238 



NATURE 



\yan. 5, 



be performed, and various corrections introduced by calculation 

 from extraneous data. 



The more closely we examine work of the highest accuracy the 

 more convinced we become of the truth of the statement of 

 Thomson and Tait (p. 333) : " Few measurements of any kind 

 are correct to more than six significant figures." Thus the 

 number of inches in a metre was found by Capt. Kater in 182 1 

 to be 39'37079, and by General Clarke in 1866 to be 39-37043 ; 

 this fundamental datum therefore is affected by a doubt of nearly 

 1/100,000, which of course affects all results dependent on it. 

 In 1856 Miller found that a cubic foot of water at 62° F. weighs 

 62-321 lbs. From Kater's result a cubic foot contains 28-3153 

 cubic decimetres, and the mean of a large number of experi- 

 n^ents, especially those ofLefevre Gineau, and Kupfifer, make 

 the cubic decimetre of water at 4° C. to weigh a kilo 

 = 2-20462125 lbs. according to Miller. Hence a cubic foot of 

 water at 4° C. weighs 62-4255 lbs. ; and taking the expansion 

 from Fdrster (1870), which is nearly identical with that used by 

 Miller, the weight at i6°-67 C. becomes 62-355 lbs. ; or about 

 1/2000 heavier than Miller's determination. But these are the 

 results obtained by picked men under all conditions to insure 

 the greatest accuracy. Results which agree to two or three in the 

 fourth figure show an exceptionally good chemist, while a 

 physicist must be careful indeed to obtain numbers concordant 

 to the fifth figure. 



_ For practical purposes, then, calculations in science may be 

 divided into two classes. The great majority of experiments in 

 physics, chemistry, biology, geodesy, mensuration, navigation, 

 and crystallography are not to be trusted beyond the fourth or 

 fifth figure. Hence a similar accuracy in calculation is all which 

 IS required. Some few experiments in each branch— such as the 

 work of Kater, Regnault, Stas, some observations in astronomy, 

 and a few reductions in sociology — may require six or eight 

 figures to be accurately dealt with. 



In pure mathematics, of course, numerical results may be pushed 

 to any extent compatible with even the partial sanity of the 

 calculator. 



The following suggestions are intended to assist such of my 

 readers as are not mathematicians in working sums of each class 

 by the aid of tables. 



Mechanical aids, such as slide rules, arithmometers, and the 

 like, are purposely omitted, since they would require a paper to 

 themselves. The objection to the larger and more powerful is 

 that they are expensive and complicated ; that they require a good 

 deal of practice on the part of the operator to give accurate 

 results ; and that they are not readily adapted to work shorter 

 sums than they are intended for. On the other hand, a slide 

 rule IS an almost indispensable servant when once one has learnt 

 the use of it for dealing rapidly with comparatively small 

 numbers ; for large numbers it becomes very cumbrous. 



The two cardinal points in approximate working are the short 

 methods of multiplying and dividing decimals suggested by 

 Oughtred in 1631, and strengthening the last figure retained 

 when the first omitted is above 4. For greater accuracy it is 

 well to mark all strengthened figures, and to allow for an excess 

 or defect of them ; as a further security one figure beyond what ■ 

 is required may be calculated. | 



Tables of the multiples from i to 9 of numbers which fre- 

 quently occur are of great assistance especially when the 

 calculator is tired. They are easily made by repeated additions 

 or by the use of the convenient "automatic multiplier" of 

 Mr. Sawyer, which is merely a modern adaptation of Napier's 

 bones. 



J ''i!^^-^."^ "^^ °f complements and reciprocals saves a good 

 deal of time in subtraction and division. 



Tables for general use and special purposes are very numerous. 

 For our present purpose they fall naturally into three classes. 

 l<ive kinds of tables should be in the hands of all calculators — 



1. Multiplication tables such as those of Crelle, by the aid of 

 which three figures may be dealt with at once with greater 

 certainty than is usually the case with one. Tables of primes 

 and factors are not much required for scientific purposes. 



2. Reciprocals, which reduce division to the short multiplica- 

 tion of decimals, render the addition of fractions easy, and 

 assist chemists in percentage compositions. 



