This is the name for that part of mechanical philosophy which treats of the motion of bodies anyhow projected from the surface of this earth, and influenced by the action of terrestrial gravity.

It is demonstrated in the physical part of astronomy, that a body so projected must describe a conic section, having the centre of the earth in one focus; and that it will describe round that focus areas proportional to the times. And it follows from the principles of that science, that if the velocity of projection exceeds 36700 feet in a second, the body (if not resisted by the air) would describe a hyperbola; if it be just 36700, it would describe a parabola; and if it be less than this, it would describe an ellipse. If projected directly upwards, in the first case, it would never return, but proceed for ever; its velocity continually diminishing, but never becoming less than an assignable portion of the excess of the initial velocity above 36700 feet in a second; in the second case, it would never return, its velocity would diminish without end, but never be extinguished. In the third case, it would proceed till its velocity was reduced to an assignable portion of the difference between 36700 and its initial velocity; and would then return, regaining its velocity by the same degrees, and in the same places, as it lost it. These are necessary consequences of a gravity directed to the centre of the earth, and inversely proportional to the square of the distance. But in the greatest projections that we are able to make, the gravitations are so nearly equal, and in directions so nearly parallel, that it would be ridiculous affectation to pay any regard to the deviations from equality and parallelism. A bullet rising a mile above the surface of the earth loses only 1/20th of its weight, and a horizontal range of 4 miles makes only 4' of deviation from parallelism.

Let us therefore affirm gravitation as equal and parallel. The errors arising from this affirmation are quite insensible in all the uses which can be made of this theory.

The theory itself will ever be regarded with some veneration and affection by the learned. It was the first fruits of mathematical philosophy. Galileo was the first who applied mathematical knowledge to the motions of free bodies, and this was the subject on which he exercised his fine genius.

Gravity must be considered by us as a constant or uniform accelerating force, according as it produces the descent, or retards the ascent, of a body. A constant or invariable accelerating force is one which produces an uniform acceleration; that is, which in equal times produces equal increments of velocity, and therefore produces increments of velocity proportional to the times in which they are produced. Forces are of themselves imperceptible, and are seen only in their effects; and they have no measure but the effect, or what measures the effect; and every thing which we can discover with regard to those measures, we must affirm with regard to the things of which we affirm them as the measures. Therefore, The motion of a falling body, or of a body projected directly downwards, is uniformly accelerated; and that of a body projected directly upwards is uniformly retarded: that is, the acquired velocities are as the times in which they are acquired by falling, and the extinguished velocities are as the times in which they are extinguished.

Cor. 1. If bodies simply fall, not being projected drawn from downwards by an external force, the times of the falls are proportional to the final velocities; and the times of ascents, which terminate by the action of gravity alone, are proportional to the initial velocities.

2. The spaces described by a heavy body falling from rest are as the squares of the acquired velocities; and the differences of these spaces are as the differences of the squares of the acquired velocities: and, on the other hand, the heights to which bodies projected upwards will rise, before their motions are extinguished, are as the squares of the initial velocities.

3. The spaces described by falling bodies are proportional to the squares of the times from the beginning of the fall; and the spaces described by bodies projected directly upwards are as the squares of the times of the ascents.

4. The space described by a body falling from rest is one half of the space which the body would have uniformly described in the same time, with the velocity acquired by the fall.—And the height to which a body will rise, in opposition to the action of gravity, is one half of the space which it would uniformly describe in the same time with the initial velocity.

In like manner the difference of the spaces which a falling or rising body describes in any equal successive parts of its fall or rise, is one half of the space which it would uniformly describe in the same time with the difference of the initial and final velocities.

This proposition will be more conveniently expressed for our purpose thus:

A body moving uniformly during the time of any fall with the velocity acquired thereby, will in that time describe a space double of that fall; and a body projected directly upwards will rise to a height which is one half of the space which it would, uniformly continued, describe in the time of its ascent with the initial velocity of projection.

These theorems have been already demonstrated in a popular way, in the article GUNNERY. But we would recommend to our readers the 39th prop. of the first book of Newton's Principia, as giving the most general investigation of this subject; equally easy with these more loose methods of demonstration, and infinitely superior to them, by being equally applicable to every variation of the accelerating force. See an excellent application of this proposition by Mr Robins, for defining the motion of a ball discharged from a cannon, in the article GUNNERY, No. 15.

5. It is a matter of observation and experience, that a heavy body falls 16 feet and an inch English measure in a second of time; and therefore acquires the velocity of 32 feet 2 inches per second. This cannot be ascertained directly, with the precision that is necessary. A second is too small a portion of time to be exactly measured and compared with the space described; but it is done with the greatest accuracy by comparing the motion of a falling body with that of a pendulum. The time of a vibration is to the time of falling through half the length of the pendulum, as the circumference of a circle is to its diameter. The length of a pendulum can be ascertained with great precision; and it can be lengthened or shortened till it makes just 86,400 vibrations in a day: and this is the way in which the space fallen through in a second has been accurately ascertained.

As all other forces are ascertained by the accelerations which they produce, they are conveniently measured by comparing their accelerations with the acceleration of gravity. This therefore has been assumed by all the later and best writers on mechanical philosophy, as the unit by which every other force is measured. It gives us a perfectly distinct notion of the force which retains the moon in its orbit, when we say it is the 3600th part of the weight of the moon at the surface of the earth. We mean by this, that if a bullet were here weighed by a spring steelyard, and pulled it out to the mark 3600; if it were then taken to the distance of the moon, it would pull it out only to the mark 1. And we make this assertion on the authority of our having observed that a body at the distance of the moon falls from that distance 1/3600 part of 16 feet in a second. We do not, therefore, compare the forces, which are imperceptible things; we compare the accelerations, which are their indications, effects, and measures.

