&c. = 1000 s, and by subtracting the former equation from the latter, 123=999 s;

therefore s =

123

999

41

333 PROHIBITION, in law, is a writ properly issuing only out of the Court of King's Bench, being the King's prerogative writ; but, for the furtherance of justice, it may now also be had in some cases out of the Court of Chancery, Common Pleas, or Exchequer, directed to the judge and parties of a suit in an inferior court, commanding them to cease from the prosecution thereof, upon a suggestion, that either the cause originally, or some collateral matter arising therein, does not belong to that jurisdic. tion, but the cognizance of some other court. Upon the court being satisfied that the matter alleged by the suggestion is sufficient, the writ of prohibition immediately issues.

PROJECTILES, are such bodies as being put in a violent motion by any great force, are then cast off or let go from the place where they received their quantity of motion; as a stone thrown from a sling, an arrow from a bow, a bullet from a gun, &c. It is usually taken for granted, by those who treat of the motion of projectiles, that the force of gravity near the earth's surface is every where the same, and acts in parallel directions; and that the effect of the air's resistance upon very heavy bodies, such as bombs and cannon-balls, is too small to be taken into consideration.

Sir Isaac Newton has shown, that the gravity of bodies which are above the superficies of the earth, is reciprocally as the squares of their distances from its centre; but the theorems concerning the descent of, heavy bodies, demonstrated by Gallileo, and Huygens, and others, are built upon this foundation, that the action of gravity is the same at all distances; and the consequences of this hypothesis are found to be very nearly agreeable to experience. For it is obvious, that the error arising from the supposition of gravity's acting uniformly, and in parallel lines, must be exceedingly small ; becanse even the greatest distance of a projectile above the surface of the earth, is inconsiderable, in comparison of its distance from the centre, to which the gravitation tends. But then, on the other hand, it is very certain, that the resistance of the air to very swift motions, is much greater than it has been commonly represented. Nevertheless, (in the application of this doctrine to gunnery)

if the amplitude of the projection, answering to one given elevation, be first found by experiment (which we suppose) the amplitudes in all other cases, where the elevations and velocities do not very much differ from the first, may be determined, to a sufficient degree of exactness, from the foregoing hypothesis; because, in all such cases, the effects of the resistance will be nearly as the amplitudes themselves; and were they accurately so, the proportions of the amplitndes, at different elevations, would then be the very same as in vacuo.

Now, in order to form a clear idea of the subject here proposed, the path of every projectile is to be considered as depending on two different forces; that is to say, on the impellant force, whereby the motion is first begun, (and would be continued in a right line) and on the force of gravity, by which the projectile, during the whole time of its flight, is continually urged downwards, and made to deviate more and more from its first direction. As whatever relates to the track and flight of a projectile, or ball, (neglecting the resistance of the air) is to be determined from the action of these two forces, it will be proper, before we proceed to consider their joint effects, to premise something concerning the nature of the motion produced by each, when supposed to act alone, independently of the other; to which end we have premised the two following lemmata.

Lemma I. Every body, after the impressed force whereby it is put in motion ceases to act, continues to move uniformly in a right line; unless it be interrupted by some other force or impediment.

This is a law of nature, and has its demonstration from experience and matter of fact.

Corollary. It follows from hence, that a ball, after leaving the mouth of the piece, would continue to move along the line of its first direction, and describe spaces therein proportional to the times of their description, were it not for the action of gravity; whereby the direction is changed, and the motion interrupted.

Lemma II. The motion, or velocity, acquired by a ball, in freely descending from rest, by the force of an uniform gravity, is as the time of the descent; and the space fallen through, as the square of that time.

The first part of this lemma is extremely obvious: for since every motion is proportional to the force whereby it is ge

nerated, that generated by the force of an uniform gravity, must be as the time of the descent; because the whole effort of such a force is proportional to the time P- TA of its action; that is, as the time of the descent.

d

To demonstrate that the distances descended are proportional to the squares of the times, let the time of falling through any proposed dis tance A B, be represented by the right line PQ; which conceive to be divided into an indefinite number of very small, equal, particles, represented each, by the symbol m; and let the distance descended in the first of them be A c; in the second cd; in the third de; and so ou. Then the velocity acquired being always as the time from the beginning of the descent, it will at the middle of the first of the said particles be represented by one-half m; at the middle of the second, by 1 m; at the middle of the third, by 31 m, &c. which values constitute the series m 3m 5m 7m 9m &c.

2'

2

B

2 2

But since the velocity, at the middle of any one of the said particles of time, is an exact mean between the velocities of the two extremes thereof, the corres ponding particle of the distance, A B, may be therefore considered as described with that mean velocity and so, the spaces Ac, cd, de, ef, &c. being respectively equal to the above-mentioned

:

quantities
&c. it follows,
by the continual addition of these, that the
space A c, A d, Ae, Af, &c. fallen through
from the beginning, will be expressed by

which the ball falls, in any given time n, is just the half of that (nm) which might be uniformly described with the last, or greatest celerity in the same time.

