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 \( \frac{1}{2000} \) of its weight, and a horizontal range of 4 miles makes only 4' of deviation from parallelism.

Let us therefore assume gravitation as equal and parallel. The errors arising from this assumption 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 or retarding force, according as it produces the deflection, 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 assume 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.

Corollaries

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 be 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 \( \frac{1}{3600} \) 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 two modes with precision, the fall of heavy bodies, in order to have determined the 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 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 those 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 H a, H b, H c, and their respective ordinates A a, B b, C c; and he measures the deflections by the changes made on the increments of the ordinates. Thus the increment of the ordinate A a, 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 BG, that is, it would have been EF; but in consequence of the deflection, it is only CF: therefore he takes EC 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, 1689. 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 1706, having been criticised 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 Fatio 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 Fatio 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 Hayée 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 take by J. Bernoulli, 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 as 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 Innocent Bernoulli published an elegant dissertation on this subject in the Leipzig Acts in 1713; in which he charges Newton (though with many protestations of 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 Newtonus, 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 ascribing this letter to this author. For, after praising John Bernoulli as summus geometra, notus- 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 first 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 $16 \times \frac{t^2}{s}$ feet, if we measure it by the space, or the velocity $32 \times \frac{t}{s}$ 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 formulae, which are of easy recollection, and will serve, without tables, to answer all questions relative to projectiles.

I. $v = 8\sqrt{\frac{h}{s}}$, $= 8 \times \frac{t^2}{s} = 32t$

II. $t = \frac{\sqrt{h}}{4} = \frac{v}{32}$

III. $\sqrt{h} = \frac{v}{8} = 4t$

IV. $h = \frac{v^2}{64} = 16t^2$.

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. $h = 625$, $\sqrt{h} = 25$, $8 \sqrt{h} = 200$ 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.

In bodies projected upwards,

$$v = 56 \times \frac{56}{8} = 7, \quad \sqrt{h} \cdot 7 = h, = 49 \text{ feet.}$$

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 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 $z$ of falling through the height $296 + x$, and the velocity $v$ acquired by this fall. The time of describing the 296 feet will be $z - y$, 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 formulae will stand thus:

I. $v = \sqrt{2gh}$, i.e. $\sqrt{2g} \sqrt{h} = gt$,

II. $t = \frac{v}{\sqrt{2g}} = \frac{\sqrt{4h}}{2g} = \frac{\sqrt{2h}}{g}$

III. $h = \frac{v^2}{2g} = \frac{gt^2}{2}$.

In all these equations, gravity, or its accelerating power, is estimated, 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:

I. $v = 2\sqrt{\frac{gh}{s}} = 2gt$

II. $t = \sqrt{\frac{h}{g}} = \frac{v}{2g}$

IV. $h = \frac{v^2}{4g} = gt^2$, and $\sqrt{h} = \frac{v}{2\sqrt{g}}$.

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 \( \sqrt{2g} \), 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 BH is uniformly accelerated, we have \( BE : BK = T^2, BE : T^2, BK = BC^2 : BH^2 = EV^2 : KG^2 \); 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 d, 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 \( EV^2 = 4BE^2 = 4BE \times AB = BE \times 4AB \), 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 b, f g; let the points a, f, continually approach A and d, and ultimately coincide with them. It is evident that B b will ultimately be to G, 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 B b is to G G 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, BH is to TG 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 dG.

These few simple propositions contain all the theory of the motion of projectiles in vacuo, or independent bolic theory on the resistance of the air; and being a very easy and 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, flies 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 masters of the language of the art.

Let OB (fig. 3.) be a vertical line. About the Fig. 3. 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-ellipses 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 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 lie 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 AFB is bisected by BC, and therefore the focus lies in the line BP; 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 θ, λ, ν. Then it is plain that λ is half of θ, and ν is half of Cθ; 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 O of the parabola APS in z, and the directrix L of the parabola BVM in δ, then PB = Pz; but BF = BA, = AO, = zδ: 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 zPB; 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 extraordinary 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 float point blank (pointer 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 poit, 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 him 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 carcases are therefore thrown upwards, so as to get over the defences and produce their effect.

These shells are of very great weight, frequently exceeding 200 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.

Fig. 4. 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 of the range r.

It is evident that \( AZ : AD = S, ADZ : S, AZD = S, DBA : S, DAB = S, p : S, e \)

And \( AD : DB = S, DBA : S, DAB = S, p : S, e \)

And \( DB : AB = S, DAB : S, ADB = S, e : S, z \)

Therefore \( AZ : AB = S^2, p : S^2, e \times S, z ; = S^2, p : S, e \times S, z \)

Or \( 4h : r = S^2, p : S, e \times S, z \), and \( 4h \times S, e \times S, z = r \times S^2, p \)

Hence we obtain the relations wanted.

Thus \( h = \frac{r \times S^2, p}{4S, e \times S, z} \) and \( r = \frac{4h \times S, e \times S, z}{S^2, p} \)

And \( S, z = r \times S^2, p \) and \( S, e = \frac{r \times S^2, p}{4h \times S, z} \)

The only other circumstance in which we are interested is the time of the flight. A knowledge of this is unnecessary for the bombardier, that he may cut the fuzes late of his shells to such lengths as that they may burst at the instant of their hitting the mark.

Now \( AB : DB = Sin, ADB : Sin, DAB = S, z \)

\( S, e \), 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: \( DB = \frac{1}{2} : t^{1/2} \). Hence \( t^{1/2} = \frac{r \times S}{16 S_{1/2}} \), and we have the following easy rule.

From the sum of the logarithms of the range, and of the sine of elevation, subtract the sum of the logarithms of 16, and of the sine 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 fuzes; 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 448 yards at 9°. 45°. 33°

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 90312, or 17½ 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 artilleryman. 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 effay 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 artilleryman. 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 causes of the causes of such a prodigious deviation from a well-established 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. Whatever motion it acquires must be taken from the bullet. 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 4½ inches in diameter, moving with the velocity of 25 feet in a second, sustained a resistance of 315 grains, nearly ¼ 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 \( R \), 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 \( r'M \), 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 \( g \times 1 - \frac{\pi}{p} \); 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, \( \frac{\pi}{p} \) 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 \( 32 \sqrt{\frac{\pi}{p}} \) which it should be.

Let \( e \) be the time of this ascent in opposition to gravity. The same theorems give us \( e = \frac{u^2}{2a} \); 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 \( u \).

Since the resistance are as the squares of the velocities, and the resistance to the velocity \( u \) is \( \frac{u^2}{2a} \), \( R \) will be \( \frac{1}{2a} \). Moreover, the times in which the same velocity will be extinguished by different forces, acting uniformly, are inversely as the forces, and gravity would extinguish the velocity \( v \) in the time \( \frac{1}{g} \) (in these measures) to \( \frac{1}{u^2} = \frac{2a}{u^2} \). Therefore we have the following proportion \( \frac{1}{2a} (= R) : \frac{u^2}{2a} (= g) = \frac{2a}{u^2} : 2a \), and \( 2a \) is equal to \( E \), the time in which the velocity \( v \) will be extinguished by the uniform action of the resistance competent to this velocity.

The velocity \( v \) would in this case be extinguished after a motion uniformly retarded, in which the space described is one-half of what would be uniformly described during the same time with the constant velocity \( v \). Therefore the space thus described by a motion which begins with the velocity \( v \), and is uniformly retarded by the resistance competent to this velocity, is equal to the height through which this body must fall in vacuo in order to acquire its terminal velocity in air.

All these circumstances may be conceived in a manner which, to some readers, will be more familiar and palpable. The terminal velocity is that where the resistance of the air balances and is equal to the weight of the body. The resistance of the air to any particular body is as the square of the velocity; therefore let \( R \) be the whole resistance to the body moving with the velocity \( v \), and \( r \) the resistance to its motion with the terminal velocity \( u \); we must have \( R = R \times u^2 \), and this must be \( W \) the weight. Therefore, to obtain the terminal velocity, divide the weight by the resistance to the velocity \( v \), and the quotient is the square of the terminal velocity, or \( \frac{W}{R} = u^2 \): And this is a very expeditious method of determining it, if \( R \) be previously known.

Then the common theorems give \( a \), the fall necessary for producing this velocity in vacuo \( = \frac{u^2}{2g} \), and the time of the fall \( = \frac{u}{g} = e \), and \( eu = 2a \), the space uniformly described with the velocity \( u \) during the time of the fall, or its equal, the time of the extinction by the uniform action of the resistance \( r \); and, since \( r \) extinguishes it in the time \( e \), \( R \), which is \( u^2 \) times smaller, will extinguish it in the time \( u^2e \), and \( R \) will extinguish the velocity \( v \), which is \( u \) times less than \( u \), in the time \( ue \), that is, in the time \( 2a \); and the body, moving uniformly during the time \( 2a \), \( = E \), with the velocity \( v \), will describe the space \( 2a \); and, if the body begin to move with the velocity \( v \), and be uniformly opposed by the resistance \( R \), it will be brought to rest when it has described the space \( a \); and the space in which the resistance to the velocity \( v \) will extinguish that velocity by its uniform action, is equal to the height through which that body must fall in vacuo in order to acquire its terminal velocity in air. And thus every thing is regulated by the time \( E \) in which the velocity \( v \) is extinguished by the uniform action of the corresponding resistance, or by \( 2a \), which is the space uniformly described during this time, with the velocity \( v \). And \( E \) and \( 2a \) must be expressed by the same number. It is a number of units, of time, or of length.

Having ascertained these leading circumstances for an unit of velocity, weight, and bulk, we proceed to deduce the similar circumstances for any other magnitude; and, to avoid unnecessary complications, we shall always suppose the bodies to be spheres, differing only in diameter and density.

First, then, let the velocity be increased in the ratio of \( 1 \) to \( v \).

The resistance will now be \( \frac{v^2}{2a} = r \).

The extinguishing time will be \( \frac{E}{v} = e \), \( = \frac{2a}{v} \), and \( ev = 2a \); so that the rule is general, that the space along which any velocity will be extinguished by the uniform action of the corresponding resistance, is equal to the height necessary for communicating the terminal velocity to that body by gravity. For \( ev \) twice the space through which the body moves while the velocity \( v \) is extinguished by the uniform resistance.

In the 2d place, let the diameter increase in the proportion of \( 1 \) to \( d \). The aggregate of the resistance changes in the proportion of the surface similarly reflected, that is, in the proportion of \( 1 \) to \( d^2 \). But the quantity of matter, or number of particles among which this resistance is to be distributed, changes in the proportion of \( 1 \) to \( d^3 \). Therefore the retarding power of the resistance changes in the proportion of \( 1 \) to \( \frac{1}{d} \). When the diameter was \( 1 \), the resistance to a velocity \( v \) was \( \frac{1}{2a} \). It must now be \( \frac{1}{2ad} \). The time in which this diminished resistance will extinguish the velocity \( v \) must increase in the proportion of the diminution of force, and must now be \( Ed \), or \( 2ad \), and the space uniformly described during this time with the initial velocity \( v \) must be \( 2ad \); and this must still be twice the height necessary for communicating the terminal velocity \( w \) to this body. We must still have \( g = \frac{w^2}{2ad} \); and therefore \( w^2 = 2gad \), and \( w = \sqrt{2gad} = \sqrt{2gad} \).

But \( u = \sqrt{2ga} \). Therefore the terminal velocity \( w \) for this body is \( = u' \sqrt{\frac{d}{n}} \); and the height necessary for communicating it is \( ad \). Therefore the terminal velocity varies in the subduplicate ratio of the diameter of the ball, and the fall necessary for producing it varies in the simple ratio of the diameter. The extinguishing time for the velocity \( v \) must now be \( \frac{Ed}{v} \).

If, in the 3d place, the density of the ball be increased in the proportion of \( 1 \) to \( m \), the number of particles among which the resistance is to be distributed is increased in the same proportion, and therefore the retarding force of the resistance is equally diminished; and if the density of the air is increased in the proportion of \( 1 \) to \( n \), the retarding force of the resistance increases in the same proportion: hence we easily deduce these general expressions.

The terminal velocity \( = a \sqrt{\frac{dm}{n}} = \sqrt{2gadm} \).

The producing fall in vacuo \( = ad \frac{m}{n} \). The retarding power of resistance to any velocity \( r' = \frac{v^2}{2ad} \).

The extinguishing time for any velocity \( v = \frac{Edm}{vn} \).

And thus we see that the chief circumstances are regulated by the terminal velocity, or are conveniently referred to it.

To render the deductions from these premises perspicuous, and for communicating distinct notions or ideas, it will be proper to assume some convenient units, by which all these quantities may be measured; and, as this subject is chiefly interesting in the case of military projectiles, we shall adapt our units to this purpose. Therefore, let a second be the unit of time, a foot the unit of space and velocity, an inch the unit of diameter of a ball or shell, and a pound avoirdupois the unit of pressure, whether of weight or of resistance; therefore \( g \) is 32 feet.

The great difficulty is to procure an absolute measure of \( r \), or \( u \); any one of these will determine the others.

Sir Isaac Newton has attempted to determine \( r \) by theory, and employs a great part of the second book of the Principia in demonstrating, that the resistance to a sphere moving with any velocity is to the force which would generate or destroy its whole motion in the time that it would uniformly move over \( \frac{1}{4} \) of its diameter with this velocity as the density of the air is to the density of the sphere. This is equivalent to demonstrating that the resistance of the air to a sphere moving through it with any velocity, is equal to half the weight of a column of air having a great circle of the sphere for its base, and for its altitude the height from which a body must fall in vacuo to acquire this velocity. This appears from Newton's demonstration; for, let the specific gravity of the air be to that of the ball as 1 to \( m \); then, because the times in which the same velocity will be extinguished by the uniform action of different forces are inversely as the forces, the resistance to this velocity would extinguish it in the time of describing \( \frac{1}{4} md \), \( d \) being the diameter of the ball. Now \( r \) is to \( m \) as the weight of the displaced air to the weight of the ball, or as \( \frac{1}{4} \) of the diameter of the ball to the length of a column of air of equal weight. Call this length \( a \); \( a \) is therefore equal to \( \frac{1}{4} md \). Suppose the ball to fall from the height \( a \) in the time \( t \), and acquire the velocity \( u \). If it moved uniformly with this velocity during this time, it would describe a space \( = 2at \), or \( \frac{1}{4} md \). Now its weight would extinguish this velocity, or destroy this motion, in the same time, that is, in the time of describing \( \frac{1}{4} md \); but the resistance of the air would do this in the time of describing \( \frac{1}{4} md \); that is, in twice the time, The resistance therefore is equal to half the weight of the ball, or to half the weight of the column of air whose height is the height producing the velocity. But the resistances to different velocities are as the squares of the velocities, and therefore, as their producing heights; and, in general, the resistance of the air to a sphere moving with any velocity, is equal to the half weight of a column of air of equal section, and whose altitude is the height producing the velocity. The result of this investigation has been acquiesced in by all Sir Isaac Newton's commentators. Many faults have indeed been found with his reasoning, and even with his principles; and it must be acknowledged that His result although this investigation is by far the most ingenious just, but of any in the Principia, and sets his acuteness and adroitness in the most conspicuous light, his reasoning is liable to serious objections, which his most ingenious commentators have not completely removed. However, the conclusion has been acquiesced in, as we have already stated, but as if derived from other principles, or by more logical reasoning. We cannot, however, say that the reasonings or assumptions of these mathematicians are much better than Newton's; and we must add, that all the causes of deviation from the duplicate ratio of the velocities, and the causes of increased resistance, which the later authors have valued themselves for discovering and introducing into their investigations, were pointed out by Sir Isaac Newton, but purposely omitted by him, in order to facilitate the discussion in re difficilima. (See Schol. prop. 37. book ii.).

It is known that the weight of a cubic foot of water is \( \frac{62}{8} \) pounds, and that the medium density of the air is \( \frac{8}{845} \) of water; therefore, let \( a \) be the height producing the velocity (in feet), and \( d \) the diameter of the ball (in inches), and \( \pi \) the periphery of a circle whose diameter is \( r \); the resistance of the air will be \( \frac{62}{845} \times \frac{\pi}{4} \times \frac{1}{144} \times \frac{a}{2} \times d^2 = \frac{a d^2}{49284} \) pounds, very nearly, \( = \frac{v^2}{49284} \times \frac{d^2}{64} = \frac{v^2}{315417} \) pounds.

