&c. then the results of the successive operations will be as in the following table. N = I = √FXH = 4.958069 0.6953125 K = Ꮮ = ✔HXI = 5.002865 0.6992187 = VIXK 4.980416 0.6972656 M= √KXL: = 4.991627 √KXM = 4.997240 0.6982421 0.6987304 As the last of these means, viz. Z, agrees with 5, the proposed number, as far at least as the 6th place of decimals, we may safely consider them as very nearly equal. Therefore their logarithms will also be very nearly equal, that is, the logarithm of 5 will be 0.6989700 nearly. By a like process, the logarithm of any number may be found. From the logarithm of 5, and the logarithm of 10, which in Briggs' system is 1, we may find the logarithm of 2, the quotient of 10 divided by 5; we have only to subtract the logarithm of 5 from the logarithm of 10, and the remainder is 0.3010300, the logarithm of 2. The Logarithms of 20, 200, 2000, &c. are obtained from the logarithms of 2 and 10, 100, 1000, &c. by adding the logarithms of 2 and 10 for the logarithm of 20, and so on. It is evident that the logarithm of any number, and the logarithm of the product of th number by any power of 10, will be the same in all the gures after the decimal point. If, in addition to the logarithms of the numbers 2 and 5, the logarithms of 3 and 7 were found, each by a series of interpolations, we might find the logarithm of many more, such as could be produced from them by multiplication, as 4=2X2, 6=2×· 3,8=2x2x2, 9=3×3, 12=2×2×3, 14=2x7, 15=3x 5, &c. In fact, the chief labor of constructing a table, is the finding of the logarithms of the prime numbers; for these once known, the logarithms of the composite numbers are obtained by addition of the logarithms of their factors. The first computers of logarithms had a prodigious labor to perform, in comparison to what would have been necessary in the present improved state of analytical science. They at first found the logarithms of the prime numbers by the simple operation of interpolating geometrical means between given numbers; and in this way, no fewer than 26 interpolations, each requiring a multiplication of 8 or 10 figures by as many, an extraction of the square root, were wanted for finding the logarithm of the number 2, to seven or eight figures. Briggs, by various contrivances, and particularly by his Differential Method, greatly shortened the process. Gregory, of St. Vincent, observed, in his Quadratura Circuli et Sectionum Coni, (1647,) that if one asymptote of an hyperbola be divided into parts in geometrical progression, and from the points of division, ordinates be drawn parallel to the other asymptote, these will divide the space between the asymptote, and the other curve into equal portions: Hence it was shown by Mersennus, that by taking the continual sum of these parts, there would be obtained areas in arithmetical progression, and which, therefore, were analagous to a system of logarithms. Thus the theory of logarithms, which had at first been regarded as belonging to Arithmetic, was now connected with the sister science, Geometry; and the calculation of these useful numbers was identified with the quadrature of the hyperbola. Having dwelt with some degree of particularity on the history, theory, and construction of logarithms, we now enter upon the solution of the examples in the treatise of Day, where the more modern methods of construction will receive further attention. It will be observed that recourse has sometimes been had to tables which are not in common use, but have recently been prepared to accompany this volume. In some instances, six or seven places of decimals may be sufficient; but in no case is the result, when arrived at, objectionable on the score that logarithms have been employed in the operation, which have been carried to a greater degree of exactness. Three examples under article 28. Example 2. Required the logarithm of 78264. Natural No. 78270 Log. 4.8935953338 Natural No. 78260 Log. 4.8935398436 78264 78260 4.89356202968, the log. of 78264. Example 3. Required the Logarithm of 143542. Natural No. 143600 Log. 5.1571544399 Natural No. 143500 Log. 5.1568519011 143542 143500 42 0012101552 100).0127066296 .000127066296 5.1568519011 5.156978967396, the log. of 143542. Example 4. Required the logarithm of 1129535. Natural No. 1130000 Log. 6.0530784435 1000 1000.0003845016::535: 1129535 1129000 23783.18, the natural number belonging to 4.37627. Required the natural number belonging to 3.69479. .00001×1=1÷9=0.11, and 4952+0.11=4952.11, the natural number belong to 3.69479. Required the natural number belonging to 1.73698. 1.73703 54.58 1.73698 1.73695 54.57 1.73695 8: .01 :: .00003 .00003 × 01=.33, and .33÷8=.00375, and 54.57+00375 =54.57375, the natural number belonging to 1.73698. Required the natural number belonging to 1.09214. 155 .00012 35 000035, and .1236+.000035=.123635, the natural number belonging to 1.09214. Note. In finding by proportion, the logarithm of a number or the natural number belonging to a logarithm, the operation may sometimes be carried on to a little better advantage, by considering x the fourth term in the proportion, and then converting the proportion into an equation. Multiplication by Logarithms. Art. 37. 11x=3, x=.2727, and 329.2727=the answer. Example 6. Article 38. Natural number, .00845 Log. 3.92686 or 7.92686 3.02857 5x=.003, x=.0006, and 9.0246=the answer. 0.036225.38=2.55871+1.40449-1.96320=0.9188. |