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Euler mentions in his preface of the book "Foundations of Differential Calculus" (Translated version of Blanton):

....Even now there is more that remains obscure than what we see clearly. As differential calculus is extended to all kinds of functions, no matter how they are produced, it is not immediately known that method is to be used to the vanishing increments of absolutely all kinds of functions. Gradually this discovery has progressed to more and more complicated functions, the ultimate ratio that the vanishing increments attain could be assigned long before the time of Newton and Leibniz, so that the differential calculus applied to only these rational functions must be held to have been invented long before that time. However, there is no doubt that Newton must be given credit for that part of differential calculus concerned with irrational functions...

I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational fuctions long before (Newton and Leibniz), without knowing application of the method to the vanishing icrements; if it was already invented, how does that differ from that of the study-of-ultimate ratio (differentail calculus) for irrational functions, it must be the same method (??).

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The differential calculus for rational functions can be reduced to finding roots of polynomials. Such methods were used by Fermat and Descartes. See problems 12.3-12.5 of my history of mathematics notes for a summary of their methods. These methods are not based on infinitesimals but rather on the idea that a tangent line is a line for which two points of intersection with a curve have degenerated into one. This translates into a double-root algebraic property for a certain polynomial, from which the tangent can be found. Such methods do not work for "irrational functions," meaning functions that cannot be reduced to a polynomial equation.

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  • $\begingroup$ "History of Mathematics Reader" gives a best bird-eye-view on the entire analysis of infinitesimals; I like it, thank you once again $\endgroup$
    – Sensebe
    Oct 6, 2015 at 11:11
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    $\begingroup$ Use of double or triple roots: tangent lines, osculating circles, etc., were possible without use of "infinitesimal" methods. $\endgroup$
    – Gerald Edgar
    Oct 6, 2015 at 18:29

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