We may take an example. A ball of cast iron weighing 12 pounds, is \( 4\frac{1}{2} \) inches in diameter. Suppose this ball to move at the rate of \( 25\frac{1}{8} \) feet in a second (the reason of this choice will appear afterwards). The height which will produce this velocity in a falling body is \( 9\frac{1}{2} \) feet. The area of its great circle is \( 0.11044 \) feet, or \( \frac{1044}{10000} \) of one foot. Suppose water to be 840 times heavier than air, the weight of the air incumbent on this great circle, and \( 9\frac{1}{2} \) feet high, is \( 0.081151 \) pounds; half of this is \( 0.0405755 \) or \( \frac{405755}{8000000} \), or nearly \( \frac{1}{20} \) of a pound. This should be the resistance of the air to this motion of the ball.

In all matters of physical discussion, it is prudent to confront every theoretical conclusion with experiment. This is particularly necessary in the present instance, because the theory on which this proposition is founded is extremely uncertain. Newton speaks of it with the most cautious diffidence, and secures the justness of the conclusions by the conditions which he assumes in his investigation. He describes with the greatest precision the state of the fluid in which the body must move, so as that the demonstrations may be strict, and leaves it to others to pronounce, whether this is the real constitution of our atmosphere. It must be granted that it is not; and that many other suppositions have been introduced by his commentators and followers, in order to fuit his investigation (for we must assert that little or nothing has been added to it) to the circumstances of the case.

Newton himself, therefore, attempted to compare his propositions with experiment. Some were made by dropping balls from the dome of St Paul's cathedral; and all these showed as great a coincidence with his theory as they did with each other; but the irregularities... ties were too great to allow him to say with precision what was the resistance. It appeared to follow the proportion of the squares of the velocities with sufficient exactness; and though he could not say that the resistance was equal to the weight of the column of air having the height necessary for communicating the velocity, it was always equal to a determinate part of it; and might be stated \(=n \cdot a\), \(n\) being a number to be fixed by numerous experiments.

One great source of uncertainty in his experiments seems to have escaped his observation: the air in that dome is almost always in a state of motion. In the summer season there is a very sensible current of air downwards, and frequently in winter it is upwards; and this current bears a very great proportion to the velocity of the defecates. Sir Isaac takes no notice of this.

He made another set of experiments with pendulums; and has pointed out some very curious and unexpected circumstances of their motions in a resisting medium. There is hardly any part of his noble work in which his address, his patience, and his astonishing penetration, appear in greater lustre. It requires the utmost intensities of thought to follow him in these disquisitions; and we cannot enter on the subject at present: some notice will be taken of these experiments in the article RESISTANCE of Fluids. Their results were much more uniform, and confirmed his general theory; and, as we have said above, it has been acquiesced in by the first mathematicians of Europe.

But the deductions from this theory were so inconsistent with the observed motions of military projectiles, when the velocities are prodigious, that no application could be made which could be of any service for determining the path and motion of cannon shot and bombs; and although Mr John Bernoulli gave, in 1718, a most elegant determination of the trajectory and motion of a body projected in a fluid which resists in the duplicate ratio of the velocities (a problem which even Newton did not attempt), it has remained a dead letter. Mr Benjamin Robins, equally eminent for physical science and mathematical genius, was the first who suspected the true cause of the imperfection of the usually received theories; and in 1737 he published a small tract, in which he showed clearly, that even the Newtonian theory of resistance must cause a cannon ball, discharged with a full allotment of powder, to deviate farther from the parabola, in which it would move in vacuo, than the parabola deviates from a straight line. But he farther asserted, on the authority of good reasoning, that in such great velocities the resistance must be much greater than this theory assigns; because, besides the resistance arising from the inertia of the air which is put in motion by the ball, there must be a resistance arising from a condensation of the air on the anterior surface of the ball, and a rarefaction behind it: and there must be a third resistance, arising from the statical pressure of the air on its anterior part, when the motion is so swift that there is a vacuum behind. Even these causes of disagreement with the theory had been foreseen and mentioned by Newton (see the Scholium to prop. 37, book ii. Principi;) but the subject seems to have been little attended to. The eminent mathematicians had few opportunities of making experiments; and the professional men, who were in the service of princes, and had their countenance and aid in this matter, were generally too deficient in mathematical knowledge to make a proper use of their opportunities. The numerous and splendid volumes which these gentlemen have been enabled to publish by the patronage of sovereigns are little more than prolix extensions of the simple theory of Galileo. Some of them, however, such as St Remy, Antonini, and Le Blond, have given most valuable collections of experiments, ready for the use of the profound mathematician.

Two or three years after this first publication, Mr Observations of Mr Robins hit upon that ingenious method of measuring the great velocities of military projectiles, which has handed down his name to posterity with great honour, and result. And having ascertained these velocities, he discovered once, the prodigious resistance of the air, by observing the diminution of velocity which it occasioned. This made him anxious to examine what was the real resistance to any velocity whatever, in order to ascertain what was the law of its variation; and he was equally fortunate in this attempt. His method of measuring the resistance has been fully described in the article GUNNERY, No 9, &c.

It appears (Robins's Math. Works, vol. i. page 205,) that a sphere of 4½ inches in diameter, moving at the rate of 25½ feet in a second, sustained a resistance of 0.04914 pounds, or \( \frac{4}{8} \) of a pound. This is a greater resistance than that of the Newtonian theory, which gave \( \frac{4}{8} \) in the proportion of 1000 to 1211, or very nearly in the proportion of five to six in small numbers. And we may adopt as a rule in all moderate velocities, that the resistance to a sphere is equal to \( \frac{6}{100} \) of the weight of a column of air having the great circle of the sphere for its base, and for its altitude the height through which a heavy body must fall in vacuo to acquire the velocity of projection.

This experiment is peculiarly valuable, because the ball is precisely the size of a 12 pound shot of cast iron; and its accuracy may be depended on. There is but one source of error. The whirling motion must have occasioned some whirl in the air, which would continue till the ball again passed through the same point of its revolution. The resistance observed is therefore probably somewhat less than the true resistance to the velocity of 25½ feet, because it was exerted in a relative velocity which was less than this, and is, in fact, the resistance competent to this relative and smaller velocity.

Accordingly, Mr Smeaton, a most sagacious naturalist, places great confidence in the observations of a De Borda, Mr Roufe of Leicestershire, who measured the resistance by the effect of the wind on a plane properly exposed to it. He does not tell us in what way the velocity of the wind was ascertained; but our deference for his great penetration and experience disposes us to believe that this point was well determined. The resistance observed by Mr Roufe exceeds that resulting from Mr widely in Robins's experiments nearly in the proportion of 7 to 10, their conclusions. Chevalier de Borda made experiments similar to those of Mr Robins, and his results exceed those of Robins in the proportion of 5 to 6. These differences are so considerable, that we are at a loss what measure to abide by. It is much to be regretted, that in a subject so interesting both to the philosopher and the man of the world, experiments have not been multiplied. Nothing would tend so much to perfect the science of gunnery; and indeed till this be done, all the labours of mathematicians are of no avail. Their investigations must remain an unintelligible cipher, till this key be supplied. It is to be hoped that Dr Charles Hutton of Woolwich, who has so ably extended Mr Robins's Examination of the Initial Velocities of Military Projectiles, will be encouraged to proceed to this part of this subject. We should wish to see, in the first place, a numerous set of experiments for ascertaining the resistances in moderate velocities; and, in order to avoid all error from the resistance and inertia of the machine, which is necessarily blended with the resistance of the ball, in Mr Robins's form of the experiment, and is separated with great uncertainty and risk of error, we would recommend a form of experiment somewhat different.

Let the axis and arm which carries the ball be connected with wheelwork, by which it can be put into motion, and gradually accelerated. Let the ball be so connected with a bent spring, that this shall gradually compress it as the resistance increases, and leave a mark of the degree of compression; and let all this part of the apparatus be screened from the air except the ball. The velocity will be determined precisely by the revolutions of the arm, and the resistance by the compression of the spring. The best method would be to let this part of the apparatus be made to slide along the revolving arm, so that the ball can be made to describe larger and larger circles. An intelligent mechanician will easily contrive an apparatus of this kind, held at any distance from the axis by a cord, which passes over a pulley in the axis itself, and is then brought along a perforation in the axis, and comes out at its extremity, where it is fitted with a swivel, to prevent it from snapping by being twisted. Now let the machine be put into motion. The centrifugal force of the ball and apparatus will cause it to fly out as far as it is allowed by the cord; and if the whole is put in motion by connecting it with some mill, the velocity may be most accurately ascertained. It may also be fitted with a bell and hammer like Gravefane's machine for measuring centrifugal forces. Now by gradually veering off more cord, the distance from the centre, and consequently the velocity and resistance increase, till the hammer is disengaged and strikes the bell.

Another great advantage of this form of the experiment is, that the resistance to very great velocities may be thus examined, which was impossible in Mr Robins's way. This is the great desideratum, that we may learn in what proportion of the velocities the resistances increase.

In the same manner, an apparatus, consisting of Dr Lynd's Anemometer, described in the article PNEUMATICS, No 311, &c. might be whirled round with prodigious rapidity, and the fluid on it might be made clammy, which would leave a mark at its greatest elevation, and thus discover the resistance of the air to rapid motions.

Nay, we are of opinion that the resistance to very rapid motions may be measured directly in the conduit pipe of some of the great cylinder bellows employed in blast furnaces: the velocity of the air in this pipe is ascertained by the capacity of the cylinder and the strokes of the piston. We think it our duty to point out to such as have the opportunities of trying them methods which promise accurate results for ascertaining this most desirable point.

We are the more puzzled what measure to abide by, The result because Mr Robins himself, in his Practical Propo- of Robins's tions, does not make use of the result of his own experiments, but takes a much lower measure. We must yet most content ourselves, however, with this experimental measure to be definite, because it is as yet the only one of which any account can be given, or well-founded opinion formed.

Therefore, in order to apply our formulae, we must apply to reduce this experiment, which was made on a ball of the formula $4\frac{1}{2}$ inches diameter, moving with the velocity of $25\frac{1}{2}$ feet per second, to what would be the resistance to a ball of one inch, having the velocity 1 foot. This will evidently give us $R = \frac{0.04914}{4\frac{1}{2} \times 25\frac{1}{2}}$, being diminished in the duplicate ratio of the diameter and velocity. This gives us $R = 0.0000381973$ pounds, or $\frac{3.81973}{1000000}$ of a pound. The logarithm is 4.58204. The resistance here determined is the same whatever substance the ball be of; but the retardation occasioned by it will depend on the proportion of the resistance to the vis inertia of the ball; that is, to its quantity of motion. This in similar velocities and diameters is as the density of the ball. The balls used in military service are of cast iron or of lead, whose specific gravities are 7.207 and 11.37 nearly, water being 1. There is considerable variety in cast iron, and this density is about the medium. These data will give us

| For Iron | For Lead | |----------|----------| | W, or weight of a ball 1 inch in diameter | lbs. 0.13048 | 0.21533 | | Log. of W | 9.13509 | 9.33310 | | E'' | 116"6 | 176"6 | | Log. of E | 3.04790 | 3.24591 | | u, or terminal velocity | 189.03 | 237.43 | | Log. u | 2.27653 | 2.37553 | | a, or producing height | 558.3 | 880.8 |

These numbers are of frequent use in all questions on this subject.

Mr Robins gives an expeditious rule for readily finding $a$, which he calls F (see the article GUNNERY), by which it is made 900 feet for a cast iron ball of an inch diameter. But no theory of resistance which he professes to use will make this height necessary for producing the terminal velocity. His F therefore is an empirical quantity, analogous indeed to the producing height, but accommodated to his theory of the trajectory of cannon-shot, which he promised to publish, but did not live to execute. We need not be very anxious about this; for all our quantities change in the same proportion with $R$, and need only a correction by a multiplier or divisor, when $R$ shall be accurately established.

We may illustrate the use of these formulae by an example or two.

1. Then, to find the resistance to a 24 pound ball Examples moving with the velocity of 1670 feet in a second, of their use, which is nearly the velocity communicated by 16lbs. of powder. The diameter is 5.603 inches. But it is found, by unequivocal experiments on the retardation of such a motion, that it is 504 lbs. This is owing to the causes often mentioned, the additional resistance to great velocities, arising from the condensation of the air, and from its pressure into the vacuum left by the ball.

2. Required the terminal velocity of this ball?

Log. R = +4.58204 Log. \(d^2\) = +1.49674

Log. resift. to veloc. \(r\) = 6.07878 Log. W = 1.38021 Diff. of \(a\) and \(b\) = log. \(u^2\) = 5.30143 Log. 447.4 = \(u\) = 2.65071

As the terminal velocity \(u\), and its producing height \(a\), enter into all computations of military projectiles, we have inserted the following Table for the usual sizes of cannon-shot, computed both by the Newtonian theory of resistance, and by the resistances observed in Robins's experiments.

| No. Ball | Term. Vel. | \(a\) | Term. Vel. | \(a\) | Diam. Inch. | |----------|-----------|------|-----------|------|-------------| | 1 | 289.9 | 2626.4 | 263.4 | 2168.6 | 1.94 | | 2 | 324.9 | 3298.5 | 295.2 | 2723.5 | 2.45 | | 3 | 348.2 | 3788.2 | 316.4 | 3127.9 | 2.80 | | 4 | 365.3 | 4170.3 | 331.9 | 3442.6 | 3.08 | | 6 | 390.8 | 4472.7 | 355.1 | 3940.7 | 3.52 | | 9 | 418.1 | 5463.5 | 379.9 | 4511.2 | 4.04 | | 12 | 438.6 | 6010.6 | 398.5 | 4962.9 | 4.45 | | 18 | 469.3 | 6883.3 | 426.5 | 5683.5 | 5.09 | | 24 | 492.4 | 7576.3 | 447.4 | 6255.7 | 5.61 | | 32 | 512.6 | 8024.8 | 465.8 | 6780.4 | 6.21 | | | 540.5 | 9129.9 | 491.5 | 7538.3 | 6.75 |

Mr Muller, in his writings on this subject, gives a much smaller measure of resistance, and consequently a much greater terminal velocity; but his theory is a mistake from beginning to end (See his Supplement to his Treatise of Artillery, art. 150, &c.). In art. 148, he affirms an algebraic expression for a principle of mechanical argument; and from its consequence draws erroneous conclusions. He makes the resistance of a cylinder one third less than Newton supposes it; and his reason is false. Newton's measure is demonstrated by his commentators Le Seur and Jaquier to be even a little too small, upon his own principles, (Not. 277, Prop. 36, B. II.). Mr Muller then, without any seeming reason, introduces a new principle, which he makes the chief support of his theory, in opposition to the theories of other mathematicians. The principle is false, and even absurd, as we shall have occasion to show by and by. In consequence, however, of this principle, he is enabled to compare the results with many experiments, and the agreement is very flattering. But we shall soon see that little dependence can be had on such comparisons. We notice these things here, because Mr Muller being head of the artillery school in Britain, his publications have become a sort of text-books. We are miserably deficient in works on this subject, and must have recourse to the foreign writers.

We now proceed to consider these motions through their whole course; and we shall first consider them actions concerned by the resistance only; then we shall consider the perpendicular ascents and descents of heavy bodies through the air; and, lastly, their motion in a curvilinear trajectory, when projected obliquely. This must be done by the help of the abstruse parts of fluxionary mathematics. To make it more perspicuous, we shall, by way of introduction, consider the simply resisted rectilinear motions geometrically, in the manner of Sir Isaac Newton. As we advance, we shall quit this track, and prosecute it algebraically, having by this time acquired distinct ideas of the algebraic quantities.

We must keep in mind the fundamental theorems of varied motions.

