PROJECTILES

P R O J E C T I L E S.

THIS is the name for that part of mechanical philosophy which treats of the motion of bodies any how 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 to affect to pay any regard to the deviations from equality and parallelism. A bullet rising a mile above the surface of the earth loses only $\backslash frac\{1\}\{1000\}$ of its weight, and a horizontal range of 4 miles makes only $\backslash frac\{1\}\{4\}$ 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

PROGRESSION, in general, denotes a regular advancing, or going forward, in the same course and manner.

PROGRESSION, in mathematics, is either arithmetical or geometrical. Continued arithmetic proportion is, where the terms do increase and decrease by equal differences, and is called arithmetic progression:

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 descent, or retards the ascent, of a body. A constant or invariable accelerating force is one which produces an uniform acceleration; that is, which in equal times produces equal increments of velocity, and therefore produces increments of velocity proportional to the times in which they are produced. Forces are of themselves imperceptible, and are seen only in their effects; and they have no measure but the effect, or what measures the effect; and every thing which we can discover with regard to those measures, we must affirm with regard to the things of which we 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.

Cor. 1. If bodies simply fall, not being projected downwards by any 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

quired 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 MECHANICS, sect. vi. § 14, 15, 16, &c. and in 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. See another in OPTICS, no 127. for defining the motion of light in refraction, &c.

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 steel-yard, 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 $\backslash frac\{1\}\{3600\}$ 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 with precision the fall of heavy bodies, in order to have of deter. an exact value of the accelerating power of terrestrial gravity. Now we must here observe, that this measure may be taken in two ways: we may take the space through which the heavy body falls in a second; or we may take the velocity which it acquires in consequence of gravity having acted on it during a second. The last is the proper measure; for the last is the immediate effect on the body. The action of gravity has changed the state of the body—in what way? by giving it a determination to motion downward: 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 bodies 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 recollect, that this is but one half of the true measure of the accelerating force. Mathematicians of the first rank 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 Ha, Hb, Hc, and their respective ordinates Aa, Bb, Cc; and he measures the deflections by the changes made on the increments of the ordinates. Thus the increment of the ordinate Aa, while the body describes the arch AB of the curve, is BG. If the deflecting force were to

cease

cause 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 algebraic (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 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 Leipic Acta, 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 inventions 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 Leipic Acta. 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 Haye 1714, and to John Bernoulli in 1719. Newton has been accused of a similar oversight by John Bernoulli, (who indeed calls it a mistake in prin-

ciple) in his Proposition X. Book 2. on the very subject we are now considering. But Dr Keill has shown it to be only an oversight, in drawing the tangent on the wrong side of the ordinate. For in this very proposition Newton exhibits, in the strictest and most beautiful manner, the difference between the geometrical and algebraical manner of considering the subject; and expressly warns the reader, that his algebraical symbol expresses the deflection only, and not the variation of the increment of the ordinate. It is therefore in the last degree improbable that he would make this mistake. He most expressly does not; and 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 Bernoulli published an elegant dissertation on this subject in the Leipic Acta 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 Leipic Acta, 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 summi geometra, natus ad summorum geometrarum paralogismos corrigendos, summi candoris ut et modestie, he betrays himself by an unguarded warmth, when defending J. B.'s demonstration of the inverse problem of centripetal forces, by calling it MEAM demonstrationem.

Let our readers now consider the scope and intention of this dissertation on projectiles, and judge whether the author's aim was to instruct the world, or to acquire fame, by correcting Newton. The dissertation does not contain one theorem, one corollary, nor one step of argument, which is not to be found in Newton's first edition; nor has he gone farther than Newton's single proposition the Xth. To us it appears an exact companion to his proposition on centripetal forces, which he boasts of having first demonstrated, although it is in every step a transcript of the 42d of the 1st Book of Newton's Principia, the geometrical language of Newton being changed into algebraic, as he has in the present case changed Newton's algebraic analysis into a very elegant geometrical one.

We hope to be forgiven for this long digression. It is a very curious piece of literary history, and shows the combination which envy and want of honourable principle had formed against the reputation of our illustrious

illustrious countryman; and we think it our duty to embrace any opportunity of doing it justice.—To return to our subject:

¹³ Accurate measure of the accelerative power of gravity. The accurate measure of the accelerative power of gravity, is the fall $16\backslash frac\{1\}\{2\}$ feet, if we measure it by the space, or the velocity of $32\backslash frac\{1\}\{2\}$ 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.

¹⁴ General formula deduced. 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 $b$ 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.

¹⁵ Examples of their use, red, in falling bodies.

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

1. To find the time of falling through 256 feet. Here $b\; =\; 256$, $\backslash sqrt\{256\}\; =\; 16$, and $\backslash frac\{16\}\{4\}\; =\; 4$. Answer 4".

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

3. To find the velocity acquired by falling 625 feet. $b\; =\; 625$, $\backslash sqrt\{b\}\; =\; 25$, $8\backslash sqrt\{b\}\; =\; 200$ feet per second.

¹⁶ In bodies projected upwards. 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.

¹⁷ And directed downwards. 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 $4\; \backslash times\; 32$, 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 $s$, through which it must fall to acquire the velocity of projection, 10 feet, and the time

of falling from this height. Then find the time $x$ of falling through the height $296\; +\; s$, 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.

¹⁸ Writers on the higher parts of mechanics always 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:

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

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 $\backslash sqrt\{v\}$. To reduce such an expression of a velocity to numbers, we must multiply it by $\backslash sqrt\{2g\}$, or by $2\backslash sqrt\{g\}$, according as we make $g$ to be the generated velocity, or the space fallen through in the unit of time.

¹⁹ Bodies. This will suffice for the perpendicular ascents or descents of heavy bodies, and we proceed to consider their projected 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,

vity, 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, $BC\; :\; BH\; =\; T\; :\; T$, $BC\; :\; T,\; BH\; =\; T,\; BE\; :\; T,\; BK$. But, because the motion along BK 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, 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 $EV^2\; =\; 4\; BE^2\; =\; 4\; BE\; \backslash times\; AB\; =\; BE\; \backslash times\; 4\; AB$, and $4\; AB$ 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 ab, fg; let the points a, f, continually approach A and d, and ultimately coincide with them. It is evident that Bb will ultimately be to gG, in the ratio of the velocity at B to the velocity at G; for the portions of the tangent ultimately coincide with the portions of the curve, and are described in equal times; but Bb is to gG as BH to TG: therefore the velocity at B is to that at G as BH to TG. But, by the properties of the parabola, $BH^2$ is to

$TG^2$ as AB to dG; and AB is to dG as the square of the velocity acquired by falling through AB to the square of the velocity acquired by falling through dG; and the velocity in BH, or in the point B of the parabola, is the velocity acquired by falling along AB; therefore the velocity in TG, or in the point G of the parabola, is the velocity acquired by falling along dG.

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

We do not propose, therefore, to dwell much on this 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 centres A and B, with the distance AB, describe the semicircles ODB, AHK, and with the axis AB, and semiaxis GE, equal to AB, describe the semi-ellipse AEB: with the focus B, vertex A, diameter AB, and tangent AD, parallel to the horizon, describe the parabola APS.

Let a body be projected from B, in any direction BC, with the velocity acquired by falling through AB. By what has already been demonstrated, it will describe a parabola BVPM. Then,

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

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

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

If C is in the upper quadrant of ODB, F is in the upper

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 locus of the vertex of all the parabolas, and the vertex V of any one of them BVM is in the intersection of this ellipse with the vertical CF: for let this vertical cut the horizontal lines AD, GE, BN, in $s$, $\backslash lambda$, $\backslash kappa$. Then it is plain that $N\backslash lambda$ is half of $Ns$, and $\backslash lambda\; V$ is half of $Cs$; 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 $P\backslash lambda$ parallel to AB, cutting the directrix O$\backslash lambda$ of the parabola APS in $\backslash lambda$, and the directrix AL of the parabola BVM in $\backslash lambda$, then $PB=PA$; but $BF=BA$, $=AO$, $=\backslash lambda$: therefore $P\backslash lambda=PF$, and the point P is in the parabola BVM. Also the tangents to both parabolas in P coincide, for they bisect the angle $\backslash lambda\; PB$; 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 locus of the greatest ranges 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 $45^\backslash circ$.

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 LFs, it will cut the circle AHK in two points F and $f$, which are evidently the foci of two parabolas BVM, B$\backslash circ$M, 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 $OBt\; =\; \backslash frac\{1\}\{2\}OBS$, 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 arcs Cc, $tC$ are equal; and therefore the angles CBt, $tBt$ are equal.

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

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

But we must observe, that this theory is of little service for directing us in the practice of cannonading. Here it is necessary to come as near as we can to the object aimed at, and the hurry of service allows no time for geometrical methods of pointing the piece after each discharge. The gunner either points the cannon directly to the object, when within 200 or 300 yards of it, in which case he is said to shoot point blank (point 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 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 post, 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 carcasses are therefore thrown upwards, so as to get over the defences and produce their effect.

These shells are of very great weight, frequently exceeding 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.

²⁴ The moving force in theory different from that in practice.
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 40,000 feet or 1333 yards.

Plate CCCCVII. 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 b.

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

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

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

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

The relations between all the circumstances of velo-

city, distance, position, elevation, and time, may be included in the following propositions.

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

Make ZA = 4AC, and draw BD perpendicular to the horizon. Describe on ZA an arch of a circle ZDA, containing an angle equal to DBA, and draw AD to the intersection of this circle with DB; then will a body projected from A, in the direction AD, with the velocity acquired by falling through CA, hit the mark B.

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

Therefore BD : DA = DA : AZ,

And DA² = BD × AZ;

Therefore BF = AF × AZ, = AF × 4AC.

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

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

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 = S², p × S, e : S², e × S, z; = S², p : S, e × S, z
Or 4b : r = S², p : S, e × S, z, and 4b × S, e × S, z = r × S², p

Hence we obtain the relations wanted.

Thus $b\; =\; \backslash frac\{r\; \backslash times\; S^2\; \backslash cdot\; p\}\{4S,\; e\; \backslash times\; S,\; z\}$, and $r\; =\; \backslash frac\{4b\; \backslash times\; S,\; e\; \backslash times\; S,\; z\}\{S^2\; \backslash cdot\; p\}$

And $S,\; z\; =\; \backslash frac\{r\; \backslash times\; S^2\; \backslash cdot\; p\}\{4b\; \backslash times\; S,\; e\}$, and $S,\; e\; =\; \backslash frac\{r\; \backslash times\; S^2\; \backslash cdot\; p\_z\}\{4b\; \backslash times\; S,\; z\}$

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

Now AB : DB = Sin, ADB : Sin, DAB, = S, z : S, e, and DB = $\backslash frac\{r\; \backslash times\; S,\; e\}\{S,\; z\}$. But the time of the flight is the same with the time of falling through DB, and 16 feet : DB = 1' : 1". Hence $t^2\; =\; \backslash frac\{r\; \backslash times\; S,\; e\}\{16S,\; z\}$, 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 fines of twice the elevation.

A ball discharged at the elev. 19°. 5' ranged 448 yards at 9.45 330

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.

²⁸ This comparison shows the deficiency of the theory. Such facts show incontrovertibly how deficient the parabolic theory is, and how unfit for directing the practice of the artillerist. 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 essay of skill. They are tempted to write a book on the subject; and it finds readers among other young mathematicians, 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 artillerist. 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.

²⁹ Causes of this deficiency. But it is the business of a philosopher to inquire into the causes of such a prodigious deviation from a well founded theory, and having discovered them, to ascertain precisely the deviations they occasion. Thus we shall obtain another theory, either in the form of the parabolic

theory corrected, or as a subject of independent discussion. This we shall now attempt.

³⁰ Effect of the atmosphere. The motion of projectiles is performed in the atmosphere. The air is displaced, or put in motion. What ever 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 ½ of an ounce. Suppose this ball to move 800 feet in a second, that is 32 times faster, its resistance would be 32×32 times ½ 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.

³¹ Compared with that of gravity. This resistance, operating constantly and uniformly on the ball, must take away four times as much from its velocity as its gravity would do in the same time. 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, 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.

³² And considered as a retarding force. 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 $R\text{'}$ on each particle, and is equivalent to $R\text{'}M$: and as $g$ is equal to $\backslash frac\{W\}\{M\}$, so $R\text{'}$ is equal to $\backslash frac\{R\}\{M\}$. We shall keep this distinction in view, by adding the differential mark ' to the letter $R$ or $r$, which expresses the aggregate resistance.

³³ The resistance of the air not uniform. 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 fillyard, and exposed to the impulse of the wind. This will compress the fillyard 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 theo-

rems for perpendicular ascents give us $g\; =\; \backslash frac\{u^2\}\{2a\}$, $u$ and $a$ being both numbers representing units of space; therefore, in the present case, we have $r\; =\; \backslash frac\{u^2\}\{2a\}$. For the whole resistance $r$, or $r/M$, is supposed equal to the weight, or to $gM$; and therefore $r\text{'}$ is equal to $g$, $=\; \backslash frac\{u^2\}\{2a\}$ and $2a\; =$

$\backslash frac\{u^2\}\{g\}$. 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 $\backslash frac\{1\}\{32\}$ 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 $\backslash rho$ be the weight of the body and $\backslash sigma$ that of an equal bulk of air: the accelerative force of relative gravity on each particle will be $g\; \backslash times\; 1\; -\; \backslash frac\{\backslash sigma\}\{\backslash rho\}$; and this relative accelerating force might be distinguished by another symbol $\backslash gamma$. But in all cases in which we have any interest, and particularly in military projectiles, $\backslash frac\{\backslash sigma\}\{\backslash rho\}$ 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 $32\; \backslash frac\{1\}\{32\}$ which it should be.

Let $e$ be the time of this ascent in opposition to gravity. The same theorems give us $e\; u\; =\; 2a$; 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 $1$.

Since the resistances are as the squares of the velocities, and the resistance to the velocity $u$ is $\backslash frac\{u^2\}\{2a\}$, $R$ will be $\backslash frac\{1\}\{2a\}$. 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 $1$ in the time $\backslash frac\{1\}\{g\}$, $=$ (in these measures) to $\backslash frac\{1\}\{u^2\}\; =\; \backslash frac\{2a\}\{u^2\}$. Therefore we have the following proportions $\backslash frac\{1\}\{2a\}\; (=R)\; :\; \backslash frac\{u^2\}\{2a\}\; (=g)\; =\; \backslash frac\{2a\}\{u^2\}\; :\; 2a$, and $2a$ is equal to $E$, the time in which the velocity $1$ will be extinguished by the uniform action of the resistance competent to this velocity.

The velocity $1$ 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 $1$. Therefore the space thus described by a motion which begins with the velocity $1$, 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

$r$, and $r$ the resistance to its motion with the terminal velocity $u$; we must have $r\; =\; R\; \backslash times\; u^2$, and this must be $=\; W$ the weight. Therefore, to obtain the terminal velocity, divide the weight by the resistance to the velocity $1$, and the quotient is the square of the terminal velocity, or $\backslash frac\{W\}\{R\}\; =\; u^2$: 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 $=\; \backslash frac\{u^2\}\{2g\}$, and the time of the fall $=\; \backslash frac\{u\}\{g\}\; =\; \backslash epsilon$, and $ru\; =\; 2a$, $=$ 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 $\backslash epsilon$, $R$, which is $u^2$ times smaller, will extinguish it in the time $u^2\backslash epsilon$, and $R$ will extinguish the velocity $1$, which is $u$ times less than $u$, in the time $u\backslash epsilon$, that is, in the time $2a$; and the body, moving uniformly during the time $2a$, $=\; E$, with the velocity $1$, will describe the space $2a$; and, if the body begin to move with the velocity $1$, 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 $1$ 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 $1$ 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 $1$. 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 $\backslash frac\{v^2\}\{2a\}\; =\; r$.

