### Part I. Section I. Chapter 23. “Of the Method of expressing Logarithms.”

242 We have seen that the logarithm of 2 is greater than ³⁄₁₀, and less than ⅓, and that, consequently, the exponent of 10 must fall between those two fractions, in order that the power may become 2. Now, although we know this, yet whatever fraction we assume on this condition, the power resulting from it will always be an irrational number, greater or less than 2; and, consequently, the logarithm of 2 cannot be accurately expressed by such a fraction: therefore we must content ourselves with determining the value of that logarithm by such an approximation as may render the error of little or no importance; for which purpose, we employ what are called decimal fractions, the nature and properties of which ought to be explained as clearly as possible.

243 It is well known that, in the ordinary way of writing numbers by means of the ten figures, or characters,

0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

the first figure on the right alone has its natural signification ; that the figures in the second place have ten times the value which they would have had in the first; that the figures in the third place have a hundred times the value; and those in the fourth a thousand times, and so on: so that as they advance towards the left, they acquire a value ten times greater than they had in the preceding rank. Thus, in the number 1765, the figure 5 is in the first place on the right, and is just equal to 5; in the second place is 6; but this figure, instead of 6, represents 10 · 6, or 60: the figure 7 is in the third place, and represents 100 · 7, or 700; and lastly, the 1, which is in the fourth row, becomes 1000; so that we read the given number thus;

One thousand, seven hundred, and sixty-five.

244 As the value of figures becomes always ten times greater, as we go from the right towards the left, and as it consequently becomes continually ten times less as we go from the left towards the right; we may, in conformity with this law, advance still farther towards the right, and obtain figures whose value will continue to become ten times less than in the preceding place: but it must be observed, that the place where the figures have their natural value is marked by a point. So that if we meet, for example, with the number 36 · 54892, it is to be understood in this manner: the figure 6, in the first place, has its natural value; and the figure 3, which is in the second place to the left, means 30. But the figure 5, which comes after the point, expresses only ⁵⁄₁₀; and the 4 is equal only to ⁴⁄₁₀₀; the figure 8 is equal to ⁸⁄₁₀₀₀; the figure 9 is equal to ⁹⁄₁₀₀₀₀; and the figure 2 is equal to ²⁄₁₀₀₀₀₀. We see then, that the more those figures advance towards the right, the more their values diminish, and at last, those values become so small, that they may be considered as nothing.

245 This is the kind of numbers which we call decimal fractions, and in this manner logarithms are represented in the Tables. The logarithm of 2, for example, is expressed by 0.3010300; in which we see, first, That since there is 0 before the point, this logarithm does not contain an integer; second, that its value is

³⁄₁₀ + ⁰⁄₁₀₀ + ¹⁄₁₀₀₀ + ⁰⁄₁₀₀₀₀ + ³⁄₁₀₀₀₀₀ + ⁰⁄₁₀₀₀₀₀₀ + ⁰⁄₁₀₀₀₀₀₀₀.

We might have left out the two last ciphers, but they serve to show that the logarithm in question contains none of those parts, which have 1000000 and 10000000 for the denominator. It is however to be understood, that, by continuing the series, we might have found still smaller parts; but with regard to these, they are neglected, on account of their extreme minuteness.

246 The logarithm of 3 is expressed in the Table by 0.4771213; we see, therefore, that it contains no integer, and that it is composed of the following fractions:

⁴⁄₁₀ + ⁷⁄₁₀₀ + ⁷⁄₁₀₀₀ + ¹⁄₁₀₀₀₀ + ²⁄₁₀₀₀₀₀ + ¹⁄₁₀₀₀₀₀₀ + ³⁄₁₀₀₀₀₀₀₀.

But we must not suppose that the logarithm is thus expressed with the utmost exactness; we are only certain that the error is less than ¹⁄₁₀₀₀₀₀₀₀; which is certainly so small, that it may very well be neglected in most calculations.

247 According to this method of expressing logarithms, that of 1 must be represented by 0.0000000, since it is really = 0: the logarithm of 10 is 1.0000 000, where it evidently is exactly = 1: the logarithm of 100 is 2.0000000, or 2. And hence we may conclude, that the logarithms of all numbers, which are included between 10 and 100, and consequently composed of two figures, are comprehended between 1 and 2, and therefore must be expressed by 1 plus a decimal fraction, as $$\log 50=1.6989700$$; its value therefore is unity, plus

⁶⁄₁₀ + ⁹⁄₁₀₀ + ⁸⁄₁₀₀₀ + ⁹⁄₁₀₀₀₀ + ⁷⁄₁₀₀₀₀₀:

and it will be also easily perceived, that the logarithms of numbers, between 100 and 1000, are expressed by the integer 2 with a decimal fraction: those of numbers between 1000 and 10000, by 3 plus a decimal fraction: those of numbers between 10000 and 100000, by 4 integers plus a decimal fraction, and so on. Thus, $$\log 800$$, for example, is 2.9030900; that of 2290 is 3.3598355, etc.