1. The momentary variation of the velocity is proportional to the force and the moment of time jointly, and may therefore be represented by \(\frac{dv}{dt} = f\), where \(v\) is the momentary increment or decrement of the velocity \(v\), \(f\) the accelerating or retarding force, and \(t\) the moment or increment of the time \(t\).

2. The momentary variation of the square of the velocity is as the force, and as the increment or decrement of the space jointly; and may be represented by \(\frac{dv}{dt} = \frac{f}{v}\). The first proposition is familiarly known. The second is the 39th of Newton's Principia, B. I. It is demonstrated in the article Optics, and is the most extensively useful proposition in mechanics.

These things being premised, let the straight line AC (fig. 5) represent the initial velocity \(V\), and let CO, perpendicular to AC, be the time in which this velocity would be extinguished by the uniform action of the resistance. Draw through the point A an equilateral hyperbola \(AeB\), having OF, OCD for its asymptotes; then let the time of the resisted motion be represented by the line CB, C being the first instant of the motion. If there be drawn perpendicular ordinates \(x, y, z, DB, &c.\) to the hyperbola, they will be proportional to the velocities of the body at the instants \(x, y, z, &c.\) and the hyperbolic areas \(AC \times e, AC \times f, ACDB, &c.\) will be proportional to the spaces described during the times \(C \times x, C \times y, CB, &c.\)

For, suppose the time divided into an indefinite number of small and equal moments, \(C \times c, D \times d, &c.\) draw the ordinates \(a \times c, b \times d\), and the perpendiculars \(b \times a, a \times c\). Then, by the nature of the hyperbola, \(AC : ac = OC : OC\); and \(AC - ac : ac = OC - OC : OC\), that is, \(Ae : ac = Ce : OC\), and \(Ae : ac = Ce : OC = AC : ac = AC : OC\); in like manner, \(B \times b : D \times d = BD : bD : BD : OD\). Now \(D \times D = C \times c\), because the moments of time were taken equal, and the rectangles \(AC \times CO, BD \times DO\), are equal, by the nature of the hyperbola; therefore \(Ae : B \times b = AC : ac : BD : bD\); but as the points \(c, d\) continually approach, and ultimately coincide with \(C, D\), the ultimate ratio of \(AC : ac\) to \(BD : bD\) is that of \(AC^2\) to \(BD^2\); therefore the momentary decrements of \(AC\): AC and BD are as AC^2 and BD^2. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC; therefore the velocities at all the other instants \(x, g, D\), are properly represented by the corresponding ordinates. Hence,

1. Since the abscissae of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and AC \(x\) is to A \(c g f\) as the space described in the time \(C x\) to the space described in the time \(C g\) (1st Theorem on varied motions).

2. The rectangle ACOF is to the area ACDB as the space formerly expressed by \(2a\), or E to the space described in the resisting medium during the time CD: for AC being the velocity V, and OC the extinguishing time e, this rectangle is \(=eV\), or E, or \(2a\), of our former disquisitions; and because all the rectangles, such as ACOF, BDOG, &c., are equal, this corresponds with our former observation, that the space uniformly described with any velocity during the time in which it would be uniformly extinguished by the corresponding resistance is a constant quantity, viz. that in which we always had \(v = E\), or \(2a\).

3. Draw the tangent \(A z\); then, by the hyperbola \(C x = CO\): now \(C x\) is the time in which the resistance to the velocity AC would extinguish it; for the tangent coinciding with the elemental arc \(A a\) of the curve, the first impulse of the uniform action of the resistance is the same with the first impulse of its varied action. By this the velocity AC is reduced to \(a c\). If this operated uniformly like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle \(AC x\).

This triangle, therefore, represents the height through which a heavy body must fall in vacuo, in order to acquire the terminal velocity.

4. The motion of a body resisted in the duplicate ratio of the velocity will continue without end, and a space will be described which is greater than any assignable space, and the velocity will grow less than any that can be assigned; for the hyperbola approaches continually to the asymptote, but never coincides with it. There is no velocity BD so small, but a smaller ZP will be found beyond it; and the hyperbolic space may be continued till it exceeds any surface that can be assigned.

5. The initial velocity AC is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion, is to the extinguishing time alone: for \(AC : BD = OD (\text{or } OC + CD) : OC\); or \(V : v = e : e + t\).

6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles AFOC, BDOG are equal; and therefore AVG F and BVCD are equal, and \(VC : VA = VG : VB\); therefore \(t = \frac{v}{v}\), and \(e = \frac{v}{v - v}\).

7. Any velocity is reduced in the proportion of \(m\) to \(n\) in the time \(e \frac{m-n}{n}\). For, let \(AC : BD = m : n\); then \(DO : CO = m : n\), and \(DC : CO = m - n : n\), and \(DC = \frac{m-n}{n} CO\), or \(t = \frac{m-n}{n}\). Therefore any velocity is reduced to one half in the time in which the initial resistance would have extinguished it by its uniform action.

Thus may the chief circumstances of this motion be determined by means of the hyperbola, the ordinates mode of and abscissa exhibiting the relations of the times and determining this velocities, and the areas exhibiting the relations of both motion, to the spaces described. But we may render the conception of these circumstances infinitely more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line ML, parallel to OD for its abscissa, and of such a nature, that if the ordinates to the hyperbola AC, \(x\), \(f g\), BD, &c. be produced till they cut this curve in L, p, n, K, &c. and the abscissa in L, s, h, d, &c. the ordinates \(p, h, n, K, &c.\) may be proportional to the hyperbolic areas \(e A C x, f A C g, d A c K\). Let us examine what kind of curve this will be.

Make OC : O = O : O ; then (Hamilton's Conics, IV. 14. Cor.), the areas AC \(x\), \(c x g f\) are equal; therefore drawing \(p s, n t\) perpendicular to OM, we shall have (by the assumed nature of the curve LpK), \(M s = s t\); and if the abscissa OD be divided into any number of small parts in geometrical progression (reckoning the commencement of them all from O), the axis Vi of this curve will be divided by its ordinates into the same number of equal parts; and this curve will have its ordinates LM, ps, nt, &c. in geometrical progression, and its abscissa in arithmetical progression.

Also, let KN, MV touch the curve in K and L, and let OC be supposed to be to Oc, as OD to Od, and therefore Ce to Dd as OC to OD; and let these lines Ce, Dd be indefinitely small; then (by the nature of the curve) Lo is equal to Kr; for the areas a AC c, b BD d are in this case equal. Also kv is to kr, as LM to Kl, because \(c C : a D = CO : DO\):

\[ \begin{align*} \text{Therefore } IN : IK &= r K : r k \\ IK : ML &= r k : o l \\ ML : MV &= o l : o L \\ \text{and } IN : MN &= r K : o L \end{align*} \]

That is, the subtangent IN, or MV, is of the same magnitude, or is a constant quantity in every part of the curve.

Lastly, the subtangent IN, corresponding to the point K of the curve, is to the ordinate Kd as the rectangle BDOG or ACOF to the parabolic area BDCA.

For let \(f g h n\) be an ordinate very near to BD \(k\); and let \(h n\) cut the curve in \(n\), and the ordinate Kl in q; then we have

\[ \begin{align*} Kq : qn &= KI : IN, \text{ or } \\ Dg : gn &= DO : IN; \\ \text{but } BD : AC &= CO : DO; \\ \text{therefore } BD : Dg : AC, qn &= CO : IN. \end{align*} \]

Therefore the sum of all the rectangles BD, Dg is to the sum of all the rectangles AC, qn, as CO to IN; but but the sum of the rectangles BD.Dg is the space ACDB; and, because AC is given, the sum of the rectangles AC.gz is the rectangle of AC and the sum of all the lines qz; that is, the rectangle of AC and RL; therefore the space ACDB : AC.RL = CO : IN, and ACDB.XN = AC.CO.RL; and therefore IN.RL = AC.CO.ACDB.

Hence it follows that QL expresses the area BVA, and in general, that the part of the line parallel to OM, which lies between the tangent KN and the curve LpK, expresses the corresponding area of the hyperbola which lies without the rectangle BDOG.

And now, by the help of this curve, we have an easy way of convincing and computing the motion of a body through the air. For the subtangent of our curve now represents twice the height through which the ball must fall in vacuo, in order to acquire the terminal velocity; and therefore serves for a scale on which to measure all the other representatives of the motion.

But it remains to make another observation on the curve LpK, which will save us all the trouble of graphical operations, and reduce the whole to a very simple arithmetical computation. It is of such a nature, that when MI is considered as the abscissa, and is divided into a number of equal parts, ordinates are drawn from the points of division, the ordinates are a series of lines in geometrical progression, or are continual proportions. Whatever is the ratio between the first and second ordinate, there is the same between the second and third, between the third and fourth, and so on; therefore the number of parts into which the abscissa is divided is the number of these equal ratios which is contained in the ratio of the first ordinate to the last: For this reason, this curve has got the name of the logistic or logarithmic curve; and it is of immense use in the modern mathematics, giving us the solution of many problems in the most simple and expeditious manner, on which the genius of the ancient mathematicians had been exercised in vain. Few of our readers are ignorant, that the numbers called logarithms are of equal utility in arithmetical operations, enabling us not only to solve common arithmetical problems with astonishing dispatch, but also to solve others which are quite inaccessible in any other way. Logarithms are nothing more than the numerical measures of the abscissa of this curve, corresponding to ordinates, which are measured on the same or any other scale by the natural numbers; that is, if ML be divided into equal parts, and from the points of division lines be drawn parallel to MI, cutting the curve LpK, and from the points of intersection ordinates be drawn to MI, these will divide MI into portions, which are in the same proportion to the ordinates that the logarithms bear to their natural numbers.

In constructing this curve we were limited to no particular length of the line LR, which represented the space ACDB; and all that we had to take care of was, that when OC, OZ, OG were taken in geometrical progression, MR, MT should be in arithmetical progression. The abscissa having ordinates equal to p, n, t, &c., might have been twice as long, as is shown in the dotted curve which is drawn through L. All the lines which serve to measure the hyperbolic spaces would then have been doubled. But NI would also have been doubled, and our proportions would have still held good; because this subtangent is the scale of measurement of our figure, as E or 2a is the scale of measurement for the motions.

Since then we have tables of logarithms calculated for every number, we may make use of them instead of this geometrical figure, which still requires considerable trouble to suit it to every case. There are two sets of logarithmic tables in common use. One is called a table of hyperbolic or natural logarithms. It is suited to such a curve as is drawn in the figure, where the subtangent is equal to that ordinate rv which corresponds to the side πO of the square πθλO inserted between the hyperbola and its asymptotes. This square is the unit of surface, by which the hyperbolic areas are expressed; its side is the unit of length, by which the lines belonging to the hyperbola are expressed; rv = 1, or the unit of numbers to which the logarithms are suited, and then IN is also 1. Now the square πθλO being unity, the area BACD will be some number; πO being also unity, OD is some number: Call it x. Then, by the nature of the hyperbola, OB : Oπ = πθ : DB: That is, x : 1 = 1 : x, so that DB = x.

Now calling D d x, the area BD db, which is the fluxion (ultimately) of the hyperbolic area, is x/x. Now in the curve LpK, MI has the same ratio to NI that BACD has to πθλO: Therefore, if there be a scale of which NI is the unit, the number on this scale corresponding to MI has the same ratio to 1 which the number measuring BACD has to 1; and i, which corresponds to BD db, is the fluxion (ultimately) of MI: Therefore, if MI be called the logarithm of x, x is properly represented by the fluxion of MI. In short, the line MI is divided precisely as the line of numbers on a Gunter's scale, which is therefore a line of logarithms; and the numbers called logarithms are just the lengths of the different parts of this line measured on a scale of equal parts. Therefore, when we meet with such an expression as x/x viz. the fluxion of a quantity divided by the quantity itself, we consider it as the fluxion of the logarithm of that quantity, because it is really so when the quantity is a number; and it is therefore strictly true that the fluent of x/x is the hyperbolic logarithm of x.

Certain reasons of convenience have given rise to another set of logarithms; these are suited to a logistic curve whose subtangent is only 414290 of the ordinate rv, which is equal to the side of the hyperbolic square, and which is assumed for the unit of number. We shall suit our applications of the preceding investigation to both these, and shall first use the common logarithms whose subtangent is 0.43429.

The whole subject will be best illustrated by taking an example of the different questions which may be proposed.

Recollect that the rectangle ACOF is = 2a, or z^2 or E. E, for a ball of cast-iron one inch diameter, and if it has the diameter \(d\), it is \(\frac{u^2d}{g}\), or \(2ad\), or \(Ed\).

I. It may be required to determine what will be the space described in a given time \(t\) by a ball setting out with a given velocity \(V\), and what will be its velocity \(v\) at the end of that time.

Here we have \(NI : MI = ACOF : BDC\); now \(NI\) is the subtangent of the logistic curve; \(MI\) is the difference between the logarithms of \(OD\) and \(OC\); that is, the difference between the logarithms of \(e + t\) and \(e\);

ACOF is \(2ad\), or \(\frac{u^2d}{g}\), or \(Ed\).

Therefore by common logarithms \(0.43429 : \log_e + t - \log_e = 2ad : S\), \(S\) space described,

or \(0.43429 : \log_e + t = 2ad : S\),

and \(S = \frac{2ad}{0.43429} \times \log_e + t\),

by hyperbolic logarithms \(S = 2ad \times \log_e + t\).

Let the ball be a 12 pounder, and the initial velocity be 1600 feet, and the time 20 seconds. We must first find \(e\), which is \(\frac{2ad}{V}\).

Therefore, \(\log_2a = +3.03236\)

\(\log_d(45) = +0.63321\)

\(\log_V(1600) = -3.20415\)

Log. of \(3^{1/3} = e = 0.48145\)

And \(e + t\) is \(23^{1/3}\), of which the log. is \(1.36229\)

from which take the log. of \(e = 0.48145\)

remains the log. of \(\frac{e + t}{e} = 0.88084\)

This must be considered as a common number by which we are to multiply \(\frac{2ad}{0.43429}\).

Therefore add the logarithms of \(2ad = +3.68557\)

\(\log_e + t = +9.94490\)

\(\log_0.43429 = -9.63778\)

Log. \(S = 9833\) feet \(= 3.99269\)

For the final velocity,

\(OD : OC = AC : BD\), or \(e + t : e = V : v\).

\(23^{1/3} : 3^{1/3} = 1600 : 210\), \(v = \frac{210}{1600} = 0.13125\).

The ball has therefore gone 3278 yards, and its velocity is reduced from 1600 to 210.

It may be agreeable to the reader to see the gradual progress of the ball during some seconds of its motion.

| T. | S. | Diff. | V. | Diff. | |----|----|-------|----|------| | 1" | 1383 | 1073 | 1203 | 397 | | 2" | 2456 | 880 | 964 | 160 | | 3" | 3336 | 804 | 804 | 114 | | 4" | 4080 | 744 | 690 | 86 | | 5" | 4725 | 645 | 604 | 67 | | 6" | 5294 | 537 | 537 | |

The first column is the time of the motion, the second is the space described, the third is the differences of the spaces, showing the motion during each successive second; the fourth column is the velocity at the end of the time \(t\); and the last column is the differences of velocity, showing its diminution in each successive second.

We see that at the distance of 1000 yards the velocity is reduced to one half, and at the distance of less than a mile it is reduced to one-third.

II. It may be required to determine the distance at which the initial velocity \(V\) is reduced to any other quantity \(v\). This question is solved in the very same manner, by substituting the logarithms of \(V\) and \(v\) for those of \(e + t\) and \(e\); for \(AC : BD = OD : OC\), and therefore \(\log_e + t = \log_e + t\).

Thus it is required to determine the distance in which the velocity 1780 of a 24 pound ball (which is the medium velocity of such a ball discharged with 16 pounds of powder) will be reduced to 1500.

Here \(d = 5.68\), and therefore the logarithm of \(2ad\) is \(+3.78671\)

\(\log_e + t = 0.07433\), of which the log is \(+8.87116\)

\(\log_0.43429 = -9.63778\)

Log. \(1047.3\) feet, or 349 yards \(= 3.02009\)

This reduction will be produced in about \(\frac{1}{3}\) of a second.

