SCHOL. 4.

Wheel carriages are used, to avoid friction as much as possible.

A wheel

turns round upon its axis, because the several points of its circumference are retarded in succession by attrition, whilst the opposite points move freely. Large wheels meet with less resistance than smaller from external obstacles, and from the friction of the axle, and are more easily drawn, having their axles level with the horses. But in uneven roads, small wheels are used, that in ascents the action of the horse may be nearly parallel with the plane of ascent, and therefore may have the greatest effect: small wheels are also more conveniently turned. The greater part of the load should be laid on the hinder part of a wheel carriage.

Plate 3.

Fig. 10,

11.

CHAP. VII.

Of Motion as produced by the united Forces of PROJECTION and
GRAVITATION.

SECT. I.

Of Projectiles.

PROP. LVIII. Bodies thrown horizontally or obliquely, have a curvilinear motion, and the path which they describe is a parabola ; the air's resistance not being considered.

If a body be thrown in the direction AF, and acted upon by the projectile force alone, it will continue to move on uniformly in the right line AF, and would describe equal parts of the line AF in equal times, as AC, CD, DE, &c. But if, in any indefinitely small portion of time, in which the body would by the projectile force move from A to C, it would, by the force of gravity, have fallen from A to G; by the composition of these forces (Prop. XVI.) it will, at the end of that time, be found in H, the opposite angle of the parallelogram ACGH. In two such portions of time, whilst it would have moved from A to D by the projectile force, it would (Prop. XXVI.) by gravitation fall through four times AG, that is, AM; and therefore, these forces being combined, it will be found at the end of that time in I, the opposite angle of the parallelogram DM. In like manner, at the end of the third portion of time, it would by the projectile force be carried through three equal divisions to E, and by the force of gravitation over nine times AG to N; and cons quently by both these forces acting jointly it will be carried to K, the opposite angle of the parallelogram EN. Therefore the lines CH, DI, EK, that is, AG, AM, AN, which are to each other as the numbers 1, 4, 9, are as the squares of the lines AC, AD AE,

that

that is, GH, MI, NK, which are as 1, 2, 3. And because the action of gravitation is continual, the body in passing from A to H, &c. is perpetually drawn out of the right line in which it would move if the force of gravitation were suspended, and therefore moves in a And H, I and K, are any points in this curve in which lines let fall from points equally distant from A in the line AB meet the curve. Therefore the body moves in a parabola, the property of which is (Simpson's Conick Sections, Book I. Prop. XII. Cor) that the abscissa AG, AM, AN, are to each other as the squares of the ordinates GH, MI, NK.

curve.

REMARK. Very dense bodies moving with small velocities describe the parabolick track so nearly, that any deviation is scarcely discoverable; but with very considerable velocities the resistance of the air will cause the body projected to describe a path altogether different from a parabola, which will not appear surprising when it is known that the resistance of the air to a cannon ball of two pounds weight, with the velocity of 2000 feet per second, is more than equivalent to 60 times the weight of the ball. See Hutton's Dict. Art. Resistance.

PROP. LIX. The path which a body thrown perpendicularly upwards describes in rising and falling is a parabola.

A stone lying upon the surface of the earth, partaking of the motion of the earth (here supposed) round its axis, this motion which it has with the earth will not be destroyed by throwing it in a direction perpendicular to the surface of the earth. After the projection, therefore, the stone will be moved by two forces, one horizontal, the other perpendicular, and will rise in a direction which may be shewn, as in the last proposition, to be the parabolick curve; in which it will continue till it reaches the highest point, from whence it might be shewn, as in the last proposition, that it will descend through the other side of the parabola.

PROP. LX. The velocity with which a body ought to be projected to make it describe a given parabola, is such as it would acquire by falling through a space equal to the fourth part of the parameter belonging to that point of the parabola from which it is intended to be projected.

The velocity of the projectile at the point A (by Prop. LVIII.) is such as would carry it from A to É, in the same time in which it would descend by its gravity from A to N. And the velocity acquired in falling from A to N (by Prop. XXVII.) is such as in the same time by an uniform motion would carry the body through a space double of AN. Therefore the velocity which is acquired by the body in falling to N is to that with which the body is projected at A, and uniformly carried forwards to E, as twice AN is to AE. But since, from the nature of the parabola, (Simpson's Conick Sections, Book I. Prop. XIII.) AE'

AN

is equal to the parameter of the point A, one fourth part of this parameter will be
expressed

Plate 3.
Fig. 13.

expressed by

1 A2
AN

And because the velocities acquired by falling bodies are (by Prop.

XXVI. Cor. 1.) as the square roots of the spaces they fall through, the velocity acquired by a body in failing through AN is to the velocity acquired in falling through

AE

or one

AN

fourth part of the parameter of A, as the square root of AN to the square root of AE

AN AE, that is, as AN to or AN to AE, or twice AN to AE. Therefore the ✔ AN velocity acquired by a body in falling from A to N has the same ratio to the velocity with which the body is projected or the line AE described, and to the velocity acquired by a body in falling through a fourth part of the parameter belonging to the point A (El. V. 11.) these velocities are equal.

consequently

COR. Hence may be determined the direction in which a projectile from a given point, with a given velocity, must be thrown to strike an object in a given situation.

Let A be the place from which the body is to be thrown and K the situation of the object. Raise AB perpendicular to the plane of the horizon, and equal to four times the height from which a body must fall to acquire the given velocity. Bisect AB in G: through G draw HG perpendicular to AB: at the point A raise AC perpendicular to AK, and meeting HG in C: on C as a centre with the radius CA describe the circle ABD; and through K draw the right line KEI perpendicular to the pane of the horizon, and cutting the circle ABD in the points E and I. AE, or AI, will be the direction required.

