In the present state of the world the improvements of science feldom die with individuals. The accumulation of knowledge by leading the understanding, and by furnishing tools to the fenfes, promotes the discovery of farther truths, and the inexhauftible

will be all equal, and will be described in equal times. Consequently two or three, or any number of them, will be defcribed in two or three, or the like number of particles of time, viz. they are as the times.

Now imagine that those triangles are infinitely increased in number, and diminished in fize; then the polygonal path ABCDEF, will become a continuate curve; for the conftant action of the centre of attraction will be continually drawing the body away from the direction of the tangent at every point of the curve. And it is evident that the sectoral areas of the faid curve, or number of infinitely small triangles, must be proportional to the times in which they are defcribed, and that the curve muft lie in one immoveable plain.

Corollary 1. The velocities in different parts of the orbit are inverfely as the perpendiculars dropped from the centre of attraction on the tangents to the orbit at those parts or points. For fince the velocities are as the bafes AB, BC, CD, &c. of equal triangles, they must be inversely as the heights of thofe triangles, (Eucl. p. 15, B. VI. and p. 38, B. I.) which are the fame as the perpendiculars dropped from the centre N, on the tangents to the orbit at those points.

Corollary 2. The times in which equal parts, or arches of the orbit are defcribed, are directly as thofe perpendiculars to the tangents. For when the arches, or bafes of the triangles, are equal, the triangles are as their altitudes; that

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1

exhauftible fund of nature offers on all fides innumerable objects of investigation to the inquifitive mind.

is, as the above-mentioned perpendiculars. But they are like wife as the times; therefore, &c.

Corollary 3. If, by drawing lines parallel to the chords AB, BC, of any two contiguous and evanefcent arches defcribed in equal times, the parallelogram be completed, the diagonal BG, when produced, will pafs through the centre of attraction N, which proves the converfe of the propofition; viz. that when the areas, which are described by aftraight line, connecting a moving body and a certain point, are proportional to the times in which they are defcribed, then the body is under the influence of a centripetal force tending to that point.

Corollary 4. In every point of the orbit the centripetal force is as the fagitta, or verfed fine, of the indefinitely fma arch at that point. The centripetal force at B is as BG, becaufe BG is equal to CH, and CH is the deviation from the straight direction AH, which has been occafioned by the centripetal force. And the half of BG, viz. B O, is the fagitta, or verfed fine, of the indefinitely fmall arch ABC.

OF THE DESCENT OF BODIES UPON INCLINED

PLANES; AND THE DOCTRINE OF PENDU

LUMS.

Prop. I.

W

HEN a body is placed upon an inclined plane, the force of gravity which urges that body downwards, acts with a power fo much less, than if the body defcended freely and perpendicularly downwards, as the elevation of the plane is less than its length.

If BD, fig. 1, Plate IV. be an horizontal plane, and a body A be laid upon it, this body will remain motionless; for though the power of gravity, or (which is the fame thing) its own weight, draws it towards the centre of the earth, yet the plane DB fupports it exactly in that direction; hence no motion can arife.

But if the plane be inclined a little to the horizon, as in fig. 2, Plate IV. then the body will defcend gently towards the lower end D. And if the inclination of the plane be increased, as in fig. 3, Plate IV. the body will run down towards D with greater quickness.

In the two laft cafes; or, in general, whenever the plane is inclined to the horizon, the action of gravity is not entirely but partially counteracted by the plane. For if, from the centre A of the body, in the figures 2 and 3, you draw two lines, viz. AG

perpendicular to the horizon, and AF perpendicular to the plane; the whole force of gravity, which is reprefented by the line AE, is refolved into two forces; viz. AF and EF, whereof AF being perpendicular to the plane, is that part of the gravitating power which is counteracted by the inclined plane; or that part of the weight of the body which is supported by the plane BD; and EF represents the other part of the gravitating power, which urges the body downwards along the furface of the plane. Therefore the force of gravity which moves the body, is diminished in the proportion of AE to EF, But the triangles AFE, EDG, and BDC, are equiangular, and of course fimilar (because the angles at F, C, and G are right, and the angle AEF is equal to the angle DEG, by Eucl. p. 15. B. I.; as alfo equal to the angle DBC, by Eucl. p. 29. B. I.) Hence we have AE to EF, as DB to BC; viz. as the length of the plane is to its elevation, or as the whole force of gravity is to that part of it which urges the body down along the inclined plane*.

Prop. II. The space which is defcribed by a body defcending freely from reft towards the earth, is to the Space which it will defcribe upon the furface of an in

* If (by trigonometry) DB be made radius, BC becomes the fine of the angle of inclination BDC; therefore the whole force of gravity is faid to be to that part of it which urges a body down an inclined plane, as radius is to the fine of the plane's inclination to the horizon.

clined plane in the fame time as the length of the plane is to its elevation, or as radius is to the fine of the plane's inclination to the horizon.

The force of gravity, which urges a body down along the furface of an inclined plane, is diminished by the partial counteraction of the inclined plane; but its nature is not otherwise changed; viz. it acts conftantly and unremittedly. Hence the velocity of the body is continually accelerated, and the spaces it runs over are alfo proportional to the fquares of the times; though those spaces will not be fo long as if the body defcended freely and perpendicularly towards the ground.

Now in order to ascertain how much the space, which is defcribed by a body running down an inclined plane in a certain time, is fhorter than the space through which it would defcend freely and perpendicularly in the fame time, we muft recollect what has been proved in page 64, relatively to the spaces,which are described in the fame time, by bodies that are acted upon by different central forces; namely, that in equal times, the fpaces are as the forces'; then, fince the whole force of gravity is to that force which draws a body down the inclined plane, as radius is to the fine of the plane's inclination. Therefore the space defcribed by a body which defcends freely, is to the space which a body will describe on an inclined plane, in the fame time as radius is to the fine of the plane's inclination, or as the length of the plane is to its altitude.

Example.