This has made philosophers so anxious to determine with precision, the fall of heavy bodies, in order to have determined an exact value of the accelerating power of terrestrial gravity. Now we must here observe, that this measure may be taken in two ways: we may take the space through which the heavy body falls in a second; or we may take the velocity which it acquires in consequence of gravity having acted on it during a second. The last is the proper measure; for the last is the immediate effect on the body. The action of gravity has changed the state of the body—in what way? by giving it a determination to motion downwards this both points out the kind and the degree or intensity of the force of gravity. The space described in a second by falling, is not an invariable measure; for, in the successive seconds, the body falls through 16, 48, 80, 112, &c. feet, but the changes of the body's state in each second is the same. At the beginning it had no determination to move with any appreciable velocity; at the end of the first second it had a determination by which it would have gone on for ever (had no subsequent force acted on it) at the rate of 32 feet per second. At the end of the second second, it had a determination by which it would have moved for ever, at the rate of 64 feet per second. At the end of the third second, it had a determination by which it would have moved for ever, at the rate of 96 feet per second, &c. &c. The difference of these determinations is a determination to the rate of 32 feet per second. This is therefore constant, and the indication and proper measure of the constant or invariable force of gravity. The space fallen through in the first second is of use only as it is one half of the measure of this determination; and as halves have the proportion of their wholes, different accelerating forces may be safely affirmed to be in the proportion of the spaces through which they uniformly impel bodies in the same time. But we should always remember, that this is but one half of the true measure of mathematicians on the accelerating force. Mathematicians of the first rank have have committed great mistakes by not attending to this; and it is necessary to notice it just now, because cases will occur in the prosecution of this subject, where we shall be very apt to confound our reasonings by a confusion in the use of these measures. Those mathematicians who are accustomed to the geometrical consideration of curvilinear motions, are generally disposed to take the actual deflection from the tangent as the measure of the deflecting force; while those who treat the same subject algebraically, by the assistance of fluxions, take the change of velocity, which is measured by twice the deflection. The reason is this: when a body passes through the point B of a curve ABC, fig. 1, if the deflecting force were to cease at that instant, the body would describe the tangent BD in the same time in which it describes the arch BC of the curve, and DC is the deflection, and is therefore taken for the measure of the deflecting force. But the algebraist is accustomed to consider the curve by means of an equation between the abscissae Aa, Hb, He, and their respective ordinates Aa, Bb, Cc; and he measures the deflections by the changes made on the increments of the ordinates. Thus the increment of the ordinate Aa, while the body describes the arch AB of the curve, is BG. If the deflecting force were to cease when the body is at B, the next increment would have been equal to LG, that is, it would have been EF; but in consequence of the deflection, it is only CF; therefore he takes FC for the measure of the deflection, and of the deflecting force. Now EC is ultimately twice DC; and thus the measure of the algebraist (derived solely from the nature of the differential method, and without any regard to physical considerations) happens to coincide with the true physical measure. There is therefore particularly great danger of mixing these measures. Of this we cannot give a more remarkable instance than Leibnitz's attempt to demonstrate the elliptical motion of the planets in the Leipzig Acts, 1659. He first considers the subject mechanically, and takes the deflection or DC for the measure of the deflecting force. He then introduces his differential calculus, where he takes the difference of the increments for the measure; and thus brings himself into a confusion, which luckily compensates for the false reasoning in the preceding part of his paper, and gives his result the appearance of a demonstration of Newton's great discovery, while, in fact, it is a confused jumble of assumptions, self-contradictory, and inconsistent with the very laws of mechanics which are used by him in the investigation. Seventeen years after this, in 1766, having been criticized for his bad reasoning, or rather accused of an envious and unsuccessful attempt to appropriate Newton's invention to himself, he gives a correction of his paralogism, which he calls a correction of language. But he either had not observed where the paralogism lay, or would not let himself down by acknowledging a mistake in what he wished the world to think his own calculus (fluxions); he applied the correction where no fault had been committed, for he had measured both the centrifugal force and the solicitation of gravity in the same way, but had applied the fluxionary expression to the last and not to the first, and, by so doing, he completely destroyed all coincidence between his result and the planetary motions. We mention this instance, not only as a caution to our mathematical readers, but also as a very curious literary anecdote. This dissertation of Leibnitz is one of the most obscure of his obscure writings, but deserves the attention of an intelligent and curious reader, and cannot fail of making an indelible impression on his mind, with relation to the modesty, candour, and probity of the author. It is preceded by a dissertation on the subject which we are now entering upon, the motion of projectiles in a resisting medium. Newton's Principia had been published a few years before, and had been reviewed in a manner shamefully slight, in the Leipzig Acts. Both these subjects make the capital articles of that immortal work. Mr Leibnitz published these dissertations, without (says he) having seen Newton's book, in order to show the world that he had, some years before, discovered the same theorems. Mr Nicholas Pato carried a copy of the Principia from the author to Hanover in 1686, where he expected to find Mr Leibnitz; he was then absent, but Pato saw him often before his return to France in 1687, and does not say that the book was not given him. Read along with these dissertations Dr Keill's letter to John Bernoulli and others, published in the Journal Littéraire de la Haye 1714, and to John Bernoulli in 1719.

