278 CALCULATING PRODIGIES [CH. XIII 



a certain line of print chosen at random in a quarto page he 

 instantly gave the correct number (63); shown twelve digits 

 he had in half a second memorized them and their positions so 

 as to be able to name instantly the particular digit occupying 

 any assigned place. It is to be regretted that we do not know 

 more of these performances. Those who are acquainted with 

 the delightful autobiography of Robert-Houdin will recollect 

 how he cultivated a similar power, and how valuable he found 

 it in the exercise of his art. 



Dase's calculations, when also allowed the use of paper and 

 pencil, were almost incredibly rapid, and invariably accurate. 

 When he was sixteen years old Strasznicky taught him the use 

 of the familiar formula tt/4 = tan -1 (^) + tan -1 (£) + tan -1 (£), and 

 asked him thence to calculate ir. In two months he carried 

 the approximation to 205 places of decimals, of which 200 

 are correct*- Dase's next achievement was to calculate the 

 natural logarithms of the first 1,005,000 numbers to 7 places of 

 decimals ; he did this in his off-time from 1844 to 1847, when 

 occupied by the Prussian survey. During the next two years 

 he compiled in his spare time a hyperbolic table which was 

 published by the Austrian Government in 1857. Later he 

 offered to make tables of the factors of all numbers from 

 7,000 000 to 10,000,000 and, on the recommendation of Gauss, 

 the Hamburg Academy of Sciences agreed to assist him so that 

 he might have leisure for the purpose, but he lived only long 

 enough to finish about half the work. 



Truman Henry Safford, born in 1836 at Eoyalton, Vermont, 

 U.S.A., was another calculating prodigy. He was of a some- 

 what different type for he received a good education, graduated 

 in due course at Harvard, and ultimately took up astronomy in 

 which subject he held a professional post. I gather that though 

 always a rapid calculator, he gradually lost the exceptional 

 powers shown in his youth. He died in 1901. 



Safford never exhibited his calculating powers in public, and 

 I know of them only through the accounts quoted by Scripture 



* The result was published in Crelle's Journal, 1844, vol. xxvn, p. 198 : oa 

 closer approximations and easier formulae, see below chapter xiii. 



