56 H. S. CARSLAW. 
25. “* Whence a geometrically moving point approaching a fixed 
one has its velocities proportionate to its distances from the fixed 
one.” 
26. ‘The logarithm of a given sine is that number which has 
increased arithmetically with the same velocity throughout as that 
with which radius began to decrease geometrically, and in the 
same time as radius has decreased to the given sine.” 
The discussion which Napier gives in sections 22 —26 of 
the Constructio, of which we have quoted the headings 
only, we shall replace by the following argument in our 
modern notation, using the methods of the Calculus, of 
course unknown in Napier’s time. 
A E B 
| 
aaa: | Y 
C x F D 
His definitions of logarithms can be put in the following 
terms :—Imagine two straight lines AB and CD, AB being 
of length equal to the radius r, and CD of infinite length. 
Let two particles start from A and C at the same time and 
with the same initial velocity, and move along these two 
lines. But while the velocity of the particle which starts 
at C is to remain constant, that of the particle which 
starts from A is to decrease in such a way that at any 
stage of its journey from A towards B, say at EH, the 
velocity at E: the velocity at A = HB: AB. 
When one particle is at H on AB, let the other particle 
be at Fon CD. Then the number which measures CF is 
called the logarithm of the number which measures EB. 
With the notation of the Calculus, the relation between 
a number and its logarithm can easily be found :— . 
Let CF = x, and BE = y at the time t from the start 
from A and B. 