Newton has been accused of a similar oversight by John Bernoulli, (who indeed calls it a mistake in principle) in his Proposition x. book 2. on the very subject we are now considering. But Dr Keill has shown it to be only an oversight, in drawing the tangent on the wrong side of the ordinate. For in this very proposition Newton exhibits, in the strictest and most beautiful manner, the difference between the geometrical and algebraical manner of considering the subject; and expressly warns the reader, that his algebraical symbol expresses the deflection only, and not the variation of the increment of the ordinate. It is therefore in the last degree improbable that he would make this mistake. He most expressly does not; and so to the real mistake, which he corrected in the second edition, the writer of this article has in his possession a manuscript copy of notes and illustrations on the whole Principia, written in 1693 by Dr David Gregory, Savilian professor of astronomy at Oxford, at the desire of Mr Newton, as preparatory for a new edition, where he has rectified this and several other mistakes in that work, and says that Mr Newton had seen and approved of the amendments. We mention these particulars, because Mr Bernoulli published an elegant dissertation on this subject in the Leipzig Acts in 1713; in which he withdraws charges Newton (though with many protestations of respect to admiration and respect) with this mistake in principle; and says, that he communicated his correction to Mr Newton, by his nephew Nicholas Bernoulli, that it might be corrected in the new edition, which he heard was in the press. And he afterwards adds, that it appears by some sheets being cancelled, and new ones substituted in this part of the work, that the mistake would have continued, had he not corrected it. We would desire our readers to consult this dissertation, which is extremely elegant, and will be of service to us in this article; and let them compare the civil things which is here said of the vir incomparabilis, the omni laude major, the summus Newionus, with what the same author, in the same year, in the Leipzig Acts, but under a borrowed name, says of him. Our readers will have no hesitation in attributing this letter to this author. For, after praising John Bernoulli as summus geometra, natus Let our readers now consider the scope and intention of this dissertation on projectiles, and judge whether the author's aim was to instruct the world, or to acquire fame, by correcting Newton. The dissertation does not contain one theorem, one corollary, nor one step of argument, which is not to be found in Newton's first edition; nor has he gone farther than Newton's single proposition the xth. To us it appears an exact companion to his proposition on centripetal forces, which he boasts of having first demonstrated, although it is in every step a transcript of the 42d of the 1st Book of Newton's Principia, the geometrical language of Newton being changed into algebraic, as he has in the present case changed Newton's algebraic analysis into a very elegant geometrical one.

We hope to be forgiven for this long digression. It is a very curious piece of literary history, and shows the combination which envy and want of honourable principle had formed against the reputation of our illustrious countryman; and we think it our duty to embrace any opportunity of doing it justice.—To return to our subject:

The accurate measure of the accelerative power of gravity, is the fall 16 feet, if we measure it by the space, or the velocity of 32 feet per second, if we take the velocity. It will greatly facilitate calculation, and will be sufficiently exact for all our purposes, if we take 16 and 32, supposing that a body falls 16 feet in a second, and acquires the velocity of 32 feet per second. Then, because the heights are as the squares of the times, and as the squares of the acquired velocities, a body will fall one foot in one fourth of a second, and will acquire the velocity of eight feet per second. Now let \( h \) express the height in feet, and call it the producing height; \( v \) the velocity in feet per second, and call it the produced velocity, the velocity due; and \( t \), the time in seconds.—We shall have the following formulæ, which are of easy recollection, and will serve, without tables, to answer all questions relative to projectiles.

\[ \begin{align*} I. & \quad v = 8 \sqrt{\frac{h}{t}} = 8 \times 4 t = 32 t \\ II. & \quad t = \frac{\sqrt{h}}{4} = \frac{v}{32} \\ III. & \quad \sqrt{h} = \frac{v^2}{8} = 4 t \\ IV. & \quad h = \frac{v^2}{64} = 16 t^2. \end{align*} \]

To give some examples of their use, let it be required,

1. To find the time of falling through 256 feet. Here \( h = 256, \sqrt{256} = 16, \) and \( \frac{16}{4} = 4. \) Answer 4".

2. To find the velocity acquired by falling four seconds. \( t = 4, 32 \times 4 = 128 \) feet per second.

3. To find the velocity acquired by falling 625 feet. \( h = 625, \sqrt{625} = 25, 8 \sqrt{h} = 200 \) feet per second.

4. To find the height to which a body will rise when projected with the velocity of 56 feet per second, or the height through which a body must fall to acquire this velocity.

\[ v = 56, \frac{56}{8} = 7, \sqrt{h} \cdot 7 = h = 49 \text{ feet}. \]

or \( 56^2 = 3136, \frac{3136}{64} = 49 \) feet.

5. Suppose a body projected directly downwards with the velocity of 10 feet per second; what will be its velocity downwards after four seconds? In four seconds it will have acquired, by the action of gravity, the velocity of \( 4 \times 32, \) or 128 feet, and therefore its whole velocity will be 138 feet per second.

6. To find how far it will have moved, compound its motion of projection, which will be 40 feet in four seconds, with the motion which gravity alone would have given it in that time, which is 256 feet; and the whole motion will be 296 feet.

7. Suppose the body projected as already mentioned, and that it is required to determine the time it will take to go 296 feet downwards, and the velocity it will have acquired.

Find the height \( x, \) through which it must fall to acquire the velocity of projection, 10 feet, and the time \( y \) of falling from this height. Then find the time \( x \) of falling through the height 296 + \( x, \) and the velocity \( v \) acquired by this fall. The time of describing the 296 feet will be \( x - y, \) and \( v \) is the velocity required.

From such examples, it is easy to see the way of answering every question of the kind.