III. Another question may be to determine the time which a ball, beginning to move with a certain velocity, employs in passing over a given space, and the diminution of velocity which it sustains from the resistance of the air.

We may proceed thus:

\(2ad : S = 0.43429 : \log_e + t = t\). Then to log.

\(\frac{e + t}{e} + \log_e\), and we obtain \(\log_e + t\), and \(e + t\); from which if we take \(e\) we have \(t\). Then to find \(v\), say \(e + t : e = V : v\).

We shall conclude these examples by applying this Application last rule to Mr Robins's experiment on a musket bullet of an experiment of \(\frac{1}{4}\) of an inch in diameter, which had its velocity reduced from 1670 to 1425 by passing through 100 feet of air. This we do in order to discover the resistance which it sustained, and compare it with the resistance to a velocity of 1 foot per second.

We must first ascertain the first term of our analogy. The ball was of lead, and therefore \(2a\) must be multiplied by \(d\) and by \(m\), which expresses the ratio of the density of lead to that of cast iron. \(d = 0.75\), and \(m\) is

\(\frac{11.37}{7.21} = 1.577\). Therefore \(\log_2a = 3.03236\)

\(\frac{d}{m} = \frac{9.87506}{0.19782}\)

Log. \(2adm = 3.10524\)

and \(2adm = 1274.2\).

Now \(1274.2 : 100 = 0.43429 : 0.03408 = \log_e + t\)

But \(\frac{2adm}{V} = 0.763\), and its logarithm \(= 9.88252\),

which, added to \(0.03408\), gives \(9.91660\), which is the log. of \(e + t = 0.825\), from which take \(e\), and there remains followed the proportion of the hyperbolic areas, we showed the nature of another curve, where lines could be found which increase in the very same manner as the path of the projectile increases; so that a point describing the abscissa MI of this curve moves precisely as the projectile does. Then, discovering that this line is the same with the line of logarithms on a Gunter's scale, we showed how the logarithm of a number really represents the path or space described by the projectile.

Having thus, we hope, enabled the reader to conceive distinctly the quantities employed, we shall leave the geometrical method, and prosecute the rest of the subject in a more compendious manner.

We are, in the next place, to consider the perpendicular ascents and descents of heavy projectiles, where the resistance of the air is combined with the action of gravity; and we shall begin with the descents.

Let \( u \), as before, be the terminal velocity, and \( g \) the accelerating power of gravity: When the body moves with the velocity \( u \), the resistance is equal to \( g \); and in every other velocity \( v \), we must have \( u^2 : v^2 = g : \frac{g}{u^2} \), for the resistance to that velocity. In the descent the body is urged by gravity \( g \), and opposed by the resistance \( \frac{g}{u^2} \): therefore the remaining accelerating force, which we shall call \( f \), is \( g - \frac{g}{u^2} \), or \( \frac{g}{u^2} \).

Now the fundamental theorem for varied motions is

\[ f = u \cdot \dot{v}, \quad s = \frac{u \cdot \dot{v}}{f}, \quad \frac{u^2}{g} \times \frac{\dot{v}}{u^2 - v^2}, \quad \text{and} \quad s = \frac{u^2}{g} \times \frac{\dot{v}}{u^2 - v^2} + C. \]

Now the fluent of \( \frac{u \cdot \dot{v}}{u^2 - v^2} \) is

\[ \sqrt{u^2 - v^2} + C. \]

For the fluxion of \( \sqrt{u^2 - v^2} \) divided by the quantity \( \sqrt{u^2 - v^2} \), of which it is the fluxion, gives precisely \( \frac{u \cdot \dot{v}}{u^2 - v^2} \), which is therefore the fluxion of its hyperbolic logarithm. Therefore \( S = \frac{u^2}{g} \times L \sqrt{u^2 - v^2} + C \). Where \( L \) means the hyperbolic logarithm of the quantity annexed to it, and \( \lambda \) may be used to express its common logarithm. (See article Fluxions).

The constant quantity \( C \) for completing the fluent is determined from this consideration, that the space described is \( o \), when the velocity is \( o \): therefore \( C = \frac{u^2}{g} \times L \sqrt{u^2} \), and the complete fluent \( S = \frac{u^2}{g} \times L \sqrt{u^2} - L \sqrt{u^2 - v^2} \),

\[ = \frac{u^2}{g} \times L \sqrt{u^2} - \frac{u^2}{g} \times \frac{u^2}{g} \times \lambda \sqrt{u^2 - v^2}. \]

or (putting \( M \) for \( 0.43429 \), the modulus or subtangent of the common logistic curve)

\[ = \frac{u^2}{M g} \times \lambda \sqrt{u^2 - v^2}. \] This equation establishes the relation between the space fallen through, and the velocity acquired by the fall. We obtain by it \( \frac{gS}{u^2} = L \sqrt{\frac{u^2}{u^2 - v^2}} \), and

\[ \frac{2gS}{u^2} = L \frac{u^2}{u^2 - v^2}, \]

or, which is still more convenient for us,

\[ \frac{M \times 2gS}{u^2} = \lambda \frac{u^2}{u^2 - v^2}, \]

that is, equal to the logarithm of a certain number; therefore having found the natural number corresponding to the fraction \( \frac{M \times 2gS}{u^2} \), consider it as a logarithm, and take out the number corresponding to it: call this \( n \). Then, since \( n \) is equal to \( \frac{u^2}{u^2 - v^2} \), we have \( n u^2 - n v^2 = u^2 \),

and \( n u^2 - u^2 = n v^2 \), or \( n v^2 = u^2 \times n - v \), and \( v^2 = \frac{n^2 - 1}{n} \).

To expedite all the computations on this subject, it will be convenient to have multipliers ready computed for \( M \times 2g \), and its half,

viz. 27,794, whose log. is 1.44396

and 13,897 1.14293.

But \( v \) may be found much more expeditiously by observing that \( \frac{u^2}{u^2 - v^2} \) is the secant of an arch of a circle whose radius is \( u \), and whose fine is \( v \); or whose radius is unity and fine \( = \frac{v}{u} \); therefore, considering the above fraction as a logarithmic secant, look for it in the tables, and then take the fine of the arc of which this is the secant, and multiply it by \( u \); the product is the velocity required.

We shall take an example of a ball whose terminal velocity is 689 feet, and ascertain its velocity after a fall of 1848 feet. Here,

\[ u^2 = 475200 \quad \text{and its log.} = 5.67688 \] \[ u = 689 \quad \text{log.} = 2.83844 \] \[ g = 32 \quad \text{log.} = 1.50515 \] \[ S = 1848 \quad \text{log.} = 3.26670 \]

Then log. 27,794 + 1.44396 log. \( S \) + 3.26670 log. \( u^2 \) = 5.67688

Log. of 0.10809 = log. \( n \) = 9.03378. 0.10809 is the logarithm of 1,2826 = \( n \), and \( n - 1 = 0.2826 \), and \( \frac{u^2 \times n - 1}{n} = 323.6 \), \( v^2 \), and \( v = 323.6 \).

In like manner, 0.054045 (which is half of 0.10809) will be found to be the logarithmic secant of 28°, whose fine 0.46947 multiplied by 689 gives 324 for the velocity.

The process of this solution suggests a very perspicuous manner of conceiving the law of descent; and it may be thus expressed:

\( M \) is to the logarithm of the secant of an arch whose fine is \( \frac{v}{u} \), and radius \( r \), as 2 \( a \) is to the height through which the body must fall in order to acquire the velocity \( v \). Thus, to take the same example,

1. Let the height \( h \) be sought which will produce the velocity 323.6, the terminal velocity of the ball being 689.44. Here 2 \( a \), or \( \frac{u^2}{g} \) is 14850, and \( \frac{323.6}{689.44} = 0.46947 \), which is the sine of 28°. The logarithmic secant of this arch is 0.05407. Now \( M \) or 0.43429 : 0.05407 = 14850 : 1848, the height wanted.

2. Required the velocity acquired by the body by falling 1848 feet. Say 14850 : 1848 = 0.43429 : 0.05407. Look for this number among the logarithmic secants. It will be found at 28°, of which the logarithmic sine is 9.67161.

Add to this the log. of \( u \) = 2.83844.

The sum = 2.51005 is the logarithm of 323.6, the velocity required.

We may observe, from these solutions, that the acquired velocity continually approaches to, but never equals, the terminal velocity. For it is always expressed by the fine of an arch of which the terminal velocity is the radius. We cannot help taking notice here of a very strange assertion of Mr Muller, late professor of mathematics and director of the royal academy at Woolwich. He maintains, in his Treatise on Gunnery, his Treatise of Fluxions, and in many of his numerous works, that a body cannot possibly move through the air with a greater velocity than this; and he makes this a fundamental principle, on which he establishes a theory of motion in a resisting medium, which he affirms with great confidence to be the only just theory; saying, that all the investigations of Bernoulli, Euler, Robins, Simpson, and others, are erroneous. We use this strong expression, because, in his criticisms on the works of those celebrated mathematicians, he lays aside good manners, and taxes them not only with ignorance, but with dishonesty; saying, for instance, that it required no small dexterity in Robins to confirm by his experiments a theory founded on false principles; and that Thomas Simpson, in attempting to conceal his obligations to him for some valuable propositions, by changing their form, had ignorantly fallen into gross errors.

Nothing can be more palpably absurd than this assertion of Mr Muller. A blown bladder will have but a small terminal velocity; and when moving with this velocity, or one very near it, there can be no doubt that it will be made to move much swifter by a smart stroke. Were the assertion true, it would be impossible for a portion of air to be put into motion through the rest, for its terminal velocity is nothing. Yet this author makes this assertion a principle of argument, saying, that it is impossible that a ball can issue from the mouth of a cannon with a greater velocity than this; and that Robins and others are grossly mistaken, when they give them velocities three or four times greater, and resistances which are 10 or 20 times greater than is possible; and by thus compensating his small velocities by still smaller resistances, he confirms his theory by many experiments adduced in support of the others. No reason whatever can be given for the assertion. Newton, or perhaps Huygens, was the first who observed that there was a limit to the velocity which gravity could communicate to a body; and this limit was found by his commentators to be a term to which it was vastly convenient to refer all its other motions. It therefore became became an object of attention; and Mr Muller, through inadvertency, or want of discernment, has fallen into this mistake, and with that arrogance and self-conceit which mark all his writings, has made this mistake a fundamental principle, because it led him to establish a novel set of doctrines on this subject. He was fretted at the superior knowledge and talents of Mr Simpson, his inferior in the academy, and was guilty of several mean attempts to hurt his reputation. But they were unsuccessful.

We might proceed to consider the motion of a body projected downwards. While the velocity of projection is less than the terminal velocity, the motion is determined by what we have already said: for we must compute the height necessary for acquiring this velocity in the air, and suppose the motion to have begun there. But if the velocity of projection be greater, this method fails. We pass it over (though not in the least more difficult than what has gone before), because it is of mere curiosity, and never occurs in any interesting case. We may just observe, that since the motion is swifter than the terminal velocity, the resistance must be greater than the weight, and the motion will be retarded. The very same process will give us for the space described

\[ S = \frac{u^2}{g} \times L \sqrt{\frac{v^2 - u^2}{v^2 - u^2}}, \]

\( V \) being the velocity of projection, greater than \( u \). Now as this space evidently increases continually (because the body always falls), but does not become infinite in any finite time, the fraction

\[ \frac{V^2 - u^2}{v^2 - u^2} \]

does not become infinite; that is, \( v^2 \) does not become equal to \( u^2 \); therefore although the velocity \( V \) is continually diminished, it never becomes so small as \( u \). Therefore \( u \) is a limit of diminution as well as of augmentation.

We must now ascertain the relation between the time of the descent and the space described, or the velocity acquired. For this purpose we may use the other fundamental proposition of varied motions \( f' = v \), which, in the present case, becomes

\[ \frac{u^2}{g} \times \frac{v}{u^2 - v^2} = \frac{u}{g} \times \frac{u}{u^2 - v^2}, \]

and \( t = \frac{u}{g} \times \int \frac{u}{u^2 - v^2} \). Now

\[ \int \frac{u}{u^2 - v^2} = L \sqrt{\frac{u + v}{u - v}}. \]

Therefore

\[ \frac{u}{g} \times L \sqrt{\frac{u + v}{u - v}} = \frac{u}{g} \times \lambda \sqrt{\frac{u + v}{u - v}}. \]

This fluent needs no constant quantity to complete it, or rather \( C = 0 \); for \( t \) must be \( = 0 \) when \( v = 0 \). This will evidently be the case: for then

\[ L \sqrt{\frac{u + v}{u - v}} = L \sqrt{\frac{u}{u}}, \]

\( L_1 = 0 \).

But how does this quantity

\[ \frac{u}{g} \times \lambda \sqrt{\frac{u + v}{u - v}} \]

signify a time? Observe, that in whatever numbers, or by whatever units of space and time, \( u \) and \( g \) are expressed,

\[ \frac{u}{g} \] expresses the number of units of time in which the velocity \( u \) is communicated or extinguished by gravity;

and \( L \sqrt{\frac{u + v}{u - v}} \) or \( \lambda \sqrt{\frac{u + v}{u - v}} \), is always an abstract number, multiplying this time.

We may illustrate this rule by the same example. In what time will the body acquire the velocity \( 323,62 \)? Here \( u + v = 1012,96 \), \( u - v = 365,72 \); therefore

\[ \lambda \sqrt{\frac{u + v}{u - v}} = 0.22122, \]

and \( \frac{u}{g} \) (in feet and seconds) is \( 21'' \), \( 542 \). Now, for greater perspicuity, convert the equation

\[ \frac{u}{g} \times \lambda \sqrt{\frac{u + v}{u - v}} \]

into a proportion: thus

\[ M : \lambda \sqrt{\frac{u + v}{u - v}} = \frac{u}{g} : t, \]

and we have \( 0.43429 : 0.22122 = 21'' \), \( 542 : 10'' \), \( 973 \) the time required.

This is by far the most distinct way of conceiving the subject; and we should always keep in mind that the numbers or symbols which we call logarithms are really parts of the line MJ in the figure of the logistic curve, and that the motion of a point in this line is precisely similar to that of the body. The Marquis Poleni, in a dissertation published at Padua in 1725, has with great ingenuity constructed logarithmics suited to all the cases which can occur. Herman, in his Phoronomia, has borrowed much of Poleni's methods, but has obscured them by an affectation of language geometrically precise, but involving the very obscure notion of abstract ratios.

It is easy to see that \( \sqrt{\frac{u + v}{u - v}} \) is the cotangent of the \( \frac{\pi}{2} \) complement of an arch, whose radius is 1, and whose sine is \( \frac{v}{u} \); For let \( KC \) (fig. 6.) be \( = u \), and Fig. & BE \( = v \); then \( KD = u + v \), and \( DA = u - v \). Join KB and BA, and draw CG parallel to KB. Now GA is the tangent of \( \frac{\pi}{2} \) BA, \( = \frac{\pi}{2} \) complement of HB. Then, by similarity of triangles, \( GA : AC = AB : BK = \sqrt{AD} : \sqrt{DK} = \sqrt{u - v} : \sqrt{u + v} \) and \( \frac{AC}{GA} (= \cotan \frac{\pi}{2} BA) = \sqrt{\frac{u + v}{u - v}} \); therefore look for \( \frac{v}{u} \) among the natural sines, or for log. \( \frac{v}{u} \) among the logarithmic sines, and take the logarithmic cotangent of the half complement of the corresponding arch. This, considered as a common number, will be the second term of our proportion. This is a shorter process than the former.

By reversing this proportion we get the velocity corresponding to a given time.

To compare this descent of 1848 feet in the air with the fall of the body in vacuo during the same time, say \( 21'' \), \( 542'' : 10'' \), \( 973'' = 1848 : 1926,6 \), which compared makes a difference of 79 feet.