For, drawing BI, BE, since AK is a tangent to the circle, and BA, IK, are parallel to each other, the angle ABE (El. III. 32.) is equal to the angle EAK; and the alternate angles BAE, AEK, are equal: Therefore the triangles ABE, AEK, are similar; and AB is AE' to AE, as AE to EK. Therefore AB x EK = AE1; and AB = In like manner, EK Since, then, AB is equal to four times the height from which a body must fall to acquire the velocity with which it is AE2 to be thrown; EK

the triangles BAI, KAI, being similar, BA is equal to

AI2
IK

A12 (or its equal) is the same. Consequently (by this Prop.) the IK

point K will be in the parabola which the body will describe, which is thrown with the given velocity in the directions AE, or AI, and the body will strike an object placed at K.

SCHOL. If the velocity with which a projectile is thrown be required, it may be determined from experiments in the following manner. By the help of a pendulum or any other exact chronometer, let the time of the perpendicular flight be taken; then, since the times of the ascent and descent are equal, the time of the descent must be equal to one half of the time of the flight, consequently, that time will be known: and, since a heavy body descends

from

from a state of rest at the rate of 161 feet in the first second of time, and that the spaces through which bodies descend are as the squares of the times; if we say, as one second is to 16'1 feet, so is the square of the number of seconds which express the time of the descent of the projectile, to a fourth proportional, we shall have the number of feet through which the projectile fell, which being doubled, will give us the number of feet which the projectile would describe in the same time with that of the fall, supposing it moved with an uniform velocity, equal to that which it acquired by the end of the fall; which last found number of feet, being divided by the number of seconds which express the time of the projectile's descent, will give a quotient, expressing the number of feet, through which the projectile would move in one second of time with a velocity equal to that which it acquired in its descent, which velocity is equal to the velocity with which the projectile was thrown up; consequently, this velocity is discovered.

PROP. LXI. The squares of the velocities of a projectile in different points of its parabola, are as the parameters belonging to those points.

For (by the last Prop.) the velocities in the several points of the parabola, are equal to the velocities acquired in falling through the fourth parts of the parameters of the points. Therefore the squares of these velocities being (by Prop. XXVI.) as the spaces described, the squares of the velocities in the several points of the parabola are as the fourth parts of the parameters of those points: but the whole parameters are as their fourth parts: therefore the squares of the velocities at the several points of the parabola are as the parameters of those points.

COR. Hence, setting aside any difference which may arise from the resistance of the air, a projectile will strike a mark as forcibly at the end as at the beginning of its course, if the two points be equally distant from the principal vertex: for, the parameters belonging to these points being equal, the velocities in these points must also be equal.

PROP. LXII. When a body is thrown obliquely with a given velocity, if the space through which it must have fallen perpendicularly to acquire that velocity is made the diameter of a circle, the height to which the body will rise is equal to the versed sine of double the angle of elevation.

Let a body be thrown in the direction BE, with the same velocity which any body would Plate 3. Fig. 12. acquire by falling perpendicularly through AB; if AB is made the diameter of a circle, the greatest height to which the body will rise will be BD.

-Plate 3.

Fig. 12.

Let IL be a right line drawn in the plane of the horizon, touching the circle in B, and making with the line BE, which is the direction in which the body is thrown, the angle IBE, or angle of elevation. Because L touches the circle, and EB drawn in the circle meets it in the point of contact, (El. III. 32.) the angle EBI is equal to the angle EAB. And ECB is double of EAB, (El. III. 20) therefore ECB is double of EBI, the angle of elevation. And BD is the versed sine of ECB, that is, of double the angle of elevation.

Then since this velocity

Let BE represent the velocity with which the body is thrown. is, by supposition, such as might be acquired by falling down AB, if the body was thrown perpendicularly upwards with the same velocity BE, it would rise to the height BA Let the oblique motion BE be resolved into two others, one in the direction BD perpendicular to the horizon, and the other in the direction DE parallel to it: then the ascending velocity will be to the horizontal velocity, as BD to DE, and to the whole velocity, as BD to BE. But the part of the velocity BD is the only part which is employed in raising the body, since the other part DE is parallel to the plane of the horizon. Now, the height of a body ascending perpendicularly with the whole velocity BE, will be to the height when it ascends with the part BD (compare Prop. XXVI. and Prop XXVIII.) as the square of BE to the square of BD. But because (El. VI. 8 ) the triangle EDB is similar to the triangle AEB, BD is to EB, as EB is to BA; and BD, BE, BA, being continued proportionals, BD is to BA, as the square of BD is to the square of BE. And the perpendicular heights to which the velocities BE and BD will make the body ascend have been shewn to be as the square of BE to the square of BD; the heights are therefore as BA to BD. Since therefore the first velocity BE would make the body ascend through BA, the other velocity BD, which is the part of the whole velocity which acts to make the body thrown in the direction DE to ascend, will carry it to the height BD, which is the versed sine of double the angle of elevation. The same might be shewn in any other direction of the body, as BF, or BG.

DEF. XV. The Random of a projectile is the horizontal distance to which a heavy body is thrown.

PROP. LXIII. When a body is thrown obliquely with a given velocity, if the space through which it must have fallen perpendicularly to acquire that velocity is made the diameter of a circle, the random will be equal to four times the sine of double the angle of elevation.

If EBI be the angle of elevation, and ECB double that angle, DE will be the sine of double the angle of elevation. Let a body be thrown from the point B in the direction BE, with the velocity which it would acquire in falling through AB; the random, or horizontal distance at which the body will fall, is equal to four times DE.

For,