others, of introducing the phraseology of infinite quantity, and estimating the values which algebraical expressions assume in different cases, when we suppose a term involved in them to become infinitely great or infinitely small. In his investigations we may trace the germs of those methods which, very soon after, in the hands of Newton, received an expansion fitting them for the purposes of analysing the most complex laws of physical phenomena. But the principles which Wallis had adopted, were not yet so far developed as to be understood in the full extent, or to exhibit the power which they really possessed. He was, however, enabled to pursue the subject of quadrature upon much more general grounds than any of his predecessors. Hitherto, as we have seen, geometers had succeeded in assigning in finite terms the value of the area of the spaces bounded by curve lines, only in a few limited and very simple cases. Des Cartes had generalised the mode of conceiving the construction of all curves; and it hence followed, that by certain algebraical processes, those applications of the method of limits which had been devised by Kepler and Cavalieri for expressing the areas, might be extended in a manner equally general, by virtue of the relation subsisting between the abscissa and ordinate, the two variables, which enter Des Cartes equation.

Wallis found, that in all cases where the value of one of these could be expressed in terms of the other, without involving fractional or negative indices, he could exhibit the value of the area, or quadrature of the curve, in finite

terms.

A partial extension was given to this method by Nicholas Kauffman, (more commonly known by his Latinised name, Mercator,) who devised a method of reducing some of the expressions into a continued series of terms. In this way he obtained the quadrature of the hyperbola, in 1667.

Wallis, however, was anxious to extend his solutions beyond this limited portion of the subject. He saw a

wide field open before him, and in this new region of research he perceived that many geometrical truths of great value and utility were to be discovered; the quadrature of the circle, and of many other curves, together with an incalculable variety of applications of such results bearing upon a number of points of physical enquiry, were all inviting research, promising a rich harvest of further discoveries, and stimulating the attempts of the enquirer with the prospect of an immortality of fame.

It occurred to him, that if the equations of the curves which he had squared were ranged in a regular series, from the simpler to the more complex, their areas would constitute another corresponding series, the terms of which were all known. He further remarked, that in the first of these series, the equation to the circle might be introduced, and would occupy the middle place between the first and second terms of the series, or between the equation of a straight line, and that of the parabola. He concluded, therefore, that if in the second series he could interpolate a term in the middle, between its first and second terms, this term must necessarily be no other than the area of the circle. But when he proceeded to pursue this very refined and philosophical idea, he was not so fortunate; and his attempt towards the requisite interpolation, though it did not entirely fail, and made known a curious property of the area of the circle, did not lead to an indefinite quadrature of that curve.

In these researches Wallis was closely associated with sir C. Wren, who in his early youth had evinced high mathemetical talents; he gave a rectification of the cycloid and several other mathemetical investigations, which Wallis published in his treatise on the cycloid. He devoted himself much to astronomy, and became professor of that science at Oxford in 1670, as well as in Gresham college; he also entered largely into the dynamical questions then discussed by the English philosophers and Huyghens. But ultimately his mag

nificent architectural labours withdrew him entirely from the pursuits of abstract science.

mena.

The Cartesian System.

Des Cartes, having brought geometry under the dominion of a comprehensive principle, seems to have been misled into the splendid but visionary notion, that the system of the world and the philosophy of mechanics might, in like manner, be established upon a theory arising out of a few first assumed axioms. These, according to his view, were to be found in certain metaphysical ideas of the Deity and his attributes; from these he affected to reason downwards, and to deduce the laws of nature: to show why things are constituted as they are, and to explain the causes of material phenoIn this way he pretended that, by a long train of consequences, he could always determine, at last, what ought to be the laws and modifications to which material agents would be subjected; and reason from the first cause to secondary causes, and from secondary causes to their visible effects. At the same time (though, it would seem, with some inconsistency) he did not wholly reject' experiment and induction; and he seems to be tacitly admitting the fundamental deficiency of his whole system when he says, that the number of different shapes, which effects might assume, is so great, that he could not determine without experiment which of them nature had preferred to the rest. "We em

ploy experiment, not as a reason by which any thing is proved; for we wish to deduce effects from their causes, and not, conversely, causes from their effects. We appeal to experience only, that out of innumerable effects, which may be produced from the same cause, we may direct our attention to one rather than another."

In the use, however, which he did make of induction, Des Cartes appears to have acknowledged the truth of Bacon's principles. He was certainly but little disposed to recognise the claims of any preceding philosophers;

and it has been said by some writers that he did not treat Bacon with more respect than the rest. But it appears from his correspondence with Mersenne (published in 1642), that in several letters he distinctly refers to the works of "Verulam" with a respect which he yielded to no other author, and in a way which shows that he had both studied them and approved the method they deliver. This sufficiently accounts for some remarkable coincidences observed by Mr. D. Stewart in the writings of Des Cartes with the ideas, and even the very words, of Bacon; although he was led to assert that, if Des Cartes ever read the works of Bacon, he has nowhere alluded to them; and in this opinion Professor Playfair also coincides. Indeed, before this time, Bacon's writings seem to have been well known on the Continent, and justly esteemed. A letter is extant from him to Baranzon, who lectured on natural philosophy at Annecy, in Savoy, in 1621 (and, it appears, had consulted him on the introduction of the inductive method), containing a very perspicuous summary of his views, and showing the authority and influence his writings had obtained at that period: his suggestions to Baranzon are almost identical with some of those referred to by Des Cartes in the letters above mentioned.*

It must be confessed, however, that we find very little of the influence of Bacon's principles in the system of Des Cartes, however he might own their general truth, and follow their guidance, when he did condescend to any experimental enquiry: this, as we have seen, he only resorted to as a subordinate means of assisting his theories; and it was very seldom that he thought it necessary to have recourse to it.

The preliminary positions on which his system rested, led him extensively into mechanical speculations. He laid down as an original view the estimate of forces by the momenta, which is no other than that before proposed by Galileo. He also introduced into the

* See Rev. W. V. Harcourt's Address, British Assoc, Reports, note, p. 24.

theory of motion the inertia of matter, regarded as a real active power, and not merely a passive indifference to motion or rest. With respect to curvilinear motion, he pointed out distinctly the necessity of supposing a deflecting force, which being removed, the motion would be rectilinear and in the direction of the tangent. He laid it down as a general principle, that there is always the same quantity of motion in the universe, which, it would seem, is the clue to his notions of inertia, &c. just mentioned. He appears to have regarded motion as a sort of quality superinduced upon matter, but which might, in some cases, be, as it were, latent. All this arose out of the theoretical principle of the permanence of such qualities; and this he deduced from the immutability of the divine attributes. The inherent fallacy of such reasoning must be sufficiently apparent; and we shall not be surprised to find that though, in several cases, he has brought out true results, yet, in others, as in the whole theory of the collision of bodies, he has run into palpable errors.

This last subject, indeed, is one which long remained without a completely satisfactory investigation. Such investigations were first given by Dr. Wallis and Sir C. Wren, nearly at the same time, in 1668; and, also independently of these, by Huyghens in 1669. They each founded their reasonings on the principle, then first fully developed, that action and reaction are equal and in opposite directions.

In 1633, Des Cartes had completed his "System of the World," having previously broached the metaphysical doctrines on which he founded his entire method of philosophical reasoning.

With regard to this system, he must be allowed the credit of having been the first who attempted to suggest any one physical principle to explain and connect all the planetary motions. For we can hardly class under the designation of philosophical theories the crystalline spheres of Ptolemy, or the vitality assigned to the earth by Kepler. Des Cartes proceeded upon principles