Page 82 - Foundations Of Differential Calculus

4. On the Nature of Differentials of Each Order 65
order, must be represented, and for this the English mode is completely
inapt.
117. We have used both the words and notations that have been accepted
in our countries; they are both more familiar and more convenient. Still, it
is not beside the point that we have spoken about the English usage and
notation, since those who peruse their books will need to know this if they
are to be intelligible. The English are not so wedded to their ways that they
refuse to read writings that use our methods. Indeed, we have read some
of their works with great avidity, and have taken from them much profit. I
have also often remarked that they have profited from reading works from
our regions. For these reasons, although it is greatly to be desired that
everywhere the same mode of expression be used, still it is not so difficult
to accustom ourselves to both methods, so that we can profit from books
written in their way.
118. Since up to this time we have used the letter ω to denote the differ-
ence or the increment by which the variable x is understood to increase,
now we understand ω to be infinitely small, so that ω is the differential of x,
and for this reason we use our method of writing ω = dx. From now on, dx
will be the infinitely small difference by which x is understood to increase.
In like manner the differential of y we express as dy.If y is any function
of x, the differential dy will indicate the increment that y receives when x
changes to x + dx. Hence, if we substitute x + dx for x in the function y
I
I
and we let y be the result, then dy = y − y, and this is understood to be
the first differential, that is, the differential of the first order. Later we will
consider the other differentials.
119. We must emphasize the fact that the letter d that we are using here
does not denote a quantity, but is used to express the word differential,
in the same way that the letter l is used for the word logarithm when the
theory of logarithms is being discussed. In algebra we are used to using
√
the symbol for a root. Hence dy does not signify, as it usually does
in analysis, the product of two quantities d and y, but rather we say the
2
differential of y. In a similar way, if we write d y, this is not the square of
a quantity d, but it is simply a short and apt way of writing the second
differential. Since we use the letter d in differential calculus not for some
quantity, but only as a symbol, in order to avoid confusion in calculations
when many different constant quantities occur, we avoid using the letter d.
Just so we usually avoid the letter l to designate a quantity in calculations
where logarithms occur. It is to be desired that these letters d and l be
altered to give a different appearance, lest they be confused with other
letters of the alphabet that are used to designate quantities. This is what
has happened to the letter r, which first was used to indicate a root; the r
has been distorted to √ .