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62 H. S. CARSLAW. 
The fourth, or middle column, contains the differences 
between the logarithms of the sines and the logarithms of 
the tangents. 
Since we have, according to the notation of the time, 
sin? : cos? = tan?@ : radius, 
log tan 9 — log radius = log sin 9 — log cos @. 
But log radius = 0. 
Therefore log tan 6 = log sin @ —- log cos 8. 
It will be noticed that the logarithms of tangents of 
- angles between 0° and 45° are positive, but, between 45° 
and 90°, they are negative. 
Further, since sin @ : radius = radius : cosec 0, and 
cos : radius = radius: sec 9, we have 
log cosec 9 = — log sin 0, and log sec 9 = — log cos 9. 
Thus the logarithms of the trigonometrical ratios can all 
be derived from his table. | 
To make it answer the purpose of a table of logarithms 
of common numbers, the author states that the following 
procedure should be adopted :— 
A number being given, find that number in any table of 
natural sines, or tangents, or secants, and note the degrees 
and minutes in itsarc. Then in his table find the logarithm 
of the corresponding sine, tangent or secant, and this wilt 
be the logarithm of the required number. 
After the definitions and descriptions of logarithms, 
Napier explains, in the Descriptio, the tables which it 
contains, and shows how to take out the logarithms of 
Sines, tangents, secants, and common numbers; as also 
how to add and subtract logarithms. He then goes on 
to teach the uses of these numbers. Firstly, in finding 
any of the terms of three or four proportionals, showing 
how to multiply and divide, and to find powers and roots, 
by logarithms. And secondly, developing the uses of the 
