5A H. S. CARSLAW. 
In this Third Table Napier obtained a set of numbers lying 
between the radius (10’) and half the radius (4 x 10’). 
They are close enough to one another to allow the table 
to be used in dealing with the sines of angles from 90° to 
30° at intervals of 1’, when the logarithms of these sines, 
according to Napier’s definition of logarithms, are to be 
determined. They are not exactly ina G.P., but each of the 
69 columns of the Third Table is a G.P. of 21 terms. 
If we had proceeded from the one G.P. to the other, as 
in the case denoted above by the terms— 
Bo By ++» Besos 
we would have had 
O01 Bo ais 5000 
Ya = Yohss ==) Gori = Aso00 y B 
i} 8 Tes) 7, ge UG ELL 1 50 
Y2 YoP3 071 10000 5g eden 
Pea Pom Tees 
Is 20 __ 100000 
Yo= YoPs = %Ny 
instead of the series 
= 400000 
Co, Ci, Co, 3,9 C29) 
which forms the first column of the Third Table. 
Also the terms at the top of the columns of the Third 
Table would have been 
Oo Orem pas 
where 6, =-Vy) == 40) — 4, 
1 = Yo = OoP4 = Ay o0000 y 
= oe 2 20 __ >. 100000 
de = OoP4 = Aaoo000 Ps = = fa . 
0 
an 68 
Ogs = OoP4 = egoooo0 
The last term in the 69th column would, in this case, 
be the same as the first in the 70th column, namely 
6900000 
69 
Ay P4,, OF Ar, OF Aggoo000 
Now the word logarithm is due to Napier. From its 
derivation he means by it the number of the ratios. 
In the series 
Oe a 2, ioe 
10’, 10(1- Foy, i sean ee TELE. 
1 Jn 
10° io 