3. Squares, cubes, square roots, cube roots. For most purposes 

 m chemistry and physics a small table up to 100 is sufficient, 

 especially when aided by the following convenient method of 

 approximating to a cube root. If a? be the nearest exact cube to 

 the given number N, N= (a ± bf = a^± ^aH + ^aP ± P or if 



d be small, ± l> = ?L-|'. Thus to find ^/Jg, a ^ 3, d = ^^7 

 = '037> ■'■ y/^8 = 3 "037 instead of 3-0366. 



De Morgan's edition of Barlow is very convenient, and suffices 

 for all ordinary purposes. 



4- Common logarithms to four and five figures. Four-figure 

 tables ^ are perhaps most convenient on one face of a card. 

 Hoiiel's reprint of Lalande, with some changes and many valu- 

 able additions, is cheap and most convenient in form ; it quite 

 suffices for all common work. 



For the reasons already mentioned seven-figure tables are 

 unnecessarily cumbrous and expensive for ordinary work. They 

 should never be put into the hands of beginners, as is now the 

 usual practice. Experience shows that boys learn the method of 

 using and appreciate the value of logarithms far more readily 

 than IS generally supposed. 



_ 5. Gauss's sum and difference logarithms are valuable in deal- 

 ing with certain trigonometrical formulae and with questions of 

 expansion. 



In the second class may be placed those general tables which 

 are less commonly required, 5uch as : — 



1. Powers of 2 and other numbers. Cohn tells us that 

 some varieties of Bacterium multiply by fission every hour, hence 

 by the end of a day one individual would increase to 2^^ = 

 16,777,216. We may therefore cease to wonder at the rapid 

 spread of some forms of infection. 



2. Factorials are required in solving permutations and com- 

 binations, and therefore in all questions relating to probabilities. 

 Hatchett recommended that a systematic examination of all 

 possible alloys of all the metals should be undertaken. He 

 forgot to remind anyone who attempted to follow his advice that 

 if only one proportion of each of thirty common metals were 

 considered, the number of binary alloys would be 435, of ternary 

 4060, and of quaternary 27,405. If four multiples of the atomic 

 weight of each of the thirty metals be taken, the binary com- 

 pounds are 5655, ternary 247,660, quaternary 1,013,985. 



3. The sums of arithmetical series are so readily obtained 

 that they are rarely tabulated. 



4. Geometrical series are required in certain social questions, 

 such as the increase of population and the output of coal. 

 Tables of the sums of these series when the ratio is nearly one 

 are common, and of considerable use in some scientific problems. 



5. For some purposes it is convenient to express numbers in 

 a scale different from the common decimal one. 



Thus (Clerk Maxwell, " Elementary Electricity," p. 180) a 

 series of resistance coils are best arranged according to the 

 powers of 2, since the smallest number of separate coils is 

 required, and they are most readily tested. The same is true for 

 a set of weights. Thus to express from i to 100 gm. 9 weight 

 are ordinarily provided ; 9 weights in the scale of 2 will expre> 

 up to 511 gm., while 7 weights suffice for 100 gm., since 100 in 

 the scale of 2 is expressed by iiooioo. 



6. The curious theory of trees due to Profs. Cayley and 

 Sylvester (B. A. Report, 1875) seems to promise the possibility 

 of computing the number of possible compounds formed by 

 elements of given valency. Thus x atoms of tetravalent carbon 

 will combine with monad hydrogen to form N compounds. 



9 

 10 



N 

 18 

 42 

 96 

 229 

 549 



X N. 



11 1346 



12 3326 



13 8329 



If of the first thirteen paraffin hydrocarbons alone there are 

 13)952 possible forms each with its own series of derivatives, 

 there seems little chance of chemists having nothing to do for 

 some time to come. 



7. Natural logarithms are required by some formulae, and ari 

 at times more convenient than common logarithms. 



According to Haughton ("Animal Mechanics," p. 282), th.. 

 study of the action of certain muscles requires the use of natural 

 logarithms. 



The ratio of the mean absolute pressure P to the initial 

 absolute pressure / in a steam-cylinder at the given rate of 



expansion r is expressed by - r= " "*" "^^- ^^ ^ . 

 p r 



Weldon supposed (B. A. Report, 1881), that some power of 