Writers on the higher parts of mechanics always more generally compute the actions of other accelerating and retarding forces by comparing them with the acceleration of gravity, and in order to render their expressions more general, use a symbol, such as \( g \) for gravity, leaving the reader to convert it into numbers. Agreeably to this view, the general formulæ will stand thus:

\[ \begin{align*} I. & \quad v = \sqrt{2gh}, \text{i.e. } \sqrt{2g} \sqrt{h} = gt \\ II. & \quad t = \frac{\sqrt{4h}}{2g} = \sqrt{\frac{4h}{2g}} = \sqrt{\frac{2h}{g}} \\ III. & \quad h = \frac{v^2}{2g} = \frac{gt^2}{2} \end{align*} \]

In all these equations, gravity, or its accelerating power, is eliminated, as it ought to be, by the change of velocity which it generates in a particle of matter in an unit of time. But many mathematicians, in their investigations of curvilinear and other varied motions, measure it by the deflection which it produces in this time from the tangent of the curve, or by the increment by which the space described in an unit of time exceeds the space described in the preceding unit. This is but one half of the increment which gravity would have produced, had the body moved through the whole moment with the acquired addition of velocity. In this sense of the symbol \( g, \) the equations stand thus:

\[ \begin{align*} I. & \quad v = 2\sqrt{gh} = 2gt \\ II. & \quad t = \sqrt{\frac{h}{g}}, \frac{v}{2g} \\ IV. & \quad h = \frac{v^2}{4g} = gt^2, \text{ and } \sqrt{h} = \frac{v}{2\sqrt{g}} \end{align*} \]

It is also very usual to consider the accelerating force... of gravity as the unit of comparison. This renders the expressions much more simple. In this way, \( v \) expresses not the velocity, but the height necessary for acquiring it, and the velocity itself is expressed by \( \sqrt{v} \). To reduce such an expression of a velocity to numbers, we must multiply it by \( \sqrt{\frac{g}{2}} \), or by \( 2\sqrt{g} \), according as we make \( g \) to be the generated velocity, or the space fallen through in the unit of time.

This will suffice for the perpendicular ascents or descents of heavy bodies, and we proceed to consider their motions when projected obliquely. The circumstance which renders this an interesting subject, is, that the flight of cannon shot and shells are instances of such motion, and the art of gunnery must in a great measure depend on this doctrine.

Let a body B (fig. 2.), be projected in any direction BC, not perpendicular to the horizon, and with any velocity. Let AB be the height producing this velocity; that is, let the velocity be that which a heavy body would acquire by falling freely through AB. It is required to determine the path of the body, and all the circumstances of its motion in this path?

1. It is evident, that by the continual action of gravity, the body will be continually deflected from the line BC, and will describe a curve line BVG, concave towards the earth.

2. This curve line is a parabola, of which the vertical line ABE is a diameter, B the vertex of this diameter and BC a tangent in B.

Through any two points V, G of the curve draw VC, GH parallel to AB, meeting BC in C and H, and draw VE, GK parallel to BC, meeting AB in E, K. It follows, from the composition of motions, that the body would arrive at the points V, G of the curve in the same time that it would have uniformly described BC, BH, with the velocity of projection; or that it would have fallen through BE, BK, with a motion uniformly accelerated by gravity; therefore the times of describing BC, BH, uniformly, are the same with the times of falling through BE, BK. But, because the motion along BH is uniform, BC is to BH as the time of describing BC to the time of describing BH, which we may express thus, BC : BH = T, BC : T, BH = T, BE : T, BK. But, because the motion along BK is uniformly accelerated, we have BE : BK = T², BE : T², BK = BC² : BH² = EV² : KG²; therefore the curve BVG is such, that the abscissae BE, BK are as the squares of the corresponding ordinates EV, KG; that is, the curve BVG is a parabola, and BC, parallel to the ordinates, is a tangent in the point B.

3. If through the point A there be drawn the horizontal line AD, it is the directrix of the parabola.

Let BE be taken equal to AB. The time of falling through BE is equal to the time of falling through AB; but BC is described with the velocity acquired by falling through AB; and therefore by No. 4, of perpendicular descents, BC is double of AB, and EV is double of BE; therefore EV² = 4BE² = 4BE × AB = BE × 4AB, and 4AB is the parameter or latus rectum of the parabola BVG, and AB being one-fourth of the parameter, AD is the directrix.

4. The times of describing the different arches BV, VG of the parabola are as the portions BC, BH of the tangent, or as the portions AD, A'd of the directrix, intercepted by the same vertical lines AB, CV, HG; for the times of describing BV, BVG are the same with those of describing the corresponding parts BC, BH of the tangent, and are proportional to these parts, because the motion along BH is uniform; and BC, BH are proportional to AD, A'd.

Therefore the motion estimated horizontally is uniform.

5. The velocity in any point G of the curve is the same with that which a heavy body would acquire by falling from the directrix along dG. Draw the tangent GT, cutting the vertical AB in T; take the points a, f, equidistant from A and d, and extremely near them, and draw the verticals a, f, g; let the points a, f, continually approach A and d, and ultimately coincide with them. It is evident that Bb will ultimately be to gG, in the ratio of the velocity at B to the velocity at G; for the portions of the tangent ultimately coincide with the portions of the curve, and are described in equal times; but Bb is to gG as BH to TG; therefore the velocity at B is to that at G as BH to TG. But, by the properties of the parabola, BH² is to TG² as AB to dG; and AB is to dG as the square of the velocity acquired by falling through AB to the square of the velocity acquired by falling through dG; and the velocity in BH, or in the point B of the parabola, is the velocity acquired by falling along AB; therefore the velocity in TG, or in the point G of the parabola, is the velocity acquired by falling along dG.

