The silence of Newton was at last broken, and in the second lemma of the second book of the Principia, he explained the fundamental principle of the fluxionary calculus. His explanation, which occupied only three pages, was terminated with the following scholium:—“In a correspondence which took place about ten years ago between that very skilful geometer, G. G. Leibnitz, and myself, I announced to him that I possessed a method of determining maxima and minima, of drawing tangents, and of performing similar operations which was equally applicable to rational and irrational quantities, and concealed the same in transposed letters involving this sentence, (_data equatione quotcunque fluentes quantitates involvente, fluxiones invenire et vice versa_). This illustrious man replied that he also had fallen on a method of the same kind, and he communicated to me his method which scarcely differed from mine except in the notation [and in the idea of the generation of quantities.”][63] This celebrated scholium, which is so often referred to in the present controversy, has, in our opinion, been much misapprehended. While M. Biot considers it as “eternalizing the rights of Leibnitz by recognising them in the Principia,” Professor Playfair regards it as containing “a highly favourable opinion on the subject of the discoveries of Leibnitz.” To us it appears to be nothing more than the simple statement of the fact, that the method communicated by Leibnitz was nearly the same as his own; and this much he might have said, whether he believed that Leibnitz had seen the fluxionary calculus among the papers of Collins, or was the independent inventor of his own. It is more than probable, indeed, that when Newton wrote this scholium he regarded Leibnitz as a second inventor; but when he found that Leibnitz and his friends had showed a willingness to believe, and had even ventured to throw out the suspicion, that he himself had borrowed the doctrine of fluxions from the differential calculus, he seems to have altered the opinion which he had formed of his rival, and to have been willing in his turn to retort the charge.