NA TURE 



471 



THURSDAY, DECEMBER 31, 1914. 



A^ 



CALCULATING DEVICES. 

 Modern Instruments and Methods of Calculation. 

 A handbook of the Napier Tercentenary Ex- 

 hibition. Edited by E. M. Horsburgh. Pp. 

 vii + 343. (London: G. Bell and Sons, Ltd., 

 and The Royal Society of Edinburgh, n.d.) 

 Price 65. net. 



LL who are interested in the history or the 

 methods of calculation owe a debt of grati- 

 tude to the editor and the committee who have 

 produced this valuable book. It is in the recollec- 

 tion of everyone concerned in mathematical opera- 

 tions that the Royal Society of Edinburgh held 

 a great celebration in July last, three hundred 

 years after the publication, by John Napier, of his 

 admirable Canon of Logarithms. This was 

 attended by learned delegates from many distant 

 countries, as well as by a number of our own 

 countrymen. Greatly as many must have re- 

 gretted their inability to be present on account 

 of duties elsewhere, this regret will not be lessened 

 by a perusal of the book now under review, for 

 from it they will learn what a magnificent collec- 

 tion was available for inspection and discussion. 

 This collection of tables, books, portraits, and 

 instruments of various kinds, which must have 

 appealed to the historic as well as the utilitarian 

 and mechanical instincts of those who were for- 

 tunate to be able to attend the celebration, form 

 the basis upon which the present work is con- 

 structed, for it is in minor part catalogue and in 

 major part a series of descriptive articles by 

 experts in the several branches. 



Seeing that the price asked for a large octavo 

 book of 343 pages and containing a very large 

 number of excellent illustrations is only six shil- 

 lings, those who are interested should feel that, 

 like the Nautical Almanack, the work is in effect 

 a gift, the price doubtfully covering the cost of 

 production. 



The first section, by Prof. G. A. Gibson, deals 



with the life of Napier, of his great invention of 



logarithms, of his meeting with Briggs, and 



matters mainly of personal and historical interest. 



Here we read how Napier formulated his ideas 



of the logarithm which was derived, not from 



algebraic methods, as is now found to be most 



convenient, but upon the relative values of the 



portions of two lines determined by the motion 



of two points, one moving uniformly and the 



!• other, starting at the same, but moving with 



^ diminishing velocity such as to be proportional 



k in amount to the length of the part untraversed. 



He thus made his logarithms without reference to 



NO. 2357, VOL. 94] 



a base, and the logarithm of " the whole sine " 

 (the sine of 90°) becomes zero, the logarithm of 

 positive quantities less than unity is positive, and 

 of quantities greater than unity is negative. 

 Curiously, hyperlx)lic logarithms to the base e are 

 not those that Napier calculated, but logarithms 

 to the base i/e. It is difficult to realise now that 

 highly convergent series for logarithms are uni- 

 versally understood how Napier could have calcu- 

 lated as he did the logarithmic series and tangents 

 of all angles from 0° to 90° by intervals of one 

 minute of arc, and this long before the days of 

 the binomial theorem. In this chapter we learn 

 incidentally that Napier invented the decimal 

 point, and we find also a description of the well- 

 known "bones." 



The next two sections are very largely in the 

 nature of catalogues of the articles in the loan 

 collection and of the collection of mathematical 

 tables, but descriptive articles are included, of 

 which one on j)ortable sun-dials, by J. R. Findlay, 

 and an account, written by Dr. Knott, of the great 

 manuscripts by Dr. Sang, given by his daughters 

 to the Royal Society of Edinburgh, may in par- 

 ticular be mentioned, as also the concluding 

 article, by W. G. Smith, on the special develop- 

 ment of calculating ability in prodigies or "calcu- 

 lating boys." 



The further chapters are as follows : " Calcu- 

 lating Machines," by F. J. W. Whipple, but in- 

 cluding special articles by P. E. Ludgate and 

 T. C. Hudson; "The Abacus," by Dr. Knott; 

 "Slide Rules," by G. D. C. Stokes; "Other 

 Mathematical Laboratory Instruments," by a 

 number of specialists. As this chapter includes 

 such varied and elaborate instruments as inte- 

 graphs, planimeters and their use in naval archi- 

 tecture, harmonic analysers, tide predictors, 

 machines for drawing conic sections, for solving 

 equations and precision plotting, many of which 

 are well illustrated and explained by a number 

 of authors having special knowledge, it will be 

 evident that this is one of the most technically 

 difficult and illuminating in the book. Among 

 remaining chapters may be mentioned that on 

 ruled papers and nomograms, which latter are 

 graphical devices by means of which numerical 

 solutions may be found for equations involving 

 several variables. For instance, in the general 

 equation for a spherical triangle showing the rela- 

 tionship between any angle and the three sides. 

 Prof. D'Ocagne has given a nomogram from 

 which, if three of the six quantities are given, the 

 other three can be determined by the aid of a 

 stretched thread. 



The chapters relating to calculating mechanism 

 contain descriptions of numerous recent additions 



T 