The extinguishing time will be $\backslash frac\{E\}\{v\}\; =\; \backslash epsilon$, $=\; \backslash frac\{2a\}\{v\}$, and $\backslash epsilon\; v\; =\; 2a$; 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 $\backslash epsilon\; v$ is 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 resisted, 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 $\backslash frac\{1\}\{d\}$. When the diameter was $1$, the resistance to a velocity $1$ was $\backslash frac\{1\}\{2a\}$. It must now

be $\backslash frac\{1\}\{2ad\}$. The time in which this diminished resistance will extinguish the velocity $1$ 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 $1$ 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 $g\; =\; \backslash frac\{w^2\}\{2ad\}$, and therefore $w^2\; =\; 2gad$, and $w\; =\; \backslash sqrt\{2gad\}\; =\; \backslash sqrt\{2ga\}\; \backslash sqrt\{d\}$. But $u\; =\; \backslash sqrt\{2ga\}$. Therefore the terminal velocity $w$ for this body is $u\backslash sqrt\{d\}$; 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 $\backslash frac\{Ed\}\{v\}$.

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 $=\; u\backslash sqrt\{d\}\; \backslash frac\{m\}\{n\}\; =\; \backslash sqrt\{2gad\}\; \backslash frac\{m\}\{n\}$.

The producing fall in vacuo $=\; ad\; \backslash frac\{m\}\{n\}$.

The retarding power of resistance to any velocity $=$

The extinguishing time for any velocity $v\; =\; \backslash frac\{Edm\}\{vn\}$.

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 $u$, or $a$; any one of these will determine the others.

Sir Isaac Newton has attempted to determine $r$ by Sir Isaac's 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 $\backslash frac\{1\}\{2\}$ 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

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 $\backslash frac\{1\}\{2\}\; m\; d$, $d$ being the diameter of the ball. Now 1 is to $m$ as the weight of the displaced air to the weight of the ball, or as $\backslash frac\{1\}\{2\}$ 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 $\backslash frac\{1\}\{2\}\; m\; d$. 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 $=\; 2a$, or $\backslash frac\{1\}\{2\}\; m\; d$. Now its weight would extinguish this velocity, or destroy this motion, in the same time, that is, in the time of describing $\backslash frac\{1\}\{2\}\; m\; d$; but the resistance of the air would do this in the time of describing $\backslash frac\{1\}\{4\}\; m\; d$; 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 although this investigation is by far the most ingenious of any in the Principia, and sets his acuteness and address 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 difficillima. (See Schol. prop. 37. b. ii.)

It is known that the weight of a cubic foot of water is 62$\backslash frac\{1\}\{2\}$ pounds, and that the medium density of the air is $\backslash frac\{1\}\{144\}$ of water; therefore let $a$ be the height producing the velocity (in feet), and $d$ the diameter of the ball (in inches), and $\backslash pi$ the periphery of a circle whose diameter is 1; the resistance of the air will be $=\; \backslash frac\{62\backslash frac\{1\}\{2\}\}\{840\}\; \backslash times\; \backslash frac\{\backslash pi\}\{4\}$

We may take an example. A ball of cast iron weighing 12 pounds, is 4$\backslash frac\{1\}\{2\}$ inches in diameter. Suppose this

ball to move at the rate of 25$\backslash frac\{1\}\{2\}$ 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$\backslash frac\{1\}\{2\}$ feet. The area of its great circle is 0.11044 feet, or $\backslash frac\{11044\}\{10000\}$ of one foot. Suppose water to be 840 times heavier than air, the weight of the air incumbent on this great circle, and 9$\backslash frac\{1\}\{2\}$ feet high, is 0.081151 pounds: half of this is 0.0405755 or $\backslash frac\{405755\}\{1000000\}$, or nearly $\backslash frac\{1\}\{2\}$ 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 39
confront every theoretical conclusion with experiment. Necessity of 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 suit 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 40
propositions with experiment. Some were made by experiment.
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 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 $=\; na$, $n$ 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 descents. 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 intenseness 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 incon- 41
sistent with the observed motions of military projectiles, the theory in practice.
when the velocities are prodigious, that no application could be made which could be of any service for determining

mining the path and motion of cannon shot and bombs; and although Mr John Bernoulli gave in 1738 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 real pressure of the air on its anterior part, 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. Princip.); 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 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 having ascertained these velocities, he discovered 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, n^o 9, &c.

It appears (Robins's Math. Works, vol. i. page 205.) that a sphere of 4$\backslash frac\{1\}\{2\}$ inches in diameter, moving at the rate of 25$\backslash frac\{1\}\{2\}$ feet in a second, sustained a resistance of 0,4914 pounds, or $\backslash frac\{4914\}\{100000\}$ of a pound. This is a greater resistance than that of the Newtonian theory, which gave $\backslash frac\{4914\}\{100000\}$ 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 $\backslash frac\{6\}\{100\}$ 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 25$\backslash frac\{1\}\{2\}$ 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 Rouse and Mr Rouse 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 Rouse exceeds that resulting from Mr Robins's experiments nearly in the proportion of 7 to 10.

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 the 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 in 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 mechanic 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 in 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 Gravefande'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, n° 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.

⁴⁷ The result of Robins's experiments as yet most to be depended on. We are the more puzzled what measure to abide by, because Mr. Robins himself, in his Practical Propositions, does not make use of the result of his own experiments, but takes a much lower measure. We must content ourselves, however, with this experimental measure, because it is as yet the only one of which any account can be given, or well-founded opinion formed.

⁴⁸ Applied to the formula. Therefore, in order to apply our formula, we must reduce this experiment, which was made on a ball of $4\backslash frac\{1\}\{2\}$ inches diameter, moving with the velocity of $25\backslash frac\{1\}\{2\}$ 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 $R\; =\; \backslash frac\{0.04914\}\{4.5\; \backslash times\; 25.2^2\}$, being diminished in

the duplicate ratio of the diameter and velocity. This gives us $R\; =\; 0.0000381973$ pounds, or $\backslash frac\{3.81973\}\{100000\}$ 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

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 ball⁴⁹ moving with the velocity of 1670 feet in a second, of their which is nearly the velocity communicated by 16 lbs. use. of powder. The diameter is 5.603 inches.

Log. 334.4 lbs. = $r$ - 2.52426

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?

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

Mr. Muller, in his writings on this subject, gives a much

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 assumes an algebraic expression for a principle of mechanical argument; and from its consequence draws erroneous conclusions. He makes the resistance of a cylinder $\backslash frac\{1\}\{2\}$ 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-book. 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 as affected 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 abstruser 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 $\backslash dot\{v\}\; =\; f/t$, where $v$ is the momentary increment or decrement of the velocity, $f$ 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 $\backslash dot\{v^2\}\; =\; f\; s$. The first proposition is familiarly known. The second is the 39th of Newton's Principia, B. I. It is demonstrated in the article OPRES, p. 281. and is the most extensively useful proposition in mechanics.

These things being premised, let the straight line $AC$ (fig. 6.) 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 $OF$, $OC$ 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 to the hyperbola, they will be pro-

portional to the velocities of the body at the instants and the hyperbolic areas will be proportional to the spaces described during the times

For, suppose the time divided into an indefinite number of small and equal moments, draw the ordinates $ae,\; bd$, and the perpendiculars $b\backslash beta,\; a\backslash alpha$. Then, by the nature of the hyperbola, $AC:ae=OC:OC$; and $AC-ae:ae=OC-OC:OC$, that is, $Aa:ae=Cc:OC$, and $Aa:Cc=ae:OC=AC:ae$; $AC:OC$; in like manner, $B\backslash beta:Dd=BD:BD$; $BD:OD$. Now $Dd=Cc$, because the moments of time were taken equal, and the rectangles $AC:CO,\; BD:DO$, are equal, by the nature of the hyperbola; therefore $Aa:B\backslash beta=AC:ae:BD:bd$; but as the points $e,\; d$ continually approach, and ultimately coincide with $C,\; D$, the ultimate ratio of $AC:ae$ to $BD:bd$ is that of $AC$ to $BD$; therefore the momentary decrements of $AC$ and $BD$ are as $AC$ and $BD$. 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 $ACae$ is to $ACgf$ as the space described in the time $Cx$ to the space described in the time $Cg$ (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 $t$, this rectangle is $=EV$, or $E$, or $2a$, of our former disquisitions; and because all the rectangles, such as 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 $ev\; =\; E$, or $2a$.

3. Draw the tangent $Ax$; then, by the hyperbola $Cx=CO$: now $Cx$ is the time in which the resistance to the velocity $AC$ would extinguish it; for the tangent coinciding with the elemental arc $Aa$ 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 $ae$. If this operated uniformly like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle $ACx$.

This triangle, therefore, represents the height thro' 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

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 $AC\; :\; BD\; =\; OD$ (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 AVGF and BVCD are equal and $VC\; :\; VA\; =\; VG\; :\; VB$; therefore $t\; =\; \backslash frac\{V-v\}\{v\}$, and $e\; =\; \backslash frac\{v\}\{V-v\}$.

7. Any velocity is reduced in the proportion of $m$ to $n$ in the time $e\; \backslash frac\{m-n\}\{n\}$. For, let $AC\; :\; BD\; =\; m\; :\; n$; then $DO\; :\; CO\; =\; m\; :\; n$, and $DC\; :\; CO\; =\; m\; -\; n\; :\; n$, and $DC\; =\; \backslash frac\{m-n\}\{n\}\; CO$, or $t\; =\; e\; \backslash frac\{m-n\}\{n\}$. 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 and abscissæ exhibiting the relations of the times and velocities, and the areas exhibiting the relations of both 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, $e$, $fg$, BD, &c. be produced till they cut this curve in L, $p$, $n$, K, &c. and the abscissæ in L, $t$, $h$, $s$, &c. the ordinates $p$, $h$, $n$, $s$, K, &c. may be proportional to the hyperbolic areas $eAC$, $fgA$, $sAK$. Let us examine what kind of curve this will be.

Make $OC\; :\; Oe\; =\; O\; :\; Og$; then (Hamilton's Conics, IV. 14. Cor.), the areas $AC\; \backslash cdot\; e$, $e\; \backslash cdot\; fg$ are equal: therefore drawing $ps$, $nt$ perpendicular to OM, we shall have (by the assumed nature of the curve LpK), $Mt\; =\; st$; 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 abscissæ in arithmetical progression.

Also, let KN, MV touch the curve in K and L, and let OC be supposed to be to Oe, as OD to Od, and therefore Ce to Dd as OC to OD; and let these lines C, Dd be indefinitely small; then (by the nature of the curve) Lo is equal to Kr: for the areas aACe, bBDd are in this case equal. Also lo is to kr, as LM to KI, because $eC\; :\; dD\; =\; CO\; :\; DO$:

That is, the subtangent IN, or MV, is of the same magnitude as the ordinate KI.

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 KI as the rectangle BDOG or ACOF to the parabolic area BDCA.

For let $fgbn$ be an ordinate very near to BD; K; and let $bn$ cut the curve in $n$, and the ordinate KI in $q$; then we have

Therefore the sum of all the rectangles BD.Dg is to the sum of all the rectangles AC.qn, as CO to IN; but the sum of the rectangles BD.Dg is the space ACDB; and, because AC is given, the sum of the rectangles AC.qn is the rectangle of AC and the sum of all the lines qn; that is, the rectangle of AC and RL: therefore the space ACDB : AC.RL = CO : IN, and $ACDB\; \backslash times\; IN\; =\; AC\; \backslash cdot\; CO\; \backslash cdot\; RL$; and therefore $IN\; :\; RL\; =\; AC\; \backslash cdot\; CO\; :\; ACDB$.

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 conceiving 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, and ordinates are drawn from the points of division, the ordinates are a series of lines in geometrical progression, or are continual proportionals. 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,

MI, cutting the curve $L\backslash rho\; K$, 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, O$\backslash times$, O$\backslash gamma$ were taken in geometrical progression, M$\backslash rho$, M$\backslash iota$ should be in arithmetical progression. The abscissæ having ordinates equal to $\backslash rho$, $\backslash iota$, &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 $\backslash tau\; u$ which corresponds to the side O of the square $\backslash pi\; \backslash times\; O$ 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; $\backslash tau\; u$ is = 1, or the unit of numbers to which the logarithms are suited, and then IN is also 1. Now the square $\backslash pi\; \backslash times\; O$ being unity, the area BACD will be some number; $\backslash pi\; \backslash times\; O$ being also unity, OD is some number: Call it $x$. Then, by the nature of the hyperbola, $OB\; :\; O\backslash pi\; =\; \backslash pi\; \backslash times\; O\; :\; DB$: That is, $x\; :\; 1\; =\; 1\; :\; \backslash frac\{1\}\{x\}$, so that $DB$ is $\backslash frac\{1\}\{x\}$.

Now calling $Dd$ $x$, the area $BDdb$, which is the fluxion (ultimately) of the hyperbolic area, is $\backslash frac\{x\}\{x\}$. Now in the curve $L\backslash rho\; K$, MI has the same ratio to NI that BACD has to $\backslash pi\; \backslash times\; O$: 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 $Ii$, which corresponds to $BDdb$, is the fluxion (ultimately) of MI: Therefore, if MI be called the logarithm of $x$, $\backslash frac\{x\}\{x\}$ 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 $\backslash frac\{x\}\{x\}$ 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 $\backslash frac\{x\}\{x\}$ 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 $\backslash frac\{u^2\}\{g\}$ of the ordinate $\backslash tau\; u$, 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 ⁵⁷ an example of the different questions which may be proposed. ^{by examples.}

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

1. 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 $NI\; :\; MI\; =\; ACOF\; :\; BDCA$; 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 $\backslash frac\{u^2\; d\}\{g\}$, or Ed.

Therefore by common logarithms 0.43429 : $\backslash log.\; e+t\; -\; \backslash log.\; e\; =\; 2ad\; :\; S$, = space described,

or 0.43429 : $\backslash log.\; \backslash frac\{e+t\}\{e\}\; =\; 2ad\; :\; S$,

and $S\; =\; \backslash frac\{2ad\}\{0.43429\}\; \backslash times\; \backslash log.\; \backslash frac\{e+t\}\{e\}$,

by hyperbolic logarithms $S\; =\; 2ad\; \backslash times\; \backslash log.\; \backslash frac\{e+t\}\{e\}$.

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 $\backslash frac\{2ad\}\{V\}$.

And $e+t$ is 23", 03, of which the log. is 1.36229 from which take the log. of $e$

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

For the final velocity,
OD : OC = AC : BD, or $e+t\; :\; e\; =\; V\; :\; v$.
$23",\; 03\; :\; 3",\; 03\; =\; 1600\; :\; 210,\; =\; v$.

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.

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 $\backslash log\; \backslash frac\{AC\}\{BD\}\; =\; \backslash log\; \backslash frac\{OD\}\{OC\}$, or $\backslash log\; \backslash frac\{V\}\{v\}\; =\; \backslash log\; \backslash frac\{e+t\}\{e\}$.

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 $d$ is 5.68, and therefore the logarithm of $2ad$ is

This reduction will be produced in about $\backslash frac\{1\}\{2\}$ 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:

$\backslash frac\{e+t\}\{e\}$ add 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 $e+t\; :\; e\; =\; V\; :\; v$.

58 Application of an experiment of Mr Robins. See Robins's Math. Works, vol. i. p. 135.

We shall conclude these examples by applying this last rule to Mr Robins's experiment on a musket bullet of $\backslash frac\{1\}\{4\}$ 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. $d$ is 0.75, and $m$ is

But $e\; =\; \backslash frac\{2adm\}\{V\}\; =\; 0.763$, 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 $t\; =\; 0.062$, or $\backslash frac\{62\}\{1000\}$ of a second, for the time of passage. Now, to find the remaining velocity, say $825\; :\; 763\; =\; 1670\; :\; 1544\; =\; v$.

But in Mr Robins's experiment the remaining velocity was only 1425, the ball having lost 245; whereas by this computation it should have lost only 126. It appears, therefore, that the resistance is double of what it would have been if the resistance increased in the duplicate proportion of the velocity. Mr Robins says it is nearly triple. But he supposes the resistance to flow motions much smaller than his own experiment, so often mentioned, fully warrants.

The time $e$, in which the resistance of the air would extinguish the velocity is 0.763. Gravity, or the weight of the bullet, would have done it in $\backslash frac\{1670\}\{32\}$ or 52;

therefore the resistance is $\backslash frac\{52\}\{0.763\}$ times, or nearly 68 times its weight, by this theory, or 5.97 pounds. If we calculate from Mr Robins's experiment, we must say log.

proportion of the resistance to the weight, and makes the resistance 12.07 pounds, fully double of the other.

It is to be observed, that with this velocity, which greatly exceeds that with which the air can rush into a void, there must be a statical pressure of the atmosphere equal to 61 pounds. This will make up the difference, and allows us to conclude that the resistance arising solely from the motion communicated to the air follows very nearly the duplicate proportion of the velocity.

