LECTURE XXXI.

OF THE MECHANICAL POWERS.

THE

HE knowledge of mechanics is one of thofe things that contribute to diftinguish civilized nations from barbarians. From it the works of art derive much of their beauty and value; without it we can make very little progress in the knowledge of the works of nature. By this fcience we are enabled to improve every power and force in nature, and render the motions of the elements water, air, and fire, fubfervient to the purposes of life.

A weight greater than the natural ftrength of man could manage, without fome mechanical contrivance, could never be lifted from the earth: he is obliged, therefore, to feek affiftance, and call in other forces, to make fuch alterations as his pleafures or neceffities may require. By means of levers he lifts weights much greater than his ftrength could overcome; with the wheel and axle, pullies, &c. he lifts them to confiderable heights, and produces fuch effects, as would, to an unexperienced favage, appear the work of enchant

ment.

Nor were the ancients without a great knowledge in this art of increafing ftrength by machinery. The ftones which are laid upon the tops of the pyramids of Egypt, each of which is as big as a fmall houfe, create even the wonder of a modern mechanic, and teach him to reverence the fuperior arts of antiquity.

VOL. III.

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Practical

Practical mechanics will fhew you how to employ a given force, fo as to produce a proposed effect by the aid of fome machine or engine. It alfo fhews how to modify by machines the action of a given power, or the quantity of a known

effect.

Three things are therefore always confidered in treating of the mechanical engines; a weight to be raised, the power by which it is to be raised, and the inftrument, or engine, by which it is to be effected.

Mechanics may be confequently reduced to two problems: 1. To determine the proportion which the power and weight ought to have to each other, that they may juft fuftain each other in equilibrio. 2. To determine what ought to be the proportion of the power and weight to each other in a given engine, that it may produce the greatest effect in a given time.

Machines, or engines, may be confidered as different means employed to facilitate the action of a power against a refifting obftacle.

The number and nature of mechanical engines, vary according to the object for which they are defigned; but, however varied or multiplied, you will find them to be only a combination of a fmall number of fimple machines, commonly called mechanical powers.

I fhall confider fix fimple machines; the lever, the pulley, the wheel and axis, the inclined plane, the wedge, and the fcrew.

Thefe are all calculated to communicate motion to bodies, and sustain a preffure, for which the power, unaffifted by them, is incompetent; and the artifice in all confifts in diftributing the weight among fuch a number of agents, that the part fuftained by the power may bear a fmall ratio to the whole.

Thus,

Thus, a power incapable of communicating motion to, or fupporting the preffure of a body, without mechanical affiftance, may effect it's purpose by tranferring a part of the weight upon the fupport, or fulcrum, diftributing it among a number of pullies, or placing it upon an inclined plane, or fcrew.

POSTULATA.

The following poftulata are neceffary for the mathematical confideration of the mechanical powers.

1. That a small portion of the earth's surface may be considered as a plane.

2. That heavy bodies defcend in lines that are parallel to each other: for though all bodies tend to the center of the earth, yet the distance from which they fall is fo fmall, when compared to their distance from the center of the earth, that their inclination is inconfiderable.

3. That the effort of any given power, or weight, is the fame in all points of it's direction; or if a body be acted on by any power in a given direction, the action will be the fame, in whatever part of that direction it be applied.

4. That though all matter be rough, all machines imperfect, &c. yet, in order to make calculations more eafy and elegant, and to render theory more perfect, we fuppofe all planes perfectly even, all bodies quite fmooth, all lines ftrait and inflexible, without weight, and without thickness, all cords to be extremely pliable, and all machines without friction.

These fuppofitions go no further than to fhew, that the reasonings are concerning perfect inftruments; but as there are none really fo, the difference between theory and practice is to be after

wards inveftigated, and, when difcovered, allowed for.

5. The effort of the power, and of the weight, is equal in all points of their directions; i.e. if you push or draw any body with a ftick (fuppofed inflexible, and without gravity), this body is pufhed or drawn with the fame force, whether the stick be long or fhort. Or if a weight be fufpended by a long or fhort string, it will neither weigh more or lefs, the weight of the ftring excepted; for though gravity varies according to the different diftances of a body from the earth, this variation is totally infenfible in the length of a cord by which a weight is fufpended from a machine.

Here it may be proper to recall to your mind what I have before fhewed you, that the momentum of any power or weight, is that force wherewith it either moves, or endeavours to move; and that it is always proportional to the product arifing by multiplying the power or weight into the velocity wherewith it moves, or would move, if it were not hindered by fome oppofite power or weight. If the product arifing from the multiplication of one weight or power into it's velocity, be equal to the product arifing from the like multiplication of any other weight or power into it's velocity, the momenta of those two powers or weights must be equal. And this will always happen when the weights or powers are to each other reciprocally as their velocities.

Hence two weights or powers will ballance, when the one exceeds the other in weight, as much as it furpaffes the other in velocity; and herein confifts the force and efficacy of mechanical engines; for they are fo contrived, as to diminish the velocity of one weight or power, and to increafe that of the other; by which means a very fmall weight or power my ballance a very large one.

In general, therefore, there is always an equilibrium, when the fum of any number of acting powers is equal, and directly oppofed to any one power, or to the fum of the forces of any number of powers; and reciprocally, if there be an equilibrium among the acting powers, the fum of the acting powers on one fide, are equal to the fum of the powers acting on the oppofite fide.

Effects are always in proportion to their adequate caufe; or the change of motion, in any body, is always in proportion to the force which produced it, and in the direction wherein that force acts for instance, if a certain force affects a certain motion, double that force will occafion double that motion.

It follows, from this axiom, that if an effect depends on feveral heterogeneous caufes, or if feveral different circumftances concur in producing any effect, this effect will be as the quotient re-. fulting from dividing the product of the circumftances that contribute to augment the effect, by the product of thofe that concur to diminish it.*

Or, in other words, an effect produced by feveral diffimilar causes, is in a ratio compounded of the direct ratio of the quantities which must be increafed in order to augment the effect, and the inverfe ratio of thofe that must be diminished to increase the effect.

To illuftrate this by an example, fuppofe a waggon to be fent to any particular place; it is clear that the facility of conveying it, or effect, depends on the weight, the number of horfes employed, the ftrength of the horfes, length of the way, goodness of the road, and time to be employed.

Now it is easy to fee, that the facility of moving

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De la Caille's Elements de Mechanique, p. 5.