Scholium. It is found by experiment, that any heavy body, near the earth's sur face (where the force of gravity may be considered as uniform) descends about 16 feet from rest, in the first second of time. Therefore, as the distances fallen through, are proved above to be in proportion as the squares of the time. It follows that, as the square of one second is to the square of any given number of seconds, so is 16 feet to the number of feet, a heavy body will freely descend in the said number of seconds. Whence the number of feet

descended in any given time will be found, by multiplying the square of the number of seconds by 16. Thus the distance descended in 2, 3, 4, 5, &c. seconds, will appear to be 64, 144, 256, 400 feet, &c. respectively. Moreover, from hence, the time of the descent through any given distance will be obtained, by dividing the said distance in feet, by 16, and extracting the square root of the quotient; or, which comes to the same thing, by extracting the square root of the whole distance, and then taking onehalf of that root for the number of seconds required. Thus, if the distance be supposed 2,640 feet; then, by either of the two ways, the time of the descent will come out 12.84, or 12.50 seconds.

It appears also (from the corol.) that the velocity per second (in feet) at the end of the fall, will be determined by multiplying the number of seconds in the fall by 32. Thus it is found that a ball, at the end of ten seconds, has acquired a velocity of 320 feet per second. After the same manner, by having any two of the four following quantities, viz. the force, the times, the velocity, and distance, the other two may be determined: for let the space freely descended by a ball, in the first second of time (which is as the accelerating force) be denoted by F; also let T denote the number of seconds wherein any distance, D, is descended; and let V be the velocity per second, at the end of the descent; then

will

V=2 FT=2√FD=2D

Τ

4 D

TT 2T All which equations are very easily deduced from the two original ones, D= FT T, and V2 F T, already demonstrated; the former in the proposition it self, and the latter in the corollary to it; by which it appears that the measure of the velocity at the end of the first second is 2F; whence the velocity (V) at the end of (T) seconds must consequently be expressed by 2 F x T or 2 FT.

Theorem 1. A projected body, whose line of direction is parallel to the plane of the horizon, describes by its fall a parabola. If the heavy body is thrown by any extrinsical force, as that of a gun or the like, from the point A, (Plate Perspective, &c. fig. 7.) so that the direction of its projection is the horizontal line, AD; the path of this heavy body will be a semi-parabola. For if the air did not resist it, nor was it acted on by its gravity, the projectile would proceed with an equable motion, always in the same direction; and the times wherein the parts of space, A B, A C, AD, A E, were passed over, would be as the spaces, AB, AC, A D, &c. respectively. Now if the force of gravity is supposed to take place, and to act in the same tenour, as if the heavy body were not impelled by any extrinsical force, that body would constantly decline from the right line, AE; and the spaces of descent, or the deviations from the horizontal line, A E, will be the same as if it had fallen perpendicularly. Wherefore if the body falling perpendicularly by the force of its gravity, passed over the space A K in the time A B descended through A L, in the time A C, and through AM in the time AD; the spaces, AK, AL, AM, will be as the squares of the times, that is, as the squares of the right lines, AB, A C, AD, &c. or K F, LG, MH. But since the impetus in the direction parallel to the horizon always remains the same (for the force of gravity, that only solicits the body downwards, is not in the least contrary to it); the body will be equally promoted forwards in the direction parallel to the plane of the horizon, as if there was no gravity at all. Wherefore, since in the

time, AB, the body passes over a space equal to A B; but being compelled by the force of gravity, it declines from the right line, A B, through a space equal to A K; and B F being equal and parallel to AK, at the end of the time, A B, the body will be in F, so in the same manner, at the end of the time, A E, the body will be in I; and the path of the projectile will be in the curve, right lines, K.F, LG, MH, N I, are proAFGHI; but because the squares of the portionable to the abseisses, A K, À L, AM, A N. The curve, AFGHI, will be a semi-parabola. The path, therefore, direction, A E, will be a semi-parabolical of a heavy body projected according to the curve, QED.

scribed by a heavy body projected ob Theorem 2. The curve line, that is deliquely and upwards, according to any direction, is a parabola.

