The Differential Calculus
The invention of the differential calculus
marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level. These contrasted periods in the progress of the history of thought are compared by Shelley to the formation of an avalanche.
The sun-awakened avalanche! whose mass, Thrice sifted by the storm, had gathered there Flake after flake,–-in heaven-defying minds As thought by thought is piled, till some great truth Is loosened, and the nations echo round,
The comparison will bear some pressing. The final burst of sunshine which awakens the avalanche is not necessarily beyond comparison in magnitude with the other powers of nature which have presided over its slow
formation. The same is true in science. The genius who has the good fortune to produce the final idea which transforms a whole region of thought, does not necessarily excel all his predecessors who have worked at the preliminary formation of ideas. In considering the history of science, it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances towards a new epoch.
In the particular instance before us, the
subject had a long history before it assumed its final form at the hands of its two inventors. There are some traces of its methods even among the Greek mathematicians, and finally, just before the actual production of the subject, Fermat (born 1601 ,
and died 1665 ), a distinguished French mathematician, had so improved on previous ideas that the subject was all but created by him. Fermat, also, may lay claim to be the joint inventor of coordinate geometry in company with his contemporary and countryman, Descartes. It was, in fact,
Descartes from whom the world of science received the new ideas, but Fermat had certainly arrived at them independently.
We need not, however, stint our admiration either for Newton or for Leibniz. Newton was a mathematician and a student of physical science, Leibniz was a mathematician
and a philosopher, and each of them in his own department of thought was one of the greatest men of genius that the world has known. The joint invention was the occasion of an unfortunate and not very creditable dispute. Newton was using the methods of Fluxions, as he called the subject,