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HISTORICAL VIEW OF THE

the most comprehensive representation of the generalform in which functions of all kinds may be developed;and though doubts have been raised as to its affording,in all cases, a finite representation of the value whichthe function assumes corresponding to a given increasein the variable, yet its essential excellence lies in indi-cating the relation which invariably subsists betweenthe developed form of the function and the successiveorders of its differential coefficients.

Nor ought we here to omit the names of Stirling, theacute commentator on Newton’s “ lines of the thirdorder,” and the real inventor of the analytical formulaknown as Maclaurin ’s theorem; nor the last-namedmathematician, as an acute and able expositor of thedoctrines of analysis; which he also successfully appliedin pursuing some of the problems arising out of thetheory of gravitation.

We have noticed the early attacks made on the prin-ciple of the calculus; the fluxional system had nowto sustain another and more formidable attack, frombishop Berkeley, in 1734. Though eminent as a me-taphysician, his mathematical qualifications were notof the highest order: yet the argument contained in“ The Analyst” is earned on with great acuteness andplausibility. The author contends, that the fundamentalidea of supposing a finite ratio to exist between termsabsolutely evanescent,—“ the ghosts of departed quan-tities,” as he, with inimitable humour, calls them,—iscompletely absurd and unintelligible : and it must beconfessed, that the language in which the fluxional prin-ciple was stated really laid it open to objections of thiskind. A reply was written by Dr. Jurin, under thename of Philalethes. Several other publications alsoappeared on both sides, llobins and Maclaurin areconsidered to be those who made the most satisfactorydefence of the principles of limiting ratios. But thoughthe subject was extensively discussed, and the “ Re-sidual Analysis” of Lauden, in 1758, exhibited ananalogous principle, in some measure divested of the