Contrary to Andrej Bauer’s contention, seventeenth-century calculus looks very little like SDG. Unlike in SDG, the integrals were construed as infinite sums, the intermediate value theorem was assumed to hold for continuous curves and, more to the point, for the most part the infinitesimals that were employed were invertible rather than nilpotent. For a while, the Dutch mathematician, Bernard Nieuwentijt, in his debate with Leibniz, argued in favor of the use of nilpotent infinitesimals, but eventually came to believe that his attack on Leibniz was ill-founded and returned to the then standard use of invertible infinitesimals. Of course, I’m not suggesting that nilpotent infinitesimals were not used—they were from time to time—but only that their use was not the main view. After all, following Leibniz, most mathematicians wanted their infinitesimals to behave like real numbers.
Nilpotent infinitesimals along with invertible infinitesimals were employed by a number of differential geometers in the nineteenth century and entered mainstream mathematics around the turn of the twentieth-century (in systems of dual numbers), when geometers such as Hjelmslev and Segre became interested in geometries in which two points need not determine a unique straight line, and Grothendieck (and others) later employed them in algebraic geometry.
I suspect that the misconception that seventeenth-century calculus looks like SDG can be traced in part to John Bell’s wonderful expository writings on SDG. Bell was taken to task for this by the historian-mathematician Detlef Laugwitz in his otherwise very positive review (for Mathematical Reviews MR1646123 (99h:00002)) of the first edition of Bell's A Primer of Infinitesimal Analysis (1998). Moreover, I am not aware of any of the many serious writings on the history of the calculus that supports the view suggested by (my friend) John.
Response to Mikhail Katz:
Mikhail: Fermat’s work was one I had in mind when I said nilpotent infinitesimals were used from time to time. However, his work, which was largely concerned with tangent constructions and lacked generality, predates the work of Newton and Leibniz, never caught on, and is not characteristic of the mainstream approaches to the calculus of the 17th century, which is what I said I was talking about. Moreover, Fermat’s work is notoriously unclear and, by my lights, the similarities with SDG are vague at best.
Many thanks, however, for the reference to Cifoletti’s work, which I will take a look at. I hasten to add, however, that the following passage from the Mathematical Reviews review of the work, which you yourself cite, does not inspire confidence.
“In the second part of the book, the author embarks on an investigation of the link between modern synthetic differential geometry, originally proposed by F. W. Lawvere in 1967 and afterwards largely developed by Lawvere and other mathematicians, and Fermat's mathematics.
In many situations, for the most part informal ones, Lawvere himself and other mathematicians working in this research field expressed their feelings that there had to be some kind of affinity between synthetic differential geometry and seventeenth-century mathematical practice. The author has tried to make explicit these general feelings, but this part of the book is mathematically weak and somewhat naive.
The best example is footnote 29, page 208, where the author claims to have established a direct connection between Fermat and synthetic differential geometry, on the basis of having been able to convince G. Rejes, during a talk she had with him about Fermat's work, to name a particular axiom of one possible formulation of the theory after Fermat.”