60 H. S. CARSLAW. 
column is radius, so its logarithm is zero. The logarithm 
of the second term is given as 5,001°250,417. This number 
is the common difference of the logarithms of the numbers 
in that column. From the last term in it, we pass, as 
before, to the first term of the second column. Its log- 
arithm is found to be 100,503°358. The numbers at the top 
of the columns of this Third Table form a G.P., so we are 
now able to write down their logarithms. The second 
term in each column differs only slightly from the first 
term. Thus by Theorem II., above, its logarithm can be 
found. Also from the first and second terms, the others 
in these columns are got by addition of the proper common 
difference. 
These are the steps taken by Napier in calculating the 
logarithms of the terms in his Third Table. The results 
are contained in his Radical Table, an extract from which 
is appended. 
The Radical Table. 
First column. Second column. 3 Siatyninth column. 
Natural Logar- Natural Logar- | & Natural : 
ember! whiss. Nambeee. ithind: a Numbers Tae amnriaiiia 
10000000-0000 “0 19900000-0000 |100503°3 4 5048858°8900 | 68384225°8 
9995000°0000 | 5001°2 }9895050:0000 |105504°6 | © 15046334°4605 | 68392271 
9990002°5000 | 10002°5 ]9890102°4750 |110505°8 4 5043811°2932 | 6844228°3 
9985007°4987 | 15003°7 |9885157°4287 |115507°1 J. |5041289°3879 | 6849229°8 
up to up to up to upto [3 up to up to 
9900473°5780 |100025:0 | 9801468°8423 !200528:2 4.998609°40384 | A934250°8 
The numbersinthe Radical Table for which the logarithms 
have been found, form a set dense enough to allow of the 
logarithms of the sines from 90° to 30° at differences of 1’ 
to be calculated. 
To obtain the logarithms of the sines of the angles from 
0° to 30°, Napier indicates two separate methods. In the 
first the given sine is to be multiplied by some number 2, 
4, 8, 10, 20, 40, 80, 100, 200, or any other proportional 
number contained in a small table he has calculated in 
