1 he were asked to explain the difference between a magnitude infinitely small and a magnitude infinitely diminished. If he saith, there is no difference, then let him be farther asked, how he dares to explain the method of fluxions, by the ratio of magnitudes infinitely diminished (p. 9), when Sir Isaac Newton hath expressly excluded all consideration of quantities infinitely small? If this able vindicator should say that quantities infinitely diminished are nothing at all, and consequently that, according to him, the first and last ratios are proportions between nothings, let him be desired to make sense of this, or explain what he means by "proportion between nothings." If he should say, the ultimate proportions are the ratios of mere limits, then let him be asked how the limits of lines can be proportioned or divided? After all, who knows but this gentleman, who hath already complained of me for an uncommon way of treating mathematics and mathematicians (p. 5), may (as well as the "Cantabrigian ") cry out "Spain and the Inquisition!" when he finds himself thus closely pursued and beset with interrogatories? That we may not, therefore, seem too hard on an innocent man, who probably meant nothing, but was betrayed by following another into difficulties and straits that he was not aware of, I shall propose one single expedient, by which his disciples (whom it most concerns) may soon satisfy themselves whether this Vindicator really understands what he takes upon him to vindicate. It is, in short, that they would ask him to explain the second, third, or fourth fluxions upon his principles. Be this the touchstone of his "Vindication." If he can do it, I shall own myself much mistaken: if he cannot, it will be evident that he was much mistaken in himself, when he presumed to defend fluxions without so much as knowing what they are. So, having put the merits of the cause on this issue, I leave him to be tried by his scholars. 1 See his Introduction to the "Quadratures.” [WALTON returned to the fray in his "Catechism of the Author of the Minute Philosopher fully considered " (1735). Berkeley answered him with the following tract. It is an interesting work, with an air of briskness that comes with relief after the querulous tone of the "Defence." Walton replied with an Appendix to his "Catechism," but elicited no response from Berkeley. The "Reasons" appeared in 1735, and closed the controversy so far as Berkeley himself is concerned. The shower of pamphlets continued for seven more years.] |