"(2) \int which is merely a long S... may be called, if you like, "the sum of." Thus \int dx means the sum of all the little bits of x or \int dt means the sum of all the little bits of t. Ordinary mathematicians call this symbol "the integral of."
Calculus Made Easy is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson. The original text continues to be available as of 2008 from Macmillan and Co., but a 1998 update by Martin Gardner is available from St. Martin's Press which provides an introduction; three preliminary chapters explaining functions, limits, and derivatives; an appendix of recreational calcu
"Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of x by 1 (for example in the last case reducing x4 to x3), and at the same time multiplying by a number (the same number in fact which originally appeared as the power). Now, when you have once seen this, you might easily conjecture how the others will run. You would expect that differentiating x5 would give 5x4 or differentiating x6 would give 6x5."
"Now any fool can see that if x is considered as made up of a lot of little bits, each of which is called dx, if you add them all up together you get the sum of all the dxs (which is the same thing as the whole of x). The word "integral" simply means "the whole."
"Now in the calculus we write dx for a little bit of x. These things such as dx, and du, and dy, are called "differentials," the differential of x, or of u, or of y, as the case may be. [You read them as dee-eks or dee-you or dee-wy.] If dx be a small bit of x, and relatively small of itself, it does not follow that such quantities as x\cdot dx or x^2 dx or a^x dx are negligible. But dx\times dx would be negligible, being a small quantity of the second order."
"Let us think of x as a quantity that can grow by a small amount so as to become x + dx, where dx is the small increment added by growth. The square of this is x^2 + 2x\cdot dx + dx^2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2."
"The process of finding the value of dy/dx is called "differentiating." But, remember, what is wanted is the value of this ratio when both dy and dx are themselves indefinitely small. The true value of the differential coefficient is that to which it approximates in the limiting case when each of them is considered as infinitesimally minute."
"The smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become. Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third (or higher)—orders, if only we take the small quantity of the first order small enough in itself."