"All the sciences which have for their end investigations concerning order and measure are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics. And a proof that it far surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides." Descartes.
"The idea of co-ordinates which forms the indispensable scheme for making all processes visible, with its many-sided and stimulating applications in all branches of daily life — whether medicine, physical geography, political economy, statistics, insurance, the technical sciences — the first beginnings of the calculus in their historical evolution, the development of the ideas of function and limit in connection with the elementary theory of curves, these are things without which in the present day not the slightest comprehension of the phenomena of nature can be attained, of which, however, the knowledge enables us by magic to gain an insight with which in depth and range, but above all in certainty, scarcely any other can be compared." Voss.
The differential and integral calculus is a branch of mathematics which treats problems involving variable quantities. Such problems are found regularly in geometry, physics, and many other branches of science. Whenever a quantity changes according to some continuous law, and most things in nature do so change, the differential calculus enables us to measure its rate of change. The integral calculus is really the inverse of the differential, and by it we can find the original quantity if we know the rate of increase or decrease. It is popularly believed that the calculus which is a general system of rules for treating such problems was an individual discovery, but it was really the result of a succession of discoveries by different men. Pope's couplet
"Nature and Nature's laws lay hid in night,
God said, "Let Newton be" and all was light."
is really poetic fallacy, because although the influence of Newton's discoveries on the subsequent development of science has been profound, the way was well prepared for him, especially for his method of variable quantities, and much was not "hid in night" to his predecessors. Nor was it the work of one century, for glimmerings of this powerful instrument can be found in the geometry of the ancients. Special procedures were used for separate cases of much that is now reducible to one rule which covers all cases alike. Much that is now calculated very easily by integration, for example the summation of series, the determination of areas of surfaces and volumes of solids involved a great deal of work for the ancient geometers and it is in their methods that we find the beginnings of the calculus. The differential calculus treats primarily of the relative increments of mutually related quantities when such increments are infinitely small, and there are not so many geometrical results which anticipated this branch of the calculus. Both branches, however, deal with infinitely small quantities and we shall first see how the Greeks treated such quantities.