50 H. S. CARSLAW. 
OF aE, 2, 5h, a BS 
abe 1 1 
7 7 en 7 “ 2 7 = 
10%, 10(1-— 2), 107 (1-104 
l — 
10° 10° 
By calculating a sufficient number of terms in these two 
series a set of numbers was obtained, dense enough to be 
used in dealing with the sines of angles between 0° and 
90°, at differences of a minute. 
\o ee 
But as a matter of fact, though this is the idea at the 
root of Napier’s work, he made to it a very remarkable 
addition, placing his tables in a class quite apart from 
that in which those of Burgi stand. We shall see later 
that when he comes to define logarithms, it is not from the 
correspondence of the above arithmetical and geometrical 
series that they are obtained, and that in his definition he 
approaches very closely to the principles on which Newton 
50 years later founded the Differential Calculus. 
The construction of his tables is explained fully in the 
Constructio, written before the Descriptio but not published 
till 1619, after his death. Napier proceeded as follows :— 
He formed first of alla G.P. of 101 terms in which the 
first term was 10’ and the common ratio (1 -= . 
We shall denote the terms of this series by do, a1, do, 
etc., and take r, for the common ratio. 
Thus a,.= 10’ \ 
: l 
t; =, dt ra? = ay 
6 1 
69 OE a) ea ee SP 
Ay = 10%(1 -" = a, 7'° 
These successive numbers were easily calculated, when 
the notation for the decimal fractions was introduced, and 
the approximation carried only as far as was necessary 