Cor. i. The time in which the body acquires the velocity \( u \) by falling through the air, is to the time of acquiring the same velocity by falling in vacuo, as \( u \).

\[ L \sqrt{\frac{u + v}{u - v}} \text{ to } v; \text{ for it would acquire this velocity in } \]

vacuo. vacuo during the time \( \frac{v}{g} \) and it acquires it in the air in the time \( \frac{u}{g} \sqrt{\frac{u+v}{u-v}} \).

2. The velocity which the body acquires by falling through the air in the time \( \frac{u}{g} \sqrt{\frac{u+v}{u-v}} \), is to the velocity which it would acquire in vacuo during the same time, as \( v \) to \( u \). For the velocity which it would acquire in vacuo during the time \( \frac{u}{g} \) \( L \sqrt{\frac{u+v}{u-v}} \) must be \( u \) \( L \sqrt{\frac{u+v}{u-v}} \) (because in any time \( \frac{v}{g} \) the velocity \( w \) is acquired).

In the next place, let a body, whose terminal velocity is \( u \), be projected perpendicularly upwards, with any velocity \( V \). It is required to determine the height to which it ascends, so as to have any remaining velocity \( v \), and the time of its ascent; as also the height and time in which its whole motion will be extinguished.

We have now \( g(u^2 + v^2) \) for the expression of \( f \); for both gravity and resistance act now in the same direction, and retard the motion of the ascending body:

\[ \begin{align*} \text{therefore } & \frac{g(u^2 + v^2)}{u^2} = -v \cdot \dot{v}, \\ & s = -\frac{u^2}{g} \times \frac{v \cdot \dot{v}}{u^2 + v^2}, \\ & \text{and } s = -\frac{u^2}{g} \times \int \frac{v \cdot \dot{v}}{u^2 + v^2} + C, \\ & = -\frac{u^2}{g} \times L \sqrt{u^2 + v^2} + C \quad \text{(see art. Fluxions). This must be } = 0 \text{ at the beginning of the motion, that is, when } v = V, \text{ that is,} \\ & = -\frac{u^2}{g} \times L \sqrt{u^2 + V^2} + C = 0, \\ & \text{or } C = \frac{u^2}{g} \times L \sqrt{u^2 + v^2}, \end{align*} \]

and the complete fluent will be \( s = \frac{u^2}{g} \times L \sqrt{u^2 + v^2} - L \sqrt{u^2 + v^2} = \frac{u^2}{g} \times L \sqrt{u^2 + V^2} \).

Let \( h \) be the greatest height to which the body will rise. Then \( s = h \) when \( v = 0 \); and \( h = \frac{u^2}{g} \times L \sqrt{u^2 + V^2} \).

We have

\[ \lambda \sqrt{\frac{u^2 + V^2}{u^2 + v^2}} = \frac{m_g}{u^2}; \]

therefore \( \lambda \left( \frac{u^2 + V^2}{u^2 + v^2} \right) = \frac{2Mgs}{u^2} \).

Therefore let \( n \) be the number whose common logarithm is \( \frac{2Mgs}{u^2} \); we shall have \( n = \frac{u^2 + V^2}{u^2 + v^2} \) and \( v = \frac{u^2 + V^2}{n} - u^2 \); and thus we obtain the relation of \( s \) and \( v \), as in the case of descents: but we obtain it still easier by observing that \( \sqrt{u^2 + V^2} \) is the secant of an arch whose radius is \( u \), and whose tangent is \( V \), and that \( \sqrt{u^2 + v^2} \) is the secant of another arch of the same circle, whose tangent is \( v \).

Let the same ball be projected upwards with the velocity 411.05 feet per second. Required the whole height to which it will rise?

Here \( \frac{V}{u} \) will be found the tangent of 30.48°, the logarithmic secant of which is 0.96606. This, multiplied by \( \frac{u^2}{Mg} \), gives 2259 feet for the height. It would have risen 2640 feet in a void.

Suppose this body to fall down again. We can compare the velocity of projection with the velocity projection with which it again reaches the ground. The ascent compared with that and descent are equal: therefore \( \sqrt{\frac{u^2 + V^2}{u^2}} \), which it reaches the ground, multiplies the constant factor in the ascent, is equal to \( \sqrt{\frac{u^2}{u^2 - v^2}} \), the multiplier in the descent. The first is the secant of an arch whose tangent is \( V \); the other is the secant of an arch whose fine is \( v \). These secants are equal, or the arches are the same; therefore the velocity of projection is to the final returning velocity as the tangent to the fine, or as the radius to the cosine of the arch. Thus suppose the body projected with the terminal velocity, or \( V = u \); then \( v = \frac{u}{\sqrt{2}} \). If \( V = 689, v = 487 \).

We must in the last place ascertain the relation of the space and the time.

Here \( \frac{g(u^2 + v^2)}{u^2} = -v \cdot \dot{v} \), and \( i = -\frac{u^2}{g} \times \frac{v \cdot \dot{v}}{u^2 + v^2} = -\frac{u}{g} \times \frac{v \cdot \dot{v}}{u^2 + v^2} \) and \( t = \frac{u}{g} \times \int \frac{v \cdot \dot{v}}{u^2 + v^2} + C \). Now (art. Fluxions) \( \int \frac{v \cdot \dot{v}}{u^2 + v^2} \) is an arch whose tangent \( = \frac{v}{u} \) and radius 1; therefore \( t = \frac{u}{g} \times \text{arc.tan.} \frac{v}{u} + C \).

This must be \( = 0 \) when \( v = V \), or \( C = \frac{u}{g} \times \text{arc.tan.} \frac{V}{u} \).

\( \frac{V}{u} = 0 \), and \( C = \frac{u}{g} \times \text{arc.tan.} \frac{V}{u} \), and the complete fluent is \( t = \frac{u}{g} \times \left( \text{arc.tan.} \frac{V}{u} - \text{arc.tan.} \frac{v}{u} \right) \). The quantities within the brackets express a portion of the arch of a circle whose radius is unity; and are therefore abstract numbers, multiplying \( \frac{u}{g} \), which we have shown to be the number of units of time in which a heavy body falls in vacuo from the height \( a \), or in which it acquires the velocity \( u \).

We learn from this expression of the time, that how ever great the velocity of projection, and the height ascent limited, to which this body will rise, may be, the time of its ascent is limited. It never can exceed the time of falling from the height \( a \) in vacuo in a greater proportion than that of a quadrant arch to the radius, nearly the proportion of 8 to 5. A 24 pound iron ball cannot continue rising above 14 seconds, even if the resistance to quick motions did not increase faster than the square of the velocity. It probably will attain its greatest height in less than 12 seconds, let its velocity be ever so great.

In the preceding example of the whole ascent, \( v = 0 \), and and the time \( t = \frac{u}{g} \times \text{arc. tan. } \frac{V}{u} \), or \( \frac{u}{g} \times \text{arc. } 30^\circ. 48' \).

Now \( 30^\circ. 48' = 1848' \), and the radius \( r \) contains 3438'; therefore the arc \( \theta = \frac{1848}{3438'} = 0.5376 \); and \( \frac{u}{g} = 21'' . 54 \).

Therefore \( t = 21'' . 54 \times 0.5376 = 11'' . 58 \), or nearly 11 seconds. The body would have risen to the same height in a void in 10\(\frac{1}{2}\) seconds.

Cor. 1. The time in which a body, projected in the air with any velocity \( V \), will attain its greatest height, is to that in which it would attain its greatest height in vacuo, as the arch whose tangent expresses the velocity is to the tangent; for the time of the ascent in the air is \( \frac{u}{g} \times \text{arch} \); the time of the ascent in vacuo is \( \frac{V}{g} \). Now \( \frac{V}{u} \) is \( \tan \), and \( V = \frac{u}{g} \times \tan \).

It is evident, by inspecting fig. 6, that the arch \( AI \) is to the tangent \( AG \) as the sector \( ICA \) to the triangle \( GCA \); therefore the time of attaining the greatest height in the air is to that of attaining the greatest height in vacuo (the velocities of projection being the same), as the circular sector to the corresponding triangle.

If therefore a body be projected upwards with the terminal velocity, the time of its ascent will be to the time of acquiring this velocity in vacuo as the area of a circle to the area of the circumscribed square.

2. The height \( H \) to which a body will rise in a void, is to the height \( h \) to which it would rise through the air when projected with the same velocity \( V \) as \( M \cdot V^2 \) to \( u^2 \times \lambda \frac{u^2 + V^2}{u^2} \); for the height to which it will rise in vacuo is \( \frac{V^2}{2g} \), and the height to which it rises in the air is \( \frac{u^2}{Mg} \times \sqrt{\frac{u^2 + V^2}{u^2}} \); therefore \( H : h = \frac{V^2}{2g} : \frac{u^2}{Mg} \times \sqrt{\frac{u^2 + V^2}{u^2}} = V^2 : \frac{u^2}{M} \times \lambda \frac{u^2 + V^2}{u^2} = M \cdot V^2 : u^2 \times \lambda \frac{u^2 + V^2}{u^2} \).

Therefore if the body be projected with its terminal velocity, so that \( V = u \), the height to which it will rise in the air is \( \frac{30103}{43429} \) of the height to which it will rise in vacuo, or \( \frac{5}{7} \) in round numbers.

We have been thus particular in treating of the perpendicular ascents and descents of heavy bodies through the air, in order that the reader may conceive distinctly the quantities which he is thus combining in his algebraic operations, and may see their connection in nature with each other. We shall also find that, in the present state of our mathematical knowledge, this simple state of the case contains almost all that we can determine with any confidence. On this account it were to be wished that the professional gentlemen would make many experiments on these motions. There is no way that promises so much for afflicting us in forming accurate notions of the air's resistance. Mr Robins's method with the pendulum is impracticable with great shot; and the experiments which have been generally resorted to for this purpose, viz. the ranges of shot and shells on a horizontal plane, are so complicated in themselves, that the utmost mathematical skill is necessary for making any inferences from them; and they are subject to such irregularities, that they may be brought to support almost any theory whatever on this subject. But the perpendicular flights are affected by nothing but the initial velocity and the resistance of the air; and a considerable deviation from their intended direction does not cause any sensible error in the consequences which we may draw from them for our purpose.

But we must now proceed to the general problem, of obtaining the motion of a body projected in any direction, and with any velocity. Our readers will believe beforehand that this must be a difficult subject, when they see the simplest cases of rectilinear motion abundantly abstruse: it is indeed so difficult, that Sir Isaac Newton has not given a solution of it, and has thought himself well employed in making several approximations, in which the fertility of his genius appears to have been employed in great lustre. In the tenth and subsequent propositions of the second book of the Principia, he shows what state of density in the air will comport with the motion of a body in any curve whatever: and then, by applying this discovery to several curves which have some similarity to the path of a projectile, he finds one which is not very different from what we may suppose to obtain in our atmosphere. But even this approximation was involved in such intricate calculations, that it seemed impossible to make any use of it. In the second edition of the Principia, published in 1713, Newton corrects some mistakes which he had committed in the first, and carries his approximations much farther, but still does not attempt a direct investigation of the path which a body will describe in our atmosphere. This is somewhat surprising. In prop. 14. &c., he shows how a body, actuated by a centripetal force, in a medium of a density varying according to certain laws, will describe an eccentric spiral, of which he assigns the properties, and the law of description. Had he supposed the density constant, and the difference between the greatest and least distances from the centre of centripetal force exceedingly small in comparison with the distances themselves, his spiral would have coincided with the path of a projectile in the air of uniform density, and the steps of his investigation would have led him immediately to the complete solution of the problem. For this is the real state of the case. A heavy body is not acted on by equal and parallel gravity, but by a gravity inversely proportional to the square of the distance from the centre of the earth, and in lines tending to that centre nearly; and it was with the view of simplifying the investigation, that mathematicians have adopted the other hypothesis.

Soon after the publication of this second edition of the Principia, the dispute about the invention of the fluxionary calculus became very violent, and the great foreign promoters of that calculus upon the continent were in the habit of proposing difficult problems to exercise the talents of the mathematician. Challenges of this kind frequently passed between the British and foreigners. Dr Keill of Oxford had keenly espoused the claim of Sir Isaac Newton to this invention, and had engaged in a very acrimonious altercation with the celebrated John Bernoulli of Balle. Bernoulli had published in the Acta Eruditorum Lighae an investigation of the law of forces, by which a body moving in a resisting medium might describe any proposed curve, reducing the whole to the simplest geometry. This is perhaps the most elegant specimen which he has given of his great talents. Dr Keill proposed to him the particular problem of the trajectory and motion of a body moving through the air, as one of the most difficult. Bernoulli very soon solved the problem in a way much more general than it had been proposed, viz. without any limitation either of the law of resistance, the law of the centripetal force, or the law of density, provided only that they were regular, and capable of being expressed algebraically. Dr Brook Taylor, the celebrated author of the Method of Increments, solved it at the same time, in the limited form in which it was proposed. Other authors since that time have given other solutions. But they are all (as indeed they must be) the same in substance with Bernoulli's. Indeed they are all (Bernoulli's not excepted) the same with Newton's first approximations, modified by the steps introduced into the investigation of the spiral motions mentioned above; and we still think it most strange that Sir Isaac did not perceive that the variation of curvature, which he introduced in that investigation, made the whole difference between his approximations and the complete solution. This we shall point out as we go along. And now proceed to the problem itself, of which we shall give Bernoulli's solution, restricted to the case of uniform density and a resistance proportional to the square of the velocity.

This solution is more simple and perspicuous than any that has since appeared.

**Problem.** To determine the trajectory, and all the circumstances of the motion of a body projected through the air from A (fig. 7.) in the direction AB, and resisted in the duplicate ratio of the velocity.

Let the arch AM be put = z, the time of describing it t, the abscissa AP=x, the ordinate PM=y. Let the velocity in the point M=v, and let MN=z, be described in the moment t; let r be the resistance of the air, g the force of gravity, measured by the velocity which it will generate in a second; and let a be the height through which a heavy body must fall in vacuo to acquire the velocity which would render the resistance of the air equal to its gravity: so that we have

\[ r = \frac{v^2}{2a} \]

because, for any velocity u, and producing height h, we have \( g = \frac{u^2}{2h} \).

Let Mm touch the curve in M; draw the ordinate pNn, and draw Mo, Nn perpendicular to Np and Mm. Then we have MN=z, and Mo=m, also mo is ultimately = y and Mm is ultimately = MN or z.

Lastly, let us suppose x to be a constant quantity, the elementary ordinates being supposed equidistant.

The action of gravity during the time t may be measured by mN, which is half the space which it would cause the body to describe uniformly in the time t with the velocity which it generates in that time.

Let this be resolved into N, by which it deflects the body into a curvilinear path, and mn, by which it retards the ascent and accelerates the descent of the body along the tangent. The resistance of the air acts solely in retarding the motion, both in ascending and descending, and has no deflective tendency. The whole action of gravity then is to its accelerating or retarding tendency as mN to mn, or (by similarity of triangles) as mM to mo. Or \( \frac{z}{y} = \frac{g}{z} \), and the whole retardation in the ascent will be \( r + \frac{gy}{z} \). The same fluxionary symbol will express the retardation during the descent, because in the descent the ordinates decrease, and y is a negative quantity.

The diminution of velocity is \( -\dot{v} \). This is proportional to the retarding force and to the time of its action jointly, and therefore \( -\dot{v} = r + \frac{gy}{z} \times t \); but the time t is as the space z divided by the velocity v; therefore \( -\dot{v} = r + \frac{gy}{z} \times \frac{z}{v} = -\frac{rz + gy}{v} \), and \( -\dot{v} = -\frac{rz - gy}{v} \).

Because mN is the deflection by gravity, it is as the force g and the square of the time t jointly (the momentary action being held as uniform). We have therefore mN, or \( -\frac{gy}{z} \times t^2 \). (Observe that mN is in fact only the half of \( -\frac{gy}{z} \); but g being twice the fall of a heavy body in a second, we have \( -\frac{gy}{z} \) strictly equal to \( \frac{g}{z} \)). But \( \frac{z}{v} = \frac{z}{v} \); therefore \( \frac{z}{v} = \frac{g}{z} \), and \( \frac{v^2}{2a} = \frac{g}{z} \). The fluxion of this equation is \( -\frac{v^2}{2a} = -\frac{v^2}{2a} \times \frac{z}{v} = -\frac{v^2}{2a} \times \frac{z}{v} \); but, because \( \frac{z}{v} = \frac{g}{z} \); therefore \( \frac{v^2}{2a} = \frac{g}{z} \), and finally \( \frac{z}{v} = \frac{g}{z} \), or \( \frac{z}{v} = \frac{g}{z} \), for the fluxionary equation of the curve.

