An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

WHAT A TANGENT TO A CURVE IS.

11

differ from tan' -- as little as we please by making x'
2m
approach x, or, what is the same thing, by making s approach
zero; hence (as in the first example) tan-1- and no other
quantity, is the limiting value of f(s) when s approaches
zero.
In a similar manner we might shew, in the case of other
curves, that f(s) has a definite limiting value when s approaches zero.
It appears therefore in this case, that when the actual value
ceases to be a definite quantity the limiting value does not.
24. This last example leads us to the best and most What a
accurate conception of what a tangent is. For draw the agurve is.
line SPT making the angle tan- - with the axis of x; then
2m
by what has been proved z PTX is the limiting value of
z PRX when Q approaches P; i.e. z PRX may be made
to differ from z PTX as little as we please by making Q
approach P without actually coming up to it; or what is
the same thing, z QPS may be made as small as we please
by making Q approach P without actually coming up to it.
Now this being the case, it is natural to say that the line
SPT just touches the curve at the point P, or that it is the
tangent to the curve at P. Hence we define a tangent in the
following manner.
25. If SPT be that line to which the secant RPQ may Definition
be made to approach nearer and nearer so as to make with it of a tangent.
an angle as small as we please, by making Q approach P
without actually coming up to it; then SPT is said to be the
line touching the curve at the point P, or the tangent at P.
Or to speak more briefly; If SPT be the limiting position of
the secant RPQ when Q approaches P, it is said to be the
tangent at P*.
* This definition of a tangent seems to me to be the only accurate one that
can be given, so as to apply to all cases of contact, such as contact at a point of con~