Jan. 5, 1888] 



NATURE 



237 



THE ART OF COMPUTATION FOR THE 



PURPOSFS OF SCIENCE. 



I. 



HTHE art of computation as distinguished from the science of 

 arithmetic it so generally neglected in our ordinary 

 courses of education, that most men and almost all women 

 , feel the greatest difficulty and repugnance in dealing with 

 figures. The causes of and remedies for this deficiency are 

 discussed at some length in a paper "On teaching Arith- 

 metic " (Journal of Education, May 1885), and the following 

 remarks refer specially to the requirements of students of 

 science. 



I must apologize for the use in proving my case of some 

 names of high and well-deserved repute. Instances are given, 

 as far as possible, which have been publicly acknowledged or 

 corrected, with the full admission that this paper is itself a house 

 of glass, and that any stone may impinge even upon Newton, 

 since, as Lord Lytton tells us, "that great master of calculations 

 the most abstruse could not accurately cast up a sum in 

 addition. Nothing brought him to an end of his majestic 

 tether like dot and carry one." 



In 1867 Mr. Stone pointed out two numerical errors in 

 Leverrier's determination of the solar parallax. 



Prof. J. D. Van der Plats writes {Chemical Neivs, July 30, 

 1886) : — " The verification to which I have submitted the calcula- 

 tions of M. Stas seems superfluous seeing that it deals with the 

 experiments of a savant who has never had an equal in 

 exactitude. It may perhaps astonish some as much as it did me 

 to find that the original memoirs contain numerous arithmetical 

 mistakes, as well as typographical errors, of which some are 

 considerable. " 



Mr. J. Y. Buchanan writes (Nature, vol. xxxv. p. 76) : — 

 " There is a statement in Nature for November il that the 

 weight of the column of water between 20 fathoms and 70 

 fathoms from the surface under the westerly equatorial current 

 is only 88 per cent, of the weight of the same column under 

 the easterly counter equatorial current. I regret that a serious 

 arithmetical error occurs in the calculation on which this state- 

 ment was founded. There is no such considerable difference of 

 weight in the two columns of water." Suppose at the equator 

 the Guinea current flows froai west to east at the rate of 40 knots 

 in twenty-four hours, and that the equatorial current flows at 

 30 knots in twenty-four hours in the opposite direction. The 

 opposite directions of the two currents cause an additive and 

 subtractive difference in the tangential velocity of the particles 

 of water due to the rotation of the earth, and therefore an 

 apparent difference in the acceleration due to gravity of about 

 1/46000, or a pressure equal to that of an additional 1/13 of an 

 inch of water on the column of 50 fathoms. 



On page 84 of i\Y& first edition (the second has been corrected) 

 of Prof Huxley's admirable "Physiography," we read : — " The 

 weight of air on a square mile is about 590, 129,971,200 lbs., and 

 the carbonic acid which it contains weighs not less than 

 3,081,870,106 lbs., or about 1,375,834 tons. The weight of 

 the carbon in this carbonic acid is 371,475 tons." 



This short statement contains excellent examples of many 

 of the common arithmetical slips and errors. 



The first number is ten times too great, and not quite 

 accurately calculated from the data (5280)^ x 144 x 1473 



= 59,133,431,808. Multiplying this by -5^^^, the proportion 



by mass of carbonic acid in the air, we obtain 31,464,899 ; here, 

 besides a slip, the number is again multiplied by ten. The 

 pounds are reduced to tons correctly, but there is a slip in the 

 reduction to carbon, since 



1.375,834 X 3 _ 



3.'5,-27- 



Many more instances might easily be brought forward, but 

 the above will suffice to prove that even the highest attainments 

 in science are too often accompanied by inaccuracy in arithmetic. 

 The causes of this defect have been frequently discussed, but, 

 with the exception of De Morgan and his pupils, little advance 

 in the methods of teaching arithmetic seems to have been made 

 since the days of Recorde and Cocker. 



