This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.

George Berkeley wrote in 1734 with reference to the early calculus that such a method is "a most inconsistent way of arguing, and such as would not be allowed in Divinity". This passage is quoted by William Dunham in 2004. Dunham concludes: "Bishop Berkeley had made his point. Although the results of the calculus seemed to be valid ... none of this mattered if the foundations were rotten". See page 72 of http://books.google.co.il/books?id=QnXSqvTiEjYC&source=gbs_navlinks_s

On the other hand, Peter Vickers in 2007 challenged "The ubiquitous assertion that the early calculus of Newton and Leibniz was an inconsistent theory" at http://philsci-archive.pitt.edu/3477/ (soon to appear in book form at Oxford University Press), and concluded that this only holds in a limited sense and "can only be imputed to a small minority of the relevant community".

Was the early calculus consistent as far as most practitioners were concerned, as Vickers contended, or was it a most inconsistent way of arguing, as did Berkeley and Dunham?

Note 1. Berkeley claimed that calculus was based on an inconsistency that can be expressed in modern notation as $(dx\not=0)\wedge(dx=0)$. Thus he was using the term "inconsistent" in much the same sense it is used in modern logic.

Note 2. For a closely related thread, see https://math.stackexchange.com/questions/445166/is-mathematical-history-written-by-the-victors

Note 3. There is a related thread at the history SE: http://hsm.stackexchange.com/questions/3301