1", 2"
2", 3",
4", &c.
321, 641, 96, 128, &c.
16, 643, 144, 2571, &c.
16, 481, 805, 112,7, &c.
times, veloci-

namely, the times in seconds
the velocities in feet will be
the spaces in the whole times
and the space for each second
71. These relations, of the
ties, and spaces, may be aptly represented
by certain lines and geometrical figures.
Thus, if the line AB denote the time of any
body's descent, and BC, at right angles to it,
the velocity gained at the end of that time;
by joining AC, and dividing the time AB into
any number of parts at the points a, b, c;
then shall ad, be, cf, parallel to BC, be the velocities at the
points of time a, b, c, or at the ends of the times, aa, ab,
Ac; because these latter lines, by similar triangles are pro-
portional to the former ad, be, ef, and the times are propor-
tional to the velocities. Also, the area of the triangle ABC
will represent the space descended through by the force of
gravity in the time AB, in which it generates the velocity BC;
because that area is equal to AB XBC, and the space descend-
ed is sv. or half the product of the time and the last
velocity. And, for the same reason, the less triangles aad,
Abe, Acf, will represent the several spaces described in the
corresponding times aa, ab, ac, and velocities ad, be, cf; those
triangles or spaces being also as the squares of their like sidest
aa, ab, ac, which represent the times, or of ad, be, ef, which
represent the velocities.

72. But as areas are rather unnatural
representations of the spaces passed over
by a body in motion, which are lines, the
relations may better be represented by
the abscisses and ordinates of a parabola. k
Thus, if ra be a parabola, PR its axis,
and Ro its ordinate; and pα, rb, pc, &c.
parallel to RQ, represent the times from R

the beginning, or the velocities, then ae, bf, cg, &c. parallel
to the axis PR, will represent the spaces described by a fall-
ing body in those times; for, in a parabola, the abscisses rh,
Pi, rk, &c. or ae, bf, cg, &c. which are the spaces described,
are as the squares of the ordinates, he, if, kg, &c. or Fa, rb,
rc, &c. which represent the times or velocities.

73. And because the laws for the destruction of motion,

are the same as those for the generation of it, by equal forces, but acting in a contrary direction; therefore,

1st, A body thrown directly upward, with any velocity will lose equal velocities in equal times.

2d, If a body be projected upward, with the velocity it acquired in any time by descending freely, it will lose all its velocity in an equal time, and will ascend just to the same height from which it fell, and will describe equal spaces in equal times, in rising and falling, but in an inverse order; and it will have equal velocities at any one and the same point of the line described, both in ascending and descending.

3d, If bodies be projected upward, with any velocities, the height ascended to, will be as the squares of those velocities, or as the squares of the times of ascending, till they lose all their velocities.

74. To illustrate now the rules for the natural descent of bodies by a few examples, let it be required.

1st, To find the space descended by a body in 7 seconds of time, and the velocity acquired.

Ans. 7881 space; and 2251 velocity. 2d, To find the time of generating a velocity of 100 feet per second, and the whole space descended.

192

Ans. 3" time; 155,8 space. 3d, To find the time of descending 400 feet, and the velocity at the end of that time.

Ans. 4′′71⁄2 time; and 16032 velocity.

PROPOSITION XIX.

75. If a Body be projected in Free Space either Parallel to the
Horizon, or in an Oblique Direction, by the Force of Gun-
Powder, or any other Impulse; it will by this Motion, in
Conjunction with the Action of Gravity describe the Curve
Line of a Parabola.

LET the body be projected from the point a, in the direction AD, with any uniform velocity: then, in any equal por

tions

t

1

tions of time, it would by prop. 4, describe the equal spaces AB, BC, CD, &c. in the line AD, if it were not drawn continually down below that line by the action of gravity. Draw BE, CF, DG, &c. in the direction of gravity, or perpendicular to the horizon, and equal to the spaces through which the body would descend by its gravity in the same time in which it would uniformly pass over the corresponding spaces AB, ac, ad, &c. by the projectile motion. Then, since by these two motions. the body is carried over the space AB, in the same time as over the space BE, and the space ac in the same time as the space cr, and the space AD in the same time as the space DC, &c.; therefore, by the composition of motions, at the end of those times, the body will be found respectively in the points E, F, G, &c.; and consequently the real path of the projectile will be the curve line AEFG, &c. But the spaces AB, AC, AD, &c. described by uniform motion, are as the times of description; and the spaces BE, CF, DG, &c. described in the same times by the accelerating force of gravity, are as the squares of the times; consequently the perpendicular descents are as the squares of the spaces in AD, that is BE, CF, DG, &c. pectively proportional to Aɛ2, ac2, ad2, &c.; which is the property of the parabola by theor. 8, Con. Sect. Therefore the path of the projectile is the parabolic line AEFG, &c. to which AD is a tangent at the point A.

curve;

are res

76. Corol. 1. The horizontal velocity of a projectile, is always the same constant quantity, in every point of the because the horizontal motion is in a constant ratio to the motion in AD, which is the uniform projectile motion. And the projectile velocity is in proportion to the constant horizontal velocity, as radius to the cosine of the angle DAM, or angle of elevation or depression of the piece above or be low the horizontal line AH.

