i 9 4 SCIENCE PROGRESS 



have begun these calculations very much as Buergi some years 

 later began his. He formed two series of numbers, the one in 

 arithmetical progression, and the other in geometrical pro- 

 gression. Buergi began with 100,000,000 and called its " red 

 number " zero ; Napier began with 10,000,000 and called its 

 logarithm zero. Napier, however, having in view a table of 

 trigonometrical sines of angles in a circle of radius 10,000,000, 

 proceeded to construct his geometrical progression by multiply- 

 ing by a ratio slightly less than unity, the ratio being in the first 

 instance unity diminished by the ten-millionth part. He also 

 secured accuracy in the last unit by continuing the calculations 

 as far as the seventh decimal place beyond the unit, using the 

 very decimal notation which we employ to-day. So far it might 

 be said there was not much difference between the two methods, 

 except that by use of a ratio less than unity Napier got rid of 

 the imperfection of a gradually increasing interval between 

 successive numbers in the geometrical progression. 



To have gone on diminishing the numbers in succession by 

 the ten-millionth part until he had filled in the whole range from 

 10,000,000 to 300 (approximately the sine of one minute of arc 

 on this scale) was of course an impossible undertaking. But 

 when Napier had carried out this simple arithmetical operation 

 a hundred times in succession he found that the 10,000,000 had 

 been reduced to a number very slightly greater than 9,999,900, 

 that is, less than the original number by the hundred-thousandth 

 part. He was then able to make a new start with 10,000,000 and 

 build upon it a new geometrical progression in which each new 

 number was formed from its predecessor by subtracting the 

 hundred-thousandth part of the latter. This very simple 

 arithmetical operation was carried out fifty times in succession ; 

 and then it was found that the last term could be very nearly 

 obtained from the original 10,000,000 by subtracting the two- 

 thousandth part, which is equivalent to multiplying by the ratio 

 °f 9)995 to 10,000. Starting with this new ratio, Napier formed 

 a third geometrical progression, which after proceeding for sixty- 

 nine terms ended with a number slightly greater than 9,900,000. 

 By repetition of this process the interval between 10,000,000 and 

 5,000,000 was filled in with numbers in geometrical progression, 

 whose numbered positions or logarithms in the corresponding 

 arithmetical progression were known. 



It was no doubt while engaged in the laborious calculations 



