the product and the sum will be two terms that will correspond with each other in these progressions. For it is evident that the ratio will be factor in the product as many times as it is in both the terms multiplied; and that the ratio in the arithmetical progression will be contained in the sum as many times as it is in both the terms added. 225. Therefore, we can, by the addition alone of two terms of the arithmetical progression, find the product of the two corresponding terms of the geometrical progression, supposing these two progressions to be sufficiently extended. For example, in adding the two terms 8 and 24, which correspond with 9 and 729, I have 32, which answers to 6561; whence I conclude that the product of 729 by 9 is 6561, which it is in effect. 226. Therefore, since the natural numbers which compose the first column of the table of logarithms, have been taken from a geometrical progression which commences with a unit; and since their logarithms are the corresponding terms of an arithmetical progression which commences with a cipher, we shall conclude that in adding the logarithms of two numbers, we have the logarithm of their product. USE OF LOGARITHMS. 227. To perform a multiplication by logarithms, we must add the logarithm of the multiplicand to the logarithm of the multiplier; the sum will be the logarithm of the product; wherefore, seeking this sum among the logarithms of the table, we shall find the product opposite; for example, if we propose to multiply 14 by 13, I find in the table that the logarithm of 14 is and that the logarithm of 13 is The sum total 1,146128 answers in the same table to the number 182, which is in effect the product. 228. To square a number, it is sufficient then to double its logarithm, since we should add this logarithm to itself in order to multiply the number by itself. 229. For the same reason, to cube a number, we munt sign which shows that the subtraction has not been entirely performed. This is the sign minus —. Thus the logarithm of the fraction is -0,917648.† 236. This sign is intended to show in the calculation, that the logarithms of fractions should be employed according to a rule entirely opposed to that which we have prescribed for the logarithms of whole numbers, or of whole numbers joined to fractions, that is to say, that if we have to multiply by a fraction, we must subtract the logarithm of this fraction; if, on the contrary, we have to divide by a fraction, we must add its logarithm. The reason of this is, for the multiplication, that to multiply by a fraction is to multiply by the numerator, and to divide afterward by the denominator; therefore, when we operate by logarithms, we should add the logarithm of the numerator, and afterward subtract that of the denominator, or, which amounts to the same thing, we should only subtract the excess of the logarithm of the denominator above that of the numerator: now, this excess is precisely the logarithm of the fraction. With regard to the division, the reason is also easy to comprehend. In effect, to divide by is (109) to multiply by; therefore, in operating by logarithms, we must add the logarithm of, that is to say,(234) the difference between the logarithm of 4 and the logarithm of 3, or between the logarithm of the denominator of the proposed fraction and the logarithm of its numerator. 237. It often happens, that in converting to a single fraction the whole number and the fraction, the logarithm of which we seek, the numerator is a number which passes the limits of the table. For example, if we would have the logarithm of 53, this number reduced to a fraction becomes 302133, the numerator of which passes the limits of the most extensive tables. It is then proper to know how we can find the logarithm of a number which passes these limits. To consider negative numbers as numbers less than nothing, is to have very false idea, as there can be no number less than nothing. The method we are about to give is not rigorously exact, but it is more than sufficiently so for ordinary purposes. Before we expose it, let us observe: 238. 1. That in adding 1, 2, 3, etc. units to the index of the logarithm of a number, we multiply this number by 10, 100, 1000, etc., since this is to add the logarithm of 10, of 100, or 1000, etc.(219 and 227.) 