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Proceedings of the Poyal Society 
which gives, at once, the correct value 
ab + ba . 
Now Berkeley, Hegel, Stirling, and others, have all in turn 
censured this process as a mere trick (or in terms somewhat similar) 
and say, in effect, that it is essentially erroneous. The fact, how- 
ever, is that, as in far greater matters, Newton here shows his 
profound knowledge of the question in hand ; and adopts, without 
any parade, a method which gives the result true to the second order 
of small quantities. The metaphysicians cannot see this, and Dr 
Stirling speaks with enthusiastic admiration of the clear sighted- 
ness and profundity of Hegel in detecting this blunder, and for it 
“harpooning” Newton ! 
What Newton seeks is the rate of increase of a quantity at a 
particular instant. Instead of measuring it by the rate of increase 
after that instant (as the metaphysicians would require) he measures 
it by observing, as it were, for equal intervals of time before and 
after the instant in question. 
Any one who is not a metaphysician can see at once the superior 
accuracy of Newton’s method, by applying both methods to the 
case of a rapidly varying velocity ; such as that of a falling stone, 
or of a railway train near a station. 
In reference to what Professor Tait had said, Mr Sang remarked, 
that the line of argument attributed to Newton had been used by 
John Nepair before Newton’s birth. Nepair’s definition of a loga- 
rithm runs thus (Descriptio, lib. i. cap. i. def. 6), (Constructio, 
23, 25) that if two points move synchronously along two lines, the 
one with a uniform velocity (arithmetice), the other (geometrice) 
with a velocity proportional to its distance from a fixed point, the 
distance passed over by the first point is the logarithm of the dis- 
tance of the second from the fixed point. In order to compare this 
variable velocity at any instant with the constant velocity, he takes 
a small interval of time preceding, and another succeeding the 
given instant, shows that the true velocity is included between 
the two velocities thus obtained, and (28, 31) takes the arith- 
metical mean as better than either, and as true ( inter terminos). 
It may be added, that Nepair devotes several sections of his 
