.

Example. Let the length BD of the inclined plane, fig. 2, Plate IV. be 10 feet, and its elevation BC, 4 feet. It is known from experiment, that in the first second of time, a body will defcend freely from reft through 16,087 feet. Therefore, by the rule of three, we fay, as 10 feet are to 4 feet, so are 16,087 feet to a fourth proportional, viz. 10: 4::

16,087 : (4×16,087 =) 6,435 feet, which shews

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that a body running down the inclined plane BD, would pass over little less than fix feet and a half, or 6,435 feet, in the firft fecond of time.

Prop. III. If upon the elevation BC, fig. 4, Plate IV. of the plane BD, as a diameter, the femicircle BEGC be deferibed, the part BE of the inclined plane, which is cut off by the femicircle, is that part of the plane over which a body will defcend, in the fame time that another body will defcend freely and perpendicu larly along the diameter of the circle, viz. from B to C, which is the altitude of the plane, or fine of its inclination to the horizon.

The triangle BEC is equiangular, and of course fimilar, to the triangle BDC (for the angle at B is common to both, and the angle BEC is by Eucl. p. 31. B. III. a right angle, and therefore equal to the right angle BCD) hence we have BD to BC as BC is to BE. But, by the preceding propofition, the space defcended freely and perpendicularly, is to the fpace run over an inclined plane in the fame time, as the length of

the plane is to its elevation, viz. as BD is to BC; therefore, the space run freely and perpendicularly, is to the space run over the inclined plane, likewife as BC is to BE. And fince BC is the fpace freely defcended by a body in a certain time, BE must be the fpace which is run down by a body on the inclined plane in the fame time.

Cor. A very useful and remarkable consequence is derived from this propofition, namely, that a body will defcend from В over any chord whatsoever as BE, or BF, or BG, of the femicircle BEFC, exactly in the fame time, viz. in the fame time that it would defcend freely from B to C. For if you imagine the inclined plane to be BH instead of BD; then by this propofition, the body will defcend either from B to F, or from B to C in the fame time; and again, if you imagine the inclined plane to be BI, then by this propofition, the body will defcend either from B to G, or from B to C, in the same time. And, in short, the fame thing may be proved of any other chord of the femicircle.

Prop. IV. The time of a body's defcending along the whole length of an inclined plane, is to the time of its defcending freely and perpendicularly along the altitude of the plane, as the length of the plane is to its altitude; or as the whole force of gravity is to that part of it which acts upon the plane.

The spaces run over the plane being as the fquares of the times, we have the fquare of the time of paffing over BD, fig. 4, Plate IV. to the fquare of

the

the time of paffing over BE, as BD is to BE. But BD is to BC as BC is BE, viz, BD, BC, and BE are three lines in continuate geometrical proportion; therefore (Eucl. p. 20, B. VI.) BD is to BE, as the fquare of BD is to the fquare of BC. It has been shewn above, that the square of the time of paffing over BD, is to the fquare of the time of paffing over BE, as BD to BE; therefore those fquares of the times are to each other as the square of BD to the fquare of BC; and of course the fquare roots of these four proportional quantities are likewife proportional (Eucl. p. 22, B. VI.) viz. the time of a body's defcending from B to D is to the time of its defcending freely and perpendicularly from B to E, or from B to C, as BD is to BC, or as the length of the plane is to its altitude; or (by the ft propofition of this chapter) as the whole force of gravity is to that part of it which the plane.

acts upon

Prop. V. A body by defcending from a certain height to the fame horizontal line, will acquire the fame velocity whether the defcent be made perpendicularly, or ob-. liquely, over an inclined plane, or over many fucceffive ́inclined planes, or laftly over a curve surface.

ift. In page 64, it has been fhewn, that the velocity of a body defcending freely towards a centre of attraction, is as the product of the attractive force multiplied by the time; and by the preceding propofition it has been proved, that on an inclined plane the force of gravity is diminished in proportion as the time of the body's running down

the

2

the whole length of the plane, is increased, viz. when the force of gravity is half as strong as it would be in free fpace, the time is doubled; and when the force is one-third as ftrong, the time is trebled, &c. therefore the product of the time by the force is always the fame; for multiplied by 2 is equal to multiplied by 3, is equal to multiplied by 4, &c. hence the velocity being as that product, muft, of course, be always the fame, or a conftant quantity. For example, suppose, that when the body defcends perpendicularly down from B to C, fig. 4, Plate IV. the whole force of gravity acts upon it. Let us call that whole force. 1, and let the time employed by the body in coming down from B to C be one minute, then the velocity acquired by that defcent is represented by the product of the time by the force, viz. 1 by 1, which makes one. Now when the body defcends from the fame altitude B, to the fame horizontal line DC, over the inclined plane BD, the force of gravity which draws it downwards is diminished; for inftance, fuppofe it to act with a quarter of its original power, then the time of the body's descending from B to D will be four minutes, and the velocity acquired by that defcent, being as the product of the force by the time, is as the product of by 4, which is one, or the fame as when 44 the body defcends perpendicularly down from B to C.

2dly. Suppose that the body defcends from the fame altitude E to the fame horizontal line DC,

fig. 5, Plate IV. along the contiguous inclined planes EF, FG, GD; by the time it arrives at D it will have acquired the fame velocity as if it had defcended perpendicularly from B to C, or from E perpendicularly down to the horizontal line DC; for, by the first part of this propofition, it will acquire the fame velocity whether it defcends from E to For from K to F, and by adding to both the plane FG, it follows that the body will acquire the fame velocity whether it defcends along the fingle plane KG, or along the two contiguous planes EF, FG. And by the like reafoning it will be proved, that the body will acquire the fame velocity whether it defcends along the fingle plane BD, or along the contiguous two planes KG, GD, or along the contiguous three planes EF, FG, GD, &c.

3dly. If the number of contiguous planes be fuppofed infinite, and their lengths infinitely small, they will conftitute a curve line, like BH; whence it follows, that a body by its descent along the curve line BH, or any other curve, will acquire the fame velocity as if it defcended perpendicularly from B to C.

Prop. VI. Let a circle be perpendicular to the horizon, and if two chords be drawn from any two points in the circumference, to the point in which the circle touches the horizon; the velocities which are acquired by the defcents of two bodies along thofe chords, will be as the lengths of the chords refpec-. tively.