340 POPULAR SCIENCE MONTHLY. 



is probably not true, but it is, perhaps, worth repeating, as it was 

 credited at the time and casts a light on the age. During his short 

 life Eegiomontanus accomplished much, and gave promise of more. 

 In particular he greatly improved the doctrine of trigonometry. Pur- 

 bach and himself were the very first Europeans to utilize the discoveries 

 of the Arabs in this science. As every astronomical calculation de- 

 pends upon the solution of spherical triangles, the tables of sines and 

 ("tangents computed by Eegiomontanus were of fundamental importance, 

 •■since they gave numerical values of these trigonometric functions 

 'Calculated once for all, and saved the computer endless special 

 reckonings. 



It is difficult for us to conceive the state of science in those days. 

 The school-boy problem : given a^ h, c, in a spherical triangle, to find 

 A, B, C, was considered operose by Eegiomontanus and his friends, 

 although the solution had been reached long before, by Albategnius. 

 Blanchini, a contemporary of note, sends Mm the following equations 

 for solution: 



a;:2/ = 5:8; x A^y = x]j. 



A star rises at Venice at 3^ 25™, and transits at 7^^ 38™, after mid- 

 ■:night ; required its longitude and latitude : is a problem addressed to 

 Blanchini, in return. The Arabs five centuries earlier would have 

 "found these questions easy. Eegiomontanus was, nevertheless, the 

 "most accomplished man of science in Europe. The ancients determined 

 the longitude of a planet somewhat as follows : The difference of longi- 

 tude between the planet and the moon was measured (J.) and next the 

 difference of longitude between the moon and the sun (5). The longi- 

 tude of the sun was calculated from the solar tables (0). The sum 

 of A, B and C gave the planet's longitude. In Walther's observatory 

 the angular distances of the planet from known stars were measured 

 and the required longitude and latitude of the planet were calculated, 

 by the formulae of spherical trigonometry, from the known longitudes 

 and latitudes of the stars. The gain in precision was considerable, and 

 the observations could be made on any clear night, whether the moon 

 was or was not above the horizon. 



Walther survived his friend for many years and carried on the 

 observations which they had begun together. It was in their observa- 

 tory that clocks (not pendulum-clocks) were first employed to meas- 

 ure short intervals of time and that observations were first corrected 

 for terrestrial refraction. A star seen through the atmosphere appears 

 higher above the horizon than if the atmosphere were absent. Its 

 apparent position must then be corrected for refraction in order to 

 obtain its true place. At an altitude greater than 45° the correction 

 is less than 1', which was inappreciable before the day of the telescope ; 

 but near the horizon the correction is large (the line of sight passing 