If we put this into the form of a proportion, we have \( a : z = y : y \). Now this evidently establishes a relation between the length of the curve and its variation of the curve itself and its evolute, which are the very circumstances introduced by Newton. Newton into his investigation of the spiral motions. And

the equation \( \frac{d^2x}{da^2} = \frac{dy}{dy} \) is evidently an equation connected

with the logarithmic curve and the logarithmic spiral.

But we must endeavour to reduce it to a lower order of

fluxions, before we can establish a relation between \( x \), \( y \),

and \( z \).

Let \( p \) express the ratio of \( y \) to \( x \), that is, let \( p \) be \( = \)

\( \frac{y}{x} \), or \( p = \frac{y}{x} \). It is evident that this expresses the

inclination of the tangent at \( M \) to the horizon, and that

\( p \) is the tangent of this inclination, radius being unity.

Or it may be considered merely as a number, multiply-

ing \( x \), so as to make it \( = y \). We now have \( y^2 = p^2 x^2 \),

and since \( x^2 = x^2 + y^2 \), we have \( x^2 = x^2 + p^2 x^2 \),

\( = 1 + p^2 x^2 \), and \( x = x \sqrt{1 + p^2} \).

Moreover, because we have supposed the abscissa \( x \)

to increase uniformly, and therefore \( x \) to be constant,

we have \( y = x p \), and \( y = x p \). Now let \( q \) express the

ratio of \( p \) to \( x \), that is, make \( \frac{p}{x} = q \), or \( q = \frac{p}{x} \).

This gives us \( x q = p \), and \( x q = x p = y \).

By these substitutions our former equation \( ay = bx \)

changes to \( a x q = x \sqrt{1 + p^2} x p \), or \( a y = p \sqrt{1 + p^2} \),

and, taking the fluent on both sides, we have

\( a q = f p \sqrt{1 + p^2} + C \), \( C \) being the constant quantity

required for completing the fluent according to the li-

miting conditions of the case. Now \( x = \frac{p}{q} \), and \( \frac{1}{q} = \)

\( f \frac{a}{p \sqrt{1 + p^2}} + C \). Therefore \( x = f \frac{a p}{p \sqrt{1 + p^2}} + C \).

Also, since \( y = p x \), \( y = \frac{pp}{q} \), we have \( y = \)

\( f \frac{a pp}{p \sqrt{1 + p^2}} + C \).

Also \( z = x \sqrt{1 + p^2} = \frac{ap \sqrt{1 + p^2}}{f p \sqrt{1 + p^2}} + C \).

The values of \( x \), \( y \), \( z \), give us

\( x = f \frac{a p}{p \sqrt{1 + p^2}} + C \),

\( y = f \frac{a pp}{p \sqrt{1 + p^2}} + C \),

\( z = f \frac{ap \sqrt{1 + p^2}}{p \sqrt{1 + p^2}} + C \).

The proofs therefore of describing the trajectory is, \( 1/4 \).

To find \( q \) in terms of \( p \) by the area of the curve whose

abscissa is \( p \) and the ordinate is \( \sqrt{1 + p^2} \).

\( 2d \), We get \( x \) by the area of another curve whose

abscissa is \( p \), and the ordinate is \( \frac{1}{q} \).

\( 3d \), We get \( y \) by the area of a third curve whose ab-

scissa is \( p \), and the ordinate is \( \frac{p}{q} \).

The problem of the trajectory is therefore comple-

tely solved, because we have determined the ordinate, ab-

scissa, and arch of the curve for any given position of

its tangent. It now only remains to compute the mag-

To com-

pounds of these ordinates and abscissae, or to draw them out

by a geometrical construction. But in this consists the magnitude

difficulty. The areas of these curves, which express the

lengths of \( x \) and \( y \), can neither be computed nor exhib-

ited geometrically, by any accurate method yet dis-

covered, and we must content ourselves with approxima-

tions. These render the description of the trajectory ex-

ceedingly difficult and tedious, so that little advantage

has as yet been derived from the knowledge we have got

of its properties. It will however greatly assist our con-

ception of the subject to proceed some length in this

construction; for it must be acknowledged that very

few distinct notions accompany a mere algebraic opera-

tion, especially if in any degree complicated, which we

confess is the case in the present question.

Let \( BmNR \) (fig. 8.) be an equilateral hyperbola, of

which \( B \) is the vertex, \( BA \) the semitransverse axis,

which we shall allow for the unity of length. Let \( AV \)

be the semiconjugate axis \( = BA \), \( = \) unity, and \( AS \) the

asymptote, bisecting the right angle \( BAV \). Let \( PN \),

\( PP \) be two ordinates to the conjugate axis, exceedingly

near to each other. Join \( BP \), \( AN \), and draw \( B \beta \), \( N \),

perpendicular to the asymptote, and \( BC \) parallel to \( AP \).

It is well known that \( BP \) is equal to \( NP \). Therefore

\( PN^2 = BA^2 + AP^2 \). Now since \( BA = 1 \), if we make

\( AP = p \) of our formulæ, \( PN \) is \( \sqrt{1 + p^2} \), and \( Pp \) is \( = \)

\( p \), and the area \( BAPNB = f \frac{p}{p \sqrt{1 + p^2}} \): That is to

say, the number \( f \frac{p}{p \sqrt{1 + p^2}} \) (for it is a number) has

the same proportion to unity of number that the area

\( BAPNB \) has to \( BCVA \), the unit of surface. This

area consists of two parts, the triangle \( APN \), and the

hyperbolic sector \( ABN \). \( APN = \frac{1}{2} AP \times PN \),

\( = \frac{1}{2} p \sqrt{1 + p^2} \), and the hyperbolic sector \( ABN = BN \),

\( = \frac{1}{2} \) which is equivalent to the hyperbolic logarithm of

the number represented by \( A \) when \( A \) is unity. There-

fore it is equal to \( \frac{1}{2} \) the logarithm of \( p + \sqrt{1 + p^2} \).

Hence we see by the bye that \( f \frac{p}{p \sqrt{1 + p^2}} = \)

\( \frac{1}{2} p \sqrt{1 + p^2} + \frac{1}{2} \) hyperbolic logarithm \( p + \sqrt{1 + p^2} \).

Now let \( AMD \) be another curve, such that its ordi-

nates \( Vm \), \( PD \), &c. may be proportional to the areas

\( ABmV \), \( ABNP \), and may have the same proportion

to \( AB \), the unity of length, which these areas have to

\( ABCV \), the unity of surface. Then \( VM : VC \),

\( VmBA : VCBA \), and \( PD : PD = NBA : VCBA \),

&c. These ordinates will now represent \( f \frac{p}{p \sqrt{1 + p^2}} \)

with reference to a linear unit, as the areas to the

hyperbola represented it in reference to a superficial

unit.

Again, Again, in every ordinate make \( PD : P = P : PO \), and thus we obtain a reciprocal to \( PD \), or to \( f' p \sqrt{1 + p^2} \), or equivalent to \( f' \frac{1}{p} \sqrt{1 + p^2} \). This will evidently be \( \frac{x}{ap} \), and \( PO \) or \( p \) will be \( \frac{x}{a} \), and the area contained between the lines \( AF, AW \), and the curve \( GEOH \), and cut off by the ordinate \( PO \), will represent \( \frac{x}{a} \).

Lastly, make \( PO : PQ = AV : AP = 1 : p \); and then \( PQ \) will represent \( \frac{y}{a} \), and the area \( ALEQP \) will represent \( \frac{y}{a} \).

But we must here observe, that the fluents expressed by these different areas require what is called the correction to accommodate them to the circumstances of the case. It is not indifferent from what ordinate we begin to reckon the areas. This depends on the initial direction of the projectile, and that point of the abscissa \( AP \) must be taken for the commencement of all the areas which gives a value of \( p \) suited to the initial direction. Thus, if the projection has been made from \( A \) (fig. 7.) at an elevation of \( 45^\circ \), the ratio of the fluxions \( x \) and \( y \) is that of equality; and therefore the point \( E \) of fig. 8., where the two curves intersect and have a common ordinate, evidently corresponds to this condition. The ordinate \( EV \) passes through \( V \), so that \( AV \) or \( p = AB, = 1, = \text{tangent } 45^\circ \), as the case requires. The values of \( x \) and \( y \) corresponding to any other point of the trajectory, such as that which has \( AP \) for the tangent of the angle which it makes with the horizon, are now to be had by computing the areas \( VEOP, VEQP \).

Another curve might have been added, of which the ordinates would exhibit the fluxions of the arch of the trajectory \( \frac{ap}{f'} \frac{\sqrt{1 + p^2}}{p} \), and of which the area would exhibit the arch itself. And this would have been very easy, for it is \( \dot{x} = a \frac{\dot{p}}{f'} \frac{\sqrt{1 + p^2}}{p} + C \), which is evidently the fluxion of the hyperbolic logarithm of \( f' p \sqrt{1 + p^2} \). But it is needless, since \( \dot{x} = \dot{x} \sqrt{1 + p^2} \), and we have already got \( \dot{x} \). It is only increasing \( PO \) in the ratio of \( BA \) to \( BP \).

And thus we have brought the investigation of this problem a considerable length, having ascertained the form of the trajectory. This is surely done when the ratio of the arch, abscissa, and ordinate, and the position of its tangent, is determined in every point. But it is still very far from a solution, and much remains to be done before we can make any practical application of it.

The only general consequence that we can deduce from the premises is, that in every case where the resistance in any point bears the same proportion to the force of gravity, the trajectory will be similar. Therefore, two balls, of the same density, projected in the same direction, will describe similar trajectories if the velocities are in the subduplicate ratio of the diameters. This we shall find to be of considerable practical importance. But let us now proceed to determine the velocity in the different points of the trajectory, and the time of describing velocity in its several portions.

Recollect, therefore, that \( v^2 = -\frac{g}{y} \), and that \( \dot{x} = \dot{x} \sqrt{1 + p^2} \) and \( \dot{y} = \dot{x} \dot{p} \). This gives \( v^2 = -\frac{g}{q} \frac{1 + p^2}{p} \).

But \( \dot{p} = q \dot{x} \). Therefore \( v^2 = -\frac{g}{q} \frac{1 + p^2}{p} \),

\[ \frac{-a \sqrt{1 + p^2}}{f' \frac{\sqrt{1 + p^2}}{p} + C} \]

and \( v = \sqrt{-\frac{g}{q} \frac{1 + p^2}{p}} \).

Alfo \( i \) was found \( = \frac{\dot{x}}{v} = \frac{\dot{x} \sqrt{1 + p^2}}{v} = \frac{\dot{p} \sqrt{1 + p^2}}{q v} \).

If we now substitute for \( v \) its value just found, we obtain \( i = \frac{\dot{p}}{\sqrt{-g q}} \), and \( l = f' \frac{\dot{p}}{\sqrt{-g q}} \),

\[ \frac{\dot{p} \sqrt{a}}{\sqrt{-g f' \frac{\sqrt{1 + p^2}}{p} + C}} = \frac{\sqrt{a}}{\sqrt{-g}} \times \frac{\dot{p}}{\sqrt{f' \frac{\sqrt{1 + p^2}}{p} + C}} \]

The greatest difficulty still remains, viz. the accommodating these formulae, which appear abundantly firm and applicable to the particular cases. It would seem at first modating flight, that all trajectories are similar; since the ratio of the fluions of the ordinate and abscissa corresponding to any particular angle of inclination to the horizon seems the same in them all; but a due attention to what has been hitherto said on the subject will show us that we have as yet only been able to ascertain the velocity in the point of the trajectory, which has a certain inclination to the horizon, indicated by the quantity \( p \), and the time (reckoned from some assigned beginning) when the projectile is in that point.

To obtain absolute measures of these quantities, the term of commencement must be fixed upon. This will be expressed by the constant quantity \( C \), which is assumed for completing the fluent of \( \dot{p} \sqrt{1 + p^2} \), which is the basis of the whole construction. We there found \( q = f' \frac{\dot{p} \sqrt{1 + p^2}}{a} \). This fluent is in general \( q = C + f' \frac{\dot{p} \sqrt{1 + p^2}}{a} \), and the constant quantity \( C \) is to be accommodated to some circumstances of the case. Different authors have selected different circumstances. Euler, Euler, in his Commentary on Robins, and in a dissertation in the Memoirs of the Academy of Berlin published in 1753, takes the vertex of the curve for the beginning of his abscissa and ordinate. This is the simplest method of any, for C must then be so chosen that the whole fluent may vanish when \( p = 0 \), which is the case in the vertex of the curve, where the tangent is parallel to the horizon. We shall adopt this method.

Therefore, let \( AP \) (fig. 9.) \( = x \), \( PM = y \), \( AM = z \). Put the quantity \( C \) which is introduced into the fluent equal to \( \frac{n}{a} \). It is plain that \( n \) must be a number; for it must be homologous with \( \frac{p}{\sqrt{1 + p^2}} \), which is a number. For brevity's sake let us express the fluent of \( \frac{p}{\sqrt{1 + p^2}} \) by the single letter \( P \); and thus we shall have \( x = a \times f \left( \frac{p}{n + P} \right) \), \( y = a \times f \left( \frac{pp}{n + P} \right) \), \( z = a \times f \left( \frac{\sqrt{1 + p^2}}{n + P} \right) \). And \( v^2 = -\frac{ag(1 + p^2)}{2g(n + P)} \). Now the height \( h \) necessary for communicating any velocity \( v \) is

\[ \frac{v^2}{2g} = -\frac{ag(1 + p^2)}{2g(n + P)} = -\frac{a(1 + p^2)}{n + P}. \]

And lastly,

\[ t = \frac{\sqrt{g}}{\sqrt{n + P}}. \]

These fluents, being all taken so as to vanish at the vertex, where the computation commences, and where \( p = 0 \) (the tangent being parallel to the horizon), we obtain in this case \( h = \frac{a}{n} \), \( = \frac{a}{2n} \), and \( n = \frac{a}{2h} \).

Hence we see that the circumstance which modifies all the curves, distinguishing them from each other, is the velocity (or rather its square) in the highest point of the curve. For \( h \) being determined for any body whose terminal velocity is \( u \), \( n \) is also determined; and this is the modifying circumstance. Considering it geometrically, it is the area which must be cut off from the area \( DMAP \) of fig. 8. in order to determine the ordinates of the other curves.

We must further remark, that the values now given relate only to that part of the area where the body is descending from the vertex. This is evident; for, in order that \( y \) may increase as we recede from the vertex, its fluxion must be taken in the opposite sense to what it was in our investigation. There we supposed \( y \) to increase as the body ascended, and then to diminish during the descent; and therefore the fluxion of \( y \) was first positive and then negative.

The same equations, however, will serve for the ascending branch \( CNA \) of the curve, only changing the sign of \( P \); for if we consider \( y \) as decreasing during the ascent, we must consider \( q \) as expressing \( \frac{p}{x} \), and therefore \( P \), or \( f \left( \frac{p}{\sqrt{1 + p^2}} \right) \), which is \( \frac{q}{a} \), must be taken negatively. Therefore, in the ascending branch, we have \( AQ \) or \( x \) (increasing as we recede from \( A \))—

\[ a \times f \left( \frac{p}{n - P} \right), QN \text{ or } y = a \times f \left( \frac{pp}{n - P} \right), AN \text{ or } z = \]

and the height producing the velocity at \( N = \frac{\frac{1}{2}a(1 + p^2)}{n - P} \).

