move at the rate of 12 feet per fecond, and another body weighing 5 pounds move at the rate of 24 feet per fecond, their momentums will be equal; that is, they will strike an obstacle with equal force, or an equal power must be exerted to stop them; for the product of 10 by 12, viz. 120, is equal to the product of 5 by 24.

The forces of bodies acting on each other by the interpofition of machines is derived from the fame principle. Thus the two bodies A and B, fig. 11, Plate V. are connected with each other by the interpofition of an inflexible rod AB (the fimpleft of all machines) which refts upon the prop or fixed point F. If the rod move out of its horizontal fituation into the oblique pofition CFE, the body A will be forced to defcribe the arch AE, whilst the body B describes the arch BC; and those arches, being described in the same time, will represent the velocities of thofe bodies refpectively; therefore, the momentum of A is to the momentum of B, as the weight of A multiplied by the arch AE, is to the weight of B multiplied by the arch BC.

The velocities of A and B are likewise represented by their distances from F; for the arches AE, BC, are as their radii FA, FB. Thofe velocities are alfo reprefented by the perpendiculars EG, CD; for fince the triangles EFG, CDF, are equiangular and fimilar, (the angles at G and D being right, and those at F being equal) we have EF to FC, as

GE

GE to CD. Therefore the refpective momentums of A and B may be reprefented either by AXAE, and BX BC; or by AXAF, and BX BF; or laftly by AXEG, and BXCD.

Note. The last expreffion is used when the motion of bodies that are fo circumftanced, refults from the action of gravity; viz. when one body recedes from, whilst the other approaches, the centre of the earth; because gravity acts in that direc

tion.

This is the principle of all forts of mechanisms; fo that in every machine the following particulars must be indispensably found. ft. One or more bodies must be moved one way, whilft one or more bodies move the contrary way. One of those bodies or fets of bodies is called the weight, and the other is called the power, or they may be called oppofite powers. 2dly. If the product of the weight of one of those powers, multiplied by the space it moves through in a certain time, be equal to the product of the weight of the oppofite power multiplied by the space it moves through in the fame time; then the oppofite momentums being equal, the machine will remain motionlefs. But if one of those products or momentums exceeds the other, then the former is faid to preponderate, and the machine will move in the direction of the preponderating power; whilft the oppofite power will be forced to move the contrary way. And the preponderance is represented by the excefs of one momentum over

the

the other; for inftance, if one of the above-mentioned products or momentums be 24, and the other 12, then the former is faid to be double the latter; or that the former is to the latter as two to one.

By a ftrict adherence to thofe particulars, the attentive reader will be enabled to eftimate the power and effect of every machine, excepting, however, the obstruction which arifes from the imperfection of materials and of workmanship; as will fully appear from the following paragraphs.

In the explanation of the properties of the mechanical powers, we fuppofe the rods, poles, planes, ropes, &c. to be destitute of weight, roughness, adhefive property, and any imperfection; for when the properties of those powers have been established, we fhall then point out the allowances proper to be made on the fcore of friction, irregularity of figure, &c.

THE LEVER.

A lever is a bar of wood, or metal, or other folid substance, one part of which is fupported by or rests against a Åteady prop, called the fulcrum, about which, as the centre of motion, the lever is moveable.

The use of this machine is to overcome a given obftacle, by means of a given power. Thus if the ftone A, fig. 12, Plate V. weighing 1000 pounds, be required to be lifted up (fo as to pass a rope un

der it, or for fome other purpofe), by means of the ordinary strength of a man; which may be reckoned equal to 100 pounds weight; a pole or lever CE is placed with one end under the stone at E; it is rested upon a stone or other fteady body at B, and the man preffes the lever down at C. In this cafe the man's ftrength is equal to the tenth part of the ftone's weight, therefore its velocity must be ten times greater than that of the `ftone; that is, the part BC of the lever must be ten times as long as the part BE, in order that the power and the weight may balance each other; and if CB is a little longer than ten times BE, then the stone will be raised. Indeed in this cafe the part CB needs not be fo long; for as the stone is not to be entirely lifted from the ground, a leffer momentum is required on the part of the power at C.

In general, to find the proper length of the lever, we need only multiply the weight by that part of the lever which is between it and the fulcrum; and divide the product by the power; for the quotient will be the length BC, which is neceffary to form an equilibrium, and of course a little more than that length will be fufficient to overcome the obftacle.

If when the length of the lever is given, you wish to find what power will be neceffary to overcome a known obftacle or weight; multiply the weight by that part of the lever which is between

VOL. I.

it

it and the fulcrum, then divide the product by the other part of the lever, and the quotient is the anfwer.

The poffible different fituations of the weight, the fulcrum, and the power, are not more than three ; hence arise three kinds of levers; to all of which, however, the preceding calculations are equally applicable. Those fpecies are, 1. when the fulcrum is placed between the weight and the power, as the one already defcribed. 2. When the fulcrum is at one end, the power at the other end, and the weight between them, as in fig. 13, Plate V. And 3. When the fulcrum is at one end, the weight at the other end, and the power between them, as in fig. 14, Plate V.

Some writers add a fourth fpecies, viz. the bent lever; but as this differs only in fhape from the others, it does not conftitute a proper difference of kind.

Hitherto we have supposed that the weight and the power act in directions perpendicular to the arms of the lever; but when this is not the case, the distances of the power and of the weight from the centre of motion must not be reckoned by the distances of the points of fufpenfion from that centre, but by the lengths of the perpendiculars, let fall from the centre of motion on the lines of the direction of the forces. For inftance, in fig. 15, Plate V, the power at. P, acts by means of the ftring PB, on the end B of the lever, in a direction,

BP,