The next experiment, with a velocity of 1690 feet, gives a resistance equal to 157 times the weight of the bullet, and this bears a much greater proportion to the former than 1690 does to 1670; which shows, that although these experiments clearly demonstrate a prodigious augmentation of resistance, yet they are by no means susceptible of the precision which is necessary for discovering the law of this augmentation, or for a good foundation of practical rules; and it is still greatly to be wished that a more accurate mode of investigation could be discovered.

Thus we have explained, in great detail, the principles and the process of calculation for the simple case.

of the motion of projectiles through the air. The learned reader will think that we have been unreasonably prolix, and that the whole might have been comprised in less room, by taking the algebraic method. We acknowledge that it might have been done even in a few lines. But we have observed, and our observation has been confirmed by persons well versed in such subjects, that in all cases where the fluxionary process introduces the fluxion of a logarithm, there is a great want of distinct ideas to accompany the hand and eye. The solution comes out by a sort of magic or legerdemain, we cannot tell either how or why. We therefore thought it our duty to furnish the reader with distinct conceptions of the things and quantities treated of. For this reason, after showing, in Sir Isaac Newton's manner, how the spaces described in the retarded motion of a projectile followed the proportion of the hyperbolic areas, we shewed 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 shewed how the logarithm of a number really represents the path or space described by the projectile. And we were the more disposed to do this, because in the articles LOGARITHMS and LOGARITHMIC CURVE, there has not been that notice taken of it which would have been proper.

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.

60
Of the perpendicular ascents of heavy projectiles.

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 $u^2\; :\; v^2\; =\; g\; :\; \backslash frac\{g\; v^2\}\{u^2\}\; =\; r$, for the resistance to that velocity. In the descent the body is urged by gravity $g$, and opposed by the resistance $\backslash frac\{g\; v^2\}\{u^2\}$: therefore the remaining accelerating force, which we shall call $f$, is $g\; -\; \backslash frac\{g\; v^2\}\{u^2\}$, or $\backslash frac\{g\; u^2\; -\; g\; v^2\}\{u^2\}$, or $\backslash frac\{g(u^2\; -\; v^2)\}\{u^2\}\; =\; f$.

Now the fundamental theorem for varied motions is $f\; s\; =\; u\; \backslash dot\{v\}$, and $s\; =\; \backslash frac\{v\; \backslash dot\{v\}\}\{f\}\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}$, and $s\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash int\; \backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}\; +\; C$. Now the fluent of $\backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}$ is $=\; -\backslash text\{hyperb.\; log\; of\; \}\; \backslash sqrt\{u^2\; -\; v^2\}$. For the fluxion of $\backslash sqrt\{u^2\; -\; v^2\}$ is $\backslash frac\{-v\; \backslash dot\{v\}\}\{\backslash sqrt\{u^2\; -\; v^2\}\}$, and this divided by the quantity $\backslash sqrt\{u^2\; -\; v^2\}$, of which it is the fluxion, gives precisely $\backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}$, which is therefore the fluxion of

its hyperbolic logarithm. Therefore $S\; =\; -\backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\; -\; v^2\}\; +\; C$. Where $L$ means the hyperbolic logarithm of the quantity annexed to it, and $\backslash lambda$ 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 $0$, when the velocity is $0$: therefore $C\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\}\; =\; 0$, and $C\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\; -\; v^2\}$, and the complete fluent $S\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\; -\; v^2\}\; =\; L\; \backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$, or (putting $M$ for $0.43429$, the modulus or subtangent of the common logarithmic curve) $=\; \backslash frac\{u^2\}\{Mg\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$.

This equation establishes the relation between the space fallen through, and the velocity acquired by the fall. We obtain by it $\backslash frac\{g\; S\}\{u^2\}\; =\; L\; \backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$, and

$\backslash frac\{2g\; S\}\{u^2\}\; =\; L\; \backslash frac\{u^2\}\{u^2\; -\; v^2\}$, or, which is still more convenient for us, $\backslash frac\{M\; \backslash times\; 2g\; S\}\{u^2\}\; =\; \backslash lambda\; \backslash frac\{u^2\}\{u^2\; -\; v^2\}$, that is, equal to

the logarithm of a certain number: therefore having found the natural number corresponding to the fraction $M\; \backslash times\; 2g\; S$, consider it as a logarithm, and take out the number corresponding to it: call this $n$. Then, since

$n$ is equal to $\backslash frac\{u^2\}\{u^2\; -\; v^2\}$, we have $n\; u^2\; -\; n\; v^2\; =\; u^2$, and $n\; u^2\; -\; u^2\; =\; n\; v^2$, or $n\; v^2\; =\; u^2\; \backslash times\; n\; -\; v^2$, and $v^2\; =\; \backslash frac\{u^2\; \backslash times\; n\; -\; u^2\}\{n\}$.

To expedite all the computations on this subject, it will be convenient to have multipliers ready computed for $M\; \backslash times\; 2g$, 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 $\backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$ is the secant of an arch of a circle whose radius is $u$, and whose sine is $v$, or whose radius is unity and sine $=\; \backslash frac\{v\}\{u\}$: therefore, considering the above fraction as a logarithmic secant, look for it in the tables, and then take the sine 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$\backslash frac\{1\}\{2\}$ feet, and ascertain its velocity after a fall of 1848 feet. Here,

Then

$\backslash log.\; \backslash text\{\; of\; \}\; 0,10809\; =\; \backslash log.\; n\; =\; 9.03378$
$0,10809$ is the logarithm of $1,2826\; =\; n$, and $n\; -\; 1\; =\; 0,2826$, and $\backslash frac\{u^2\; \backslash times\; n\; -\; 1\}\{n\}\; =\; 323,6^2\; =\; v^2$, and $v\; =\; 323,6$.

In like manner, $0,054045$ (which is half of $0,10809$) will be found to be the logarithmic secant of $28^\backslash circ$, whose sine $0,46947$ multiplied by $689\backslash frac\{1\}\{2\}$ 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 sine is $\backslash frac\{v\}\{u\}$, and radius $1$, as $2a$ 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 $323,62$, the terminal velocity of the ball being $689,34$. Here $2a$, or $\backslash frac\{u^2\}\{g\}$ is $14850$, and $\backslash frac\{323,62\}\{689,34\}\; =\; 0,46947$, which is the sine of $28^\backslash circ$. The logarithmic secant of this arch is $0,05407$. Now $M$ or $0,43429\; :\; 0,05407\; =\; 14850\; :\; 148$, 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 $28^\backslash circ$, of which the logarithmic sine is

is the logarithm of $323,62$, 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 sine 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 asserts 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 asser-

tion 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 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

Now as this space evidently increases continually (because the body always falls), but does not become infinite in any finite time, the fraction

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 between acquired. For this purpose we may use the other fundamental proposition of varied motions $f\; =\; v$, which, in the present case, becomes $\backslash frac\{g(u^2\; -\; v^2)\}\{u^2\}\; =\; v$; therefore $i\; =\; \backslash frac\{g\}\{u^2\}$.

Now (art. FLUXIONS) $\backslash int\; \backslash frac\{uv\}\{u^2\; -\; v^2\}\; =\; L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$. There-

fore $t\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}\; =\; \backslash frac\{u\}\{Mg\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$. This flu-
ent needs no constant quantity to complete it, or rather
$C=0$; for $t$ must be $=0$ when $v=0$. This will evi-
dently be the case: for then $L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ is $L\; \backslash sqrt\{\backslash frac\{u\}\{u\}\}\; =$
$L,\; 1,\; =\; 0$.

But how does this quantity $\backslash frac\{u\}\{Mg\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ signify a
time? Observe, that in whatever numbers, or by what-
ever units of space and time, $u$ and $g$ are expressed,
$\backslash frac\{u\}\{g\}$ expresses the number of units of time in which the ve-
locity $u$ is communicated or extinguished by gravity;
and $L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$, or $\backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$, 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 $u+v=1012,95$, $u-v=365,72$; therefore

$21\text{'},\; 542$. Now, for greater perspicuity, convert the
equation $t\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ into a proportion: thus

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 MI in the figure of the logitific
curve, and that the motion of a point in this line is pre-
cisely similar to that of the body. The Marquis Po-
leni, in a dissertation published at Padua in 1725, has
with great ingenuity constructed logarithms suited to
all the cases which can occur. Herman, in his Phono-
nomia, has borrowed much of Poleni's methods, but has
obscured them by an affectation of language geometri-
cally precise, but involving the very obscure notion of
abstract ratios.

It is easy to see that $\backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ is the cotangent of
the $\backslash frac\{1\}\{4\}$ complement of an arch, whose radius is 1, and
whose sine is $\backslash frac\{u\}\{u\}$: For let KC (fig. 6.) be $=u$, and
$BE=v$; then $KD=u+v$, and $DA=u-v$. Join KB
and BA, and draw CG parallel to KB. Now GA is
the tangent of $\backslash frac\{1\}\{4\}\; R\backslash hat\{A\}$, $=\backslash frac\{1\}\{4\}$ complement of HB. Then,
by similarity of triangles, $GA\; :\; AC\; =\; AB\; :\; BK,\; =$
$\backslash sqrt\{AD\}\; :\; \backslash sqrt\{DK\}\; =\; \backslash sqrt\{u-v\}\; :\; \backslash sqrt\{u+v\}$ and $\backslash frac\{AC\}\{GA\}\; (=\; \backslash text\{cotan.\}$
$\backslash frac\{1\}\{4\}\; BA)\; =\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$; therefore look for $\backslash frac\{u\}\{u\}$ among the na-
tural fines, or for $\backslash log.\; \backslash frac\{u\}\{u\}$ among the logarithmic fines,

and take the logarithmic cotangent of the half comple-
ment of the corresponding arch. This, considered as a
common number, will be the second term of our pro-
portion. This is a shorter process than the former.

By reversing this proportion we get the velocity cor-
responding to a given time.

To compare this descent of 1848 feet in the air Fall of a
with the fall of the body in vacuo during the same body in
time, say $21\text{'},\; 542\; :\; 10\text{'},\; 973\; =\; 1848\; :\; 1926,6$, which air compa-
makes a difference of 79 feet. red with
that of one

Cor. 1. 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 $u$

$L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ to $v$: for it would acquire this velocity in
vacuo during the time $\backslash frac\{v\}\{g\}$, and it acquires it in the air
in the time $\backslash frac\{u\}\{g\}\; L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$.

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

In the next place, let a body, whose terminal velo-
city is $u$, be projected perpendicularly upwards, with the ascent
any velocity $V$. It is required to determine the height of a body
to which it ascends, so as to have any remaining velocity projected
$v$, and the time of its ascent; as also the height perpendi-
cularly.
and time in which its whole motion will be extin-
guished.

We have now $\backslash frac\{g(u^2+v^2)\}\{u^2\}$ for the expression of $f$;
for both gravity and resistance act now in the same di-
rection, and retard the motion of the ascending body:
therefore $\backslash frac\{g(u^2+v^2)\}\{u^2\}\; :\; i\; =\; -v\; :\; v$, and $i\; =\; -\backslash frac\{v^2\}\{g\}\; \backslash times\; \backslash frac\{v\}\{u^2+v^2\}$,
and $i\; =\; -\backslash frac\{v^2\}\{g\}\; \backslash times\; \backslash int\; \backslash frac\{v\}\{u^2+v^2\}\; +\; C,\; =\; -\backslash frac\{v^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2+v^2\}\; +$
$C$ (see art. FLUXIONS). This must be $=0$ at the be-
ginning of the motion, that is, when $v=V$, that is,
$-\backslash frac\{v^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2+V^2\}\; +\; C\; =\; 0$, or $C\; =\; \backslash frac\{v^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2+V^2\}$, and
the complete fluent will be $i\; =\; \backslash frac\{v^2\}\{g\}\; \backslash times\; (L\; \backslash sqrt\{u^2+V^2\}\; -$

Let $b$ be the greatest height to which the body will
rise. Then $i\; =\; b$ when $v=0$; and $b\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times$

Therefore let $n$ be the number whose common logarithm is $\backslash frac\{2Mg\}\{u^2\}$; we shall have $n\; =\; \backslash frac\{u^2\; +\; V^2\}\{u^2\; +\; v^2\}$, and $v^2\; =\; \backslash frac\{u^2\; +\; V^2\}\{n\}\; -\; u^2$; 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 $\backslash sqrt\{u^2\; +\; V^2\}$ is the secant of an arch whose radius is $u$, and whose tangent is $V$, and that $\backslash sqrt\{u^2\; +\; v^2\}$ 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 $\backslash frac\{V\}\{u\}$ will be found the tangent of $30.48^\backslash circ$; the logarithmic secant of which is 0.06606. This, multiplied by $\backslash frac\{u^2\}\{Mg\}$, 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 with which it again reaches the ground. The ascent and descent are equal: therefore $\backslash sqrt\{\backslash frac\{u^2\; +\; V^2\}\{u^2\}\}$, which multiplies the constant factor in the ascent, is equal to $\backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$, 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 sine 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 sine, or as the radius to the cosine of the arch. Thus suppose the body projected with the terminal velocity, or $V\; =\; u$; then $v\; =\; \backslash frac\{u\}\{\backslash sqrt\{2\}\}$. If $V\; =\; 689$, $v\; =\; 487$.

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

Here $\backslash frac\{g(u^2\; +\; v^2)\}\{u^2\}\; \backslash dot\{t\}\; =\; -\backslash dot\{v\}$, and $\backslash dot\{t\}\; =\; -\backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash frac\{\backslash dot\{v\}\}\{u^2\; +\; v^2\}\; =\; -\backslash frac\{u\}\{g\}\; \backslash times\; \backslash frac\{u\; \backslash dot\{v\}\}\{u^2\; +\; v^2\}$; and $t\; =\; -\backslash frac\{u\}\{g\}\; \backslash times\; \backslash int\; \backslash frac\{u\; \backslash dot\{v\}\}\{u^2\; +\; v^2\}\; +\; C$. Now (art. FLUXIONS) $\backslash int\; \backslash frac\{u\; \backslash dot\{v\}\}\{u^2\; +\; v^2\}$ is an arch whose tangent is $\backslash frac\{v\}\{u\}$ and radius 1; therefore $t\; =\; -\backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\}\; \backslash frac\{v\}\{u\}\; +\; C$. This must be 0 when $v\; =\; V$, or $C\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\}\; \backslash frac\{V\}\{u\}$, and $C\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\}\; \backslash frac\{V\}\{u\}$, and the complete fluent is $t\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash left(\; \backslash text\{arc.\; tan.\}\; \backslash frac\{V\}\{u\}\; -\; \backslash text\{arc.\; tan.\}\; \backslash frac\{v\}\{u\}\; \backslash right)$. The quantities within the brackets express a portion of the arch of a circle whose radius is unity; and are therefore abstract numbers, multiplying $\backslash frac\{u\}\{g\}$, 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 however great the velocity of projection, and the height

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 quadrantal 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 the time $t\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\}\; \backslash frac\{V\}\{u\}$, or $\backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\}\; 30.48^\backslash circ$. Now $30.48^\backslash circ\; =\; 18.48$, and the radius 1 contains 3438; therefore the arch $=\; \backslash frac\{18.48\}\{3438\}\; =\; 0.5376$; and $\backslash frac\{u\}\{g\}\; =\; 21\text{'}\text{'}\; 54$. Therefore $t\; =\; 21\text{'}\text{'}\; 54\; \backslash times\; 0.5376\; =\; 11\text{'}\text{'}\; 58$, or nearly 11 $\backslash frac\{1\}\{2\}$ seconds. The body would have risen to the same height in a void in 10 $\backslash frac\{1\}\{2\}$ 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 in air and is to the tangent; for the time of the ascent in the air is $\backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\}$; the time of the ascent in vacuo is $\backslash frac\{V\}\{g\}$. Now $\backslash frac\{V\}\{u\}\; =\; \backslash tan.$ and $V\; =\; u\; \backslash times\; \backslash tan.$ and $\backslash frac\{V\}\{g\}\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash tan.$

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

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 $\backslash frac\{30103\}\{43429\}$ of the height to which it will rise in vacuo, or $\backslash frac\{5\}\{7\}$ 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 assisting 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, to determine 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 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 eq. 1 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 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 Basle. Bernoulli had published in the Acta Eruditorum Lipsiae 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 Brooke 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 we 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 thro' 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 $=x$, 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

$r\; =\; \backslash frac\{v^2\}\{2a\}$; because, for any velocity $v$, and producing

height $h$, we have $g\; =\; \backslash frac\{u^2\}\{2h\}$

Let Mm touch the curve in M; draw the ordinate P N m,

to $Mm$, and draw $Mo$, $Nn$ perpendicular to $N\backslash rho$ and $Mm$. Then we have $MN\; =\; \backslash infty$, and $Mo\; =\; \backslash infty$, also $Mo$ is ultimately $=\; y$ and $Mm$ is ultimately $=\; MN$ or $x$. Lastly, let us suppose $x$ to be a constant quantity, the elementary ordinates being supposed equidistant.