2. On the contrary, if we subtract 1, 2, 3, etc. units from the index of a logarithm, we divide the corresponding number by 10, 100, 1000, etc. 239. This established, let it be required to find the logarithm of 357859. I shall separate, by a comma, upon the right of this number, as many figures as are necessary, so that the remainder may be found in the table. Here, for instance, I shall separate two, this will give me 3578,59, which (28) is a hundred times less than the proposed number 357859. I seek in the table the logarithm of 3578, and opposite 357, and under 8 I find that the decimal part of this logarithm is ,553640, to which, because the number consists of four figures, I prefix the index 3, and I have 3,553640; I take at the same time by the side of this logarithm,† the difference 122, between this logarithm and that of 3579; after which I make this rule of three: if, for 1, unit of difference between the two numbers 3579 and 3578, we have 122 difference between their logarithms, how much for 0,59 difference between the two numbers 3578,59 and 3578, shall we have of difference between their logarithms? That is to say, I seek the fourth term of a proportion, of which the three first are: 1: 122: : 0,59: This fourth term is 71,98, or 72 adding a unit to 71 because the decimals separated exceed 0,5. I add then 72 to the logarithm 3,553640 of 3578, and I have 3,553712 for the logarithm of 3578,59; all that is now required to have that of 357859, is to add two units to the index of the logarithm + The differences are found by the side of the logarithms in the column marked Diff. which we have found; we shall therefore have 5,553712 for the logarithm sought, since 357859 is 100 times greater than 3578,59. If the figures which we separate on the right be all ci phers, after having found in the table the logarithm of the part which remains on the left, all that is required is to add as many units to the index as we shall have separated ciphers. 240. If we require the logarithm of a number accompa nied by decimals, we shall seek this logarithm, as if the proposed number had no comma; and having found it, whe ther immediately in the table, or by the method given, (239,) we shall subtract as many units from the index, as there are decimals in the proposed number, because having considered the number as if it had no comma, that is to say, as 10, 100, or 1000, etc. times greater than it is, we should bring it to its value by a suitable diminution of the index of its logarithm. 241. Lastly, if there be nothing but decimals in the proposed number, we shall still seek this number in the table as if it had no comma; and having taken the corresponding logarithm, we shall subtract it from as many units as there are decimals in this same number, and to the number we shall prefix the sign; for example, to have the logarithm of 0,03, I seek that of 3, which is 0,477121: Isubtract it from 2 units, and prefixing to the remainder the sign, I bave— 1,522879 for the logarithm of 0,03. In effect, 0,03 is nothing else than now, to have the logarithm of 3, we must (235) subtract the logarithm of 3, from that of 100, and prefix to the remainder the sign-. OF THE LOGARITHMS OF WHICH THE NUMBERS ARE NOT FOUND IN THE TABLE. 242. This research is no less necessary than the prece ding. For example, in division, it rarely happens that the quotient is a whole number. Now, if we perform the ope ration by logarithms, we shall not find in the table the remaining logarithm, except when the quotient shall be a whole number. There is an infinity of other cases of the same kind. 243. Let us propose to find the number answering to a proposed logarithm, whether it exceed the limits of the table, or whether it fall between the logarithms of the table. We shall subtract from the index as many units as are necessary in order to find in the table the first figures of the proposed logarithm thus prepared. If all the figures be then found in the table, the number sought will be the same number that we find opposite to this logarithm in the table; but we must place as many ciphers on the right of it as we have taken units from the index.(238.) For example, the logarithm 7,696007 is found (after baving taken 4 units from its index,) to answer to the number 4966; I therefore conclude that the proposed logarithm 7,696007, answers to 49660000. If we find in the table only the first figures of the loga. rithm, we operate as in the following example : To find the number answering to the logarithm 5,243276, I take 2 units from its index; the logarithm 3,243276 which I then have, falls between the logarithms of 1750 and 1751: the number to which it answers is then 1750 and a fraction. In order to have this fraction, I subtract from my loga. rithm 3,243276, the logarithm of 1750, and I have for the difference 238. I take also in the table the difference 248 between the logarithms of 1751 and 1750; after which I make the following Rule of Three: If 248, the difference between the logarithms of 1751 and 1750, Answers to 1, unit of difference between these numbers, To what difference of the numbers should 238 answer, which is the difference between my logarithm and that of 1750? 489 I find for the fourth term 23, or 12; therefore, the logarithm 3,243276 belongs to the number 175011, very nearly; consequently, the proposed logarithm, which belongs to a number 100 times greater, (238) has for its correspondent |