In tracing the progress of discovery concerning the mathematical theory of mechauical action, we shall have little to notice till we come to the 16th century ; for the ancients, who devoted themselves with so much ardour to the researches of pure science, almost entirely neglected the application of the latter to subjects which appeared to them to terminate in mere practical utility. It must he observed however that Aristotle, who left no department of nature untouched, has noticed, in his mechanical questions, the equilibrium of unequal weights on the unequal arms of a balanced lever, though he gives a very unphilo sophical reason for tho fact. But in his Physics' he states correctly that if two forces move with velocities reciprocally proportional to their intensities, they will exert equal efforts : this may apply to a well-known property of the lever, but it may have been meant to refer only to the effect of two unequal bodies moving with unequal velocities, and striking each other or a third body.
Sicily enjoys the honour of having given birth to the first philosopher who can properly be said to have been a theoretical mechanician : we allude to Archimedes, who died about 212 D.C., and in whose works there is direct evidence of an effort to determine the principle of equilibrium in machines. Commencing, in the treatise whose Latin title is De eEquiponderantibus,' with the axiom that two equal weights balance each other on a lever (of uniform dimensions), when at equal distances from the fulcrum, he supposes the weights to be divided into an equal number of equal parts, and that the parts are removed to equal distances from the point of support ; observing then that the equilibrium still subsists, he proceeds, by the method of exhaustion', to show that it always will take place provided the bodies are inversely proportional to their distances from the fulcrum. Archimedes thence concludes that there must exist in every one body, considered as an assemblage of smaller bodies, a centre of force (that is, a centre of gravity) corresponding to the fulcrum in the former case; and lie pro ceeds, by the analysis of that day, to investigate the seat of the centre of force in a triangle, a parabola, and a paraboloid.
This philosopher has obtained celebrity by the contrivances which he is said to have adopted for the defence of Syracuse. No precise account is given of the machinery which he employed to raise up and destroy the galleys of the enemy, and the effects are probably exagge rated. Tho vessels must. have been close to the walls, and it is con ceivable that, by hooks at ends of chains which were suspended from levers on the ramparts, the rigging, or some parts of the turrets erected as usual on the docks, in order to enable the assailants to pass over the parapets, might be caught ; then, the levers being raised by the force of men or otherwise, the vessels or the turrets would be easily overturned.
During about 1800 years, which elapsed between the time of Archl modes and that of Cardau, we have no other notices concerning the theory of mechanics (beyond those which occur in the writings of the former mathematician), than such as are contained in the Mathe matical Collections' of Pappus, which amount merely to a statement that the ancients had reduced the theory of every machine to that of the lever, and an unsuccessful attempt to explain the cause of the equilibrium of a body on an inclined plane. It is remarkable moreover that both Carden and, subsequently, the Marquis Ubaldi (the latter of whom published, in 1577, a treatise in which he explains at length the combinations of pulleys, and reduces their theory to that of the lever) should also have given erroneous solutions of the problem couceming that equilibrium. Tho discovery of the truo theory of the inclined plane was however, about the same time, made by Stevinus, a native of Flanders. This mathematician and engineer supposed a chain o uniform dimensions to be placed on a doubly inclined plane, having a common summit and base, the chain being perfectly free to slide on the planes, and its ends banging vertically to equal distances below the base ; then, in order to prove that the chain would remain at rest, he shows that if any motion should take place, it might continue for ever ; and this he concludes to be absurd. As the argument holds good when one of the two plaues is in a vertical position, Stevinus infers that, when a body is in equilibrio upon a plane, the retaining power is to the weight as the height of the plane is to its length ; and he further shows that if three forces act on any point, they hold the latter in equilibrio when they are proportional to the three sides of a triangle formed by lines drawn parallel to the directions of the forces. It should be remarked however that Stevinus demonstrates the law in that case only in which two of the forces are at right angles to each other. He died in 1635.
To Galileo we are indebted for thd first reduction of mechanical propositions to purely mathematical formulae. In order to demonstrate the equilibrium of a body on an inclined plane, be imagined the weight and the sustaining power to be applied to the ends of a bent lever whose arms were of equal length and perpendicular to the vertical and slant sides of the plane ; then reducing the lever to a straight one, between the lines of direction of the weight and 'power, it was easy to prove that the forces in equilibrio on the plane were also in equilibrio on the lever, and were to one another as the length to the height of the plane.