These few simple propositions contain all the theory of the motion of projectiles in vacuo, or independent of the resistance of the air; and being a very easy and ingenious, neat piece of mathematical philosophy, and connected with very interesting practice, and a very respectable profession, they have been much commented on, and have furnished matter for many splendid volumes. But the air's resistance occasions such a prodigious diminution of motion in the great velocities of military projectiles, that this parabolic theory, as it is called, is hardly of any use. A musket ball, discharged with the ordinary allotment of powder, issues from the piece with the velocity of 1670 feet per second: this velocity would be acquired by falling from the height of eight miles. If the piece be elevated to an angle of 45°, the parabola should be of such extent that it would reach 16 miles on the horizontal plain; whereas it does not reach above half a mile. Similar deficiencies are observed in the ranges of cannon shot.

We do not propose, therefore, to dwell much on this short theory, and shall only give such a synoptical view of it as shall make our readers understand the more general circumstances of the theory, and be matters of the language of the art.

Let OB (fig. 3.) be a vertical line. About the centres A and B, with the distance AB, describe the semicircles ODB, AHK, and with the axis AB, and semi-axis GE, equal to AB, describe the semi-ellipse AEB: with the focus B, vertex A, diameter AB, and tangent AD, parallel to the horizon, describe the parabola APS.

Let a body be projected from B, in any direction BC, PROJECTILES.

BC, with the velocity acquired by falling through AB. By what has already been demonstrated, it will describe a parabola BVPM. Then,

1. ADL parallel to the horizon is the directrix of every parabola which can be described by a body projected from B with this velocity. This is evident.

2. The semicircle AHK is the locus of all the foci of these parabolas; For the distance BH of a point B of any parabola from the directrix AD is equal to its distance BF from the focus F of that parabola; therefore the foci of all the parabolas which pass through B, and have AD for their directrix, must be in the circumference of the circle which has AB for its radius, and B for its centre.

3. If the line of direction BC cut the upper semicircle in C, and the vertical line CF be drawn, cutting the lower semicircle in F, F is the focus of the parabola BVPM, described by the body which is projected in the direction BC, with the velocity acquired by falling through BA; for drawing AC, BF, it is evident that ACFB is a rhombus, and that the angle ABF is bisected by BC, and therefore the focus lies in the line BF; but it also lies in the circumference AFK, and therefore in F.

If C is in the upper quadrant of ODB, F is in the upper quadrant of AFK; and if C be in the lower quadrant of ODB (as when BC is the line of direction) then the focus of the corresponding parabola BvM is in the lower quadrant of AHK, as at f.

4. The ellipse AEB is the focus of the vertex of all the parabolas, and the vertex V of any one of them BVPM is in the intersection of this ellipse with the vertical CF; for let this vertical cut the horizontal lines AD, GE, BN, in δ, λ, η. Then it is plain that Nλ is half of ND, and λV is half of Cδ; therefore NV is half of NC, and V is the vertex of the axis.

If the focus is in the upper or lower quadrant of the circle AHK, the vertex is in the upper or the lower quadrant of the ellipse AEG.

5. If BFP be drawn through the focus of any one of the parabolas, such as BVM, cutting the parabola APS in P, the parabola BVM touches the parabola APS in P; for drawing Pδ parallel to AB, cutting the directrix Oδ of the parabola APS in δ, and the directrix AL of the parabola BVM in δ, then PB = Pδ; but BF = BA, = AO, = δδ; therefore Pδ = PF, and the point P is in the parabola BVM. Also the tangents to both parabolas in P coincide, for they bisect the angle xPB; therefore the two parabolas having a common tangent, touch each other in P.

Cor. All the parabolas which can be described by a body projected from B, with the velocity acquired by falling through AB, will touch the concavity of the parabola APS, and lie wholly within it.

6. P is the most distant point of the line BP which can be hit by a body projected from B with the velocity acquired by falling through AB. For if the direction is more elevated than BC, the focus of the parabola described by the body will lie between F and A, and the parabola will touch APS in some point between P and A; and being wholly within the parabola APS, it must cut the line BP in some point within P. The same thing may be shown when the direction is less elevated than BC.

7. The parabola APS is the focus of the greatest range on any planes BP, BS, &c., and no point lying without this parabola can be struck.

8. The greatest range on any plane BP is produced when the line of direction BC bisects the angle OBP formed by that plane with the vertical: for the parabola described by the body in this case touches APS in P, and its focus is in the line BP, and therefore the tangent BC bisects the angle OBP.

Cor. The greatest range on a horizontal plane is made with an elevation of 45°.

9. A point M in any plane BS, lying between B and S, may be struck with two directions, BC and Bc; and these directions are equidistant from the direction Bt, which gives the greatest range on that plane: for if about the centre M, with the distance ML from the directrix AL, we describe a circle LFf, it will cut the circle AHK in two points F and f', which are evidently the foci of two parabolas BVM, BvM, having the directrix AL and diameter ABK. The intersection of the circle ODB, with the verticals FC, fc, determine the directions BC, Bc of the tangents. Draw At parallel to BS, and join tB, Cc, Ff'; then OBt = ½ GBS, and Bt is the direction which gives the greatest range on the plane BS: but because Ff' is a chord of the circles described round the centres B and M, Ff' is perpendicular to BM, and Cc to At, and the arches Ct, ct are equal; and therefore the angles CBt, cBt are equal.

Thus we have given a general view of the subject, which shows the connection and dependence of every circumstance which can influence the result; for it is evident that to every velocity of projection there belongs a set of parabolas, with their directions and ranges; and every change of velocity has a line AB corresponding to it, to which all the others are proportional. As the height necessary for acquiring any velocity increases or diminishes in the duplicate proportion of that velocity, it is evident that all the ranges with given elevations will vary in the same proportion, a double velocity giving a quadruple range, a triple velocity giving a nonuple range, &c. And, on the other hand, when the ranges are determined beforehand (which is the usual case), the velocities are in the subduplicate proportion of the ranges. A quadruple range will require a double velocity, &c.