74 Action of gravity in a given time. 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 $nN$, 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 $z\; :\; y\; =\; g\; :\; \backslash frac\{zy\}\{x\}$, and the whole retardation in

the ascent will be $r\; +\; \backslash frac\{zy\}\{x\}$. 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 $-v$. This is proportional to the retarding force and to the time of its action

jointly, and therefore $-v\; =\; r\; +\; \backslash frac\{zy\}\{x\}\; \backslash times\; t$; but the time $t$ is as the space $z$ divided by the velocity $v$; therefore $-v\; =\; r\; +\; \backslash frac\{zy\}\{x\}\; \backslash times\; \backslash frac\{z\}\{v\}\; =\; -\backslash frac\{rz\; +\; gy\}\{v\}$, and $-v\; \backslash dot\{v\}\; =\; -\backslash frac\{r\; \backslash dot\{z\}\; +\; g\; \backslash dot\{y\}\}\{v\}$.

$r\; \backslash dot\{z\}\; -\; g\; \backslash dot\{y\}\; =\; \backslash frac\{v^2\; \backslash dot\{z\}\}\{2a\}\; -\; g\; \backslash dot\{y\}$. 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 $-y\; =\; g\; t^2$. (Observe that $mN$ is in fact only the half of $-y$; but $g$ being twice the fall of a heavy body in a second, we have $-y$ strictly equal to $g\; t^2$). But $t^2\; =\; \backslash frac\{z^2\}\{v^2\}$; therefore $-y\; =\; \backslash frac\{g\; z^2\}\{v^2\}$, and $v^2\; =\; \backslash frac\{g\; z^2\}\{y\}$, and $-v^2\; \backslash dot\{y\}\; =\; g\; \backslash dot\{z\}^2$. The fluxion of

this equation is $-v^2\; \backslash dot\{y\}\; -\; 2v\; \backslash dot\{y\}\; \backslash dot\{v\}\; =\; 2g\; \backslash dot\{z\}^2$; but, because $z\; :\; y\; =\; mM\; :\; mo\; =\; mN\; :\; mn\; =\; y\; :\; z$, we have $\backslash dot\{z\}\; \backslash dot\{z\}\; =\; y\; \backslash dot\{y\}$. Therefore $2g\; \backslash dot\{y\}\; \backslash dot\{y\}\; =\; 2g\; \backslash dot\{z\}^2\; =\; -v^2\; \backslash dot\{y\}\; -\; 2v\; \backslash dot\{y\}\; \backslash dot\{v\}$, and $-2v\; \backslash dot\{y\}\; \backslash dot\{v\}\; =\; v^2\; \backslash dot\{y\}$, $-2g\; \backslash dot\{y\}\; \backslash dot\{y\}$, and $-v\; \backslash dot\{v\}\; =\; \backslash frac\{v^2\; \backslash dot\{y\}\}\{2y\}\; -\; g\; \backslash dot\{y\}$. But we have already $-v\; \backslash dot\{v\}\; =\; \backslash frac\{v^2\; \backslash dot\{z\}\}\{2a\}\; -\; g\; \backslash dot\{y\}$; therefore $\backslash frac\{v^2\; \backslash dot\{y\}\}\{y\}\; =\; \backslash frac\{v^2\; \backslash dot\{z\}\}\{a\}$, and finally $\backslash frac\{\backslash dot\{y\}\}\{y\}\; =\; \backslash frac\{\backslash dot\{z\}\}\{a\}$.

$\backslash frac\{\backslash dot\{z\}\}\{a\}$, or $a\; \backslash dot\{y\}\; =\; \backslash dot\{z\}\; y$, for the fluxionary equation of the curve.

75 If we put this into the form of a proportion, we Relation between have $a\; :\; z\; =\; \backslash dot\{y\}\; :\; y$. Now this evidently establishes a re-the length lation between the length of the curve and its variation of the of curvature; and between the curve itself and its eva-curve and its varia- luta, which are the very circumstances introduced by tion of Newton into his investigation of the spiral motions. And curvature.

the equation $\backslash frac\{\backslash dot\{z\}\}\{a\}\; =\; \backslash frac\{\backslash dot\{y\}\}\{y\}$ is evidently an equation connect-

ed 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 $z$, $x$, and $y$.

Let $p$ express the ratio of $y$ to $x$, that is, let $p$ be $=\; \backslash frac\{y\}\{x\}$, or $p\; x\; =\; y$. 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, multiplying $x$, so as to make it $=\; y$. We now have $y^2\; =\; p^2\; x^2$, and since $z^2\; =\; x^2\; +\; y^2$, we have $z^2\; =\; x^2\; +\; p^2\; x^2\; =\; (1\; +\; p^2)\; x^2$ and $x\; =\; \backslash frac\{z\}\{\backslash sqrt\{1\; +\; p^2\}\}$.

Moreover, because we have supposed the abscissa $x$ to increase uniformly, and therefore $x$ to be constant, we have $\backslash dot\{y\}\; =\; \backslash dot\{x\}\; p$, and $\backslash dot\{z\}\; =\; \backslash dot\{x\}\; \backslash sqrt\{1\; +\; p^2\}$. Now let $q$ express the ratio of $\backslash dot\{p\}$ to $\backslash dot\{x\}$, that is, make $\backslash frac\{\backslash dot\{p\}\}\{\backslash dot\{x\}\}\; =\; q$, or $q\; \backslash dot\{x\}\; =\; \backslash dot\{p\}$. This gives us $\backslash dot\{q\}\; =\; \backslash dot\{p\}\; \backslash dot\{x\}$, and $\backslash dot\{x\}^2\; \backslash dot\{q\}\; =\; \backslash dot\{x\}\; \backslash dot\{p\}\; \backslash dot\{x\}\; =\; \backslash dot\{y\}\; \backslash dot\{z\}$.

By these substitutions our former equation $a\; \backslash dot\{y\}\; =\; \backslash dot\{z\}\; y$ changes to $a\; x^2\; \backslash dot\{q\}\; =\; \backslash dot\{x\}\; \backslash sqrt\{1\; +\; p^2\}\; |\; \backslash dot\{x\}\; p$, or $a\; \backslash dot\{q\}\; =\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}$, and, taking the fluent on both sides, we have $a\; q\; =\; f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\; +\; C$, $C$ being the constant quantity required for completing the fluent according to the limiting conditions of the case. Now $\backslash dot\{x\}\; =\; \backslash frac\{\backslash dot\{p\}\}\{q\}$, and $\backslash frac\{1\}\{q\}\; =$

$f\; \backslash frac\{a\}\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\; +\; C\}$. Therefore $\backslash dot\{x\}\; =\; f\; \backslash frac\{a\; \backslash dot\{p\}\}\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\; +\; C\}$

Also, since $\backslash dot\{y\}\; =\; p\; \backslash dot\{x\}\; =\; \backslash frac\{p\; \backslash dot\{p\}\}\{q\}$, we have $y\; =$

$f\; \backslash frac\{a\; p\; \backslash dot\{p\}\}\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\; +\; C\}$
Also $\backslash dot\{z\}\; =\; \backslash dot\{x\}\; \backslash sqrt\{1\; +\; p^2\}\; =\; \backslash frac\{a\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\}\{f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\; +\; C\}$

The values of $\backslash dot\{x\}$, $\backslash dot\{y\}$, $\backslash dot\{z\}$, give us
$x\; =\; f\; \backslash frac\{a\; \backslash dot\{p\}\}\{f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\; +\; C\}\; =\; a\; f\; \backslash frac\{\backslash dot\{p\}\}\{f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; p^2\}\; +\; C\}$
4B $g\; =$

The process therefore of describing the trajectory is, 1^st. To find $q$ in terms of $p$ by the area of the curve whose abscissa is $p$ and the ordinate is $\backslash sqrt\{1+p^2\}$.

2^d, We get $x$ by the area of another curve whose abscissa is $p$, and the ordinate is $\backslash frac\{1\}\{q\}$.

3^d, We get $y$ by the area of a third curve whose abscissa is $p$, and the ordinate is $\backslash frac\{p\}\{q\}$.

The problem of the trajectory is therefore completely solved, because we have determined the ordinate, abscissa, and arch of the curve for any given position of its tangent. It now only remains to compute the magnitudes of these ordinates and abscissæ, or to draw them by a geometrical construction. But in this consists the difficulty. The areas of these curves, which express the lengths of $x$ and $y$, can neither be computed nor exhibited geometrically, by any accurate method yet discovered, and we must content ourselves with approximations. These render the description of the trajectory exceedingly 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 conception of the subject to proceed some length in this construction; for it must be acknowledged that very few distinct notions accompany a mere algebraic operation, especially if in any degree complicated, which we confess is the case in the present question.

Let $B$ be $NR$ (fig. 8.) be an equilateral hyperbola, of which $B$ is the vertex, $BA$ the semitransverse axis, which we shall assume for the unity of length. Let $AV$ be the femiconjugate axis $=\; BA$, $=$ unity, and $AS$ the asymptote, bisecting the right angle $BAV$. Let $PN$, $p$ be two ordinates to the conjugate axis, exceedingly near to each other. Join $BP$, $AN$, and draw $B^1N$, 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 formulae, $PN$ is $\backslash sqrt\{1+p^2\}$, and $Pp$ is $p$, and the area $BAPNB\; =\; f,\; \backslash sqrt\{1+p^2\}$: That is to

say, the number $f,\; \backslash sqrt\{1+p^2\}$ (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 $ABN$. $APN\; =\; \backslash frac\{1\}\{2\}\; AP\; \backslash times\; PN$, $=\; \backslash frac\{1\}\{2\}\; p\; \backslash sqrt\{1+p^2\}$, and the hyperbolic sector $ABN\; =\; BN\; \backslash cdot\; \backslash beta$, which is equivalent to the hyperbolic logarithm of the number represented by $A$, when $A$ is unity. Therefore it is equal to $\backslash frac\{1\}\{2\}$ the logarithm of $p\; +\; \backslash sqrt\{1+p^2\}$. Hence we see by the bye that $f,\; \backslash sqrt\{1+p^2\}\; =\; \backslash frac\{1\}\{2\}\; p\; \backslash sqrt\{1+p^2\}\; +\; \backslash frac\{1\}\{2\}$ hyperbolic logarithm $p\; +\; \backslash sqrt\{1+p^2\}$.

Now let $AMD$ be another curve, such that its ordinates $Vm$, $PD$, &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 $VM\; :\; VC\; =$

$Vm\; :\; BA\; :\; VCBA$ and $PD\; :\; P\_1\; =\; PNBA\; :\; VCBA$, &c. These ordinates will now represent $f,\; \backslash sqrt\{1+p^2\}$ with reference to a linear unit, as the areas to the hyperbola represented it in reference to a superficial unit.

Again, in every ordinate make $PD\; :\; P\_1\; =\; P\_1\; :\; PO$, and thus we obtain a reciprocal to $PD$ or to $f,\; \backslash sqrt\{1+p^2\}$, or equivalent to $f,\; \backslash frac\{1\}\{\backslash sqrt\{1+p^2\}\}$. This

will evidently be $\backslash frac\{x\}\{ap\}$, and $PO$ or $p$ will be $\backslash frac\{x\}\{a\}$, and the area contained between the lines $AF$, $AW$, and the curve $GEOH$, and cut off by the ordinate $PO$, will represent $\backslash frac\{x\}\{a\}$.

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

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 $45^\backslash circ$, 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 $p\; =\; AB$, $=\; 1$, $=$ tangent $45^\backslash circ$, 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 $z\; =\; \backslash frac\{ap\backslash sqrt\{1+p^2\}\}\{f,\; \backslash sqrt\{1+p^2\}\; +\; C\}$ and of which the area

would exhibit the arch itself. And this would have been very easy, for it is $z\; =\; a\; \backslash frac\{p\backslash sqrt\{1+p^2\}\}\{f,\; \backslash sqrt\{1+p^2\}\; +\; C\}$,

which is evidently the fluxion of the hyperbolic logarithm of $f,\; \backslash sqrt\{1+p^2\}$. But it is needless, since $z\; =\; x\backslash sqrt\{1+p^2\}$, and we have already got $x$. 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 knowing 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.

PROJECTILES.

Fig. 3. Plate CCCCXVII.

Fig. 2.

Fig. 4.

Fig. 5, called by mistake in the print 6 see 332 p.

Fig. 6.

Fig. 7.

Abull Print Water Sculptor facit.

the method of the Academy of Berlin published in 1753, takes the vertex of the curve for the beginning ascending branch can the inclination of the tangent be very such

Fig. 9

Fig. 8

Fig. 10

Fig. 1

Fig. 2

PROTRACTOR.
Fig. 3

Fig. 4

ses 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 its several portions.

Recollect, therefore, that $v^2\; =\; \backslash frac\{-gz^2\}\{y\}$ and that $\backslash dot\{z\}^2\; =\; \backslash dot\{x\}^2\; +\; \backslash dot\{y\}^2$ and $\backslash dot\{y\}\; =\; \backslash dot\{x\}\; \backslash dot\{p\}$. This gives $v^2\; =\; \backslash frac\{-g\; \backslash dot\{x\}^2\; (1\; +\; \backslash dot\{p\}^2)\}\{\backslash dot\{p\}\}$.

But $\backslash dot\{p\}\; =\; q\; \backslash dot\{x\}$. Therefore $v^2\; =\; \backslash frac\{-g\; \backslash dot\{x\}^2\; (1\; +\; \backslash dot\{p\}^2)\}\{q\}$, = $\backslash frac\{-ag(1\; +\; \backslash dot\{p\}^2)\}\{f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\; +\; C\}$, and $v\; =\; \backslash sqrt\{\backslash frac\{-g(1\; +\; \backslash dot\{p\}^2)\}\{q\}\}$, = $\backslash sqrt\{f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\; +\; C\}$.

Also $i$ was found = $\backslash frac\{\backslash dot\{z\}\}\{v\}$, = $\backslash frac\{\backslash dot\{x\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\}\{v\}$, = $\backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\}\{q\; v\}$. If we now substitute for $v$ its value just found, we obtain $i\; =\; \backslash frac\{\backslash dot\{p\}\}\{\backslash sqrt\{-gq\}\}$, and $t\; =\; \backslash int\; \backslash frac\{\backslash dot\{p\}\}\{\backslash sqrt\{-gq\}\}$, = $\backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{a\}\}\{\backslash sqrt\{-g\}\; f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\; +\; C\}$, = $\backslash frac\{\backslash sqrt\{a\}\}\{\backslash sqrt\{-g\}\}\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\}\{f\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\; +\; C\}$.

The greatest difficulty still remains, viz. the accommodating these formulae, which appear abundantly simple, to the particular cases. It would appear at first sight, that all trajectories are similar; since the ratio of the fluxions 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 $\backslash dot\{p\}$, 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 $\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}$, which is the basis of the whole construction. We there found $q\; =\; \backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\}\{a\}$. This fluent is in general $q\; =\; \backslash frac\{C\; +\; \backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\}\{a\}\}\{a\}$, and the constant quantity $C$ is to be accommodated to some circumstances of the case. Different authors have selected different circumstances. 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 $\backslash dot\{p\}\; =\; 0$, 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 $\backslash frac\{n\}\{a\}$. It is plain that $n$ must be a number; for it must be homologous with $\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}$, which is a number. For brevity's sake let us express the fluent of $\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}$ by the single letter $P$; and thus we shall have $x\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\}\{n\; +\; P\}$, $y\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash dot\{p\}\}\{n\; +\; P\}$, $z\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\}\{n\; +\; P\}$. And $v^2\; =\; \backslash frac\{-ag(1\; +\; \backslash dot\{p\}^2)\}\{n\; +\; P\}$. Now the height $h$ necessary for communicating any velocity $v$ is $\backslash frac\{v^2\}\{2g\}\; =\; \backslash frac\{-ag(1\; +\; \backslash dot\{p\}^2)\}\{2g(n\; +\; P)\}$, = $\backslash frac\{-\backslash frac\{1\}\{2\}a(1\; +\; \backslash dot\{p\}^2)\}\{n\; +\; P\}$. And lastly, $t\; =\; \backslash frac\{\backslash sqrt\{a\}\}\{\backslash sqrt\{g\}\}\; \backslash int\; \backslash frac\{\backslash dot\{p\}\}\{\backslash sqrt\{n\; +\; P\}\}$.

