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17. On Infinites In Geometry, And Sir Isaac Newton's
The labyrinth and abyss of infinity is also a new course Sir Isaac Newton has
gone through, and we are obliged to him for the clue, by whose assistance we
are enabled to trace its various windings.
Descartes got the start of him also in this astonishing invention. He advanced
with mighty steps in his geometry, and was arrived at the very borders of infinity,
but went no farther. Dr. Wallis, about the middle of the last century, was the first
who reduced a fraction by a perpetual division to an infinite series.
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
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