Hence we learn by the bye, that in no part of the Remark-ascending branch can the inclination of the tangent be able property of the curve or trajectory. Such that \( P \) shall be greater than \( n \); and that if we suppose \( P \) equal to \( n \) in any point of the curve, the velocity in that point will be infinite. That is to say, there is a certain assignable elevation of the tangent which cannot be exceeded in a curve which has this velocity in the vertex. The best way for forming a conception of this circumstance in the nature of the curve, is to invert the motion, and suppose an accelerating force, equal and opposite to the resistance, to act on the body in conjunction with gravity. It must describe the same curve, and this branch \( ANC \) must have an asymptote \( LO \), which has this limiting position of the tangent. For, as the body descends in this curve, its velocity increases to infinity by the joint action of gravity and this accelerating force, and yet the tangent never approaches so near the perpendicular position as to make \( P = n \). This remarkable property of the curve was known to Newton, as appears by his approximations, which all lead him to curves of a hyperbolic form, having one asymptote inclined to the horizon. Indeed it is pretty obvious: For the resistance increasing faster than the velocity, there is no velocity of projection so great but that the curve will come to deviate so from the tangent, that in a finite time it will become parallel to the horizon. Were the resistance proportional to the velocity, then an infinite velocity would produce a rectilineal motion, or rather a deflection from it less than any that can be assigned.

We now see that the particular form and magnitude of this trajectory depends on two circumstances, \( a \) and \( n \). \( a \) affects chiefly the magnitude. Another circumstance might indeed be taken in, viz. the diminution of the accelerating force of gravity by the statical effect of the air's gravity. But, as we have already observed, this is too trifling to be attended to in military projectiles.

\[ \frac{y}{z} \text{ was made equal to } \frac{p}{x}. \text{ Therefore the radius of curvature, determined by the ordinary methods, is } \frac{x(1 + p^2)(\sqrt{1 + p^2})}{p}, \text{ and, because } \frac{x}{p} \text{ is } * \text{ Simpson's Finzioni, § 68, &c.} \]

\[ = \frac{a}{n + P} \text{ for the descending branch of the curve, the radius of curvature at } M \text{ is } \frac{a \times \sqrt{1 + p^2} \times \sqrt{1 + p^2}}{n + P}, \text{ and, in the ascending branch at } N, \text{ it is } \frac{a \times 1 + p^2 \times \sqrt{1 + p^2}}{n - P}. \]

On both sides, therefore, when the velocity is infinitely great, and \( P \) by this means supposed to equal or exceed \( n \), the radius of curvature is also infinitely great. We also see that the two branches are unlike each other, and that when \( p \) is the same in both, that is, when the tangent is equally inclined to the horizon, the radius of curvature, the ordinate, the abscissa, and the arch, are all greater in the ascending branch. This is pretty obviouss. For as the resistance acts entirely in diminishing the velocity, and does not affect the deflection occasioned by gravity, it must allow gravity to incurvate the path so much the more (with the same inclination of its line of action) as the velocity is more diminished. The curvature, therefore, in those points which have the same inclination of the tangent, is greatest in the descending branch, and the motion is swiftest in the ascending branch. It is otherwise in a void, where both sides are alike. Here \( u \) becomes infinite, or there is no terminal velocity; and \( n \) also becomes infinite, being

\[ \frac{a}{2h}. \]

It is therefore in the quantity \( P \), or \( f \cdot p \cdot \sqrt{1 + p^2} \),

that the difference between the trajectory in a void and in a resisting medium consists; it is this quantity which expresses the accumulated change of the ratio of the increments of the ordinate and abscissa. In vacuo the second increment of the ordinate is constant when the first increment of the abscissa is \( a \), and the whole increment of the ordinate is \( 1 + p \). And this difference is so much the greater as \( P \) is greater in respect of \( n \). \( P \) is nothing at the vertex, and increases along with the angle \( MTP \); and when this is a right angle, \( P \) is infinite. The trajectory in a resisting medium will come therefore to deviate infinitely from a parabola, and may even deviate farther from it than the parabola deviates from a straight line. That is, the distance of the body in a given moment from that point of its parabolic path where it would have been in a void, is greater than the distance between that point of the parabola from the point of the straight line where it would have been, independent of the action of gravity. This must happen whenever the resistance is greater than the weight of the body, which is generally the case in the beginning of the trajectory in military projectiles; and this (were it now necessary) is enough to show the inutility of the parabolic theory.

Although we have no method of describing this trajectory, which would be received by the ancient geometers, we may ascertain several properties of it, which will assist us in the solution of the problem. In particular, we can assign the absolute length of any part of it by means of the logistic curve. For because \( P = f \cdot p \cdot \sqrt{1 + p^2} \), we have \( p \cdot \sqrt{1 + p^2} = P \), and therefore \( z \), which was \( a \times f \cdot p \cdot \sqrt{1 + p^2} \cdot \frac{1}{f \cdot p \cdot \sqrt{1 + p^2}} + C \), or \( a \times f \cdot p \cdot \sqrt{n + P} \), may be expressed by logarithms; or \( z = a \times \text{hyp. log. of } \frac{n + P}{n} \), since at the vertex \( A \), where \( z \) must be \( = a \), \( P \) is also \( = 0 \).

Being able, in this way, to ascertain the length \( AM \) of the curve (counted from the vertex), corresponding to any inclination \( p \) of the tangent at its extremity \( M \), we can ascertain the length of any portion of it, such as \( MM \), by first finding the length of the part \( Am \), and then of the part \( AM \). This we do more expeditiously thus: Let \( p \) express the position of the tangent in \( M \), and \( q \) its position at \( m \); then \( AM = a \times \log. \frac{n + P}{n} \) and \( Am = a \times \log. \frac{n + Q}{n} \), and therefore \( MM = a \times \log. \frac{n + Q}{n + P} \). Thus we can find the values of a great number of small portions, and the inclination of the tangents at their extremities. Then to each of these portions we can assign its proportion of the abscissa and ordinate, without having recourse to the values of \( x \) and \( y \). For the portion of abscissa corresponding to the arch \( MM \), whose middle point is inclined to the horizon in the angle \( b \), will be \( MM \times \cosine b \), and the corresponding portion of the ordinate will be \( MM \times \sin b \). Then we obtain the velocity in each part of the curve by the equation \( h = \frac{a}{n + P} \cdot \sqrt{1 + p^2} \); or, more directly the velocity \( v \) at \( M \) will be \( = \sqrt{ag} \cdot \frac{\sqrt{1 + p^2}}{\sqrt{n + P}} \). Lastly, divide the length of the little arch by this, and the quotient will be the time of describing \( MM \) very nearly. Add all these together, and we obtain the whole time of describing the arch \( AM \), but a little too great, because the motion in the small arch is not perfectly uniform. The error, however, may be as small as we please, because we may make the arch as small as we please; and for greater accuracy, it will be proper to take the \( p \) by which we compute the velocity, a medium between the \( p \) for the beginning and that for the end of the arch.

This is the method followed by Euler, who was one Euler's methods of the most expert analysts, if not the very first, in Europe. It is not the most elegant, and the methods of some other authors, who approximate directly to the areas of the curves which determine the values of \( x \) and \( y \), have a more scientific appearance; but they are not ultimately very different: For, in some methods, these areas are taken piecemeal, as Euler takes the arch; and by the methods of others, who give the value of the areas by Newton's method of describing a curve of the parabolic kind through any number of given points, the ordinates of these curves, which express \( x \) and \( y \), must be taken finitely, which amounts to the same thing, with the great disadvantage of a much more complicated calculus, as any one may see by comparing the expressions of \( x \) and \( y \) with the expression of \( z \). As to those methods which approximate directly to the areas or values of \( x \) and \( y \) by an infinite series, they all, without exception, involve us in most complicated expressions, with coefficients of sines and tangents, and ambiguous signs, and engage us in a calculation almost endless. And we know of no series which converges fast enough to give us tolerable accuracy, without such a number of terms as is sufficient to deter any person from the attempt. The calculation of the arches is very moderate, so that a person tolerably versant in arithmetical operations may compute an arch with its velocity and time in about five minutes. We have therefore no hesitation in preferring this method of Euler's to all that we have seen, and therefore proceed to determine some other circumstances which render its application more general. If there were no resistance, the smallest velocity would be at the vertex of the curve, and it would immediately increase by the action of gravity confining (in however small degree) with the motion of the body. But in a resisting medium, the velocity at the vertex is diminished by a quantity to which the acceleration of gravity in that point bears no assignable proportion. It is therefore diminished, upon the whole, and the point of smallest velocity is a little way beyond the vertex. For the same reasons, the greatest curvature is a little way beyond the vertex. It is not very material for our present purpose to ascertain the exact positions of those points.

The velocity in the descending branch augments continually; but it cannot exceed a certain limit, if the velocity at the vertex has been less than the terminal velocity; for when the curve is infinite, \( p \) is also infinite, and

\[ h = \frac{ap^2}{P}, \quad \text{because } n \text{ in this case is nothing in respect of } P, \]

which is infinite; and because \( p \) is infinite, the number hyp. log. \( p \times \sqrt{1 + p^2} \), though infinite, vanishes in comparison with \( p \times \sqrt{1 + p^2} \); so that in this case \( P = \frac{p^2}{2} \), and \( h = a \), and \( v \) is the terminal velocity.

If, on the other hand, the velocity at the vertex has been greater than the terminal velocity, it will diminish continually, and when the curve has become infinite, \( v \) will be equal to the terminal velocity.

In either case we see that the curve on this side will have a perpendicular asymptote. It would require a long and pretty intricate analysis to determine the place of this asymptote, and it is not material for our present purpose. The place and position of the other asymptote \( LO \) is of the greatest moment. It evidently distinguishes the kind of trajectory from any other. Its position depends on this circumstance, that if \( p \) marks the position of the tangent, \( n - P \), which is the denominator of the fraction expressing the square of the velocity, must be equal to nothing, because the velocity is infinite: therefore, in this place, \( P = n \), or \( n = \frac{x}{y} \times \left( \int \frac{p}{n-P} - \frac{1}{p} \int \frac{pp}{n-P} \right) \).

It is evident that the logarithms used in these expressions are the natural or hyperbolic. But the operations may be performed by the common tables, by making the value of the arch \( Mm \) of the curve \( = \frac{a}{M} \times \log. \)

\[ \frac{n+Q}{n+P}, \quad \text{where } M \text{ means the subtangent of the common logarithms, or } 0.43429; \text{ also the time of describing this arch will be expediently had by taking a medium } \mu \text{ between the values of } \frac{\sqrt{1+p^2}}{\sqrt{n+P}} \text{ and } \frac{\sqrt{1+q}}{\sqrt{n+Q}}, \]

and making the time \( = \frac{\sqrt{a}}{Ma \sqrt{g}} \times \log. \frac{n+Q}{n+P} \).

Such then is the process by which the form and magnitude of the trajectory, and the motion in it, may be determined. But it does not yet appear how this is to be applied to any question in practical artillery. In this process we have only learned how to compute the motion from the vertex in the descending branch till the ball has acquired a particular direction, and the motion to the vertex from a point of the ascending branch where the ball has another direction, and all this depending on the greatest velocity which the body can acquire by falling, and the velocity which it has in the vertex of the curve. But the usual question is, "What will be the motion of the ball projected in a certain direction with a certain velocity?"

The mode of application is this: Suppose a trajectory computed for a particular terminal velocity, produced by the fall \( a \), and for a particular velocity at the vertex, which will be characterized by \( n \), and that the velocity at that point of the ascending branch where the inclination of the tangent is \( 30^\circ \) is 900 feet per second. Then, we are certain, that if a ball, whose terminal velocity is that produced by the fall \( a \), be projected with the velocity of 900 feet per second, and an elevation of \( 30^\circ \), it will describe this very trajectory, and the velocity and time corresponding to every point will be such as is here determined.

Now this trajectory will, in respect to form, answer an infinity of cases: for its characteristic is the proportion of the velocity in the vertex to the terminal velocity. When this proportion is the same, the number \( n \) will be the same. If, therefore, we compute the trajectories for a sufficient variety of these proportions, we shall find a trajectory that will nearly correspond to any case that can be proposed; and an approximation sufficiently exact will be had by taking a proportional medium between the two trajectories which come nearest to the case proposed.

Accordingly, a set of tables or trajectories have been computed by the English translator of Euler's Compendium on Robins's Gunnery. They are in number 18, distinguished by the position of the asymptote of the ascending branch. This is given for \( 5^\circ, 10^\circ, 15^\circ, \ldots \) to \( 85^\circ \), and the whole trajectory is computed as far as it can ever be supposed to extend in practice. The following table gives the value of the number \( n \) corresponding to each position of the asymptote.

| OLB | n | |-----|-----| | 0 | 0.0000 | | 5 | 0.0876 | | 10 | 0.1772 | | 15 | 0.2771 | | 20 | 0.3718 | | 25 | 0.4826 | | 30 | 0.6079 | | 35 | 0.7538 | | 40 | 0.9291 |

Since the path of a projectile is much less incurvated, and more rapid in the ascending than in the descending branch, and the difference is so much the more remarkable in great velocities; it must follow, that the range on a horizontal or inclined plane depends most on the ascending branch: therefore the greatest range will not be made with that elevation which bisects the angle of position, but with a lower elevation; and the deviation from the bisecting elevation will be greater as the initial velocities... velocities are greater. It is very difficult to frame an exact rule for determining the elevation which gives the greatest range. We have subjoined a little table which gives the proper elevation (nearly) corresponding to the different initial velocities.

It was computed by the following approximation, which will be found the same with the series used by Newton in his Approximation.

Let \( e \) be the angle of elevation, \( a \) the height producing the terminal velocity, \( h \) the height producing the initial velocity, and \( c \) the number whose hyperbolic logarithm is 1 (i.e., the number 2,718).

Then,

\[ y = x \times \left( \tan e + \frac{a}{2h \cdot \cot e} \right) - \frac{a^2}{2h} \left( C \cdot \cot e - 1 \right), \]

&c. Make \( y = v \), and take the maximum by varying \( e \), we obtain

\[ \sin^2 e + \frac{a \cdot \tan e}{2h} = \text{hyperbol. log.} \]

\[ \left( 1 + \frac{2h}{a \cdot \tan e} \right), \]

which gives us the angle \( e \).

The numbers in the first column, multiplied by the terminal velocity of the projectile, give us the initial velocity; and the numbers in the last column, being multiplied by the height producing the terminal velocity, and by 2,3026, give us the greatest ranges. The middle column contains the elevation. The table is not computed with scrupulous exactness, the question not requiring it. It may, however, be depended on within one part of 2000.

To make use of this table, divide the initial velocity by the terminal velocity \( u \), and look for the quotient in the first column. Opposite to this will be found the elevation giving the greatest range; and the number in the last column being multiplied by 2,3026 \( \times a \) (the height producing the terminal velocity) will give the range.

**Table of Elevations giving the greatest Range.**

| Initial vel. | Elevation | Range | |--------------|-----------|-------| | 0.6909 | 43°.40' | 0.1751 | | 0.7820 | 43°.20 | 0.2169 | | 0.8645 | 42°.50 | 0.2548 | | 1.3817 | 41°.00 | 0.4999 | | 1.5641 | 40°.20 | 0.5789 | | 1.7291 | 40°.10 | 0.6551 | | 2.0726 | 39°.50 | 0.7877 | | 2.3461 | 37°.20 | 0.8967 | | 2.5936 | 35°.50 | 0.9752 | | 2.7635 | 35°.00 | 1.0319 | | 3.1881 | 34°.40 | 1.1411 | | 3.4544 | 34°.20 | 1.2298 | | 3.4581 | 34°.20 | 1.2277 | | 3.9101 | 33°.50 | 1.3371 | | 4.1452 | 33°.30 | 1.3901 | | 4.3227 | 33°.30 | 1.4274 | | 4.6921 | 31°.50 | 1.5050 | | 4.8631 | 31°.50 | 1.5341 |

Such is the solution which the present state of our mathematical knowledge enables us to give of this celebrated problem. It is exact in its principle, and the application of it is by no means difficult, or even operose.

But let us see what advantage we are likely to derive from it.