These fluents, being all taken so as to vanish at the vertex, where the computation commences, and where $\backslash dot\{p\}\; =\; 0$ (the tangent being parallel to the horizon), we obtain in this case $h\; =\; \backslash frac\{\backslash frac\{1\}\{2\}a\}\{n\}$, = $\backslash frac\{a\}\{2n\}$, and $n\; =\; \backslash frac\{a\}\{2b\}$.

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 $n$, $n$ 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 farther 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 $-\backslash frac\{\backslash dot\{p\}\}\{n\}$, and

therefore $P$, or $\backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\}\{n\}$, which is $\backslash frac\{q\}\{a\}$, must be taken negatively. Therefore, in the ascending branch, we have $AQ$ or $x$ (increasing as we recede from $A$) = $a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\}\{n\; -\; P\}$, $QN$ or $y\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash dot\{p\}\}\{n\; -\; P\}$, $AN$ or $z\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash dot\{p\}^2\}\}\{n\; -\; P\}$, $t\; =\; \backslash frac\{\backslash sqrt\{a\}\}\{\backslash sqrt\{g\}\}\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{p\}\}\{\backslash sqrt\{1\; -\; P\}\}$, and the height producing the velocity at $N\; =\; \backslash frac\{\backslash frac\{1\}\{2\}a(1\; +\; \backslash dot\{p\}^2)\}\{n\; -\; P\}$.

Hence we learn by the bye, that in no part of the ascending branch can the inclination of the tangent be

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 rectilinear 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.

$\backslash frac\{y\}\{x\}$ was made equal to $\backslash dot\{p\}$. Therefore the radius of curvature, determined by the ordinary method, is $\backslash frac\{x(1+p^2)(\backslash sqrt\{1+p^2\})\}\{\backslash dot\{p\}\}$, and, because $\backslash frac\{x\}\{\backslash dot\{p\}\}$ is $=\; \backslash frac\{a\}\{n+P\}$ for the descending branch of the curve, the radius of curvature at $M$ is $\backslash frac\{a\; \backslash times\; 1\; +\; p^2\; \backslash times\; \backslash sqrt\{1+p^2\}\}\{n+P\}$, and, in the ascending branch at $N$, it is $\backslash frac\{a\; \backslash times\; 1\; +\; p^2\; \backslash times\; \backslash sqrt\{1+p^2\}\}\{n-P\}$.

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 absciss, and the arch, are all greater in the ascending branch. This is pretty obvious. 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 $\backslash frac\{a\}\{2b\}$.

It is therefore in the quantity $P$, or $f\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$, 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 absciss. In vacuo the second increment of the ordinate is constant when the first increment of the abscissa is so, and the whole increment of the ordinate is as $1+p$. 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 utility 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 $P\; =\; f\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$, we have $\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\; =\; P$, and there-

fore $z$, which was $=\; a\; \backslash times\; \backslash frac\{f\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}\{f\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}\; +\; C$, or $=\; a\; \backslash times\; \backslash frac\{P\}\{n+P\}$, may be expressed by logarithms; or $z\; =\; a\; \backslash times\; \backslash text\{hyp.\; log.\; of\; \}\; \backslash frac\{n+P\}\{n\}$, since at the vertex $A$, where $z$ must be $=\; a$, $P$ is also $=\; a$.

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 $AM\; =\; a\; \backslash times\; \backslash log.\; \backslash frac\{n+P\}\{n\}$ and $Am\; =\; a\; \backslash times\; \backslash log.\; \backslash frac\{n+Q\}\{n\}$, and therefore $Mm$ is $=\; a\; \backslash times\; \backslash log.\; \backslash frac\{n+Q\}\{n+P\}$.

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 absciss corresponding to the arch $Mm$, whole

whose middle point is inclined to the horizon in the angle $b$, will be $Mm\; \backslash times\; \backslash cosine\backslash \; b$, and the corresponding portion of the ordinate will be $Mm\; \backslash times\; \backslash sin.\backslash \; b$. Then we obtain the velocity in each part of the curve by the equation $b\; =\; \backslash frac\{1\}\{2\}\; a\; \backslash times\; \backslash frac\{1\; +\; \backslash rho^2\}\{n\; +\; \backslash rho\}$; or, more directly the velocity

$v$ at $M$ will be $=\; \backslash sqrt\{ag\}\; \backslash frac\{\backslash sqrt\{1\; +\; \backslash rho^2\}\}\{\backslash sqrt\{n\; +\; \backslash rho\}\}$. 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 $\backslash rho$ by which we compute the velocity, a medium between the $\backslash rho$ for the beginning and that for the end of the arch.

84
Rule's method preferred.
This is the method followed by Euler, who was one 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 singly, 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.

85
Its application made 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 conspiring (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, $\backslash rho$ is also infinite, and $b\; =\; \backslash frac\{1\}\{2\}\; \backslash frac\{ap^2\}\{P\}$, because $n$ in this case is nothing in respect of $P$, which is infinite; and because $\backslash rho$ is infinite, the number $\backslash log.\; \backslash rho\; +\; \backslash sqrt\{1\; +\; \backslash rho^2\}$, though infinite, vanishes in comparison with $\backslash rho\; \backslash times\; \backslash sqrt\{1\; +\; \backslash rho^2\}$; so that in this case $P\; =\; \backslash frac\{1\}\{2\}\; \backslash rho^2$, and $b\; =\; a$, and $v\; =$ 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 $\backslash rho$ 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 $n\; =\; \backslash frac\{1\}\{2\}\; \backslash rho\; \backslash sqrt\{1\; +\; \backslash rho^2\}\; +\; \backslash log.\; \backslash rho\; +\; \backslash sqrt\{1\; +\; \backslash rho^2\}$. In order, therefore, to find the point $L$, where the asymptote $LO$ cuts the horizontal line $AL$, put $P\; =\; n$, then will $AL\; =\; x\; -\; \backslash frac\{yx\}\{y\}\; =\; a\; \backslash times\; \backslash left(\; \backslash frac\{1\}\{n\; -\; P\}\; -\; \backslash frac\{1\}\{\backslash rho\}\; \backslash right)\; \backslash frac\{f\; \backslash rho\; \backslash rho\}\{n\; -\; P\}$.

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 $=\; \backslash frac\{a\}\{M\}\; \backslash times\; \backslash log.$

$\backslash frac\{n\; +\; Q\}\{n\; +\; P\}$ &c. where $M$ means the subtangent of the common logarithms, or 0.43429; also the time of describing this arch will be expediently had by taking a medium $\backslash mu$ between the values of $\backslash frac\{\backslash sqrt\{1\; +\; \backslash rho^2\}\}\{\backslash sqrt\{n\; +\; P\}\}$ and $\backslash frac\{\backslash sqrt\{1\; +\; q^2\}\}\{\backslash sqrt\{n\; +\; Q\}\}$ and making the time $=\; \backslash frac\{\backslash sqrt\{a\}\}\{M\; \backslash mu\; \backslash sqrt\{g\}\}\; \backslash times\; \backslash log.\; \backslash frac\{n\; +\; Q\}\{n\; +\; P\}$.

86
Such then is the process by which the form and magnitude of the trajectory, and the motion in it, may be applied 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

which will be characterized by $n$, and that the velocity at that point of the ascending branch where the inclination of the tangent is $30^\backslash circ$ 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 $50^\backslash circ$, 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 Commentary on Robins's Gunnery. They are in number 18, distinguished by the position of the asymptote of the ascending branch. This is given for $5^\backslash circ$, $10^\backslash circ$, $15^\backslash circ$, &c. to $85^\backslash circ$, 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.

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 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 elevations (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 Approximations.

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

&c. Make $y\; =\; v$, and take the maximum by varying $e$, we obtain $\backslash sin.\; \backslash frac\{1\}\{2\}\; e\; +\; \backslash frac\{a\; \backslash cdot\; \backslash sin.\; e\}\{2b\}\; =\; \backslash text\{hyperbol.\; log.\}$

$\backslash left(\; 1\; +\; \backslash frac\{2b\}\{a\; \backslash sin.\; e\}\; \backslash right)$, 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 $\backslash times\; a$ (the height producing the terminal velocity) will give the range.

TABLE of Elevations giving the greatest Range.

Such is the solution which the present state of our Advantage mathematical knowledge enables us to give of this cele- to be de- brated problem. It is exact in its principle, and the rived from application of it is by no means difficult, or even ope- the solution of the- rose. But let us see what advantage we are likely to problem. 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

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 assumed 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 Robins'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 $H\; +\; \backslash frac\{H^2\}\{28845\}$. 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 assigning this precise addition. They are rather reasons of arithmetical convenience 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, 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 en-

tirely 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 1786, 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 $\backslash frac\{1\}\{4\}$ of Mr Robins's, and about $\backslash frac\{1\}\{4\}$ of Muller's.

What are we to conclude from all this? Simply this, ⁶⁰ 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 consists 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 his 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 artillerist must rely chiefly on the records of experiments contained in the books of practice.

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 falls very much short of the mark, a little more powder is added; and, in cannoneading, 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 compensations 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 physical theory is of no use to the practical artillerist. 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 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 resorted

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 pre- 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.

The next table contains the ranges in yards of a 24 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.

95
Use of the
alt table.

The initial velocities can never be pushed as far as we have calculated for 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 ve-

VOL. XV. Part II.

locity 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, $\backslash sqrt\{d\}:\backslash sqrt\{D\}\; =\; 1500$, 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 those different quantities, we have given this table, expressed in a figure (fig. 10). The abscissa, or axis DA, is the scale of the initial velocities in feet per second, measured on a scale of 400 equal parts in an inch. The ordinates to the curve ACG express the yards of the range on a scale containing 800 yards in an inch. 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

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.

PROJECTION OF THE SPHERE.

THE PROJECTION OF THE SPHERE is a perspective representation of the circles on the surface of the sphere; and is variously denominated according to the different positions of the eye and plane of projection.

There are three principal kinds of projection; the stereographic, the orthographic, and gnomonic. In the stereographic projection the eye is supposed to be placed on the surface of the sphere; in the orthographic it is supposed to be at an infinite distance; and in the gnomonic projection the eye is placed at the centre of the sphere. Other kinds of projection are, the globular, Mercator's, stereographic, &c. for which see the articles GEOGRAPHY, NAVIGATION, PERSPECTIVE, &c.

DEFINITIONS.

1. The plane upon which the circles of the sphere are described, is called the plane of projection, or the primitive circle. The pole of this circle is the pole of projection, and the place of the eye is the projecting point.

2. The line of measures of any circle of the sphere is that diameter of the primitive, produced indefinitely, which passes through the centre of the projected circle.

AXIOM.

The projection, or representation of any point, is where the straight line drawn from it to the projecting point intersects the plane of projection.

SECTION I.

Of the Stereographic Projection of the Sphere.

In the stereographic projection of the sphere, the eye is placed on the surface of the sphere in the pole of the great circle upon which the sphere is to be projected. The projection of the hemisphere opposite to the eye falls within the primitive, to which this projection is generally limited: it, however, may be extended to the other hemisphere, or that wherein the eye is placed, the projection of which falls without the primitive.

As all circles in this projection are projected either into circles or straight lines, which are easily described, it is therefore more generally understood, and by many preferred to the other projections.

PROPOSITION I. THEOREM I.

Every great circle which passes through the projecting point is projected into a straight line passing through the centre of the primitive; and every arch of it, reckoned from the other pole of the primitive, is projected into its semitangent.

Let ABCD (fig. 1.) be a great circle passing thro' A, C, the poles of the primitive, and intersecting it in the line of common section BED, E being the centre of the sphere. From A, the projecting point, let there be drawn straight lines AP, AM, AN, AQ, to any number of points P, M, N, Q, in the circle ABCD: these lines will intersect BED, which is in the same plane with them. Let them meet it in the points p, m, n, q; then p, m, n, q, are the projections of P, M, N, Q: hence the whole circle ABCD is projected into the straight line BED, passing through the centre of the primitive.

Again, because the pole C is projected into E, and the point M into m; therefore the arch CM is projected into the straight line Em, which is the semitangent of the arch CM to the radius AE. In like manner, the arch CP is projected into its semitangent Ep, &c.

COROLLARIES.

1. Each of the quadrants contiguous to the projecting point is projected into an indefinite straight line, and each of those that are remote into a radius of the primitive.

2. Every small circle which passes through the projecting point is projected into that straight line which is its common section with the primitive.

3. Every straight line in the plane of the primitive, and produced indefinitely, is the projection of some circle on the sphere passing through the projecting point.

4. The projection of any point in the surface of the sphere, is distant from the centre of the primitive, by the semitangent of the distance of that point from the pole opposite to the projecting point.

PROPOSITION II. THEOREM II.

Every circle on the sphere which does not pass through the projecting point is projected into a circle.

If the given circle be parallel to the primitive, then a straight line drawn from the projecting point to any point in the circumference, and made to revolve about the circle, will describe the surface of a cone; which being cut by the plane of projection parallel to the base, the section will be a circle. See Conic Sections.

But if the circle MN (fig. 2.) be not parallel to the primitive circle BD, let the great circle ABCD, passing through the projecting point, cut it at right angles in the diameter MN, and the primitive in the diameter BD. Through M, in the plane of the great circle, let MF be drawn parallel to BD; let AM, AN be joined, and

and meet BD in $m,\; n$. Then, because AB, AD are quadrants, and BD, MF parallel, the area AM is equal to AF, and the angle AMF or $A\_\{mn\}$ is equal to ANM. Hence the conic surface described by the revolution of AM about the circle MN is cut by the primitive in a subcontrary position; therefore the section is in this case likewise a circle.

like manner $HG=HK$; hence the triangles GHF, KHF are equal, and the angle $FGH=FKH=LMN$.

COROLLARIES.

1. 1. An angle contained by any two circles of the sphere is equal to the angle formed by their projections. For the tangents to these circles on the sphere are projected into straight lines, which either coincide with, or are tangents to, their projections on the primitive.
2. 2. An angle contained by any two circles of the sphere is equal to the angle formed by the radii of their projections at the point of intersection.

COROLLARIES.

1. 1. The centres and poles of all circles parallel to the primitive have their projections in its centre.
2. 2. The centre and poles of every circle inclined to the primitive have their projections in the line of measures.
3. 3. All projected great circles cut the primitive in two points diametrically opposite; and every circle in the plane of projection, which passes through the extremities of a diameter of the primitive, or through the projections of two points that are diametrically opposite on the sphere, is the projection of some great circle.
4. 4. A tangent to any circle of the sphere, which does not pass through the projecting point, is projected into a tangent to that circle's projection; also, the circular projections of tangent circles touch one another.
5. 5. The extremities of the diameter, on the line of measures of any projected circle, are distant from the centre of the primitive by the semitangents of the least and greatest distances of the circle on the sphere, from the pole opposite to the projecting point.
6. 6. The extremities of the diameter, on the line of measures of any projected great circle, are distant from the centre of the primitive by the tangent and cotangent of half the great circle's inclination to the primitive.
7. 7. The radius of any projected circle is equal to half the sum, or half the difference of the semitangents of the least and greatest distances of the circle from the pole opposite to the projecting point, according as that pole is within or without the given circle.

PROPOSITION III. THEOREM III.

An angle formed by two tangents at the same point in the surface of the sphere, is equal to the angle formed by their projections.

