Page:A History of Mathematics (1893).djvu/246

 error (pointed out by John Bernoulli, but not admitted by him) that an osculating circle will necessarily cut a curve in four consecutive points. Well known is his theorem on the nth differential coefficient of the product of two functions of a variable. Of his many papers on mechanics, some are valuable, while others contain grave errors.

Before tracing the further development of the calculus we shall sketch the history of that long and bitter controversy between English and Continental mathematicians on the invention of the calculus. The question was, did Leibniz invent it independently of Newton, or was he a plagiarist?

We must begin with the early correspondence between the parties appearing in this dispute. Newton had begun using his notation of fluxions in 1666.[41] In 1669 Barrow sent Collins Newton's tract, De Analysi per Equationes, etc.

The first visit of Leibniz to London extended from the 11th of January until March, 1673. He was in the habit of committing to writing important scientific communications received from others. In 1890 Gerhardt discovered in the royal library at Hanover a sheet of manuscript with notes taken by Leibniz during this journey.[40] They are headed "Observata Philosophica in itinere Anglicano sub initium anni 1673." The sheet is divided by horizontal lines into sections. The sections given to Chymica, Mechanica, Magnetica, Botanica, Anatomica, Medica, Miscellanea, contain extensive memoranda, while those devoted to mathematics have very few notes. Under Geometrica he says only this: "Tangentes omnium figurarum. Figurarum geometricarum explicatio per motum puncti in moto lati." We suspect from this that Leibniz had read Barrow's lectures. Newton is referred to only under Optica. Evidently Leibniz did not obtain a knowledge of fluxions during this visit to London, nor is it claimed that he did by his opponents.