Chapter 6 MECHANICS.

It is said that Archytas was the first to treat mechanics in a systematic way by the aid of mathematical principles; but no trace survives of any such work by him. In practical mechanics he is said to have constructed a mechanical dove which would fly, and also a rattle to amuse children and "keep them from breaking things about the house" (so says Aristotle, adding "for it is impossible for children to keep still").

In the Aristotelian Mechanica we find a remark on the marvel of a great weight being moved by a small force, and the problems discussed bring in the lever in various forms as a means of doing this. We are told also that practically all movements in mechanics reduce to the lever and the principle of the lever (that the weight and the force are in inverse proportion to the distances from the point of suspension or fulcrum of the points at which they act, it being assumed that they act in directions perpendicular to the lever). But the lever is merely "referred to the circle"; the force which acts at the greater distance from the fulcrum is said to move a weight more easily because it describes a greater circle.

There is, therefore, no proof here. It was reserved for Archimedes to prove the property of the lever or balance mathematically, on the basis of certain postulates precisely formulated and making no large demand on the faith of the learner. The treatise On Plane Equilibriums in two books is, as the title implies, a work on statics only; and, after the principle of the lever or balance has been established in Props. 6, 7 of Book I., the rest of the treatise is devoted to finding the centre of gravity of certain figures. There is no dynamics in the work and therefore no room for the parallelogram of velocities, which is given with a fairly adequate proof in the Aristotelian Mechanica.

Archimedes's postulates include assumptions to the following effect: (1) Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium, but the system in that case "inclines towards the weight which is at the greater distance," in other words, the action of the weight which is at the greater distance produces motion in the direction in which it acts; (2) and (3) If when weights are in equilibrium something is added to or subtracted from one of the weights, the system will "incline" towards the weight which is added to or the weight from which nothing is taken respectively; (4) and (5) If equal and similar figures be applied to one another so as to coincide throughout, their centres of gravity also coincide; if figures be unequal but similar, their centres of gravity are similarly situated with regard to the figures.

The main proposition, that two magnitudes balance at distances reciprocally proportional to the magnitudes, is proved first for commensurable and then for incommensurable magnitudes. Preliminary propositions have dealt with equal magnitudes disposed at equal distances on a straight line and odd or even in number, and have shown where the centre of gravity of the whole system lies. Take first the case of commensurable magnitudes. If A, B be the weights acting at E, D on the straight line ED respectively, and ED be divided at C so that A : B = DC : CE, Archimedes has to prove that the system is in equilibrium about C. He produces ED to K, so that DK = EC, and DE to L so that EL = CD; LK is then a straight line bisected at C. Again, let H be taken on LK such that LH = 2LE or 2CD, and it follows that the remainder HK = 2DK or 2EC. Since A, B are commensurable, so are EC, CD. Let x be a common measure of EC, CD. Take a weight w such that w is the same part of A that x is of LH. It follows that w is the same part of B that x is of HK. Archimedes now divides LH, HK into parts equal to x, and A B into parts equal to w, and places the w's at the middle points of the x's respectively. All the w's are then in equilibrium about C. But all the w's acting at the several points along LH are equivalent to A acting as a whole at the point E. Similarly the w's acting at the several points on HK are equivalent to B acting at D. Therefore A, B placed at E, D respectively balance about C.

Prop. 7 deduces by reductio ad absurdum the same result in the case where A, B are incommensurable. Prop. 8 shows how to find the centre of gravity of the remainder of a magnitude when the centre of gravity of the whole and of a part respectively are known. Props. 9-15 find the centres of gravity of a parallelogram, a triangle and a parallel-trapezium respectively.

Book II., in ten propositions, is entirely devoted to finding the centre of gravity of a parabolic segment, an elegant but difficult piece of geometrical work which is as usual confirmed by the method of exhaustion.

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