Instruments make a person strong, but it by no means follows from this that instruments permit one to expend a little work and obtain a lot. The law of conservation of energy convinces us that a gain in work, i.e, the creation of work out of “nothing”, is impossible.

The work obtained cannot be greater than the work performed. On the contrary, the inevitable energy loss due to friction leads to the fact that the work obtained with the aid of an instrument will always be less than that performed. In the ideal case, these works can be equal.

Properly speaking, we are wasting time by explaining this obvious truth: for the rule of torques was derived from the condition of equality of the work performed by the active and overcome forces.

If the points of application of the forces moved distances \(s_{1}\) and \(s_{2}\) the condition of equality of the work assumes the following form: \[F_{1}^{t}s_{1} = F_{2}^{t}s_{2}\] In overcoming some force \(F_{2}\) along a path of length \(s_{2}\) with the aid of a lever, we can make this by means of force \(F_{1}\) much less than \(F_{2}\). But the displacement \(s_{1}\) of our hand must be as many times greater than \(s_{2}\) as the muscular force \(F_{1}\) is less than \(F_{2}\).

This law is often expressed by the following brief sentence: the gain in force is equal to the loss in path.

The law of the lever was discovered by the greatest scientist of antiquity—Archimedes. Amazed at the strength of his proof, this remarkable scientist of antiquity wrote to King Hiero II of Syracuse: “If there were another world and I could go to it, I would move this one.” A very long lever whose pivot is near the Earth would make it possible to solve such a problem.

We shall not grieve with Archimedes over the absence of a fulcrum, which, as he thought, was all that he lacked to move the Earth.

Let us dream: take the strongest possible lever, place it on a pivot and “suspend a small sphere” of weight … \(6\times 10^{24}\,\mathrm{kgf}\) on its short end. This modest number shows how much the Earth “pressed into a small sphere” weighs. Now apply muscular force to the long end of the lever.

If the force exerted by Archimedes can be taken as \(60\,\mathrm{kgf}\), then in order to displace the “Earth nut” by \(1\,\mathrm{cm}\), Archimedes’ hand would have to cover a distance \(6\times 10^{24}/60 = 10^{23}\) times greater. But \(10^{23}\,\mathrm{cm}\) are \(10^{18}\,\mathrm{km}\), which is three billion times greater than the diameter of the Earth’s orbit!

This fantastic example clearly demonstrates the scale of the “loss in path” involved in the work of a lever.

Any of the examples considered by us above can be used to illustrate not only a gain in force but also a loss in path. The hand of the chauffeur jacking up his car covers a path which will be as many times longer than the height to which he raises it as his muscular force is less than the weight of the automobile. Moving a pair of scissors in order to cut a sheet of tin-plate, we perform work along a path which is as many times longer than the depth of the cut as our muscular force is less than the resistance of the tin-plate. The stone lifted by the crow-bar will rise to a height as many times less than that by which the hand is lowered as the muscular force is less than the weight of the stone. This rule clarifies the principle of the screw’s action. Imagine that we are screwing in a bolt, whose threading has a \(1\,\mathrm{mm}\) screw pitch, with the aid of a wrench of length \(30\,\mathrm{cm}\). The bolt will advance \(1\,\mathrm{mm}\) along its axis during a single turn, while our hand will cover a \(2\,\mathrm{m}\) path during the same time. Our gain in force is two thousand-fold, and we either safely fasten the components together or move a heavy weight with a slight effort of our hand.

Archimedes [c. (287–212) B.C.]—The greatest mathematician, physicist and engineer of antiquity. Archimedes computed the volume and the surface area of a sphere and its parts, of a cylinder and of bodies formed by rotating an ellipse, hyperbola or parabola. He was the first to compute the ratio of the circumference of a circle to its diameter with a high degree of accuracy, showing that it satisfies the inequalities \(3 \frac{10}{71} < \pi < 3 \frac{1}{7}\). In mechanics he established the laws of lever, the conditions governing floating bodies (Archimedes’ principle), the composition of parallel forces. Archimedes invented the machine for pumping water (Archimedean screw, used in our times for transporting free-flowing or viscous cargo), systems of levers and blocks for raising heavy weights, and military engines successfully employed during the siege of his native city, Syracuse, by the Romans.