On the principles now established is founded the ordinary theory of gunnery, furnishing rules which are to principally direct the art of throwing shot and shells, so as to hit the mark with a determined velocity.

But we must observe, that this theory is of little service for directing us in the practice of cannonading. Here it is necessary to come as near as we can to the object aimed at, and the hurry of service allows no time for geometrical methods of pointing the piece after each discharge. The gunner either points the cannon directly to the object, when within 200 or 300 yards of it, in which case he is said to shoot point blank (pointeur au blanc, i.e. at the white mark in the middle of the gunners target); or, if at a greater distance, he estimates to the best of his judgment the deflection corresponding to his distance, and points the cannon accordingly. In this he is aided by the greater thickness at the breech of a piece of ordnance. Or, lastly, when the intention is not to batter, but to rake along a line occupied... occupied by the enemy, the cannon is elevated at a considerable angle, and the shot discharged with a small force, so that it drops into the enemy's fort, and bounds along the line. In all these services the gunner is directed entirely by trial, and we cannot say that this parabolic theory can do any service.

The principal use of it is to direct the bombardier in throwing shells. With these it is proposed to break down or set fire to buildings, to break through the vaulted roofs of magazines, or to intimidate and kill troops by bursting among them. These objects are always under cover of the enemy's works, and cannot be touched by a direct shot. The bombs and carcasses are therefore thrown upwards, so as to get over the defences and produce their effect.

These shells are of very great weight, frequently exceeding 2000 lbs. The mortars from which they are discharged must therefore be very strong, that they may resist the explosion of gunpowder which is necessary for throwing such a mass of matter to a distance; they are consequently unwieldy, and it is found most convenient to make them almost a solid and immovable lump. Very little change can be made in their elevation, and therefore their ranges are regulated by the velocities given to the shell. These again are produced by the quantities of powder in the charge; and experience (confirming the best theoretical notions that we can form of the subject) has taught us, that the ranges are nearly proportional to the quantities of powder employed, only not increasing quite so fast. This method is much easier than by differences of elevation; for we can select the elevation which gives the greatest range on the given plane, and then we are certain that we are employing the smallest quantity of powder with which the service can be performed; and we have another advantage, that the deviations which unavoidable causes produce in the real directions of the bomb will then produce the smallest possible deviation from the intended range. This is the case in most mathematical maxima.

In military projectiles the velocity is produced by the explosion of a quantity of gunpowder; but in our theory it is conceived as produced by a fall from a certain height, by the proportions of which we can accurately determine its quantity. Thus a velocity of 1600 feet per second is produced by a fall from the height of 4000 feet, or 1333 yards.

The height CA (fig. 4.) for producing the velocity of projection is called, in the language of gunnery, the IMPETUS. We shall express it by the symbol h.

The distance AB to which the shell goes on any plane AB is called the AMPLITUDE or the RANGE r.

It is evident that \( \frac{AZ}{AD} = S_{ADZ} : S_{DBA} : S_{DAB} : S_{p} : S_{e} \)

And \( \frac{AD}{DB} = S_{DBA} : S_{DAB} : S_{p} : S_{e} \)

And \( \frac{DB}{AB} = S_{DAB} : S_{ADB} : S_{e} : S_{z} \)

Therefore \( \frac{AZ}{AB} = S_{p} : S_{e} : S_{z} : S_{x} \)

Or \( \frac{h}{r} = S_{p} : S_{e} : S_{z} : S_{x} \)

Hence we obtain the relations wanted.

Thus \( h = \frac{r \times S_{p}}{S_{e} \times S_{z}} \), and \( r = \frac{4h \times S_{e} \times S_{z}}{S_{p}} \)

And \( S_{z} = \frac{r \times S_{p}}{4h \times S_{e}} \), and \( S_{e} = \frac{r \times S_{p}}{4h \times S_{z}} \)

The only other circumstance in which we are interest-

The angle DEA, made by the vertical line and the plane AB, may be called the angle of POSITION of that plane, p.

The angle DAB, made by the axis or direction of the piece, and the direction of the object, may be called the angle of ELEVATION of the piece above the plane AB, e.

The angle ZAD, made by the vertical line, and the direction of the piece, may be called the ZENITH distance, z.

The relations between all the circumstances of velocity, distance, position, elevation, and time, may be included in the following propositions:

I. Let a shell be projected from A, with the velocity acquired by falling through CA, with the intention of hitting the mark B situated in the given line AB.

Make ZA = AC, and draw BD perpendicular to the horizon. Describe on ZA an arch of a circle ZDA containing an angle equal to DBA, and draw AD to the intersection of this circle with DB; then will a body projected from A, in the direction AD, with the velocity acquired by falling through CA, hit the mark B.

For, produce CA downwards, and draw BF parallel to AD, and draw ZD. It is evident from the construction that AB touches the circle in B, and that the angles ADZ, DBA, are equal, as also the angles AZD, DAB; therefore the triangles ZAD, ADB are similar.

Therefore \( BD : DA = DA : AZ \),

And \( DA = BD \times AZ \);

Therefore \( BF = AF \times AZ = AF \times AC \).

Therefore a parabola, of which AF is a diameter, and AZ its parameter, will pass through B, and this parabola will be the path of the shell projected as already mentioned.

Remark. When BD cuts this circle, it cuts it in two points D, d; and there are two directions which will solve the problem. If B'D' only touches the circle in D', there is but one direction, and AB' is the greatest possible range with this velocity. If the vertical line through B does not meet the circle, the problem is impossible, the velocity being too small. When B'D' touches the circle, the two directions AD' and Ad' coalesce into one direction, producing the greatest range, and bisecting the angle ZAB; and the other two directions AD, Ad, producing the same range AB, are equidistant from AD', agreeably to the general proposition.

ed is the time of the flight. A knowledge of this is necessary for the bombardier, that he may cut the fuzes of his shells to such lengths as that they may burst at the very instant of their hitting the mark.

