PROGRESS OF MATHEMATICS AND MECHANICS 245 



Kepler recognized immediately the enormous significance of the 

 new logarithmic method and addressed an enthusiastic panegyric 

 to Napier in 1620, not knowing that he had died in 1617. What if 

 logarithms had been invented in time to save Kepler his vast com- 

 putations ? 



A few years ago we have been shown in a rectorial address what the 

 telescope has meant for observational astronomy. An equally great 

 significance attached to logarithms for the computing astronomer. 



Gutzmer. 



Vlacq of Leyden soon after filled the gap in Briggs' table, and 

 this is the basis for the tables since published. The first tables to 

 base e, commonly called Napierian, were published in 1619. In 

 more recent times methods of interpolation have been employed 

 which are more powerful and less laborious, while ordinary com- 

 putation has been simplified by avoiding the use of too many deci- 

 mal places, and by the mechanical device of the slide-rule. The 

 modern computing machine naturally tends to supersede the 

 logarithmic method. Among the remarkable computations 

 characteristic of the sixteenth century may be mentioned Ludolph 

 von Ceulen's achievement in computing TT to 35 decimal places, 

 using regular polygons of 96 and 192 sides. German writers in 

 consequence have sometimes attached his name to this important 

 constant. 



In England Thomas Harriott (1560-1621) and William Oughtred 

 (1575-1660) rendered important services in introducing the most 

 recent advances in arithmetic, algebra and trigonometry. The 

 former rejected negative and imaginary roots indeed, but used 

 the signs > and <, denotes a 2 by a a, etc. Oughtred uses the 

 symbols X and : :, also the contractions for sine, cosine, etc. 



"Two NEW BRANCHES OF SCIENCE." Even after Galileo's 

 condemnation by the Inquisition, though old, infirm, and nearly 

 blind, his scientific ardor was unquenched, and in 1638 he pub- 

 lished (at Leyden) a work on mechanics under the title, Conver- 

 sations and Mathematical Demonstrations on two New Branches 

 of Science, which constituted the most notable progress in mechan- 

 ics since Archimedes. He says : 



