were always uncertain till confirmed by rigorous analysis. Notwithstanding these unavoidable defects, the importance and the generality of his discoveries respecting the system of the universe, and the most interesting points of natural philosophy, the great number of profound and original views, which have been the origin of the most brilliant discoveries of the mathematicians of the last century, which were all presented with much elegance, will insure to the Principia a lasting pre-eminence over all other productions of the human mind."
Newton's Arithmetica Universalis, consisting of algebraical lectures delivered by him during the first nine years he was professor at Cambridge, were published in 1707, or more than thirty years after they were written. This work was published by Mr. Whiston. We are not accurately informed how Mr. Whiston came in possession of it, but according to some authorities its publication was a breach of confidence on his part.
The Arithmetica Universalis contains new and important results on the theory of equations. His theorem on the sums of powers of roots is well known. Newton showed that in equations with real coefficients, imaginary roots always occur in pairs. His inventive genius is grandly displayed in his rule for determining the inferior limit of the number of imaginary roots, and the superior limits for the number of positive and negative roots. Though less expeditious than Descartes', Newton's rule always gives as close, and generally closer, limits to the number of positive and negative roots. Newton did not prove his rule. It awaited demonstration for a century and a half, until, at last, Sylvester established a remarkable general theorem which includes Newton's rule as a special case.
The treatise on Method of Fluxions contains Newton's method
