Page 10 - Foundations Of Differential Calculus

Preface ix
the magnitude of the whole terrestrial globe, it is the custom easily to grant
him an error not only of a single grain of dust, but of even many thousands
of these. However, geometric rigor shrinks from even so small an error,
and this objection would be simply too great were any force granted to it.
Then it is difficult to say what possible advantage might be hoped for in
distinguishing the infinitely small from absolutely nothing. Perhaps they
fear that if they vanish completely, then will be taken away their ratio, to
which they feel this whole business leads. It is avowed that it is impossi-
ble to conceive how two absolutely nothings can be compared. They think
that some magnitude must be left for them that can be compared. They are
forced to admit that this magnitude is so small that it is seen as if it were
nothing and can be neglected in calculations without error. Neither do they
dare to assign any certain and definite magnitude, even though incompre-
hensibly small. Even if they were assumed to be two or three times smaller,
the comparisons are always made in the same way. From this it is clear that
this magnitude gives nothing necessary for undertaking a comparison, and
so the comparison is not taken away even though that magnitude vanishes
completely.
Now, from what has been said above, it is clear that that comparison,
which is the concern of differential calculus, would not be valid unless the
increments vanish completely. The increment of the quantity x, which we
2
have been symbolizing by ω, has a ratio to the increment of the square x ,
2
which is 2xω + ω ,as1to2x + ω. But this always differs from the ratio
of1to2x unless ω = 0, and if we do require that ω = 0, then we can truly
say that this ratio is exactly as 1 to 2x. In the meantime, it must be un-
derstood that the smaller the increment ω becomes, the closer this ratio is
approached. It follows that not only is it valid, but quite natural, that these
increments be at first considered to be finite and even in drawings, if it is
necessary to give illustrations, that they be finitely represented. However,
then these increments must be conceived to become continuously smaller,
and in this way, their ratio is represented as continuously approaching a cer-
tain limit, which is finally attained when the increment becomes absolutely
nothing. This limit, which is, as it were, the final ratio of those increments,
is the true object of differential calculus. Hence, this ratio must be consid-
ered to have laid the very foundation of differential calculus for anyone who
has a mind to contemplate these final ratios to which the increments of the
variable quantities, as they continuously are more and more diminished,
approach and at which they finally arrive.
We find among some ancient authors some trace of these ideas, so that
we cannot deny to them at least some conception of the analysis of the
infinite. Then gradually this knowledge grew, but it was not all of a sudden
that it has arrived at the summit to which it has now come. Even now,
there is more that remains obscure than what we see clearly. As differential
calculus is extended to all kinds of functions, no matter how they are pro-