Let FGI and GH (fig. 3.) be the two tangents, and A the projecting point; let the plane AGF cut the sphere in the circle AGL, and the primitive in the line BML. Also, let MN be the line of common section of the plane AGH with the primitive; then the angle $FGH=LMN$. If the plane FGH be parallel to the primitive BLD, the proposition is manifest. If not, through any point K in AG produced, let the plane FKH, parallel to the primitive, be extended to meet FGH in the line FH. Then, because the plane AGF meets the two parallel planes BLD, FKH, the lines of common section LM, FK are parallel; therefore the angle $AML=AKF$. But since A is the pole of BLD, the chords, and consequently the arcs AB, AL, are equal, and the arc ABG is the sum of the arcs AL, BG; hence the angle AML is equal to an angle at the circumference standing upon AG, and therefore equal to AGI or FGK; consequently the angle $FGK=FKG$, and the side $FG=FK$. In

PROPOSITION IV. THEOREM IV.

The centre of a projected great circle is distant from the centre of the primitive; the tangent of the inclination of the great circle to the primitive, and its radius, is the secant of its inclination.

Let MNG (fig. 4.) be the projection of a great circle, meeting the primitive in the extremities of the diameter MN, and let the diameter BD, perpendicular to MN, meet the projection in F, G. Bisect FG in H, and join NH. Then, because any angle contained by two circles of the sphere is equal to the angle formed by the radii of their projections at the point of intersection; therefore the angle contained by the proposed great circle and the primitive is equal to the angle ENH, of which EH is the tangent, and NH the secant, to the radius of the primitive.

COROLLARIES.

1. 1. All circles which pass through the points M, N are the projections of great circles, and have their centres in the line BG; and all circles which pass through the points F, G are the projections of great circles, and have their centres in the line HI, perpendicular to BG.
2. 2. If NF, NH be continued to meet the primitive in L, F; then BL is the measure of the great circle's inclination to the primitive; and $MT=2BL$.

PROPOSITION V. THEOREM V.

The centre of projection of a less circle perpendicular to the primitive, is distant from the centre of the primitive, the secant of the distance of the less circle from its nearest pole; and the radius of projection is the tangent of that distance.

Let MN (fig. 5.) be the given less circle perpendicular to the primitive, and A the projecting point. Draw AM, AN to meet the diameter BD produced in G and H; then GH is the projected diameter of the less circle; bisect GH in C, and C will be its centre; join NE, NC. Then because AE, NE are parallel, the angle $INE=NEA$; but $NEA=2NMA=2NHG=NCG$; hence $ENC=INE+INC=NCG+INC$ a right angle; and therefore NC is a tangent to the primitive at N; but the arch ND is the distance of the less circle from its nearest pole D; hence NC is the tangent, and EC the secant of the distance of the less circle from its pole to the radius of the primitive.

The projection of the poles of any circle, inclined to the primitive, are, in the line of measures, distant from the centre of the primitive, the tangent, and cotangent, of half its inclination.

Let MN (fig. 6.) be a great circle perpendicular to the primitive ABCD, and A the projecting point; then P, p are the poles of MN, and of all its parallels m n, &c. Let AP, Ap meet the diameter BD in Ff, which will therefore be the projected poles of MN and its parallels. The angle BEM is the inclination of the circle MEN, and its parallels, to the primitive: and because BC and MP are quadrants, and MC common to both; therefore PC = BM: and hence PEC is also the inclination of MN and its parallels. Now EF is the tangent of EAF, or of half the angle PEC the inclination; and Ef is the tangent of the angle EAF; but EAF is the complement of EAP, hence Ef is the cotangent of half the inclination.

1. 1. The projection of that pole which is nearest to the projecting point is without the primitive, and the projection of the other within.
2. 2. The projected centre of any circle is always between the projection of its nearest pole and the centre of the primitive; and the projected centres of all circles are contained between their projected poles.

Equal arches of any two great circles of the sphere will be intercepted between two other circles drawn on the sphere through the remote poles of those great circles.

Let AGB, CFD (fig. 7.) be two great circles of the sphere, whose remote poles are E, P; through which draw the great circle PBEC, and less circle PGE, intersecting the great circles AGB, CFD, in the points B, G, and D, F; then the arch BG is equal to the arch DF.

Because E is the pole of the circle AGB, and P the pole of CFD, therefore the arches EB, PD are equal; and since BD is common to both, hence the arch ED is equal to the arch PB. For the same reason, the arches EF, PG are equal; but the angle DEF is equal to the angle BPG: hence these triangles are equal, and therefore the arch DF is equal to the arch DG.

If from either pole of a projected great circle, two straight lines be drawn to meet the primitive and the projection, they will intercept similar arches of these circles.

On the plane of projection AGB (fig. 7.) let the great circle CFD be projected into cfd, and its pole P into p; through p draw the straight lines pd, pf, then are the arches GB, fd similar.

Since pd lies both in the plane AGB and APBE, it is in their common section, and the point B is also in their common section; therefore pd passes through the point B. In like manner it may be shown that the line pf passes through G. Now the points D, F

are projected into d, f; hence the arches FD, fd are similar; but GB is equal to FD, therefore the intercepted arch of the primitive GB is similar to the projected arch fd.

Hence, if from the angular point of a projected spherical angle two straight lines be drawn through the projected poles of the containing sides, the intercepted arch of the primitive will be the measure of the spherical angle.

To describe the projection of a great circle through two given points in the plane of the primitive.

Let P and B be given points, and C the centre of the primitive.

1. When one point P (fig. 8.) is the centre of the primitive, a diameter drawn through the given points will be the great circle required.

2. When one point P (fig. 9.) is in the circumference of the primitive. Through P draw the diameter PD; and an oblique circle described through the three points P, B, D, will be the projection of the required great circle.

3. When the given points are neither in the centre nor circumference of the primitive. Through either of the given points P (fig. 10.) draw the diameter ED, and at right angles thereto draw the diameter FG. From F through P draw the straight line FPH, meeting the circumference in H: draw the diameter HI, and draw the straight line FIK, meeting ED produced in D; then an arch, terminated by the circumference, being described through the three points P, B, K, will be the great circle.

To describe the representation of a great circle about any given point as a pole.

Let P be the given pole, and C the centre of the primitive.

1. When P (fig. 8.) is in the centre of the primitive, then the primitive will be the great circle required.

2. When the pole P (fig. 11.) is in the circumference of the primitive. Through P draw the diameter PE, and the diameter AB drawn at right angles to PE will be the projected great circle required.

3. When the given pole is neither in the centre nor circumference of the primitive. Through the pole P (fig. 12.) draw the diameter AB, and draw the diameter DE perpendicular to AB; through E and P draw the straight line EPF, meeting the circumference in F. Make FG equal to FD; through E and G draw the straight line EGH, meeting the diameter AB produced if necessary in H; then from the centre H, with the radius HE, describe the oblique circle DIE, and it will be the projection of the great circle required.

Or, make DK equal to FA; join EK, which intersects the diameter AB in I; then through the three points D, I, E, describe the oblique circle DIE.

To find the poles of a great circle.

1. When the given great circle is the primitive, its centre is the pole.

2. To

2. To find the pole of the right circle ACB (fig. 11.) Draw the diameter PE perpendicular to the given circle AB; and its extremities P, E are the poles of the circle ACB.

3. To find the pole of the oblique circle DEF (fig. 13.) Join DF, and perpendicular thereto draw the diameter AB, cutting the given oblique circle DEF in E. Draw the straight line FEG, meeting the circumference in G. Make GI, GH, each equal to AD; then FI being joined, cuts the diameter AB in P, the lower pole; through F and H draw the straight line FH, meeting the diameter AB produced in P, which will be the opposite or exterior pole.

To describe a less circle about any given point as a pole, and at any given distance from that pole.

1. When the pole of the less circle is in the centre of the primitive; then from the centre of the primitive, with the semitangent of the distance of the given circle from its pole, describe a circle, and it will be the projection of the less circle required.

2. If the given pole is in the circumference of the primitive, from C (fig. 14.) the centre of the primitive, let off CE the secant of the distance of the less circle from its pole P; then from the centre E, with the tangent of the given distance, describe a circle, and it will be the less circle required. Or, make PG, PF each equal to the chord of the distance of the less circle from its pole. Through B, G, draw the straight line BGD meeting CP produced in D: bisect GD in H, and draw HE perpendicular to GD; and meeting PD in E, then E is the centre of the less circle.

3. When the given pole is neither in the centre nor circumference of the primitive. Through P (fig. 15.) the given pole, and C the centre of the primitive, draw the diameter AB, and draw the diameter DE perpendicular to AB; join EP, and produce it to meet the primitive in P; make PF, PG, each equal to the chord of the distance of the less circle from its pole; join EF which intersects the diameter AB in H; from E through G draw the straight line EGI, meeting the diameter AB produced in I; bisect HI in K: then a circle described from the centre K, at the distance KH or KI, will be the projection of the less circle.

To find the poles of a given less circle.

The poles of a less circle are also those of its parallel great circle. If therefore the parallel great circle be given, then its poles being found by Prob. III. will be those of the less circle. But if the parallel great circle be not given, let HMIN (fig. 15.) be the given less circle. Through its centre, and C the centre of the primitive, draw the line of measures IAHB; and draw the diameter DE perpendicular to it, also draw the straight line EHF meeting the primitive in F; make F equal to the chord of the distance of the less circle from its pole; join E P, and its intersection P with the diameter AB is the interior pole. Draw the diameter CI through E and L, draw EL meeting the diameter AB produced in Q; then Q is the exterior pole. Or thus: Join EI intersecting the primitive in G; join also EH, and produce it to meet the primitive in F, bisect

the arch GF in P; from E to P draw the straight line EP, and P is the pole of the given less circle.

To measure any arch of a great circle.

1. Arcs of the primitive are measured on the line of chords.

2. Right circles are measured on the line of semitangents, beginning at the centre of the primitive. Thus, the measure of the portion AC (fig. 16.) of the right circle DE, is found by applying it to the line of semitangents. The measure of the arch DB is found by subtracting that of BC from $90^\backslash circ$: the measure of the arch AF, lying partly on each side of the centre, is obtained by adding the measures of AC and CF. Lastly, To measure the part AB, which is neither terminated at the centre or circumference of the primitive, apply CA to the line of semitangents; then CB, and the difference between the measures of these arcs, will be that of AB.

Or thus, Draw the diameter GH perpendicular to DE; then from either extremity, as D, of this diameter, draw lines through the extremities of the arch intended to be measured; and the intercepted portion of the primitive applied to the line of chords will give the measure of the required arch. Thus IK applied to the line of chords will give the measure of AB.

3. To measure an arch of an oblique circle: draw lines from its pole through the extremities of the arch to meet the primitive, then the intercepted portion of the primitive applied to the line of chords will give the measure of the arch of the oblique circle. Thus, let AB (fig. 17.) be an arch of an oblique circle to be measured, and P its pole; from P draw the lines PAD, PBE meeting the primitive in B and E; then the arch DE applied to the line of chords will give the measure of the arch of the oblique circle AB.

To measure any arch of a less circle.

Let DEG (fig. 18.) be the given less circle, and DE the arch to be measured; and its internal pole P; and describe the circle AFI parallel to the primitive, and whose distance from the projecting point may be equal to the distance of the given less circle from its pole P: then join PD, PE, which produce to meet the parallel circle in A and F. Now AF applied to a line of chords will give the measure of the arch DE of the given less circle.

To measure any spherical angle.

1. If the angle is at the centre of the primitive, it is measured as a plane angle.

2. When the angular point is in the circumference of the primitive; let A (fig. 19.) be the angular point, and ABE an oblique circle inclined to the primitive. Through P, the pole of ABE, draw the line P meeting the circumference in P; then the arch EP is the measure of the angle BAD, and the arch AF is the measure of its supplement BAF: also PF is the measure of the angle BAC, and ED that of its supplement.

g. If the angular point is neither at the centre nor circumference of the primitive. Let $A$ (fig. 20.) be the angular point, and $DAH$, or $GAF$ the angle to be measured, $P$ the pole of the oblique circle $DAF$, and $\backslash rho$ the pole of $GAF$: then from $A$, through the points $P$, $\backslash rho$, draw the straight lines $APM$, $A\backslash rho\; N$, and the arch $MN$ will be the measure of the angle $DAH$; and the supplement of $MN$ will be the measure of the angle $HAF$ or $DAG$.

To draw a great circle perpendicular to a projected great circle, and through a point given in it.

Find the pole of the given circle, then a great circle described through that pole and the given point will be perpendicular to the given circle. Hence if the given circle be the primitive, then a diameter drawn through the given point will be the required perpendicular. If the given circle is a right one, draw a diameter at right angles to it; then through the extremities of this diameter and the given point describe an oblique circle, and it will be perpendicular to that given. If the given circle is inclined to the primitive, let it be represented by $BAD$ (fig. 21.), whose pole is $P$, and let $A$ be the point through which the perpendicular is to be drawn; then, by Prob. I. describe a great circle through the points $P$ and $A$, and it will be perpendicular to the oblique circle $BAD$.

Through a point in a projected great circle, to describe another great circle to make a given angle with the former, provided the measure of the given angle is not less than the distance between the given point and circle.

Let the given circle be the primitive, and let $A$ (fig. 19.) be the angular point. Draw the diameters $AE$, $DF$ perpendicular to each other; and make the angle $CAG$ equal to that given, or make $CG$ equal to the tangent of the given angle; then from the centre $G$, with the distance $GC$, describe the oblique circle $ABE$, and it will make with the primitive an angle equal to that given.

If the given circle be a right one, let it be $APB$ (fig. 22.) and let $P$ be the given point. Draw the diameter $GH$ perpendicular to $AB$; join $GP$, and produce it to $a$; make $Ha$ equal to twice $Ap$; and $Gb$ being joined intersects $AB$ in $C$. Draw $CD$ perpendicular to $AB$, and equal to the cotangent of the given angle to the radius $PC$; or make the angle $CPD$ equal to the complement of that given; then from the centre $D$, with the radius $DP$, describe the great circle $FPE$, and the angle $APF$, or $BPE$, will be equal to that given.

If $APB$ (fig. 23.) is an oblique circle. From the angular point $P$, draw the lines $PG$, $PC$ through the centres of the primitive and given oblique circle. Through $C$, the centre of $APB$, draw $GCD$ at right angles to $PG$; make the angle $GPD$ equal to that given; and from the centre $D$, with the radius $DP$, describe the oblique circle $FPE$, and the angle $APF$, or $BPE$, will be equal to that proposed.

Any great circle cutting the primitive being given, to

describe another great circle which shall cut the given one in a proposed angle, and have a given arch intercepted between the primitive and given circles.

If the given circle be a right one, let it be represented by $APC$ (fig. 24.); and at right angles thereto draw the diameter $BPM$; make the angle $BPF$ equal to the complement of the given angle, and $PF$ equal to the tangent of the given arch; and from the centre of the primitive with the secant of the same arch describe the arch $Gg$. Through $F$ draw $FG$ parallel to $AC$, meeting $Gg$ in $G$; then from the centre $G$, with the tangent $PF$, describe an arch $no$, cutting $APC$ in $I$, and join $GI$. Through $G$, and the centre $P$, draw the diameter $HK$; draw $PL$ perpendicular to $HK$, and $IL$ perpendicular to $GI$, meeting $PL$ in $L$; then $L$ will be the centre of the circle $HIK$, which is that required.

But if the given great circle be inclined to the primitive, let it be $ADB$ (fig. 25.), and $E$ its centre; make the angle $BDF$ equal to the complement of that given, and $DF$ equal to the tangent of the given arch, as before. From $P$, the centre of the primitive, with the secant of the same arch, describe the arch $Gg$, and from $E$, the centre of the oblique circle, with the extent $EF$, describe an arch intersecting $Gg$ in $G$. Now $G$ being determined, the remaining part of the operation is performed as before.

When the given arch exceeds $90^\backslash circ$, the tangent and secant of its supplement are to be applied on the line $DF$ the contrary way, or towards the right; the former construction being reckoned to the left.

Any great circle in the plane of projection being given to describe another great circle, which shall make given angles with the primitive and given circles.

Let $ADC$ (fig. 26.) be the given circle, and $Q$ its pole. About $P$, the pole of the primitive, describe an arch $mn$, at the distance of as many degrees as are in the angle which the required circle is to make with the primitive. About $Q$ the pole of the circle $ADC$, and at a distance equal to the measure of the angle which the required circle is to make with the given circle $ADC$, describe an arch $on$, cutting $mn$ in $n$. Then about $n$ as a pole, describe the great circle $EDF$, cutting the primitive and given circle in $E$ and $D$, and it will be the great circle required.

