1. That in bodies falling freely by their own gravity, the spaces defcribed in falling from reft, are as the fquares of the times of falling.

2. That the spaces defcribed by falling bodies, are as the fquares of the velocities.

3. That therefore the spaces defcribed by falling bodies, are in the compound ratio of the times and the velocities acquired by falling.

4. If a body falls through any space, and moves afterwards with the velocity gained in falling, it will defcribe twice that (pace in the time of it's falling.

The motion of a body thrown directly upwards is continually retarded, in the fame manner as the motion of a falling body is accelerated: for the action of gravity, in this cafe, continually acts contrary to the motion of the rifing body; whereas in that of falling bodies it confpires with it.

As the power of gravity communicates to the body in every equal moment equal velocities, fo the velocity of a body thrown upwards, is equally retarded in equal times; for the fame force of gravity, which generates motion in a falling body, destroys it in a rifing one. It follows, therefore, that in the fame time and manner the fame velocities are generated and destroyed in the fame times. A body thrown upwards continues to afcend till it has loft all it's motion, when it will begin to defcend, and the motion will be juft as much accelerated as it was before retarded; it therefore afcends during the time that a body in falling can acquire the velocity with which it was thrown up.

And the heights to which bodies projected directly upwards, with different velocities, can afcend, are to one another as the fquares of their first velocities.

Therefore bodies afcending defcribe spaces

which

which follow the progreffion of the odd numbers taken in a retrograde order.

To render this fubject still clearer, let us apply this theory to practice. But here I must first premife, that when a body falls freely by the force of gravity, it will defcribe about 16 feet and an inch in one fecond of time; but, for the fake of round numbers, I fhall fuppofe it to be 16 feet. This measure has been deduced from the motion of pendulums; between which and the rectilinear defcent of heavy bodies there is fo clofe a connection, that neither of them can be thoroughly understood independent of the other. Now as a body falls through 16 feet in the first fecond, it follows, that at the end of the fecond moment it will have fallen 16 multiplied by 4, equal 64 feet; the space being as the fquares of the times, and the fquare of 2 is 4. By the fame rule, at the end of the third fecond it will have fallen 144 feet, equal 16 multiplied by 9; at the end of the fourth, 256 feet, equal 16 multiplied by 16.

Hence, if the fpace be given through which a body is to fall, you may collect the time wherein it will finish it's defcent: for let the number of feet in fuch a space be divided by 16, and the quotient will exprefs the fquare root of the time fought, in feconds and parts of a fecond.

Thus, if the fpace be 144 feet, this divided by 16 gives 9, of which the fquare root is 3. The converfe of this is equally true; for if the time be given, the space through which the body hath defcended may be found. If you defire to learn the depth of a well, from the furface of the earth to the furface of the water, let a bullet of lead be dropped therein, and let us fuppofe it to ftrike the water in five feconds: the fquare of 5 is 25: which, being multiplied by 16, the product will be 400 feet for the depth of the well.

Again,

Again, were you to fee a mafs of burning matter, or a large red-hot stone, fhot upwards from the mouth of a volcano, and could obferve accurately the whole time of it's flight in the air, rifing and falling, which we will fuppofe thirty feconds, you may thence difcover the height to which it arofe, for a projected body will both rife and fall in the fame time; therefore we are to take half the above-mentioned time, or fifteen feconds: now 15 multiplied by 15, and this by 16, give 3600 feet, or 1200 yards, which is not far from three quarters of a mile.

But you are to obferve, that in the application of this theory two things are taken for granted, before we arrive at any one of thefe conclufions: first, that the theory is true in practice to a mathematical exactnefs, of which you will foon fee the proofs; fecondly, that the motion is without impediment, or in a medium that gives no refiftance: but this is falfe, as the bodies meet with great interruptions from the refiftance of the air. Upon this account, the well whereof we are finding the depth, will not be fo deep as may be imagined by feveral feet; nor the stone projected from the volcano rife so high as we have already concluded. The refiftance of the air is liable to fo many variations, as to render the finding the abfolute quantity of it at any time, a problem fo difficult and complicated, as hardly to admit of an adequate folution.

OF MR. ATWOOD'S APPARATUS FOR MAKING EXPERIMENTS on the rectilinear motion of bodies which are acted on by conftant forces. The machine before you, fig. 13, pl. 1, is the contrivance of Mr. Atwood, and renders fenfible to the eye and ear the laws of motion uniformly accelerated or retarded,

Atwood's Treatife on the Rectilinear Motion of Bodies.

tarded, as well as thofe of uniform motion, and that without employing a space more than five and a half feet, which renders it extremely convenient for a courfe of Lectures.

Mechanical experiments are of two kinds; the one relating to the quiefcence of bodies, and the other to their motion.

Among the former are included those which demonstrate, or rather make evident to the fenfes, the equilibrium of the mechanic powers, and the correfponding proportions of the weights fuftained, to the forces which fuftain them, the properties of the center of gravity, the composition and refolution of forces, &c.

By the latter, or those on motion, are shewn the laws of collifion, of acceleration, and the various effects of forces which communicate motion to bodies.

Of mechanical experiments it may be proper to obferve to you, that thofe wherein an equilibrium is formed, will frequently appear coincident with the theory, although confiderable errors are committed in their conftruction. This arifes from the effects of friction, tenacity, and other causes. The cafe is different in experiments concerning the motions of bodies; in which, whatever care be taken to render the proportion of the forces, and the weights moved, fuch as is required by the theory; yet the interference of friction, which renders the former apparently more perfect than they really are, caufes these to differ from the theory.

If the experiments are only defigned to affift the imagination, by fubftituting fenfible objects instead of abstract and ideal quantities, an apparent agreement between the theory and experiment may be fufficient to anfwer this purpose, although it may be produced from an erroneous conftruction: fuch experiments cannot, however, impress the

mind with that fatisfactory conviction that arifes from experiments accurately made.

Dr. Defaguliers tried the effect of falling bodies, by letting a leaden ball fall from the inner cupola of St. Paul's church, whofe altitude from the ground is 272 feet. The ball defcended

through this fpace in four feconds and a half; in which time, from theory, it fhould have defcended through 325,6 feet, which makes a difference of about one-fifth of the actual descent between the experiment and the theory. Dr. Defaguliers fhews, in his fifth lecture, that this difference arofe principally from the refiftance of the air. To remedy the defects of thefe experiments, Mr. Atwood contrived his apparatus.

Of the mass moved. In order to obtain an adequate idea of the laws that are obferved in the communication of motion, and obferve the effects of the moving force, the interference of all other force fhould be prevented. The bodies impelled fhould be conceived to exist in free fpace, and be void of gravity or weight; fo that to a given fubftance various degrees of force may be applied. This indeed cannot be effected in bodies falling freely near the earth's furface: we cannot abstract the natural gravity, or weight, from any fubftance whatsoever; for the fame fubftance is always impelled by the fame force of gravity, which admits not of increase or diminution.

Yet this difficulty may be obviated by ballancing two equal weights, joined by a flexible line, which goes over a pulley. The axle of the pulley must reft on wheels conftructed for the purpose of diminishing friction.

The motive force of gravity being deftroyed by the contrary and equal action of the weights, they will remain quiefcent till fome force is applied to them. When any impulfe is communicated to

them