Now \( AB : DB = Sin, ADB : Sin, DAB, = S_{z} : S_{e} \)

\( S_{z}, e \) and \( DB = \frac{r \times S_{e}}{S_{z}} \). But the time of the flight is the same with the time of falling through DB, and 16 feet: \( DB = t' : t'' \). Hence \( t'' = \frac{t'}{165} \), and we have

the following easy rule.

From the sum of the logarithms of the range, and of the time of elevation, subtract the sum of the logarithms of 16, and of the fine of the zenith distance, half the remainder is the logarithm of the time in seconds.

This becomes still easier in practice; for the mortar should be so elevated that the range is a maximum: in which case \( AB = DB \), and then half the difference of the logarithms of \( AB \) and of 16 is the logarithm of the time in seconds.

Such are the deductions from the general propositions which constitute the ordinary theory of gunnery. It remains to compare them with experiment.

In such experiments as can be performed with great accuracy in a chamber, the coincidence is as great as can be wished. A jet of water, or mercury, gives us the finest example, because we have the whole parabola exhibited to us in the simultaneous places of the succeeding particles. Yet even in these experiments a deviation can be observed. When the jet is made on a horizontal plane, and the curve carefully traced on a perpendicular plane held close by it, it is found that the distance between the highest point of the curve and the mark is less than the distance between it and the spout, and that the descending branch of the curve is more perpendicular than the ascending branch. And this difference is more remarkable as the jet is made with greater velocity, and reaches to a greater distance. This is evidently produced by the resistance of the air, which diminishes the velocity, without affecting the gravity of the projectile. It is still more sensible in the motion of bombs. These can be traced through the air by the light of their fuses; and we see that their highest point is always much nearer to the mark than to the mortar on a horizontal plane.

The greatest horizontal range on this plane should be when the elevation is 45°. It is always found to be much lower.

The ranges on this plane should be as the sines of twice the elevation.

A ball discharged at the elev. 19°. 5' ranged 438 yards at 9.45 332

It should have ranged by theory 241

The range at an elevation of 45° should be twice the impetus. Mr Robins found that a musket-ball, discharged with the usual allotment of powder, had the velocity of 1700 feet in a second. This requires a fall of 45156 feet, and the range should be 9312, or 17.5 miles; whereas it does not much exceed half a mile. A 24 pound ball discharged with 16 pounds of powder should range about 16 miles; whereas it is generally short of 3 miles.

Such facts show incontrovertibly how deficient the parabolic theory is, and how unfit for directing the practice of the artillery. A very simple consideration is sufficient for rendering this obvious to the most un instructed. The resistance of the air to a very light body may greatly exceed its weight. Any one will feel this in trying to move a fan very rapidly through the air; therefore this resistance would occasion a greater deviation from uniform motion than gravity would in that body. Its path, therefore, through the air may differ more from a parabola than the parabola itself deviates from the straight line.

It is for such cogent reasons that we presume to say, that the voluminous treatises which have been published on this subject are nothing but ingenious amusements for young mathematicians. Few persons who have been much engaged in the study of mechanical philosophy have missed this opportunity in the beginning of their studies. The subject is easy. Some property of the parabola occurs, by which they can give a neat and systematic solution of all the questions; and at this time of study it seems a considerable elay of skill. They are tempted to write a book on the subject; and it finds readers among other young mechanicians, and employs all the mathematical knowledge that most of the young gentlemen of the military profession are possessed of. But these performances deserve little attention from the practical artillery. All that seems possible to do for his education is, to multiply judicious experiments on real pieces of ordnance, with the charges that are used in actual service, and to furnish him with tables calculated from such experiments.

These observations will serve to justify us for having given so concise an account of this doctrine of the parabolic flight of bodies.

But it is the business of a philosopher to inquire into the causes of such a prodigious deviation from a well-founded theory, and having discovered them, to ascertain precisely the deviations they occasion. Thus we shall obtain another theory, either in the form of the parabolic theory corrected, or as a subject of independent discussion. This we shall now attempt.

The motion of projectiles is performed in the atmosphere. The air is displaced, or put in motion. What the atmosphere ever motion it acquires must be taken from the bullet, sphere. The motion communicated to the air must be in the proportion of the quantity of air put in motion, and of the velocity communicated to it. If, therefore, the displaced air be always similarly displaced, whatever be the velocity of the bullet, the motion communicated to it, and lost by the bullet, must be proportional to the square of the velocity of the bullet and to the density of the air jointly. Therefore the diminution of its motion must be greater when the motion itself is greater, and in the very great velocity of shot and shells it must be prodigious. It appears from Mr Robins's experiments that a globe of 4 inches in diameter, moving with the velocity of 25 feet in a second, sustained a resistance of 315 grains, nearly \( \frac{1}{4} \) of an ounce. Suppose this ball to move 800 feet in a second, that is 32 times faster, its resistance would be \( 32 \times 32 \times \frac{1}{4} \) of an ounce, or 768 ounces or 48 pounds. This is four times the weight of a ball of cast iron of this diameter; and if the initial velocity had been 1600 feet per second, the resistance would be at least 16 times the weight of the ball. It is indeed much greater than this.

