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contained the true principle of this problem. The work is "Jordanus Nemorarius De Ponderositate." The date and history of this author were probably even then unknown; for in 1599, Benedetti, correcting some of the errors of Tartalea, says they are taken "a Jordano quodam antiquo." The book was probably a kind of school-book, and much used; for an edition printed at Frankfort, in 1533, is stated to be "Cum gratia et privilegio Imperiali, Petro Apiano mathematico Ingolstadiano ad XXX annos concesso." But this edition does not contain the inclined plane. Though those who compiled the work assert in words something like the inverse proportion of weights and their velocities, they had not learnt at that time how to apply this maxim to the inclined plane; nor were they even able to render a sound reason for it. In the edition of Venice, 1565, however, such an application is attempted. The reasonings are founded on the usual Aristotelian assumption, "that bodies descend more quickly in proportion as they are heavier." To this principle are added some others; as, that "a body is heavier in proportion as it descends more directly to the centre," and that, in proportion as a body descends more obliquely, the intercepted part of the direct descent is smaller. By means of these principles, the "descending force" of bodies, on inclined planes, was compared, by a process, which, so far as it forms a line of proof at all, is a somewhat curious example of confused and vicious reasoning. When two bodies are supported

on two inclined planes, and are connected by a string passing over the junction of the planes, so that when one descends the other ascends, they must move through equal spaces on the planes; but on the plane which is more oblique (that is, more nearly horizontal,) the vertical descent will be smaller in the same proportion in which the plane is longer. Hence, by the Aristotelian principle, the weight of the body on the longer plane is less; and, to produce an equality of effect, the body must be greater in the same proportion. We may observe that the Aristotelian principle is not only false, but is here misapplied; for its genuine meaning is, that when bodies fall freely by gravity, they move quicker in proportion as they are heavier; but the rule is here applied to the motions which bodies would have, if they were moved by a force extraneous to their gravity. The proposition was supposed by the Aristotelians to be true of actual velocities; it is applied by Jordanus to virtual velocities. This confusion being made, the result is got at by taking for granted that bodies thus proved to be equally heavy, have equal powers of descent on the inclined planes; whereas, in the previous part of the reasoning, the weight was supposed to be proportional to the descent in the vertical direction. It is obvious, in all this, that though the author had adopted the false Aristotelian principle, he had not settled in his own mind whether the motions of which it spoke were actual or virtual motions;-motions in the direction of the inclined

plane, or of the intercepted parts of the vertical, corresponding to these; nor whether the "descending force" of a body was something different from its weight. We cannot doubt that, if he had been required to point out, with any exactness, the cases to which his reasoning applied, he would have been unable to do so; not possessing any of those clear fundamental ideas of pressure and force, on which alone any real knowledge on such subjects must depend. The whole of Jordanus's reasoning is an example of the confusion of thought of his period, and nothing more. It no more supplied the want of some man of genius, who should give the subject a real scientific foundation, than Aristotle's knowledge of the proportion of the weights on the lever superseded the necessity of Archimedes's proof of it.

We are not, therefore, to wonder that, though this pretended theorem was copied by other writers, as by Tartalea, in his Quesiti et Inventioni Diversi, published in 1554, no progress was made in the real solution of any one mechanical problem by means of it. Guido Ubaldi, who, in 1577, writes in such a manner as to show that he had taken a good hold of his subject for his time, refers to Pappus's solution of the problem of the inclined plane, but makes no mention of that of Jordanus and Tartalea. No progress was likely to occur, till the mathematicians had distinctly recovered the genuine idea of pressure, as a force producing equilibrium, which Archimedes had possessed, and which was soon to reappear in Stevinus.

The properties of the lever had always continued known to mathematicians, although, in the dark period, the superiority of the proof given by Archimedes had not been recognised. We are not to be surprised, if reasonings like those of Jordanus were applied to demonstrate the theories of the lever with apparent success. Writers on mechanics were, as we have seen, so vacillating in their mode of dealing with words and propositions, that they would be made to prove anything which was already known to be true.. We proceed to speak of the beginning of the real progress of mechanics in modern times.

Sect. 2.-Revival of the Scientific Idea of PressureStevinus.-Equilibrium of Oblique Forces.

THE doctrine of the centre of gravity was the part of the speculations of Archimedes which was most diligently prosecuted after his time. Pappus and others, among the ancients, had solved some new problems on this subject, and Commandinus, in 1565, published De Centro Gravitatis Solidorum. Such treatises contained, for the most part, only mathematical consequences of the doctrines of Archimedes; but the mathematicians also retained a steady conviction of the mechanical property of the centre of gravity, namely, that all the weight of the body might be collected there, without any change in the mechanical results; a conviction which is closely connected with our fundamental concep

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tions of mechanical action. Such a principle, also, will enable us to determine the result of many simple mechanical arrangements; for instance, if a mathematician of those days had been asked whether a solid ball could be made of such a form, that, when placed on a horizontal plane, it should go on rolling forwards without limit, merely by the effect of its own weight, he would probably have answered, that it could not; for that the centre of gravity of the ball would seek the lowest position it could find, and that, when it had found this, the ball could have no tendency to roll any further. And, in making this assertion, the supposed reasoner would not be anticipating any wider proofs of the impossibility of a perpetual motion, drawn from principles subsequently discovered, but would be referring the question to certain fundamental convictions, which, whether put into axioms or not, inevitably accompany our mechanical conceptions.

In the same way, if Stevinus of Bruges, in 1586, when he published his Beghinselen der Waaghconst (Principles of Equilibrium), had been asked why a loop of chain, hung over a triangular beam, could not, as he asserted it could not, go on moving round and round perpetually, by the action of its own weight, he would probably have answered, that the weight of the chain, if it produced motion at all, must have a tendency to bring it into some certain position; and that when the chain had reached this position, it would have no tendency to go any further; and thus he would have reduced the impos

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