Newton's *Principia* remains one of the few undisputed masterpieces of mathematical physics that is still read. It is generally agreed to have established the foundations of dynamics and the theory of gravity, and to have given a remarkably accurate description of the motion of planets and their satellites. But there are few accounts of its reception, and what is usually said goes along these lines: the British loved it, but more discerning Continental mathematicians distrusted its physics and held out, initially and unsuccessfully, for a revival of the vortex theory of planetary motion which Newton had refuted. What really moved things on, however, was Euler's re-writing of this entire family of ideas in the language of the calculus, after which it was much easier for mathematicians to appreciate and extend what Newton had done. As this sketch indicates, a major part in the acceptance of Newton's ideas is supposed to have involved dumping his geometric analyses in favour of the systematic use of the calculus, which Newton had done so much to invent but largely left out of the *Principia.*

Now for the first time we have a considered examination of how the *Principia* was actually read, and so one of the few detailed accounts we have of how any major theory has been taken up and adapted (string theory, any one?). It complements and extends the recent study by Berteloni Meli of Leibniz's study of the *Principia*. It begins where it must, in a careful study of the mathematical techniques of the day (1687). What did Huygens, or Leibniz, or Newton himself for that matter, have by way of mathematical methods? The first interesting point that Guicciardini establishes is that there was a significant shift in Newton's views between the 1660s, when he invented his calculus, and the 1680s, when he wrote the *Principia*. Newton became a devotee of ancient learning, which he supposed included hidden wisdom accessible in secret texts, and in matters mathematical he came to prefer traditional geometry to the formal methods generated even by his own calculus. He expressed this preference in philosophical terms: geometry referred to objects, algebra was merely a heuristic; geometry, but not algebra, was able to provide proofs. He did not renounce the calculus, and at times used it in the *Principia* out of necessity, but the *Principia* is written in geometry not (or not only) to be more easily understood by those who had not mastered the calculus, but because Newton deeply believed it fitted the context. There was, in Newton's opinion, an intimate intellectual connection between physics and geometry.

This decision caused Newton trouble later on, when he wanted to use the *Principia* as evidence that he had discovered the calculus before Leibniz, precisely because there was not much calculus in it. It placed him on the side of Huygens, whom he admired, and against Descartes, whom he detested. But in another way, Newton's mathematics was almost unintelligible to Huygens. As Guicciardini explains, Huygens wrote his mathematics in the language of proportion theory, much as Euclid had written it and with many of its limitations. In this language, lengths may be compared with lengths, but not with times. The expression *s* µ *t*^{2} is illegitimate, and the truth that it expresses must be written as s_{1} : s_{2} :: t_{1}^{2} : t_{2}^{2}. Newton broke with that part of the tradition, and wrote directly in the language of equations. This innovation in geometry permitted Newton to develop his new physics, but it troubled the older Huygens greatly.

Newton's growing dislike of algebra for its lack of ontological commitment was part of his hostility to all things Cartesian (and a break with some of his own earliest successes in geometry). Continental, largely Cartesian mathematicians, preferred algebra to geometry because of its greater flexibility. They shared with Leibniz a desire for making the differential and integral calculus as routine as possible. In the heated atmosphere of the priority dispute over the invention of the calculus, the Bernoullis combed the *Principia* for mistakes, and found very few. What they did, however, was to shift the emphasis away from geometric physics, and towards the formulation and solution of differential equations. Guicciardini discusses at length the best known part of this dispute, the claim by Johann Bernoulli that Newton had failed to understand, let alone prove, that an inverse square law of gravity implies conic orbits. He shows that Newton had expressed himself obscurely but was able to put his argument over well enough for Bernoulli eventually to withdraw the claim. More significantly, he shows that for Bernoulli and others the question belonged to an emerging branch of the calculus, according to which differential equations had solutions with appropriate numbers of arbitrary constants. Newton, in contrast, gave arguments about force and velocity.

Guicciardini argues persuasively that between say 1687 and 1740 British and Continental mathematicians could easily read each others' works and translate them from dots (denoting the derivative--or the fluxion--of *x* by *x* with a dot over it) to d's (writing *dx* for the differential of *x*). They could pass back and forth between the geometric language and the analytic language, although the Leibnizian integral sign was distinctly advantageous. This is surely correct, and demonstrated in writings of the period. It allows him to propose a more significant distinction, between Newtonian adherence to geometry and the Continental enthusiasm for notation. The dispute is between those who saw a continual need for interpretation and those who wanted mechanical manipulation of symbols. I suspect similar debates are common in mathematics and alive today; it is exciting to see them displayed so clearly here.

Guicciardini's book allows us to see a number of things more clearly. Not least of these is the *Principia* itself, which he draws on with enviable lucidity. He shows us mathematicians working hard to understand it, even to find fault with it, and to draw precise lessons from it as well as general principles. This is truly exciting. He valuably deepens the old view that what Newton could do with geometry (most of the time) lesser mortals needed the calculus to accomplish, by showing that the rival calculi of the day were not so different, but that there was a shift towards writing physics in the language of differential equations. This points up Continental doubts about the inverse square law and action at a distance, which persisted after the old vortex theory was dropped. It also leads attractively into the view that it is Euler's reformulation of mechanics that severs the Newtonian and Leibnizian traditions to the decisive advantage of the latter. But this was not successful because d's were better than dots, or because geometric foundations for the calculus are somehow inappropriate. It succeeded because of the power of the function concept, a radical innovation not known to Newton or Leibniz.

Jeremy Gray ( j.j.gray@open.ac.uk) has worked at the Open University since 1974. In 1996 he was a Resident Fellow at the Dibner Institute for the History of Science and Technology, MIT, Cambridge, USA. He is also an Affiliated Research Scholar at the Department of History and Philosophy of Science of the University of Cambridge, England. He works on the history of mathematics in the 19^{th} and 20^{th} Centuries, with a particular interest in complex function theory and geometry.