It will hence be an easy matter to construct all the various spherical triangles. The reader is, however, referred to the article Spherical Trigonometry, for the method of constructing them agreeable to this projection; and also for the application to the resolution of problems of the sphere. For the method of projecting the sphere upon the plane of the meridian, and of the horizon, according to the stereographic projection, see the article GEOGRAPHY.

THE orthographic projection of the sphere, is that in which the eye is placed in the axis of the plane of projection

Orthographic Projection of the Sphere.
jection, at an infinite distance with respect to the diameter of the sphere; so that at the sphere all the visual rays are assumed parallel, and therefore perpendicular to the plane of projection.

Hence the orthographic projection of any point is where a perpendicular from that point meets the plane of projection: and the orthographic representation of any object is the figure formed by perpendiculars drawn from every point of the object to the plane of projection.

This method of projection is used in the geometrical delineation of eclipses, occultations, and transits. It is also particularly useful in various other projections, such as the analemma. See GEOGRAPHY, &c.

PROPOSITION I. THEOREM I.

Every straight line is projected into a straight line. If the given line be parallel to the plane of projection, it is projected into an equal straight line; but if it is inclined to the primitive, then the given straight line will be to its projection in the ratio of the radius to the cosine of inclination.

Let AB (fig. 27.) be the plane of projection, and let CD be a straight line parallel thereto: from the extremities C, D of the straight line CD, draw the lines CE, DF perpendicular to AB; then by 3. of XI. of Eucl. the intersection EF, of the plane CEFD, with the plane of projection, is a straight line: and because the straight lines CD, EF are parallel, and also CE, DF; therefore, by 34. of I. of Eucl. the opposite sides are equal; hence the straight line CD, and its projection EF, are equal. Again, let GH be the proposed straight line, inclined to the primitive; then the lines GE, HF being drawn perpendicular to AB, the intercepted portion EF will be the projection of GH. Through G draw GI parallel to AB, and the angle IGH will be equal to the inclination of the given line to the plane of projection. Now GH being the radius, GI, or its equal EF, will be the cosine of IGH; hence the given line GH is to its projection EF as radius to the cosine of inclination.

COROLLARIES.

1. 1. A straight line perpendicular to the plane of projection is projected into a point.
2. 2. Every straight line in a plane parallel to the primitive is projected into an equal and parallel straight line.
3. 3. A plane angle parallel to the primitive is projected into an equal angle.
4. 4. Any plane rectilinear figure parallel to the primitive is projected into an equal and similar figure.
5. 5. The area of any rectilinear figure is to the area of its projection as radius to the cosine of its inclination.

PROPOSITION II. THEOREM II.

Every great circle, perpendicular to the primitive, is projected into a diameter of the primitive; and every arch of it, reckoned from the pole of the primitive, is projected into its fine.

Let BFD (fig. 28.) be the primitive, and ABCD a great circle perpendicular to it, passing through its poles A, C; then the diameter BED, which is their

line of common section, will be the projection of the Orthographic Projection of the Sphere. For if from any point, as G, in the primitive circle ABC, a perpendicular GH fall upon BD, it will also be perpendicular to the plane of the primitive: therefore H is the projection of G. Hence the whole circle is projected into BD, and any arch AG into EH equal to GI its fine.

COROLLARIES.

1. 1. Every arch of a great circle, reckoned from its intersection with the primitive, is projected into its versed fine.
2. 2. Every less circle perpendicular to the primitive is projected into its line of common section with the primitive, which is also its own diameter; and every arch of the semicircle above the primitive, reckoned from the middle point, is projected into its fine.
3. 3. Every diameter of the primitive is the projection of a great circle; and every chord the projection of a less circle.
4. 4. A spherical angle at the pole of the primitive is projected into an equal angle.

PROPOSITION III. THEOREM III.

A circle parallel to the primitive is projected into a circle equal to itself, and concentric with the primitive.

Let the less circle FIG (fig. 29.) be parallel to the plane of the primitive BND. The straight line HE, which joins their centres, is perpendicular to the primitive; therefore E is the projection of H. Let any radii HI and IN perpendicular to the primitive be drawn. Then IN, HE being parallel, are in the same plane; therefore IH, NE, the lines of common section of the plane IE, with two parallel planes, are parallel; and the figure IHEN is a parallelogram. Hence NE = IH, and consequently FIG is projected into an equal circle KNI, whose centre is E.

COROLLARY.

The radius of the projection is the cosine of the distance of the parallel circle from the primitive, or the sine of its distance from the pole of the primitive.

PROPOSITION IV. THEOREM IV.

An inclined circle is projected into an ellipse, whose transverse axis is the diameter of the circle.

1. 1. Let ELF (fig. 30.) be a great circle inclined to the primitive EBF, and EF their line of common section. From the centre C, and any other point K, in EF, let the perpendiculars CB, KI be drawn in the plane of the primitive, and CL, KN, in the plane of the great circle, meeting the circumference in L, N. Let LG, ND be perpendicular to CB, KI; then G, D are the projections of L, N. And because the triangles LCG, NKD are equiangular, $CL^2\; :\; CG^2\; ::\; NK^2\; :\; DK^2$; or $EC^2\; :\; CG^2\; ::\; EKF\; :\; DK^2$; therefore the points G, D are in the curve of an ellipse, of which EF is the transverse axis, and CG the semiconjugate axis.

COROLLARIES.

1. 1. In a projected great circle, the semiconjugate axis is the cosine of the inclination of the great circle to the primitive.

2. Perpendiculars to the transverse axis intercept corresponding arches of the projection and the primitive.
3. The eccentricity of the projection is the sine of the inclination of the great circle to the primitive.

Case 2. Let $AQB$ (fig. 31.) be a less circle, inclined to the primitive, and let the great circle $LBM$, perpendicular to both, intersect them in the lines $AB$, $LM$. From the centre $O$, and any other point $N$ in the diameter $AB$, let the perpendiculars $TOP$, $NQ$, be drawn in the plane of the less circle, to meet its circumference in $T$, $P$, $Q$. Also, from the points $A$, $N$, $O$, $B$, let $AG$, $NI$, $OC$, $BH$, be drawn perpendicular to $LM$; and from $P$, $Q$, $T$, draw $PE$, $QD$, $TF$ perpendicular to the primitive; then $G$, $I$, $C$, $H$, $E$, $D$, $F$, are the projections of these points. Because $OP$ is perpendicular to $LBM$, and $OC$, $PE$, being perpendicular to the primitive, are in the same plane, the plane $COPE$ is perpendicular to $LBM$. But the primitive is perpendicular to $LBM$; therefore the common section $EC$ is perpendicular to $LBM$, and to $LM$. Hence $CP$ is a parallelogram, and $EC\; =\; OP$. In like manner, $FC$, $DI$, are proved perpendicular to $LM$, and equal to $OT$, $NQ$. Thus, $ECF$ is a straight line, and equal to the diameter $PT$. Let $QR$, $DK$ be parallel to $AB$, $LM$; then $RO\; =\; NQ\; =\; DI\; =\; KC$, and $PR\; \backslash times\; RT\; =\; EK\; \backslash times\; KF$. But $AO\; :\; CG\; ::\; NO\; :\; CI$; therefore $AO^2\; :\; CG^2\; ::\; QR^2\; :\; DK^2$ and $EC^2\; :\; CG^2\; ::\; EKF\; :\; DK^2$.

1. 1. The transverse axis is to the conjugate as radius to the cosine of the circle's inclination to the primitive.
2. 2. Half the transverse axis is the cosine of half the sum of the greatest and least distances of the less circle from the primitive.
3. 3. The extremities of the conjugate axis are in the line of measures, distant from the centre of the primitive by the cosines of the greatest and least distances of the less circle from the primitive.
4. 4. If from the extremities of the conjugate axis of any elliptical projection perpendiculars be drawn (in the same direction if the circle do not intersect the primitive, but if otherwise in opposite directions), they will intersect an arch of the primitive, whose chord is equal to the diameter of the circle.

The projected poles of an inclined circle are in its line of measures distant from the centre of the primitive the sine of the inclination of the circle to the primitive.

Let $ABCD$ (fig. 32.) be a great circle, perpendicular both to the primitive and the inclined circle, and intersecting them in the diameters $AC$, $MN$. Then $ABCD$ passes through the poles of the inclined circle; let these be $P$, $Q$; and let $P\backslash rho$, $Qq$, be perpendicular to $AC$; $\backslash rho$, $q$ are the projected poles; and it is evident that $\backslash rho\; O\; =$ sine of $BP$, or $MA$, the inclination.

1. 1. The centre of the primitive, the centre of the projection, the projected poles, and the extremities of the conjugate axis, are all in one and the same straight line.
2. 2. The distance of the centre of projection from the

centre of the primitive, is to the cosine of the distance of the circle from its own pole, as the sine of the circle's inclination to the primitive is to the radius.

To describe the projection of a circle perpendicular to the primitive, and whose distance from its pole is equal to a given quantity.

Let $PA\backslash rho\; B$ (fig. 33.) be the primitive circle, and $P$, $\backslash rho$ the poles of the right circle to be projected. Then if the circle to be projected is a great circle, draw the diameter $AB$ at right angles to the axis $P\backslash rho$, and it will be that required. But if the required projection is that of a less circle, make $PE$, $PF$ each equal to the chord of the distance of the less circle from its pole; join $EF$, and it will be the projection of the less circle required.

Through a given point in the plane of the primitive to describe the projection of a great circle, having a given inclination to the primitive.

1. 1. When the given inclination is equal to a right angle, a straight line drawn through the centre of the primitive and the given point will be the projection required.
2. 2. When the given inclination is less than a right angle, and the given point in the circumference of the primitive. Let $R$ (fig. 34.) be a point given in the circumference of the primitive, through which it is required to draw the projection of a great circle, inclined to the primitive in an angle measured by the arch $QP$ of the primitive.

Through the given point $R$ draw the diameter $RCS$, and draw $GC$ at right angles to it. Make the arch $GV$ of the primitive equal to $QP$, and draw $VA$ at right angles to $GC$; and in $G$, towards the opposite parts of $C$, take $CB$ equal to $AC$; then, with the greater axis $RS$, and less axis $AB$, describe an ellipse, and it will be the projection of the oblique circle required.

1. 3. When the distance of the given point from the primitive is equal to the cosine of the given inclination.

Every thing remaining as in the preceding case; let $A$ be the given point, and $AC$ the cosine of an arch $GV$, equal to the given arch $QP$; then drawing the diameter $RCS$ at right angles to $ACB$, the ellipse described with the given axis $RS$, $AB$ will be the projection of the inclined circle.

1. 4. When the distance of the given point from the centre of the primitive is less than the semidiameter of the primitive, but greater than the cosine of the given inclination.

Let $D$ be the given point, through which draw the diameter $IC$; and at the point $D$ draw $DL$ perpendicular to $DC$ meeting the primitive in $L$; also draw $LK$, making with $LD$ the angle $DLK$ equal to the complement of the given inclination. Let $LK$ meet $DC$ in $K$; then will $DK$ be less than $DC$. On $DC$ as the diameter describe a circle, and make $DH$ equal to $DK$; through $H$ draw a diameter of the primitive $RCS$, and describe an ellipse through the points $R$, $D$, $S$, and it will be the projection of the inclined circle.

PROPOSITION VIII. PROBLEM III.

Through two given points in the plane of the primitive to describe the projection of a great circle.

1. If the two given points and the centre of the primitive be in the same straight line, then a diameter of the primitive being drawn through these points will be the projection of the great circle required.

2. When the two given points are not in the same straight line with the centre of the primitive; and one of them is in the circumference of the primitive.

Let D, R, (fig. 34.) be the two given points, of which R is in the circumference of the primitive. Draw the diameter RCS, and GC_g, FDH perpendicular to it, meeting the primitive in G_g F. Divide GC, gC, in A, B, in the same proportion as FH is divided in D; and describe the ellipse whose axes are RS, AB, and centre C; and it will be the projection required.

3. When the given points are within the primitive, and not in the same straight line with its centre.

Let D, E (fig. 35.) be the two given points; through C the centre of the primitive draw the straight lines ID, KE; draw DL perpendicular to Ii, and EO perpendicular to Kk, meeting the primitive in L, O. Through E, and towards the same parts of C, draw EP parallel to DC, and in magnitude a fourth proportional to LD, DC, OE. Draw the diameter CP meeting the primitive in R, S, and describe an ellipse through the points D and R or S, and it will also pass through E. This ellipse will be the projection of the proposed inclined circle.

PROPOSITION IX. PROBLEM IV.

To describe the projection of a less circle parallel to the primitive, its distance from the pole of the primitive being given.

From the pole of the primitive, with the fine of the given distance of the circle from its pole, describe a circle, and it will be the projection of the given less circle.

PROPOSITION X. PROBLEM V.

About a given point as a projected pole to describe the projection of an inclined circle, whose distance from its pole is given.

Let P (fig. 36.) be the given projected pole, through which draw the diameter G_g, and draw the diameter Hb perpendicular thereto. From P draw PL perpendicular to GP meeting the circumference in L; through which draw the diameter Ll. Make LT, LK each equal to the chord of the distance of the less circle from its pole, and join TK, which intersects Ll, in Q. From the points T, Q, K draw the lines FA, QS, KB, perpendicular to G_g; and make OR, OS, each equal to QT, or QK. Then an ellipse described through the points A, S, B, R will be the projection of the proposed less circle.

PROPOSITION XI. PROBLEM VI.

To find the poles of a given projected circle.

1. If the projected circle be parallel to the primitive, the centre of the primitive will be its pole.

2. If the circle be perpendicular to the primitive, then the extremities of a diameter of the primitive drawn at right angles to the straight line representing the projected circle, will be the poles of that circle.

3. When the projected circle is inclined to the primitive.

Let ARBS (fig. 36, 37.) be the elliptical projection of any oblique circle; through the centre of which, and C the centre of the primitive, draw the line of measures CBA, meeting the ellipse in B, A; and the primitive in G, g. Draw CH, BK, AT perpendicular to G_g, meeting the primitive in H, K, T. Bisect the arch KT in L, and draw LP perpendicular to G_g; then P will be the projected pole of the circle, of which ARBS is the projection.

PROPOSITION XII. PROBLEM VII.

To measure any portion of a projected circle, and conversely.

1. When the given projection is that of a great circle.

Let ADEB (fig. 38.) be the given great circle, either perpendicular or inclined to the primitive, of which the portion DE is to be measured, and let Mm be the line of measures of the given circle. Through the points D, E draw the lines EG, DF parallel to Mm; and the arch FG of the primitive will be the measure of the arch DE of the great circle, and conversely.

2. When the projection is that of a less circle parallel to the primitive.

Let DE (fig. 39.) be the portion to be measured, of the less circle DEH parallel to the primitive. From the centre C draw the lines CD, CE, and produce them to meet the primitive in the points B, F. Then the intercepted portion BF of the primitive will be the measure of the given arch DE of the less circle DEH.

3. If the given less circle, of which an arch is to be measured, is perpendicular to the primitive.

Let ADEB (fig. 40.) be the less circle, of which the measure of the arch DE is required. Through C, the centre of the primitive, draw the line of measures Mm, and from the intersection O of the given right circle, and the line of measures, with the radius OA, or OB, describe the semicircle AFGB; through the points D, E draw the lines DF, EG parallel to the line of measures, and the arch FG will be the measure of DE, to the radius AO. In order to find a similar arch in the circumference of the primitive, join OF, OG, and at the centre C of the primitive, make the angle mCH equal to FOG, and the arch mH to the radius Cm will be the measure of the arch DE.

4. When the given projection is of a less circle inclined to the primitive.

Let RDS (fig. 41.) be the projection of a less circle inclined to the primitive, and DE a portion of that circle to be measured. Through O the centre of the projected circle, and C the centre of the primitive, draw the line of measures Mm; and from the centre O, with the radius OR, or OS, describe the semicircle RGFS; through the points D, E draw the lines DF, EG parallel to the line of measures, and FG will be the measure of the arch DE to the radius OR, or OS. Join OF, OG, and make the angle mCH equal to FOG,

FOG, and the arch $m$ II of the primitive will be the measure of the arch DE of the inclined circle RDS.

The converse of this proposition, namely, to cut off an arch from a given projected circle equal to a given arch of the primitive, is obvious.

The above operation would be greatly shortened by using the line of sines in the sector.