The teachers of arithmetic in our public and higher-grade 

 schools are usually good mathematicians who, in their own 



school-days, have been hurried through the hated subject to 

 higher work, and have had no subsequent experience in the 

 practical computation required in the laboratory, worksho]), or 

 counting-house. When compelled to work out a sum for them- 

 selves, the theory is supplied by their knowledge of algebra, and 

 the practical work by a table of logarithms. When brought 

 face to face with the fact that their pupils dislike and are very 

 weak in arithmetic, they fall back upon the stock argument that 

 they teach arithmetic as a training for the mind, and not as a 

 useful art. In too many cases it is to be feared that they are 

 not teaching arithmetic at all. 



The great majority of the text-books in common use seem to 

 be defective from the point of view of a student of science in 

 at least three points. 



More than half the rules and examples are devoted to money, 

 and arithmetic is treated as though it applied only to pounds, 

 shillings, and pence. 



Secondly, few give any suggestion as to the use of tables in 

 lightening arithmetical work, and a boy leaves school disgusted 

 with long rows of figures in which he sees no utility, and 

 without any idea to how large an extent the work could be 

 lightened. 



Lastly, the various methods of dealing with approximate 

 quantities are omitted, and a painstaking boy calculates vast 

 collections of figures of which only two or three have any 

 meaning. 



Thus Prof. Huxley gives the tenth figure, 6, in the expression 

 for the amount of carbonic acid on a square mile, ignoring the 

 facts that while the percentage of carbonic acid varies in the first 

 figure, its density is not known to the fourth, and the pressure of 

 the air varies in the second. 



It is convenient to bear in mind the following simple rules, due, 

 I believe, to De Morgan. If two numbers, a and b, each true 

 to the first decimal place, are multiplied together, the result is 



true to only : a second true decimal in each number makes 



20 ^ 



the result ten times more correct, and so on. In dividing ajb 

 where each is true to the first place, the result is true to 



r^ • and so on. Any attempt at greater accuracy in calculation 



than is indicated by these results should be avoided, since it only 

 precludes the use of cheap and handy tables, tires the calculator 

 making him more liable to error in the important figures, and 

 tends to give quite a false idea of the accuracy of the experiments 

 on which the calculations are based ; unless, indeed, we take 

 seriously the answer of Dulong when asked why he always gave 

 his results to eight figures, " I don't see why I should erase the 

 last decimals, for, if the first figures are wrong, possibly the last 

 are correct." 



The natural tendency of the human mind, even if controlled 

 by mathematical and scientific training, is to exalt the accuracy 

 of one's own experiments. This is well shewn by Prof. Ramsay 

 and Dr. Young in discussing the vapour-tension of liquid benzene 

 (Proc. Phys. Soc, January 1887) : — 



" A curve was drawn to represent these (experimental) re- 

 lations, and from it three points were chosen, 0° C. 26*54 mai., 

 40° C. i8o'2 mm., and 80° C. 755 mm. The constants for the 

 formula log/= a + M are a — 472452, log b{-) = 0"5l85950, 

 log o = I '996847125." Nine places of decimals are given with 

 apparent confidence, when (i) only three of the whole num'ier 

 of experiments were made even in duplicate ; (2) the last 

 pressure, 755, was obtained not by experiment at all, but by extra- 

 polation from a freehand curve, the highest experiment being 

 79° -6 and 743*1 mm. ; (3) a difference of ^° at low temperatures 

 produced no change in pressure which was appreciable by the 

 apparatus used. With the above-mentioned constants the 

 author's calculate for 60° C. 388 "51 ram. Using their data and 

 a table of four-figure logarithms, I find a — 47239, b = -yi, 

 log a = i "99684, which gives for 60° C. 390 mm. Regnault gives 

 390T mm. 



Under suitable conditions the observation of one quantity can 

 be made with great exactness. It is possible that Sir George 

 Airy estimated i/ioo of a second in a day, or 1/8,640,000 ; that 

 a balance can be made to estimate 1/1,000,000 of the load, 

 though those of Stas were only accurate to 1/825,000 ; and that 

 Sir J. Whitworth measured the 1/1,000,000 of an inch. These 

 cases, however, are exceptional, and give quite a wrong idea 

 of the accuracy attainable in ordinary observations and ex- 

 periments, when several operations, each liable to error, have to 



