262 



NATURE 



[Jan. 



12, I 



Class III. The develop Jient might just as well be ia inverse 

 order, though we have important reasons for believing it is 

 not so. 



The astronomy of the future must decide between these two 

 alternatives. My object in undertaking this work was to facili- 

 tate this decision by giving as exact descriptions as possible of 

 the spectra presented by the different stars of Class III. in the 

 year 1880. : 



THE ART OF COMPUTATION FOR 

 PURPOSES OF SCIENCE} 



II. 



THE 



SOME few problems in astronomy and certain theories in pure 

 mathematics require more than seven figures to be calculated. 

 In these cases a large arithmometer is generally the most con- 

 venient. Ten-figure tables of logarithms may be obtained second- 

 hand ; or the required logarithms must be calculated. 



The tables of Vlacq, re edited by Vega in 1749, 1794, and 

 1797 are somewhat difficult to obtain and cumbrous to use. 

 The logarithms of numbers up to 101,000 are given to ten 

 figures with first and second differences. Thus to find log 

 10 542 482 375, from the table directly 



log 10 S42 



•0229 230 119 Aj 

 198 712 3 



5 



log required '0229 428 836 3 



The true log of 10 542 482 375 is 

 •022 942 883 626 562. 



=411 946 



482 375 



I 647 784 



329 557 + 



8 239 + 



I 236 + 



288 + 



20 - 



198 712 3 I subtracted. 

 A., = 40 

 •48C48- i)(-40 ) ^ 4.99,. 



In default of Vega, or if more places are required, the log- 

 arithm must be calculated, and this is by no means such a serious 

 affair as one is led to think by the ordinary books on algebra. 

 I am much indebted in what follows to the article by Mr. J. W. 

 L. Glaisher on logarithms in the new edition of the " Encyclo- 

 paedia Britannica," to which I refer my readers for further par- 

 ticulars in theory, restricting myself to practical details. 



The easiest way to calculate a table of logarithms absolutely 

 de novo would be by the method of differences, with some 

 mechanical assistance, such as the difference-engine of Babbage 

 or of Scheulz. It seems unlikely that larger tables will be 

 calculated than those already in existence, since the cost increases 

 with great rapidity. Mr. Sang has, however, recently calculated 

 independently the logarithms of numbers from looooo to 

 200000, where the ordinary tables are weakest. 



Briggs used at least two methods for the calculation of log- 

 arithms which depended upon the extraction of a succession of 

 roots. For instance, by taking the square root of 10 fifty-four 

 times he found log i-{py''>i 278 191 493tobe -(0)150 555 in 512. 

 Whence assuming that very small numbers vary as their log- 

 arithms, log i-(o)i°i = 555 III 512/r 278 191 493, or log 

 i-{py-^i — 0-43 429 448 = M, the modulus. And if x be small, 

 log I -{of^x = ,r X 0-43 429 448. To find log 2 he extracted 

 the square root of the tenth power, 1024/1000 forty-seven times, 

 and found v{oY^i 685 160 570, which multiplied by M gave 

 •(0)1^0731 855936. This multiplied by 2*'' gave log 1'024; 

 adding 3 and dividing by 10 gives log 2. .Another more simple 

 method was to find a series of geometrical means between two 

 numbers, such as 10 and i, the logarithms of which are known. 

 After taking 22 of these roots, log 5 is found to be 0-69897. 



It was soon found that logarithms could be more easily calcu- 

 lated by the summation of various series, and many great 

 mathematicians, such as Newton, Gregory, Halley, Cotes, 

 exercised their ingenuity in discovering those most suitable for the 

 purpose. 



Though for practical purposes the use of series has been 



^ Continued from p. 239. 



almost superseded, three very simple ones are still occasionally 

 useful : — 



log (I ± .v) = M f ± X - ■'■' ± -*"' - -l" ± :^' 

 ^ V 2345 



which converges rapidly \i x be small. M is a number depend- 

 ing upon the system of logarithms adopted, and constant for 

 each system. If M be i, the system is called the Naperian, or 

 natural one ; and if M = 0-434 &c., the system is the common 

 one. Unless otherwise stated M will be assumed to be r, or the 

 logarithms will be natural ones. 



Thus to calculate lo" l-i 



I + — , omitting M : — 

 10 



log I -I = — - 

 •^ 10 



3000 



200 3000 40000 

 01003 3534 - 00050 2517 



500000 

 0-0953 IOI7- 



Suppose X be small, log (l ± x) — ± M.r nearly. Thus if 

 log I •(o)^"9 be required to twenty decimals, it is 



•(o)"9 - 1 (-9 X lo-^T, 



or the error caused by omitting this and all subsequent terms is 

 only 4 in the twenty-first decimal place. Using common 

 logarithms the multiplication by M reduces the error by one- 

 half. This result is of great importance in calculating logarithms 

 by Flower's method, since the factors which have to be dealt 

 with are only half the number of decimal places in the required 

 logarithm. 



Writing — for x in the above series, we obtain — 



log (i -f -v) - log X — M ( _ 



I 



2.r- 



3X3 



I 



4x* 



\xfi~) 



S^ 



which converges rapidly when x is large. Various artifices may 

 he used to render x large, even when the number the logarithm 

 of which is required is small. Thus, Prof. J. C. Adams has 

 calculated (Nature, vol. xxxv. p. 381) log 2, log 3, log 5, log 7, 



— and M to 270 places of decimal?. 

 M 



Another very valuable series is — 



log(«±.r) = loT«±2M.f— :^ + -(-^^\ + H-~-\ + ^''\ 

 ^ ' ° ^2a-t-x ^ 3V2a-i-.r/ 5\2a-i-A7 j 



Thus, supposing log 219 known, to calculate log 2198 : — 



log 2198 = 7-6916 5682 2810 + 2 



■0036 4630 81 13 

 4039 



ijL^ 



3 \2i 



^V-f&c.l 



-i- = -0018 2315 40565 

 2194 



log 2198 = 76593 0313 4962 1(7^) " ■°''^°^^ 



Using common logarithms, the third term of the series is 



< -r { -\ , that is less than 5 in the ninth place when 



27-6 \ a I ^ 



- < . Hence, with a table giving the logarithms of loo-ioco 



to eight figures the third term may be neglected, or the required 



difference is ± -^±-^ , or, writing log (a + x) - log a = }', 

 2a + X 

 _ 2ay 



2M - y 



The given numbers may also be broken up into factors by the 

 aid of such a table as Burkhard's, which gives the factors of 

 all numbers up to 3,036,000, The logarithms of the factors 

 may then be found from tables and added together. Of all 

 tables for this purpose, that of Wolfram is the most valuable ; 

 it gives the natural logarithms to forty-eight places oi all num- 

 bers up to 2200, and of all which are not easily divisible up to 

 10,009. . 



The multiplication by M to convert into common logarithms 

 is tedious, and it is frequently belter to dispense with it in heavy 

 calculations. If necessary, a table of the first ninety-nine mul- 

 tiples of M should be prepared^ and Oughtred's short method of 

 multiplication used. 



If any of my readers desire to test themselves and their tables 



