## Devlin's Angle |

That first genre setting volume was Guillaume Francois Antoine de l'Hospital's *Analysis of the Infinitely Small.* Written for the mathematical community, l'Hospital's book contained no problem sets, no color-highlighted definitions, and no full-color photographs, diagrams, and illustrations. But as Dunham points out, it was a calculus textbook, designed to "spread the word" about the then new techniques of the differential calculus.

Invented (or, if you prefer, discovered) just a few years earlier by Isaac Newton and later Gottfried Leibniz, a great deal of the early development work in calculus had been done by the Bernoulli brothers, Jakob and Johann. (Among the early uses to which the calculus was put was Johann Bernoulli's discovery of the catenary curve, the shape assumed by a chain suspended between two supports.) In fact, until the appearance of l'Hospital's book, Newton, Leibniz, and the two Bernoullis were pretty well the only people on the face of the earth who knew much about calculus.

Born in 1661, l'Hospital was a French nobleman of fairly minor rank who developed a keen interest in mathematics at an early age. He met Johann Bernoulli in 1691, shortly after the latter had made his discovery of the catenary. Eager to learn about this marvellous new technique calculus, l'Hospital hired Bernoulli to teach him.

L'Hospital learned rapidly, and before long was able to assemble all he had discovered into his comprehensive expository text, much as Euclid had done with geometry some two thousand years earlier. As was the case with Euclid and *Elements,* it is not clear that l'Hospital's text contained any significant discoveries he had made himself. Most if not all the "new" results and techniques were due to Leibniz and the Bernoullis, and the treatment was largely that of Johann Bernoulli. L'Hospital's contribution was to sift, to organize, to explain, and to assemble into a cohesive whole.

In short, l'Hospital was an *expositor. * Until relatively recently, that was a derogatory word in mathematical circles. Never mind the empirically-observable fact that first-rate expositors of mathematics are as rare as first-rate (i.e., Fields Medal standard) mathematical researchers, for centuries the mathematical community regarded its primary (if not sole) task to be proving theorems and solving problems.

In his classic book *A Short Account of the History of Mathematics,* Cambridge (England) mathematics historian W. W. Rouse Ball wrote, in 1908,

Leaving for a moment the English mathematicians of the first half of the eighteenth century, we come next to a number of continental writers who barely escape mediocrity, and to whom it will be necessary to devote but few words. Their writings mark the steps by which analytical geometry and the differential calculus were perfected and made familiar to mathematicians. [p.369]When you stop and think about it, that's an extraordinarily arrogant passage. The contrast between "English mathematicians" and "continental writers" is particularly stark, and speaks volumes for the value systems of the mathematics community at the time (value systems still dominant when I learned my mathematics in the England of the 1960s, and not totally dead to this day). Rouse Ball's statement is the more remarkable when you read on. The first of those "continental writers who barely escape mediocrity" that Rouse Ball mentions is l'Hospital. Of l'Hospital's book, Rouse Ball says,

... the credit of putting together the first treatise which explained the principles and use of the method is due to l'Hospital. ... This work had a wide circulation; it brought the differential notation into general use in France, and helped make it known in Europe. [pp.371-372]For that service to mankind, I would hope that contemporary historians of mathematics give rather more credit to l'Hospital than did Rouse Ball. It is, of course, an interesting question whether, in these post-Boyer [

Personally, I would think that, as the author of the first calculus textbook, a work that was largely responsible for the initial dissemination of the techniques of calculus, l'Hospital was a fairly significant player on the mathematical scene. On the other hand, as far as Rouse Ball is concerned, l'Hospital comes off little worse than Descartes (who was born four hundred years ago this year), of whom the eminent Cambridge historian wrote:

As to his philosophical theories, it will be sufficient to say that he discussed the same problems which have been debated for the last two thousand years, and probably will be debated with equal zeal two thousand years hence. [pp.271-272]Five sentences later, Rouse Ball ends the brief paragraph summing up Descartes's entire life work in philosophy with:

Whether this is a correct historical generalization of the views which have been successively prevalent I do not care to discuss here, but the statement as to the scope of modern philosophy marks the limitations of Descartes's writings. [p.272]So there! But, alas, now I have strayed from the mathematical community's view of expository writing onto its traditional view of philosophy. (Mind you, those philosophers do write a lot, you know.) So I'd better stop.

**- Keith Devlin**

William Dunham, 1996-A Triple Anniversary, *Math Horizons,* September 1996, pp. 8-13, MAA Publications, Washiongton, D.C.

W. W. Rouse Ball, *A Short Account of the History of Mathematics, *fourth edition (1908), reprinted by Dover, New York, 1960.

Devlin's Angle is updated at the start of each month.