The Lord Brouncker employed this series to square the hyperbola.

Mercator published a demonstration of this quadrature; much about which time Sir Isaac Newton, being then twenty-three years of age, had invented a general method, to perform on all geometrical curves what had just before been tried on the hyperbola.

It is to this method of subjecting everywhere infinity to algebraical calculations, that the name is given of differential calculations or of fluxions and integral calculation.  It is the art of numbering and measuring exactly a thing whose existence cannot be conceived.

And, indeed, would you not imagine that a man laughed at you who should declare that there are lines infinitely great which form an angle infinitely little?

That a right line, which is a right line so long as it is finite, by changing infinitely little its direction, becomes an infinite curve; and that a curve may become infinitely less than another curve?

That there are infinite squares, infinite cubes, and infinites of infinites, all greater than one another, and the last but one of which is nothing in comparison of the last?

All these things, which at first appear to be the utmost excess of frenzy, are in reality an effort of the subtlety and extent of the human mind, and the art of finding truths which till then had been unknown.

This so bold edifice is even founded on simple ideas.  The business is to measure the diagonal of a square, to give the area of a curve, to find the square root of a number, which has none in common arithmetic.  After all, the imagination ought not to be startled any more at so many orders of infinites than at the so well-known proposition, viz., that curve lines may always be made to pass between a circle and a tangent; or at that other, namely, that matter is divisible in infinitum.  These two truths have been demonstrated many years, and are no less incomprehensible than the things we have been speaking of.

For many years the invention of this famous calculation was denied to Sir Isaac Newton.  In Germany Mr. Leibnitz was considered as the inventor of the differences or moments, called fluxions, and Mr. Bernouilli claimed the integral calculus.  However, Sir Isaac is now thought to have first made the discovery, and the other two have the glory of having once made the world doubt whether it was to be ascribed to him or them.  Thus some contested with Dr. Harvey the invention of the circulation of the blood, as others disputed with Mr. Perrault that of the circulation of the sap.

Hartsocher and Leuwenhoek disputed with each other the honour of having first seen the vermiculi of which mankind are formed.  This Hartsocher also contested with Huygens the invention of a new method of calculating the distance of a fixed star.  It is not yet known to what philosopher we owe the invention of the cycloid.