furface. In some respects, the first may be said to generate the latter. Both are capable of increase or diminution, and yet no increase or diminution of the one can produce the other.

That preffure, which gravity caufes bodies to D exert against any obstacle interpofed between them and the earth, is called Weight. We have already taken notice of its effect when free to produce an uniform acceleration in falling bodies (26, AB C.) If its power could be increased or diminished, it would proportionally increafe or diminish the momentaneous velocities and spaces, (28, E) and confequently the whole space paffed through in a given time; that is to say, conftant forces are as the spaces paffed ✰ through by acceleration in a given time, or as the laft acquired velocities.

When the effect of any retarding force is confidered (31,P), the force will, in a given time, be as the whole space defcribed during the retardation, till the motion is destroyed, or as the initial velocity.

Let two accelerating forces be to each other in any ratio, the last acquired velocities will (35, E) be in the fame ratio. Imagine the lefs motion to be continued till its laft acquired velocity becomes equal to that of the other, and the whole time (27, D) will then be to the former time as the greater velocity to the lefs, or inverfely as the forces; that is, the times required to produce equal velocities o are inversely as the accelerating forces, But the fpaces defcribed in equal times are as the forces (35, E). Whence the spaces described in any other D2 times.

36 ACCELERATING AND RETARDING FORCES. times will be as the forces and the fquares of the times (29, G) jointly. But when equal velocities are produced, the fquares of the times will be inverfely as the squares of the forces (35, G). Therefore the fpaces in this cafe will be as the forces directly, and the fquares of the forces inverfely, or inHverfely as the forces; that is, the spaces paffed through in producing equal final velocities are inversely as the accelerating forces.

I

When the effect of a retarding force is confidered (31, P) the spaces paffed through in destroying motion are inversely as the retarding forces, when the initial velocities are equal.

K If a body be acted upon by a conftant and invariable force, and its mass be either increased or diminished, without altering the force, the effect will be the fame with respect to acceleration, or retardation, as if the force, without changing the body, were diminished or increased in the inverted ratio of the mafs. For the force being fuppofed invariable, will always produce or deftroy the fame quantity of motion (21, ) in a given time. This quantity will be measured by the product of the mafs into the velocity (19, L). And that this product may continue unaltered, it is neceffary that the velocity should diminish in the fame proportion as the mass is increased, and the contrary.

L A body impinging with different velocities on tallow, clay, timber, and fome other fubftances, penetrates to depths in the fame fubftance which are as the fquares of the initial velocities. Whence it follows,

follows, (32, Q) that these substances oppose a conftant and invariable force of retardation against the motions of given bodies.

Suppose a body to impinge on an uniformly refifting fubftance, if the initial velocity vary only, it will penetrate to depths which are as the squares of the velocities (32, Q). But if the mafs (not magnitude) vary only, the confequence will be the fame as if the retarding force had varied in the inverted ratio of the mafs (36, K). And the depths or spaces will be inversely as the retarding forces (36, 1) or directly as the maffes: confequently, if both the mafs and м velocity vary, the depths will be in the compound ratio of the maffes and the fquares of the velocities. The difpute concerning the measure of forces, N which divided the philofophical world for confiderably more than half a century, was founded on a partial confideration of the effects of collifion. The o queftion agitated was, whether the forces of bodies in motion ought to be measured by the mass of matter multiplied by the velocity, or by the mafs multiplied by the fquare of the velocity. The former affirmation was called the old opinion, and the latter the new opinion.

Neither of these opinions are fufficiently general P o apply to every cafe of motion, neither are they repugnant to each other, as the contenders for each infifted. The chief argument urged by the main- e tainers of the new opinion was, that spheres of equal magnitude, but of different weights, being let fall into tallow, from heights that were inversely as the

weights, made pits of equal depths in the fame. Now, faid the difputants, equal caufes are those which produce equal effects; the forces of these bodies at their impact on the tallow must be equal, as being the causes of equal effects, namely, the pits in the fame fubftance. But the squares of the velocities of the impacts are (29, G) as the heights from whence the bodies fall, or in this cafe inverfely as the weights of the bodies. Therefore the product of each weight or mafs into the square of its velocity is equal to the product of the other weight into the fquare of its velocity, when the pits, or, as it is affirmed, the forces, are equal.

R All this is true, when it is confidered as a mere explanation of the meaning of the word force, which, if understood and applied in this fenfe, will s not be productive of error. But when the above is intended to ferve as a proof that the action of a body in motion cannot be measured by the mass multiplied into its velocity, it becomes neceffary to observe, that the pit in the tallow 'being equal to another pit, does not prove that they were made by Tequal powers or forces. For powers cannot be faid to

be equal, unless they produce equal effects in equal times; it being eafy to imagine, that a weaker power continued for a longer time may produce an effect equal to that of a power of greater intenfity, u though of less duration. And it is evident, that these pits are not described in equal times; for they are equal to half the spaces which would have been described with their respective initial velocities uni

formly

formly continued (32, s) in the respective times of defcription. But the initial velocities would describe fuch equal spaces in times which are inversely as the velocities themselves. And it has been already seen, that the refult of the prefent experiment is an easy confequence of the properties of retarded motion, confidered jointly with that definition which affirms the force or quantity of motion in a body to be as the product of the mafs into the velocity (37, M).

The confideration that moving bodies penetrate w obftacles to depths which are as the mass of the body multiplied by the fquare of its velocity, is of great use in almost every circumstance of this kind. It follows from hence, that the depths to x which a body of given magnitude will penetrate into any substance, may be varied to infinity, with-. out changing the quantity of motion. For the depths will always be greatest when the velocity is greatest (37, M); and the quantity of motion, or product of the mafs into the velocity, will not be changed, if the mafs be diminished proportionally while the velocity is augmented, and the con

trary.

Thus it is fhewn, that a fmall hammer, having y the fame ftriking, furface and quantity of motion, will do more work at a blow than a large one. The driving of nails, or of piles into the earth, follows nearly the fame law, though in the inftance of the engine for driving piles by the fall of a weight, nothing (37, Q) would be gained by lef