The labours of Peter Fermat, a counsellor of the parliament of Toulouse, approached still nearer to the fluxionary calculus. In his method of determining the maxima and minima of the ordinates of curves, he substitutes _x_ + _e_ for the independent variable _x_ in the function which is to become a maximum, and as these two expressions should be equal when _e_ becomes infinitely small or 0, he frees this equation from surds and radicals, and after dividing the whole by _e_, _e_ is made = 0, and the equation for the maximum is thus obtained. Upon a similar principle he founded his method of drawing tangents to curves. But though the methods thus used by Fermat are in principle the same with those which connect the theory of tangents and of maxima and minima with the analytical method of exhibiting the differential calculus, yet it is a singular example of national partiality to consider the inventer of these methods as the inventer of the method of fluxions. “One might be led,” says Mr. Herschel, “to suppose by Laplace’s expression that the calculus of finite differences had then already assumed a systematic form, and that Fermat had actually observed the relation between the two calculi, and derived the one from the other. The latter conclusion would scarcely be less correct than the former. No method can justly be regarded as bearing any analogy to the differential calculus which does not lay down a system of rules (no matter on what considerations founded, by what names called, or by what extraneous matter enveloped) by means of which the second term of the development of any function of _x_ + _e_ in powers of _e_, can be correctly calculated, ‘quæ extendet se,’ to use Newton’s expression, ‘_citra_ ullum molestum calculum in terminis surdis æque ac in integris procedens.’ It would be strange to suppose Fermat or any other in possession of such a method before any single surd quantity had ever been developed in a series. But, in point of fact, his writings present no trace of the kind; and this, though fatal to his claim, is allowed by both the geometers cited. Hear Lagrange’s candid avowal. ‘Il fait disparaitre dans cette equation,’ that of the maximum between _x_ and _e_, ‘les radicaux et les fractions s’il y en à.’ Laplace, too, declares that ‘il savoit etendre son calcul aux fonctions irrationelles en se debarrassant des irrationalités par l’elevation des radicaux aux puissances.’ This is at once giving up the point in question. It is allowing unequivocally that Fermat in these processes only took a circuitous route to avoid a difficulty which it is one of the most express objects of the differential calculus to face and surmount. The whole claim of the French geometer arises from a confusion (too often made) of the calculus and its applications, the means and the end, under the sweeping head of ‘nouveaux calculs’ on the one hand, and an assertion somewhat too unqualified, advanced in the warmth and