The credit of first giving the true theory of equilibrium on the inclined plane is usually ascribed to Stevin, although, as we shall presently show, with very little reason. Stevin supposed a chain to be placed over two inclined planes, and to hang down in the manner represented in the figure. He then urged that the chain would be in equilibrium; for otherwise, it would incessantly continue in motion, if there were any cause why it should begin to move. This being conceded, he remarks further, that the parts AD and BD are also in equilibrium, being exactly similar to each other; and therefore if they are taken away, the remaining parts AC and BC will also be in equilibrium. The weights of these parts are proportional to the lengths AC and BC; and hence Stevin concluded that two weights would balance on two inclined planes, which are to each other as the lengths of the planes included between the same parallels to the horizon.[137] This conclusion is the correct one, and there is certainly great ingenuity in this contrivance to facilitate the demonstration; it must not however be mistaken for an _à priori_ proof, as it sometimes seems to have been: we should remember that the experiments which led to the principle of virtual velocities are also necessary to show the absurdity of supposing a perpetual motion, which is made the foundation of this theorem. That principle had been applied directly to determine the same proportion in a work written long before, where it has remained singularly concealed from the notice of most who have written on this subject. The book bears the name of Jordanus, who lived at Namur in the thirteenth century; but Commandine, who refers to it in his Commentary on Pappus, considers it as the work of an earlier period. The author takes the principle of virtual velocities for the groundwork of his explanations, both of the lever and inclined plane; the latter will not occupy much space, and in an historical point of view is too curious to be omitted. "_Quæst. 10._—If two weights descend by paths of different obliquities, and the proportion be the same of the weights and the inclinations taken in the same order, they will have the same descending force. By the inclinations, I do not mean the angles, but the paths up to the point in which both meet the same perpendicular.[138] Let, therefore, _e_ be the weight upon _dc_, and _h_ upon _da_, and let _e_ be to _h_ as _dc_ to _da_. I say these weights, in this situation, are equally effective. Take _dk_ equally inclined with _dc_, and upon it a weight equal to _e_, which call 6. If possible let _e_ descend to _l_, so as to raise _h_ to _m_, and take 6_n_ equal to _hm_ or _el_, and draw the horizontal and perpendicular lines as in the figure. [Illustration] Then _nz_:_n_6 :: _db_:_dk_ and _mh_:_mx_ :: _da_:_db_