In the first place, it is very limited in its application. There are few circumstances of general coincidence, and almost every case requires an appropriated calculus. Perhaps the only general rules are the two following:

1. Balls of equal density, projected with the same elevation, and with velocities which are as the square-roots of their diameters, will describe similar curves.—This is evident, because, in this case, the resistance will be in the ratio of their quantities of motion. Therefore all the homologous lines of the motion will be in the proportion of the diameters.

2. If the initial velocities of balls projected with the same elevation are in the inverse subduplicate ratio of the whole resistances, the ranges, and all the homologous lines of their track, will be inversely as those resistances.

These theorems are of considerable use; for by means of a proper series of experiments on one ball projected with different elevations and velocities, tables may be constructed which will ascertain the motions of an infinity of others.

But when we take a retrospective view of what we have done, and consider the conditions which were variously confused in the solution of the problem, we shall find that much yet remains before it can be rendered of great practical use, or even satisfy the curiosity of the man of science. The resistance is all along supposed to be in the duplicate ratio of the velocity; but even theory points out many causes of deviation from this law, such as the pressure and condensation of the air, in the case of very swift motions; and Mr Robin's experiments are sufficient to show us that the deviations must be exceedingly great in such cases. Mr Euler and all subsequent writers have allowed that it may be three times greater, even in cases which frequently occur; and Euler gives a rule for ascertaining with tolerable accuracy what this increase and the whole resistance may amount to. Let \( H \) be the height of a column of air whose weight is equivalent to the resistance taken in the duplicate ratio of the velocity. The whole resistance will be expressed by \( H + \frac{H^2}{28845} \). This number 28845 is the height in feet of a column of air whose weight balances its elasticity. We shall not at present call in question his reasons for affixing this precise addition. They are rather reasons of arithmetical conveniency than of physical import. It is enough to observe, that if this measure of the resistance is introduced into the process of investigation, it is totally changed; and it is not too much to say, that with this complication it requires the knowledge and address of a Euler to make even a partial and very limited approximation to a solution.—Any law of the resistance, therefore, which is more complicated than what Bernoulli has assumed, namely, that of a simple power of the velocity, is abandoned by all the mathematicians, as exceeding their abilities; and they have attempted to avoid the error arising from the assumption of the duplicate ratio of the velocity, either by supposing the resistance throughout the whole trajectory to be greater than what it is in general, or they have divided the trajectory into different portions, and assigned different resistances to each, which vary,

vary, through the whole of that portion, in the duplicate ratio of the velocities. By this kind of patchwork they make up a trajectory and motion which corresponds, in some tolerable degree, with what? With an accurate theory? No; but with a series of experiments. For, in the first place, every theoretical computation that we make, proceeds on a supposed initial velocity; and this cannot be ascertained with any thing approaching to precision, by any theory of the action of gunpowder that we are yet possessed of. In the next place, our theories of the resisting power of the air are entirely established on the experiments on the flights of shot and shells, and are corrected and amended till they tally with the most approved experiments we can find. We do not learn the ranges of a gun by theory, but the theory by the range of the gun. Now the variety and irregularity of all the experiments which are appealed to are so great, and the acknowledged difference between the resistance to slow and swift motions is also so great, that there is hardly any supposition which can be made concerning the resistance, that will not agree in its results with many of those experiments. It appears from the experiments of Dr Hutton of Woolwich, in 1784, 1785, and 1796, that the shots frequently deviated to the right or left of their intended track 200, 300, and sometimes 400 yards. This deviation was quite accidental and anomalous, and there can be no doubt but that the shot deviated from its intended and supposed elevation as much as it deviated from the intended vertical plane, and this without any opportunity of measuring or discovering the deviation. Now, when we have the whole range from one to three to choose among for our measure of resistance, it is evident that the confirmations which have been drawn from the ranges of shot are but feeble arguments for the truth of any opinion. Mr Robins finds his measures fully confirmed by the experiments at Metz and at Minorca. Mr Muller finds the same. Yet Mr Robins's measure both of the initial velocity and of the resistance are at least treble of Mr Muller's; but by compensation they give the same results. The Chevalier Borda, a very expert mathematician, has adduced the very same experiments in support of his theory, in which he abides by the Newtonian measure of the resistance, which is about \( \frac{1}{3} \) of Mr Robins's, and about \( \frac{1}{2} \) of Muller's.

What are we to conclude from all this? Simply this, that we have hardly any knowledge of the air's resistance, and that even the solution given of this problem has not as yet greatly increased it. Our knowledge confines only in those experiments, and mathematicians are attempting to patch up some notion of the motion of a body in a resisting medium, which shall tally with them.

There is another essential defect in the conditions assumed in the solution. The density of the air is supposed uniform; whereas we are certain that it is less by one-fifth or one-sixth towards the vertex of the curve, in many cases which frequently occur, than it is at the beginning and end of the flight. This is another latitude given to authors in their assumptions of the air's resistance. The Chevalier de Borda has, with considerable ingenuity, accommodated his investigation to this circumstance, by dividing the trajectory into portions, and, without much trouble, has made one equation answer them all. We are disposed to think that his solution of the problem (in the Memoirs of the Academy of Paris for 1769) corresponds better with the physical circumstances of the case than any other. But this process is there delivered in too concise a manner to be intelligible to a person not perfectly familiar with all the resources of modern analysis. We therefore preferred John Bernoulli's, because it is elementary and rigorous.

After all, the practical artillery must rely chiefly on necessity of the records of experiments contained in the books of attending practice at the academies, or those made in a more public manner. Even a perfect theory of the air's resistance can do him little service, unless the force of gunpowder were uniform. This is far from being the case even in the same powder. A few hours of a damp day will make a greater difference than occurs in any theory; and, in service, it is only by trial that every thing is performed. If the first shell fall very much short of the mark, a little more powder is added; and, in cannonading, the correction is made by varying the elevation.

We hope to be forgiven by the eminent mathematicians for these observations on their theories. They by no means proceed from any disrespect for their labours. We are not ignorant of the almost insuperable difficulty of the task, and we admire the ingenuity with which some of them have contrived to introduce into their analysis reasonable substitutions for those terms which would render the equations intractable. But we must still say, upon their own authority, that these are but ingenious guesses, and that experiment is the touchstone by which they mould these substitutions; and when they have found a coincidence, they have no motive to make any alteration. Now, when we have such a latitude for our measure of the air's resistance, that we may take it of any value, from one to three, it is no wonder that compensation of errors should produce a coincidence; but where is the coincidence? The theorist supposes the ball to set out with a certain velocity, and his theory gives a certain range; and this range agrees with observation—but how? Who knows the velocity of the ball in the experiment? This is concluded from a theory incomparably more uncertain than that of the motion in a resisting medium.

The experiments of Mr Robins and Dr Hutton show, in the most incontrovertible manner, that the resistance to a motion exceeding 1100 feet in a second, is almost three times greater than in the duplicate ratio to the resistance to moderate velocities. Euler's translator, in his comparison of the author's trajectories with experiment, supposes it to be no greater. Yet the coincidence is very great. The same may be said of the Chevalier de Borda's. Nay, the same may be said of Mr Robins's own practical rules: for he makes his \( F \), which corresponds to our \( a \), almost double of what these authors do, and yet his rules are confirmed by practice. Our observations are therefore well founded.

But it must not be inferred from all this, that the theory of physical theory is of no use to the practical artillery, is still of some use in practice. It plainly shows him the impropriety of giving the projectile an enormous velocity. This velocity is of no effect after 200 or 300 yards at farthest, because it is so rapidly reduced by the prodigious resistance of the air. Mr Robins has deduced several practical maxims of the greatest importance from what we already know of this subject, and which could hardly have been even conjectured without this knowledge. See Gunnery. And it must still be acknowledged, that this branch of physical science is highly interesting to the philosopher; nor should we despair of carrying it to a greater perfection. The defects arise almost entirely from our ignorance of the law of variation of the air's resistance. Experiments may be contrived much more conducive to our information here than those commonly referred to. The oblique flights of projectiles are, as we have seen, of very complicated investigation, and ill fitted for instructing us; but numerous and well contrived experiments on the perpendicular ascents are of great simplicity, being affected by nothing but the air's resistance. To make them instructive, we think that the following plan might be pursued. Let a set of experiments be premised for ascertaining the initial velocities. Then let shells be discharged perpendicularly with great varieties of density and velocity, and let nothing be attended to but the height and the time; even a considerable deviation from the perpendicular will not affect either of these circumstances, and the effect of this circumstance can easily be computed. The height can be ascertained with sufficient precision for very valuable information by their light or smoke. It is evident that these experiments will give direct information of the air's retarding force; and every experiment gives us two measures, viz. the ascent and descent: and the comparison of the times of ascent and descent, combined with the observed height in one experiment made with a great initial velocity, will give us more information concerning the air's resistance than 50 ranges. If we should suppose the resistance as the square of the velocity, this comparison will give in each experiment an exact determination of the initial and final velocities, which no other method can give us. These, with experiments on the time of horizontal flights, with known initial velocities, will give us more instruction on this head than any thing that has yet been done; and till something of this kind is carefully done, we presume to say that the motion of bodies in a resisting medium will remain in the hands of the mathematicians as a matter of curious speculation. In the mean time, the rules which Mr Robins has delivered in his Gunnery are very simple and easy in their use, and seem to come as near the truth as any we have met with. He has not informed us upon what principles they are founded, and we are disposed to think that they are rather empirical than scientific. But we profess great deference for his abilities and penetration, and doubt not but that he had framed them by means of as scientific a discussion as his knowledge of this new and difficult subject enabled him to give it.

We shall conclude this article, by giving two or three tables, computed from the principles established above, and which serve to bring into one point of view the chief circumstances of the motion in a resisting medium. Although the result of much calculation, as any person who considers the subject will readily see, they must not be considered as offering any very accurate results; or that, in comparison with one or two experiments, the differences shall not be considerable. Let any person peruse the published registers of experiments which have been made with every attention, and he will see such enormous irregularities, that all expectations of perfect agreement with them must cease. In the experiments at Woolwich in 1735, which were continued for several days, not only do the experiments of one day differ among themselves, but the mean of all the experiments of one day differs from the mean of all the experiments of another no less than one-fourth of the whole. The experiments in which the greatest regularity may be expected, are those made with great elevations. When the elevation is small, the range is more affected by a change of velocity, and still more by any deviation from the supposed or intended direction of the shot.

The first table shows the distance in yards to which a ball projected with the velocity 1600 will go, while its velocity is reduced one-tenth, and the distance at which it drops 16 feet from the line of its direction. This table is calculated by the resistance observed in Mr Robins's experiments. The first column is the weight of the ball in pounds. The second column remains the same whatever be the initial velocity; but the third column depends on the velocity. It is here given for the velocity which is very usual in military service, and its use is to assist us in directing the gun to the mark.—If the mark at which a ball of 24 pounds is directed is 474 yards distant, the axis of the piece must be pointed 16 feet higher than the mark. These deflections from the line of direction are nearly as the squares of the distances.

| I. | II. | III. | |----|-----|------| | 2 | 92 | 420 | | 4 | 121 | 428 | | 9 | 159 | 456 | | 18 | 200 | 470 | | 32 | 272 | 479 |

The next table contains the ranges in yards of a 2 pound shot, projected at an elevation of 45°, with the different velocities in feet per second, expressed in the first column. The second column contains the distances to which the ball would go in vacuo in a horizontal plane; and the third contains the distances to which it will go through the air. The fourth column is added, to show the height to which it rises in the air; and the fifth shows the ranges corrected for the diminution of the air's density as the bullet ascends, and may therefore be called the corrected range.

| I. | II. | III. | IV. | V. | |----|-----|------|-----|----| | 200| 416 | 349 | 166 | 360 | | 400| 1664| 1121 | 338 | 1150| | 600| 3740| 1812 | 606 | 1859| | 800| 6649| 2373 | 866 | 2435| | 1000|10300| 2845 | 1138| 2919| | 1200|14961| 3259 | 1378| 3343| | 1400|20364| 3640 | 1606| 3734| | 1600|26597| 3950 | 1814| 4050| | 1800|33663| 4235 | 1992| 4345| | 2000|41559| 4494 | 2168| 4610| | 2200|50286| 4720 | 2348| 4842| | 2400|59846| 4917 | 2460| 5044| | 2600|70306| 5106 | 2630| 5238| | 2800|81729| 5293 | 2762| 5430| | 3000|94125| 5455 | 2862| 5596| | 3200|107500| 5732 | | | The initial velocities can never be pushed as far as we have calculated in this table; but we mean it for a table of more extensive use than appears at first sight. Recollect, that while the proportion of the velocity at the vertex to the terminal velocity remains the same, the curves will be similar: therefore, if the initial velocities are as the square-roots of the diameters of the balls, they will describe similar curves, and the ranges will be as the diameters of the balls.

Therefore, to have the range of a 12 pound shot, if projected at an elevation of 45°, with the velocity 1500; suppose the diameter of the 12 pounder to be \(d\), and that of the 24 pounder \(D\); and let the velocities be \(v\) and \(V\): Then say, \(\sqrt{d} : \sqrt{D} = 1500\), to a fourth proportional \(V\). If the 24 pounder be projected with the velocity \(V\), it will describe a curve similar to that described by the 12 pounder, having the initial velocity 1500. Therefore find (by interpolation) the range of the 24 pounder, having the initial velocity \(V\). Call this \(R\). Then \(D : d = R : r\), the range of the 12 pounder which was wanted, and which is nearly 3380 yards.

We see by this table the immense difference between the motions through the air and in a void. We see that the ranges through the air, instead of increasing in the duplicate ratio of the initial velocities, really increase slower than those velocities in all cases of military service; and in the most usual cases, viz. from 800 to 1600, they increase nearly as the square-roots of the velocities.

A set of similar tables, made for different elevations, would almost complete what can be done by theory, and would be much more expeditious in their use than Mr Euler's Trajectories, computed with great labour by his English translator.

The same table may also serve for computing the ranges of bomb-shells. We have only to find what must be the initial velocity of the 24 pound shot which corresponds to the proposed velocity of the shell. This must be deduced from the diameter and weight of the shell, by making the velocity of the 24 pounder such, that the ratio of its weight to the resistance may be the same as in the shell.

That the reader may see with one glance the relation of these different quantities, we have given this table, expressed in a figure (fig. 10.). The abscissa, Fig. 10., or axis \(DA\), is the scale of the initial velocities in feet per second, measured in a scale of 400 equal parts in Relation of yards of the range on a scale containing 800 yards in it. The ordinates to the curve \(ACG\) express the different quantities of the range on a scale containing 800 yards in it. The ordinates to the curve \(AXY\) express (by the same scale) the height to which the ball rises in the air.

The ordinate \(BC\) (drawn through the point of the abscissa which corresponds to the initial velocity 2000) is divided in the points 4, 9, 12, 18, 24, 32, 42, in the ratio of the diameters of cannon-shot of different weights; and the same ordinate is produced on the other side of the axis, till \(BO\) be equal to \(BA\); and then \(BO\) is divided in the subduplicate ratio of the same diameters. Lines are drawn from the point \(A\), and from any point \(D\) of the abscissa, to these divisions.

We see distinctly by this figure how the effect of the initial velocity gradually diminishes, and that in very great velocities the range is very little increased by its augmentation. The dotted curve \(APQR\), shows what the ranges in vacuo would be.

By this figure may the problems be solved. Thus, to find the range of the 12 pounder, with the initial velocity 1500. Set off 1500 from \(B\) to \(F\); draw \(FH\) parallel to the axis, meeting the line \(12A\) in \(H\); draw the ordinate \(HK\); draw \(KL\) parallel to the axis, meeting \(24B\) in \(L\); draw the ordinate \(LM\), cutting \(12B\) in \(N\). \(MN\) is the range required.

If curves, such as \(ACG\), were laid down in the same manner for other elevations, all the problems might be solved with great dispatch, and with much more accuracy than the theory by which the curves are drawn can pretend to.

Note, that fig. 10. as given on Plate CCCCXLII. is one-half less than the scale according to which it is described; but the practical mathematician will find no difficulty in drawing the figure on the enlarged scale to correspond to the description.