_ 
i « 
ea ir 
, 
58 H. S. CARSLAW. 
Thus it is clear that if a table of these logarithms could 
be calculated, multiplication and division would be reduced 
to addition and subtraction. 
It must now be seen how Napier obtained this table or 
canon of logarithms. He relies on the fact that the log- 
arithms of numbers in G.P. have a common difference ; 
and also on the two following theorems. 
I. (Constructio, Section 28). The logarithm of any 
given sine is greater than the difference between radius 
and the given sine, and less than the difference between 
radius and the quantity which exceeds it in the ratio of 
radius to the given sine. 
II. (Constructio, Section 39). The difference of the 
logarithms of two sines lies between two limits; the 
greater limit being to radius as the difference of the sines 
to the less sine, and the less limit being to radius as the 
difference of the sines to the greater sine. 
These theorems are easy to establish with the aid of the 
logarithmic series. Napier obtains them direct from his 
definition of logarithms. 
I. Let s be a sine nearly equal to the radius r. 
Then logws = r log. = 
= r log. (1+ aaa 
Bese) 
(ja 
= — rlog, (1- 
Expanding log, (1 + fS8) and log, (1 - “=*\, we have 
Ss 
(r—s)— > logws > (r-s). 
A close approximation to logws would thus be the arith- 
metical mean between 
(r—s) = and (r-s), namely->—(r—s) (r+s). 
II. Again let si and sz be two sines nearly equal to each 
other, si being the greater. 