248 On the other hand, the logarithms of numbers which are less than 10, or expressed by a single figure, do not contain an integer, and for this reason we find before the point: so that we have two parts to consider in a logarithm. First, that which precedes the point, or the integral part; and the other, the decimal fractions that are to be added to the former. The first part of a logarithm, which is usually called the integral part, is easily determined from what we have said in the preceding article. Thus, it is 0, for all the numbers which have but one figure; it is 1, for those which have two; it is 2, for those which have three; and, in general, it is always one less than the number of figures. If therefore the logarithm of 1766 be required, we already know that the first part, or that of the integers, is necessarily 3.

249 So reciprocally, we know at the first sight of the integral part of a logarithm, how many figures compose the number answering to that logarithm; since the number of those figures always exceed the integral part of the logarithm by unity. Suppose, for example, the number answering to the logarithm 6.4771213 were required, we know immediately that that number must have seven figures, and be greater than 1000000. And in fact this number is 3000000; for $$\log 3000000 = \log 3 + \log 1000000$$. Now $$\log 3 = 0.4771213$$, and $$\log 1000000 = 6$$, and the sum of those two logarithms is 6.4771213.

250 The principal consideration therefore with respect to each logarithm is, the decimal fraction which follows the point, and even that, when once known, serves for several numbers. In order to prove this, let us consider the logarithm of the number 365; its first part is undoubtedly 2; with respect to the other, or the decimal fraction, let us at present represent it by the letter $$x$$; we shall have $$\log 365 = 2 + x$$; then multiplying continually by 10, we shall have $$\log 3650 = 3 + x$$; $$\log 36500 = 4 + x$$; $$\log 365000 = 5 + x$$, and so on.

But we can also go back, and continually divide by 10; which will give us $$\log 36.5 = 1+x$$; $$\log 3.65 = 0+x$$; $$\log 0.365 = -1+x$$; $$\log 0.0365 = -2+x$$; $$\log 0.00365 = -3+x$$, and so on.

251 All those numbers then which arise from the figures 365, whether preceded, or followed, by ciphers, have always the same decimal fraction for the second part of the logarithm: and the whole difference lies in the integer before the point, which, as we have seen, may become negative; namely, when the number proposed is less than 1. Now, as ordinary calculators find a difficulty in managing negative numbers, it is usual, in those cases, to increase the integers of the logarithm by 10, that is, to write 10 instead of 0 before the point; so that instead of -1 we have 9; instead of -2 we have 8; instead of -3 we have 7, etc.; but then we must remember, that the integral part has been taken ten units too great, and by no means suppose that the number consists of 10, 9, or 8 figures. It is likewise easy to conceive, that, if in the case we speak of, this integral part be less than 10, we must write the figures of the number after a point, to show that they are decimals: for example, if the integral part be 9, we must begin at the first place after a point; if it be 8, we must also place a cipher in the first row, and not begin to write the figures till the second: thus 9.5622929 would be the logarithm of 0.365, and 8.5622929 the log of 0.0365. But this manner of writing logarithms is principally employed in Tables of sines.

252 In the common Tables, the decimals of logarithms are usually carried to seven places of figures, the last of which consequently represents the ¹⁄₁₀₀₀₀₀₀₀ part, and we are sure that they are never erroneous by the whole of this part, and that therefore the error cannot be of any importance. There are, however, calculations in which we require still greater exactness; and then we employ the large Tables of Vlacq, where the logarithms are calculated to ten decimal places.1 2

253 As the first part, or integral part, of a logarithm, is subject to no difficulty, it is seldom expressed in the Tables; the second part only is written, or the seven figures of the decimal fraction. There is a set of English Tables in which we find the logarithms of all numbers from 1 to 100000, and even those of greater numbers; for small additional Tables show what is to be added to the logarithms, in proportion to the figures, which the proposed numbers have more than those in the Tables. We easily find, for example, the logarithm of 379456, by means of that of 37945 and the small Tables of which we speak.

254 From what has been said, it will easily be perceived, how we are to obtain from the Tables the number corresponding to any logarithm which may occur. Thus, in multiplying the numbers 343 and 2401; since we must add together the logarithms of those numbers, the calculation we be as follows:

$\begin{array}{rrr} \log 343&=&=2.5352941\\ \log 2401&=&3.3803922\\ \hline \textrm{their sum}&=&5.9156863\\ \textrm{nearest tabular log} \qquad \log 823540&=&5.9156847\\ \hline \textrm{difference}&=&0.0000016 \end{array}$

which in the Table of Differences answers to 3; this therefore being used instead of the cipher, gives 823543 for the product sought: for the sum is the logarithm of the product required; and its integral part 5 shows that the product is composed of 6 figures; which are found as above.

255 But it is in the extraction of roots that logarithms are of the greatest service; we shall therefore give an example of the manner in which they are used in calculations of this kind. Suppose, for example, it were required to extract the square root of 10. Here we have only to divide the logarithm of 10, which is 1.0000000 by 2; and the quotient 0.5000000 is the logarithm of the root required. Now, the number in the Tables which answers to that logarithm is 3.16228, the square of which is very nearly equal to 10, being only one hundred thousandth part too great.

#### Editions

1. Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
2. Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.
1. Frank J. Swetz (The Pennsylvania State University), “Mathematical Treasures - Adriaan Vlacq’s Logarithms,” Convergence (December 2019)