NO. 0] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 9 



Table IV: Values of e* to 62 places of decimals at decimal intervals from 1 X 10-'° to 

 9 X 10-'. 



Table V: Values of er^ ranging from 52 to 62 places of decimals at intervals of unity 

 from to 100. 



Table VI: Values of e~^ ranging from 33 to 48 places of decimals at intervals of 0.1 from 

 0.0 to 50.0. 



Table VII: Values of e~^ to 62 places of decimals at decimal intervals from 1 x 10"'° to 

 9 X 10-'. 



Table VIII: Values of giCnWaeo) ^q 23 places of decimals or significant figures at intervals 

 of unity from n = to n = 360. 



Table IX: Values of «='='" to 25 places of decimals or significant figures for various values 

 of n. 



Table X: Values of sin x and cos x to 23 places of decimals at intervals of unity from 

 to 100. 



Table XI: Values of sin x and cos x to 23 places of decimals at intervals of 0.1 from 0.0 

 to 10.0. 



Table XII: Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from 

 0.000 to 1.600. 



Table XIII: Values of sin x and cos x to 25 places of decimals at decimal intervals from 

 1 X 10-'° to 9 X lO-*. 



Table XIV; Miscellaneous values of e^, e~^, sin x and cos x to a great number of decimals, 

 including Boorman's value of e. 



The tabular error of the preceding*tables may in some cases slightly exceed 5 units in the 

 next succeeding tabular digit as two digits only were dropped from most of the values. 



In my preliminary computations, Glaisher's ' table of the reciprocals of the factorials 

 was used. It contains all of the recurring decimals from n=l to 71=12, inclusive, and from 

 71=13 to n = 50, inclusive, the values are given to 28 significant figures. This table frequently 

 failed to give the requisite number of decimals in the vicinity of 7i=13 and upwards, so it was 

 afterwards extended rouglily to 110 decimals, and the range of the argument extended from 

 50 to 74, the results being verified by forming the summations for e and e~^, and then computing 

 the product of these two quantities, in addition to making a direct comparison with the well 

 known value of e. A further check on the value of e"' consisted in reciprocating e, written in 

 the form (a + 6)-'. 



With the table of reciprocal factorials as a basis, it was easy to compute the value of e*-' 

 from the series: and afterwards by repeated multiphcations by this factor, in accordance with 

 the formula, 



gZ+Ax=gZ. gAi (2)^ 



the values of e"', e"*, ... e were obtained and verified by comparing the last computed 

 value with the well known value of e. Similarly the value of e'" was computed from e and the 

 value of e'°° was computed from e'°. Values of the descending exponential for the same inter- 

 vals of argument were determined in the same manner, and the evaluation of both functions 

 was verified at frequent intervals by means of the product relation, e^e"*. Another check con- 

 sisted in substituting values of x and Ax in (2). Subsequent interpolations to one- tenth the 

 previous interval of interpolation provided a further complete check on the entire computation. 

 The maximum difference between any value and the corresponding value obtained by 10 

 interpolations did not exceed 15 units in the last decimal or significant figure. Practically all 

 of the computations were made with a 10-groove computing machine of the millionaire type. 



> Z. W. L. Olalsher, Tables of the exponential function. Trans. Camb. Phil. Soc., vol. 13 (1883), pp. 244-272. 



• C. F. Degen, Tabularam Enneas (Copenhagen 1824) gives IS-place values of logi« (n!) from n— 1 to n— 1200. De Morgan reprinted the same 

 to 6 places in his article on " Probabilities " in Encjclopedia MetropoUtana. J. \V. L. Glaisher gives 20-place values of n X nl and lO-place values 

 ot-log (nXn!) to n-71 in Phil. Trans. Roy. Soc., vol. 160, 1S70, p. 370. Shortrede, Tables (1849, Vol. I) contains 5-place values of log (n!) to 

 n- 1000 and 8-place values for arguments ending In 0. 