Let AF (fig. 8) be the direction horizon, gravity being supposed not to of projection, any ways inclined to the act, the moving body would always continue its motion in the same right line, and &c. proportional to the times. But by the would describe the spaces A B, A C, A D, action of gravity it is compelled continually in a curve, which will be a parabola. Let to decline from the path A F, and to move cularly in the time A B, through the space ns suppose the heavy body falling perpendiAQ, and in the time AC, through the space AR, &c. The spaces AQ, AR, AS, will be as the squares of the times, or as fest from what was demonstrated in the last the squares of A B, AC, AD. It is manithere is taken B M = A Q and the paralletheorem, that if in the perpendicular BG, logram be completed, the place of the heavy body at the end of the time A B, will be M, and so of the rest; and all the deviations BM, &c. from the right line A F, arising from the times, will be equa? to the spaces A Q, AR, A S, which are as the squares of the right lines AB, AC, right line AP, meeting the path of the AD. Through A draw the horizontal projectile in P. From P raise the perpendicular P E, meeting the line of direction in E; and by reason the triangles ABG, ACH, &c. are equiangular, the squares of the right lines A B, A C, &c. will be proportionable to the squares of A G, A H, &c. so that the deviations BM, CN, &c. will be proportionable to the squares of the right lines A G, AH, &c.

Les

=

the line L be a third proportional to EP and AP; and it will be (by 17 El. 6) LXEP AP q, but APq.: AGq. :: EP: BM : : LX EP: L x BM; whence since it is LX EPA Pq. it will be L x B MAG4. In like manner it will be LX CNA Hq, &c. But because it is BG: AG::(EP: AP:: by hypothesis) A PL; it will be L X BG AGXA PAG× AGAG × GP=AGq. + A G × G P. But it has been shown that it is L x B MAGq, wherefore it will be LX BG-LXBM = AGX GP, that is, LX M G = AG × GP. By the same way of reasoning it will be LX NH=AH x HP, &c. Wherefore the rectangle under M G and L, will be equal to the square of A G, which is the property of the parabola; and so the curve A M N O P K, wherein the projectile is moved, will be a parabola.

Cor. 1. Hence the right line L is the latus rectum or parameter of the parabola, that belongs to its axis.

Cor. 2. Let AHHP, and it will be LX CN=AHq. LX NH, whence it will be NH = CN; and consequently the right line A F being the line of direction of the projectile, will be a tangent to the parabola.

Cor. 3. If a heavy body be projected downwards, in a direction oblique to the horizon; the path of the projectile will be a parabola.

Theorem 3. The impetus of a projected body in different parts of the parabola, are as the portions of the tangents intercepted between two right lines parallel to the axis; that is, the impetus of the body projected in the points A and B (fig. 9) to which A D, and BE are tangents, will be as CD, and EB, the portions of the tangents intercepted between two right lines C B, and DE parallel to the axis.

We have here treated the path of a projected body as an exact parabola, though from the resistance of the air, the line of a projectile is not exactly parabolical, but rather a kind of hyperbola; which, if considered and applied to practice, would render the computations far more operose, and the very small difference (as experience shows in heavy shot) would, in a great measure, lessen the elegancy of the demonstrations given by accounting for it; since the common rules are sufficiently exact, and easy for practice.

PROJECTION, in mechanics, the act of communicating motion to a body, from

thence called projectile. In perspective, projection is the appearance or representa. tion of an object on the perspective plane. The projection of the sphere is either orthographic, or stereographic. The former, or orthographic projection, supposes the eye placed at an infinite distance; whereas, in the stereographic projection, it is supposed to be only 90 degrees distant from the primitive circle, or placed in its pole, and thence viewing the circies on the sphere. The primitive circle is that great circle which limits or bounds the representation or projection; and the place of the eye is called the projecting point.

PROLATE, in geometry, an epithet applied to a spheroid produced by the revolution of a semi-ellipses about its larger diameter.

PROLEGOMENA, in philology, certain preparatory observations or discourses prefixed to a book, &c. containing something necessary for the reader to be apprized of, to enable him the better to understand the book, or to enter deeper into the science, &c.

PROMISE, in law, is where, upon a valuable consideration, persons bind themselves by words to do or perform such a thing agreed on; it is in the nature of a verbal covenant, and wants only the solemnity of writing and sealing to make it absolutely the same. Yet for the breach of it the remedy is different; for instead of an action of covenant, there lies only an action upon the case, the damages whereof are to be estimated and determined by the jury. If there is no consideration it is void, and it is called a nude compact, or in Latin a nudum pactum.

PROMISSORY note. See BILLS of

Exchange.

PRONOUN, pronomen, in grammar, a declinable part of speech, which being put instead of a noun, points out some person, or thing. See GRAMMAR.

PRONUNCIATION, in grammar, the manner of articulating or sounding the words of a language. Pronunciation makes much the most difficult part of a written grammar; in regard that a book expressing itself to the eyes, in a matter that wholly concerns the ears, seems next akin to that of teaching the blind to distinguish colours; hence it is that there is no part so defective in grammar as that of the pronunciation, as the writer has frequently no term whereby to give the reader n idea of the sound he would express; f Want of a proper term