This resistance, operating constantly and uniformly compared on the ball, must take away four times as much from with that its velocity as its gravity would do in the same time, of gravity. We know that in one second gravity would reduce the velocity 800 to 768 if the ball were projected straight upwards. This resistance of the air would therefore reduce it in one second to 672, if it operated uniformly; but as the velocity diminishes continually by the resistance, and the resistance diminishes along with the velocity, city, the real diminution will be somewhat less than 128 feet. We shall, however, see afterwards that in one second its velocity will be reduced from 800 to 687. From this simple instance, we see that the resistance of the air must occasion great deviation from parabolic motion.

In order to judge accurately of its effect, we must consider it as a retarding force, in the same way as we consider gravity. The weight \( W \) of a body is the aggregate of the action of the force of gravity \( g \) on each particle of the body. Suppose the number of equal particles, or the quantity of matter, of a body to be \( M \), then \( W \) is equivalent to \( gM \). In like manner, the resistance \( R_r \) which we observe in any experiment, is the aggregate of the action of a retarding force \( R' \) on each particle, and is equivalent to \( R'M \); and as \( g \) is equal to \( \frac{W}{M} \), so \( R' \) is equal to \( \frac{R}{M} \). We shall keep this distinction in view, by adding the differential mark ' to the letter \( R \) or \( r \), which expresses the aggregate resistance.

If we, in this manner, consider resistance as a retarding force, we can compare it with any other such force by means of the retardation which it produces in similar circumstances. We would compare it with gravity by comparing the diminution of velocity which its uniform action produces in a given time with the diminution produced in the same time by gravity. But we have no opportunity of doing this directly; for when the resistance of the air diminishes the velocity of a body, it diminishes it gradually, which occasions a gradual diminution of its own intensity. This is not the case with gravity, which has the same action on a body in motion or at rest. We cannot, therefore, observe the uniform action of the air's resistance as a retarding force. We must fall on some other way of making the comparison. We can state them both as dead pressures. A ball may be fitted to the rod of a spring stillyard, and exposed to impulse of the wind. This will compress the stillyard to the mark 3, for instance. Perhaps the weight of the ball will compress it to the mark 6. We know that half this weight would compress it to 3. We account this equal to the pressure of the air, because they balance the same elasticity of the spring. And in this way we can estimate the resistance by weights, whose pressures are equal to its pressure, and we can thus compare it with other resistances, weights, or any other pressures. In fact, we are measuring them all by the elasticity of the spring. This elasticity in its different positions is supposed to have the proportions of the weights which keep it in these positions. Thus we reason from the nature of gravity, no longer considered as a dead pressure, but as a retarding force; and we apply our conclusions to resistances which exhibit the same pressures, but which we cannot make to act uniformly. This sense of the words must be carefully remembered whenever we speak of resistances in pounds and ounces.

The most direct and convenient way of stating the comparison between the resistance of the air and the accelerating force of gravity, is to take a case in which we know that they are equal. Since the resistance is here assumed as proportional to the square of the velocity, it is evident that the velocity may be so increased that the resistance shall equal or exceed the weight of the body. If a body be already moving downwards with this velocity, it cannot accelerate; because the accelerating force of gravity is balanced by an equal retarding force of resistance. It follows from this remark, that this velocity is the greatest that a body can acquire by the force of gravity only. Nay, we shall afterwards see that it never can completely attain it; because as it approaches to this velocity, the remaining accelerating force decreases faster than the velocity increases. It may therefore be called the limiting or terminal velocity by gravity.

Let \( a \) be the height through which a heavy body must fall, in vacuo, to acquire its terminal velocity in air. If projected directly upwards with this velocity, it will rise again to this height, and the height is half the space which it would describe uniformly, with this velocity, in the time of its ascent. Therefore the resistance to this velocity being equal to the weight of the body, it would extinguish this velocity, by its uniform action, in the same time, and after the same distance, that gravity would.

Now let \( g \) be the velocity which gravity generates or extinguishes during an unit of time, and let \( u \) be the terminal velocity of any particular body. The theorems for perpendicular ascents give us \( g = \frac{u^2}{2a} \); \( u \) and \( a \) being both numbers representing units of space; therefore, in the present case, we have \( r = \frac{u^2}{2a} \). For the whole resistance \( r \), or \( rM \), is supposed equal to the weight, or to \( gM \); and therefore \( r \) is equal to \( g \), \( \frac{u^2}{2a} \) and \( 2a = \frac{u^2}{g} \). There is a consideration which ought to have place here. A body descends in air, not by the whole of its weight, but by the excess of its weight above that of the air which it displaces. It descends by its specific gravity only as a stone does in water. Suppose a body 32 times heavier than air, it will be buoyed up by a force equal to \( \frac{1}{32} \) of its weight; and instead of acquiring the velocity of 32 feet in a second, it will only acquire a velocity of 31, even though it sustained no resistance from the inertia of the air. Let \( p \) be the weight of the body and \( \pi \) that of an equal bulk of air; the accelerative force of relative gravity on each particle will be \( g \times \frac{\pi}{p} \); and this relative accelerating force might be distinguished by another symbol \( y \). But in all cases in which we have any interest, and particularly in military projectiles, \( \pi \) is so small a quantity that it would be pedantic affectation to attend to it. It is much more than compensated when we make \( g = 32 \) feet instead of \( 32\pi \) which it should be.

Let \( e \) be the time of this ascent in opposition to gravity. The same theorems give us \( eu = 2a \); and since the resistance competent to this terminal velocity is equal to gravity, \( e \) will also be the time in which it would be extinguished by the uniform action of the resistance; for which reason we may call it the extinguishing time for this velocity. Let \( R \) and \( E \) mark the resistance and extinguishing time for the same body moving with the velocity \( r \).

Since the resistances are as the squares of the velocities, and the resistance to the velocity \( u \) is \( \frac{u^2}{2a} \), \( R \) will