, as often as in both the terms that are added togeth er. 225. Then, by the addition only of two terms of an arithmetical progression, we may know the product of the two corresponding terms of the geometrical progression; by supposing the progressions to be sufficiently extended. For example, by adding the terms 8 and 24, which correspond to 9 and 729, we have 32 for the sum, which corresponds to 6561. Hence we infer, that the product of 729 by 9 is 6561; which is the case. Then as the natural numbers that compose the first column of the annexed Table, have been taken from a geometrical progression, which commences with unity; and as their logarithms are the corresponding terms of an arithmetical progression that commences with cipher; we may infer, that by adding together the logarithms of two numbers, we obtain the logarithm of their product. Hence we deduce the following uses. USES OF LOGARITHMS. 227. To perform multiplication by logarithms, add the logarithm of the multiplicand to the logarithm of the multiplier, and their sum will be the logarithm of the product ; seek for this sum among the logarithms of the Table, and by the side of it will be found the product. For example, if it be required to multiply 14 by 13, we seek in the table for the logarithm of 14, which is 1,146128 and that of 13, which is 1,113943 The sum 2,260071 corresponds in the same Table to the number 182, which is the product of 14 by 13. 228. Then to square a number, we need only double its logarithm; for, to multiply any number by itself, the logarithm of that number is added to itself. 229. For the same reason, to cube a number, we triple its logarithm; and in general, to raise a number to any power whatever, we take the logarithm of that number as any times as there are units in the number, that denotes the power. That is, we multiply its logarithm by the number, that expresses the power. For example, to raise a number to the seventh power, we multiply its logarithm by the number 7. 230. Then reciprocally, to extract the square, cube, fourth, &c. root of a given number, we must divide that number by 2, 3, 4, &c.; that is, in general, by the number that expresses the degree of the root to be extracted. For example, if the square root of 144 be required, we find by the Table, that the logarithm of this number is 2,158362; then taking 1,079181 the half of this logarithm, we find that 12 is the number corresponding to it; which is the square root of 144. If the seventh root of 128 be required, we find by the Table, the logarithm of this number is 2,107210; then taking the seventh part of it, we seek in the Table what number corresponds to 0,301030, the quotient, and find it is 2; which is the seventh root of 128. 231. To find the quotient of the division of one number by another, subtract the logarithm of the divisor from the logarithm of the dividend; the number that corresponds to the logarithm of the remainder, will be the quotient. For example, if it be required to divide 187 by 17, we seek in the Table for the logarithms of these two numbers, and find for the logarithm of 187, The difference 1,041393 corresponds in the Table to 11; which is the quotient of 187 divided by 17. When division cannot be performed without a remainder, the whole logarithm of the remainder is not used. We shall show hereafter what must be done in such cases. The reason for this rule is founded on the principle established in article (74), that the quotient multiplied by the divisor, produces the dividend; then, (227) the logarithm of the quotient, added to the logarithm of the divisor will produce the logarithm of the dividend; consequently, the logarithm of the quotient is equal to the T |