77. Corol. 2. The velocity of the projectile in the direction of the curve, or of its tangent at any point a is as the secant of its angle BAI of direction above the horizon. For the motion in the horizontal direction ar is constant, and ai is to AB, as radius to the secant of the angle a; therefore the motion at ▲, in AB, is every where as the secant of the angle A.

78. Corol. 3. The velocity in the direction DG of gravity or perpendicular to the horizon, at any point & of the curve, is to the first uniform projectile velocity at A, or point of contact of a tangent, as 2GD is to AB. Fòr, the times in AD and DG being equal, and the velocity acquired by freely de

scending

scending through DG, being such as would carry the body uniformly over twice DG in an equal time, and the spaces described with uniform motions being as the velocities, therefore the space AD is to the space 2DG, as the projectile velocity at a, to the perpendicular velocity at G.

PROPOSITION XX.

79. The Velocity in the Direction of the Curve, at any Point of it as a, is equal to that which is generated by Gravity in freely descending through a Space which is equal to One-Fourth of the Parameter of the diameter of the Parabola at that Point.

LET PA OF AB be the height due to the velocity of the projectile at any point a, in the direction of the curve or fangent ac, or the velocity acquired by falling through that height; and complete the parallelogram ACDB. Then is CD =AB or AP, the height due to the velocity in the curve at A and CD is also the height due to the perpendicular velocity at D, which must be equal to the former; but by the last corol. the velocity at a is to the perpendicular velocity at D, as AC to 2CD; and as these velocities are equal, therefore AC or BD is equal to 2CD, or 2AB; and hence AB Or AP is equal to BD, or 1 of the parameter of the diameter AB, by corol. to theor. 13 of the Parabola.

A

80. Corol. 1. Hence, and from cor. 2, theor. 13 of the parabola, it appears that, if from the directrix of the parabola which is the path of the projectile, several lines Hɛ be drawn perpendicular to the directrix, or parallel to the axis; then the velocity of the projectile in the direction of the curve, at any point E, is always equal to the velocity acquired by a body falling freely through the perpendicular line HE.

81. Corol. 2. If a body, after falling through the height PA (last fig. but one), which is equal to AB, and when it arrives at A, have its course changed, by reflection from an elastic plane Ar, or otherwise, into any direction Ac, without altering the velocity; and if Ac be taken 2AP or 2AB, and the parallelogram be completed; then the body will describe the parabola passing through the point n.

82. Corol,

82. Corol. 3. Because Ac = 2AB or 2cD OF 2AP, therefore ac2 —2ap X2cd or AP. 4CD; and because all the perpendiculars EF, CD, GH, are as АE2, AC2, AG2; therefore also AP. 4EF = AE2, and AP. 4GH = AG2, &c.; and because the rectangle of the extremes is equal to the rectangle of the means of four proportionals, therefore always

it is AP AE :: AE : 4EF,
and APAC :: AC: 4CD,
and ap : ag :: AG: 4GH,
AG
and so on.

PROPOSISION XXI.

$3. Having given the Direction, and the Impetus, or Altitude due to the First Velocity of a Projectile, to determine the Greatest Height to which it will rise, and the Random ør Horizontal Range.

LET AP be the height due to the projectile velocity at A, AG the direction, and AH the horizon. On AG let fall the perpendicular PQ, and on AP the perpendicular QR; so shall AR be equal to the greatest. altitude cv, and 4QR equal to the horizontal range AH. Or, having

drawn PQ perp. to AG, take AG = 4AQ, and draw GH perp. to

AH; then AH is the range.

For, by the last corollary,

and, by similar triangles,

AP AG: AG: 4GH;

AP: AG :: AQ: GH,

AP : AG :: 4AQ : 4GH;

or

therefore aG=4AQ; and, by similar triangles, AH=4QR.

Also, if v be the vertex of the parabola, then ab or LaG— 2aq, or aq=qb; consequently AR BV, which is property of the parabola.

84. Corol. 1. Because the angle Q is a right angle, which is the angle in a semicircle, therefore if, on AP as a diameter, a semicircle be described, it will pass through the point Q.

85. Corol. 2. If the horizontal range and the projectile velocity be given,

=cv by the

the

AD C H