## Jacques Hadamard on "Who discovered the calculus"

 In An Essay on the Psychology of Invention in the Mathematical Field (Princeton University Press, 1945), Jacques Hadamard looks at the question: "Who discovered the Infinitesimal Calculus." We present here a version of his argument. Perhaps it is worth considering as we read, given the emphasis Hadamard puts on the proof that integration is the converse of differentiation, to ask where James Gregory should fit. What is certain is that Gregory was the first to publish a proof of this fundamental theorem of the calculus. Hadamard writes:

Heraclitus's profound idea that everything ought to be considered in its "devenir" i.e. in its continuous transformation, had not been understood during Antiquity. Only in the fourteenth century A.D. did one of the greatest medieval thinkers, Nicole Oresme, notice that the rate of increase or decrease of a quantity is slowest in the neighbourhood of a maximum or minimum. The bearing of such a remark was not perceived by anybody, including Oresme himself, to whom it did not appear that such a fundamental idea had to be developed.

Three centuries later the same principle was enunciated by Johannes Kepler, but Kepler went no further than Oresme had already gone before him; the discovery went only halfway.

Then, in Fermat's hands, the principle received mathematical expression. In several instances, considering one quantity in terms of another (such as time), Fermat used a mathematical operation which gave zero as the result if the quantity in question was a maximum or minimum. Moreover, the same method allowed him to find tangents to several curves considered by his contemporaries.

The operation performed by Fermat is precisely what we now call differentiation. Does this mean, as many are inclined to think, that he invented the Differential Calculus? In one sense we must answer "yes," for we see him applying his method to various problems, and even pointing out that the method could he applied to similar ones. But in another sense we must say "no," for the method he used appeared to nobody in his time, not even to himself, as a general rule for solving a whole class of problems, or as a new conception the properties of which deserved further investigation. Adapting an expression of Poincaré, we can say that things are more or less discovered, not discovered outright from complete obscurity to complete revelation. One step consists in acquiring the idea of a principle; another, if not several others, in giving a precise form to that idea and driving it far enough to be able to take it a starting point for further researches. In the present case, this was the work of Newton and Leibniz.

But the Differential Calculus is not the whole Infinitesimal Calculus. There is a second branch, the Integral Calculus, the fundamental operation of which is the valuation of plane curvilinear areas; and this implies a discovery which lay deep and had been entirely unsuspected, viz. the fact that integration is nothing else than the converse of differentiation. Who made that essential and difficult discovery? In the development of differentiation something of it was in the making. Torricelli and Fermat (perhaps even, though doubtful, Descartes) used methods which looked like the application of the principle and were indeed rather near it, but with a fundamental difference. I should say the same even of Barrow, Newton's master, in his 'Lectiones Geometricae', although the essential content of the No 11 of his xth lecture is really equivalent to that principle. Besides Descartes, who did not let the connection appear clearly, both Torricelli and Fermat treated special cases whose properties concealed the general principle; while for Barrow the true meaning of the principle was hidden by the meaning he gave to the notion of the tangent.

Thus Oresme, Keper, and Fermat failed to discover the Differential Calculus because they did not pursue their initial and fruitful ideas.

We see how psychological considerations can be illustrated by the history of science; and conversely, how they can help us to understand correctly a question which, very often, can be considered as a badly set one: Who is the author of such or such a discovery?

JOC/EFR July 2012