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.