Power to Weight Ratio

5. The ratio of power to weight on an inclined plane was for a long time a problem among modern mechanicians.

Stevin was the first to solve it.

But his solution is founded on an indirect consideration independent of the theory of the lever.

Stevin considers a solid triangle placed on its horizontal base, so that its two sides form two inclined planes.

He imagines that a necklace formed of several equal weights, strung at equal distances, or rather a chain of equal thickness, is placed on the two sides of this triangle, so that the whole upper part is applied to the two sides of the triangle, and the lower part hangs freely below the base, as if it were attached to the two ends of this base.

Stevin remarks that, supposing the chain can slide freely on the triangle, it must however remain at rest; for, if it began to slide of itself in one direction, it would have to continue sliding always, since the same cause of motion would subsist, the chain, due to the uniformity of its parts, always being placed in the same way on the triangle; from which would result a perpetual motion, which is absurd.

There is therefore necessarily equilibrium between all the parts of the chain; now one can regard the portion which hangs below the base as being already in equilibrium by itself.

Therefore the effort of all the weights leaning on one side must counterbalance the effort of the weights leaning on the other side; but the sum of the ones is to the sum of the others in the same ratio as the lengths of the sides on which they are leaning.

Therefore it will always require the same power to sustain one or more weights placed on an inclined plane, when the total weight is proportional to the length of the plane, supposing the height the same: but, when the plane is vertical, the power is equal to the weight.

Therefore, in any inclined plane, the power is to the weight as the height of the plane is to its length.

I have reported this demonstration of Stevin, because it is very ingenious and besides little known.

He:

• deduces from this theory that of the equilibrium between 3 powers which act on the same point
• finds that this equilibrium takes place when the powers are parallel and proportional to the 3 sides of any rectilinear triangle.

But this fundamental theorem of Statics was demonstrated by this author only in the case where the directions of two of the powers make a right angle with each other.

Stevin remarks with reason that a weight resting on an inclined plane, and held by a power parallel to the plane, is in the same case as if it were supported by two cords, one perpendicular, and the other parallel to the plane.

By his theory of the inclined plane, he finds that the ratio of the weight to the power parallel to the plane is as the hypotenuse to the base of a right triangle formed on the plane by two lines, one vertical and the other perpendicular to the plane.

Stevin then contents himself with extending this proportion to the case where the cord which holds the weight on the inclined plane is also inclined to this plane, by constructing an analogous triangle with the same lines, one vertical, the other perpendicular to the plane, and taking the base in the direction of the cord.

But for that he would have needed to have demonstrated that the same proportion holds in the equilibrium of a weight supported on an inclined plane by a power oblique to the plane, which cannot be deduced from the consideration of the chain imagined by Stevin.

6. Galileo’s Mechanics was first published in French by Father Mersenne in 1634.

In it, the equilibrium on an inclined plane is reduced to that of an angular lever with two equal arms, one of which is supposed perpendicular to the plane and loaded with a weight resting on the plane, and the other is horizontal and loaded with a weight equivalent to the power necessary to hold the weight on the plane.

This equilibrium is then reduced to that of a straight and horizontal lever, by regarding the weight attached to the inclined arm as suspended from a horizontal arm forming a straight lever with the horizontal arm of the angular lever.

Thus, the weight is to the power sustaining it on the inclined plane, in inverse ratio of these two arms of the straight lever, and it is easy to prove that these arms are to each other as the height of the plane to its length.

One can say that this is the first direct demonstration that has been had of equilibrium on an inclined plane. Galileo has since used it to rigorously demonstrate the equality of velocities acquired by heavy bodies, in descending from the same height on planes differently inclined, an equality which he had merely supposed in the first edition of his Dialogues.

It would have been easy for Galileo to also solve the case where the power which holds the weight has a direction oblique to the plane; but this new step was only made some time later, by Roberval, in a Treatise on Mechanics printed in 1636, in Mersenne’s Universal Harmony.

7. Roberval:

• also regards the weight resting on the inclined plane as attached to the arm of a lever perpendicular to the plane.
• considers the power as a force applied to the same arm, following a given direction
• thus has a lever with a single arm, one end of which is fixed, and the other end is pulled by two forces, that of the weight and that of the power which holds it.

He then:

• substitutes for this lever an angular lever with two arms perpendicular to the directions of the two forces and having the same fixed point as fulcrum.
• supposes the 2 forces applied to the arms of this lever following their own directions, which gives him for equilibrium the ratio of the weight to the power, in inverse ratio of the two arms of the angular lever, that is to say, of the perpendiculars drawn from the fixed point to the directions of the weight and the power.

From this, Roberval deduces the equilibrium of a weight supported by 2 cords which make any angle with each other, by substituting for the lever perpendicular to the plane a cord attached to the fulcrum of the lever, and for the power another cord pulled by a force in the direction of this power.

By various somewhat complicated constructions and analogies, he arrives at this conclusion that, if from any point taken on the vertical of the weight, one draws a parallel to one of the cords, until it meets the other cord, the triangle thus formed will have its sides proportional to the weight and to the powers which act in the direction of the same sides, which is, as one sees, the theorem given by Stevin.

Roberval’s demonstration:

• is the first rigorous demonstration of Stevin’s theorem
• has remained forgotten in a Treatise on Harmony, quite rare today

I have entered into this detail concerning the theory of the lever, only to please those who love to follow the progress of the mind in the sciences, and to know the paths which inventors have taken and the more direct paths they could have taken.

8. The Treatises on Statics which appeared after that of Roberval, until the time of the discovery of the composition of forces, added nothing to this part of Mechanics; one finds in them only the already known properties of the lever and the inclined plane.

Their application to other simple machines; there are even some which contain rather inaccurate theories, like that of Lami on the equilibrium of solids, where he gives a false proportion of the weight to the power which holds it on an inclined plane.

I do not speak here of Descartes, Torricelli, and Wallis, because they adopted for equilibrium a principle which relates to that of virtual velocities, and of which they did not have the demonstration.