It seems unnecessary to insist farther on this projection, especially as the reader will see the application of it to the projection of the sphere on the planes of the Meridian, Equator, and Horizon in the article GEOGRAPHY; and to the delineation of Eclipses in the article ASTRONOMY. The Analemma, Plate CCXII. in the article GEOGRAPHY, is also according to this projection; and the method of applying it to the solution of astronomical problems is there exemplified.

In this projection the eye is in the centre of the sphere, and the plane of projection touches the sphere in a given point parallel to a given circle. It is named gnomonic, on account of its being the foundation of dialling: the plane of projection may also represent the plane of a dial, whose centre being the projected pole, the semiaxis of the sphere will be the stile or gnomon of the dial.

As the projection of great circles are represented by straight lines, and less circles parallel to the plane of projection are projected into concentric circles; therefore many problems of the sphere are very easily resolved. Other problems, however, become more intricate on account of some of the circles being projected into ellipses, parabolas, and hyperbolas.

Every great circle is projected into a straight line perpendicular to the line of measures; and whose distance from the centre is equal to the cotangent of its inclination, or to the tangent of its nearest distance from the pole of the projection.

Let BAD (fig. 42.) be the given circle, and let the circle CBED be perpendicular to BAD, and to the plane of projection; whose intersection CF with this last plane will be the line of measures. Now since the circle CBED is perpendicular both to the given circle BAD and to the plane of projection, the common section of the two last planes produced will therefore be perpendicular to the plane of the circle CBED produced, and consequently to the line of measures: hence the given circle will be projected into that section; that is, into a straight line passing through $d$, perpendicular to $Cd$. Now $Cd$ is the cotangent of the angle $CdA$, the inclination of the given circle, or the tangent of the arch CD to the radius AC.

1. A great circle perpendicular to the plane of projection is projected into a straight line passing through the centre of projection; and any arch is projected into its correspondent tangent.

2. Any point, as D, or the pole of any circle, is

projected into a point $d$, whose distance from the pole of projection is equal to the tangent of that distance.

3. If two great circles be perpendicular to each other, and one of them passes through the pole of projection, they will be projected into two straight lines perpendicular to each other.

4. Hence if a great circle be perpendicular to several other great circles, and its representation pass through the centre of projection; then all these circles will be represented by lines parallel to one another, and perpendicular to the line of measures, for representation of that first circle.

If two great circles intersect in the pole of projection, their representations will make an angle at the centre of the plane of projection, equal to the angle made by these circles on the sphere.

For since both these circles are perpendicular to the plane of projection, the angle made by their intersections with this plane is the same as the angle made by these circles.

Any less circle parallel to the plane of projection is projected into a circle whose centre is the pole of projection, and its radius is equal to the tangent of the distance of the circle from the pole of projection.

Let the circle PI (fig. 42.) be parallel to the plane GF, then the equal arches PC, CI are projected into the equal tangents GC, CH; and therefore C, the point of contact and pole of the circle PI and of the projection, is the centre of the representation G, H.

If a circle be parallel to the plane of projection, and 45 degrees from the pole, it is projected into a circle equal to a great circle of the sphere; and therefore may be considered as the primitive circle, and its radius the radius of projection.

A less circle not parallel to the plane of projection is projected into a conic section, whose transverse axis is in the line of measures; and the distance of its nearest vertex from the centre of the plane of projection is equal to the tangent of its nearest distance from the pole of projection; and the distance of the other vertex is equal to the tangent of the greatest distance.

Any less circle is the base of a cone whose vertex is at A (fig. 43.); and this cone being produced, its intersection with the plane of projection will be a conic section. Thus the cone DAF, having the circle DF for its base, being produced, will be cut by the plane of projection in an ellipse whose transverse diameter is $df$; and $Cd$ is the tangent of the angle CAD, and $Cf$ the tangent of CAF. In like manner, the cone AFE, having the side AE parallel to the line of measures $df$, being cut by the plane of projection, the section will be a parabola, of which $f$ is the nearest ver-

Gnomonic Projection of the Sphere.
tex, and the point into which E is projected is at an infinite distance. Also the cone AFG, whose base is the circle FG, being cut by the plane of projection, the section will be a hyperbola; of which f is the nearest vertex; and GA being produced gives d the other vertex.

COROLLARIES.

1. A less circle will be projected into an ellipse, a parabola, or hyperbola, according as the distance of its most remote point is less, equal to, or greater than, 90 degrees.

2. If H be the centre, and K, k, l the focus of the ellipse, hyperbola, or parabola; then $HK\; =\; \backslash frac\{Ad\; -\; Af\}\{2\}$ for the ellipse; $Hk\; =\; \backslash frac\{Ad\; +\; Af\}\{2\}$ for the hyperbola; and $fn$ being drawn perpendicular to AE $f/l\; =\; \backslash frac\{nE\; +\; Ff\}\{2\}$ for the parabola.

PROPOSITION V. THEOREM V.

Plate ccccxxii. Let the plane TW (fig. 44.) be perpendicular to the plane of projection TV, and BCD a great circle of the sphere in the plane TW. Let the great circle BED be projected into the straight line $bck$. Draw CQS perpendicular to $bck$, and Cm parallel to it and equal to CA, and make QS equal to Qm; then any angle QSt is the measure of the arch Qt of the projected circle.

Join AQ; then because Cm is equal to GA, the angle Qcm equal to QCA, each being a right angle, and the side QC common to both triangles; therefore Qm, or its equal QS, is equal QA. Again, since the plane ACQ is perpendicular to the plane TV, and bQ to the intersection CQ; therefore bQ is perpendicular both to AQ and QS; hence, since AQ and QS are equal, all the angles at S cut the line bQ in the same points as the equal angles at A. But by the angles at A the circle BED is projected into the line bQ. Therefore the angles at S are the measures of the parts of the projected circle bQ; and S is the dividing centre thereof.

COROLLARIES.

1. Any great circle bQt is projected into a line of tangents to the radius SQ.

2. If the circle bC pass through the centre of projection, then the projecting point A is the dividing centre thereof, and Cb is the tangent of its correspondent arch CB to CA the radius of projection.

PROPOSITION VI. THEOREM VI.

Let the parallel circle GLH (fig. 44.) be as far from the pole of projection C as the circle FNI is from its pole; and let the distance of the poles C, P be bisected by the radius AO; and draw bAD perpendicular to AO; then any straight line bQt drawn through b will cut off the arches bl, Fn equal to each other in the representations of these equal circles in the plane of projection.

Let the projections of the less circles be described. Then, because BD is perpendicular to AO, the arches

BO, DO are equal; but since the less circles are equally distant each from its respective pole, therefore the arches FO, OH are equal; and hence the arch BF is equal to the arch DH. For the same reason the arches BN, DL are equal; and the angle FBN is equal to the angle LDH; therefore, on the sphere, the arches FN, HL are equal. And since the great circle BNLD is projected into the straight line bQn l, &c. therefore n is the projection of N, and l that of L; hence fn, bl, the projections of FN, HL respectively, are equal.

PROPOSITION VII. THEOREM VII.

If Fnk, blg (fig. 45.) be the projections of two equal circles, whereof one is as far from its pole P as the other from its pole C, which is the centre of projection; and if the distance of the projected poles C, p be divided in o, so that the degrees in Co, op be equal, and the perpendicular oS be erected to the line of measures gb. Then the line pn, Cl drawn from the poles C, p, through any point Q in the line oS, will cut off the arches Fn, bl equal to each other, and to the angle QCp.

The great circle AO perpendicular to the plane of the primitive is projected into the straight line oS perpendicular to gb, by Prop. i. Cor. 3. Let Q be the projection of q; and since pQ, CQ are straight lines, they are therefore the representations of the arches Pg, Cq of great circles. Now since PqC is an isosceles spherical triangle, the angles PCQ, CPQ are therefore equal; and hence the arches Pg, Cq produced will cut off equal arches from the given circles FI, GH, whose representations Fn, bl are therefore equal; and since the angle QCp is the measure of the arch bl, it is also the measure of its equal Fn.

COROLLARY.

Hence, if from the projected pole of any circle a perpendicular be erected to the line of measures, it will cut off a quadrant from the representation of that circle.

PROPOSITION VIII. THEOREM VIII.

Let Fnk (fig. 45.) be the projection of any circle FI, and p the projection of its pole P. If Cg be the cotangent of CAP, and gB perpendicular to the line of measures gC, let CAP be bisected by AO, and the line oB drawn to any point B, and also pB cutting Fnk in d; then the angle g o B is the measure of the arch Fd.

The arch PG is a quadrant, and the angle g o A = gPA + oAP = gAC + oAP = gAC + CAo = gAo; therefore gA = go; consequently o is the dividing centre of gB, the representation of GA; and hence, by Prop. v. the angle g o B is the measure of gB. But since pg represents a quadrant, therefore p is the pole of gB; and hence the great circle pdB passing through the pole of the circles gB and Fn will cut off equal arches in both, that is, Fd = gB = angle g o B.

COROLLARY.

The angle g o B is the measure of the angle g p B.
4 D 2 For

Gnomonic Projection of the Sphere. For the triangle $g\backslash rho\; B$ represents a triangle on the sphere, wherein the arch which $g\; B$ represents is equal to the angle which the angle $\backslash rho$ represents; because $g\backslash rho$ is a quadrant; therefore $g\backslash rho\; B$ is the measure of both.

To draw a great circle through a given point, and whose distance from the pole of projection is equal to a given quantity.

Plate MCCXXII. Let $ADB$ (fig. 46.) be the projection, $C$ its pole or centre, and $P$ the point through which a great circle is to be drawn: through the points $P,\; C$ draw the straight line $PCA$, and draw $CE$ perpendicular to it: make the angle $CAE$ equal to the given distance of the circle from the pole of projection $C$; and from the centre $C$, with the radius $CE$, describe the circle $CFG$: through $P$ draw the straight line $PIK$, touching the circle $CFG$ in $I$, and it will be the projection of the great circle required.

To draw a great circle perpendicular to a great circle which passes through the pole of projection, and at a given distance from that pole.

Let $ADB$ (fig. 46.) be the primitive, and $CI$ the given circle: draw $CI$ perpendicular to $CI$, and make the angle $CLI$ equal to the given distance: then the straight line $KP$, drawn through $I$ parallel to $CI$, will be the required projection.

At a given point in a projected great circle, to draw another great circle to make a given angle with the former; and, conversely, to measure the angle contained between two great circles.

Let $P$ (fig. 47.) be the given point in the given great circle $PB$, and $C$ the centre of the primitive: through the points $P,\; C$ draw the straight line $PCG$; and draw the radius of the primitive $CA$ perpendicular thereto; join $PA$; to which draw $AG$ perpendicular: through $G$ draw $BGD$ at right angles to $GP$, meeting $PB$ in $B$; bisect the angle $CAP$ by the straight line $AO$; join $BO$, and make the angle $BOD$ equal to that given; then $DP$ being joined, the angle $BPD$ will be that required.

If the measure of the angle $BPD$ be required, from the points $B,\; D$ draw the lines $BO,\; DO$, and the angle $BOD$ is the measure of $BPD$.

To describe the projection of a less circle parallel to the plane of projection, and at a given distance from its pole.

Let $ADB$ (fig. 46.) be the primitive, and $C$ its centre: set the distance of the circle from its pole, from $B$ to $H$, and from $H$ to $D$; and draw the straight line $AED$, intersecting $CE$ perpendicular to $BC$, in the point $E$: with the radius $CE$ describe the circle $EFG$, and it is the projection required.

To draw a less circle perpendicular to the plane of projection.

Let $C$ (fig. 48.) be the centre of projection, and $TI$ a great circle parallel to the proposed less circle: at $C$ make the angles $ICN,\; TCO$ each equal to the distance of the less circle from its parallel great circle $TI$: let $CL$ be the radius of projection, and from the extremity $L$ draw $LM$ perpendicular thereto: make $CV$ equal to $LM$; or $CF$ equal to $CM$: then with the vertex $V$ and asymptotes $CN,\; CO$ describe the hyperbola $WVK$; or, with the focus $F$ and $CV$ describe the hyperbola, and it will be the perpendicular circle described.

Gnomonic Projection of the Sphere. Plate MCCXXIII. See Conic Sections.

To describe the projection of a less circle inclined to the plane of projection.

Draw the line of measures $d\backslash rho$ (fig. 49.) and at $C$, the centre of projection, draw $CA$ perpendicular to $d\backslash rho$, and equal to the radius of projection: with the centre $A$, and radius $AC$, describe the circle $DCFG$; and draw $RAE$ parallel to $d\backslash rho$: then take the greatest and least distances of the circle from the pole of projection, and set them from $C$ to $D$ and $F$ respectively, for the circle $DF$; and from $A$, the projecting point, draw the straight lines $AF,\; f$, and $AD,\; d$; then $df$ will be the transverse axis of the ellipse: but if $D$ fall beyond the line $RE$, as at $G$, then from $G$ draw the line $GAD,\; d$, and $df$ is the transverse axis of an hyperbola: and if the point $D$ fall in the line $RE$, as at $E$, then the line $AE$ will not meet the line of measures, and the circle will be projected into a parabola whose vertex is $f$: bisect $df$ in $H$, the centre, and for the ellipse take half the difference of the lines $A,\; d,\; Af$, which laid from $H$ will give $K$ the focus: for the hyperbola, half the sum of $A,\; d,\; Af$ being laid from $H$, will give $k$ its focus: then with the transverse axis $df$, and focus $K$, or $k$, describe the ellipse $dMf$, or hyperbola $fM$, which will be the projection of the inclined circle: for the parabola, make $EQ$ equal to $f,\; f$, and draw $fn$ perpendicular to $AQ$, and make $fk$ equal to one half of $nQ$: then with the vertex $f$, and focus $k$, describe the parabola $fM$, for the projection of the given circle $EE$.

To find the pole of a given projected circle.

Let $DMF$ (fig. 50.) be the given projected circle, whose line of measures is $DF$, and $C$ the centre of projection; from $C$ draw the radius of projection $CA$, perpendicular to the line of measures, and $A$ will be the projecting point: join $AD,\; AF$, and bisect the angle $DAF$ by the straight line $AP$; hence $P$ is the pole. If the given projection be an hyperbola, the angle $fAG$ (fig. 49.), bisected, will give its pole in the line of measures; and in a parabola, the angle $fAE$ bisected will give its pole.

To measure any portion of a projected great circle, or to lay off any number of degrees thereon.

Let $EP$ (fig. 51.) be the great circle, and $IP$ a portion thereof to be measured: draw $ICD$ perpendicular to $IP$; let $C$ be the centre, and $CB$ the radius of projection, with which describe the circle $EBD$; make

IA equal to IB; then A is the dividing centre of EP; hence AP being joined, the angle IAP is the measure of the arch IP.
Or, if IAP be made equal to any given angle, then IP is the correspondent arch of the projection.

PROPOSITION XVII. PROBLEM IX.

To measure any arch of a projected less circle, or to lay off any number of degrees on a given projected less circle.

edccxxiii. Let $Fn$ (fig. 52.) be the given less circle, and $P$ its pole: from the centre of projection $C$ draw $CA$ perpendicular to the line of measures $GH$, and equal to the radius of projection; join $AP$, and bisect the angle $CAP$ by the straight line $AO$, to which draw $AD$ perpendicular: describe the circle $G/H$, as far distant from the pole of projection $C$ as the given circle is from its pole $P$; and through any given point $n$, in the projected circle $Fn$, draw $Dn/$, then $H/$ is the measure of the arch $Fn$.

Or let the measure be laid from $H$ to $/$, and the line $D/$ joined will cut off $Fn$ equal thereto.

PROPOSITION XVIII. PROBLEM X.

To describe the gnomonic projection of a spherical triangle, when three sides are given; and to find the measures of either of its angles.

Let $ABC$ (fig. 53.) be a spherical triangle whose three sides are given: draw the radius $CD$ (fig. 54.) perpendicular to the diameter of the primitive $EF$; and at the point $D$ make the angles $CDA$, $CDG$, $ADI$, equal respectively to the sides $AC$, $BC$, $AB$, of the spherical triangle $ABC$ (fig. 53.), the lines $DA$, $DG$ intersecting the diameter $EF$, produced if necessary in the points $A$ and $G$: make $DI$ equal to $DG$; then from the centre $C$, with the radius $CG$, describe an arch; and from $A$, with the distance $AI$, describe another arch, intersecting the former in $B$; join $AB$, $CB$, and $ACB$ will be the projection of the spherical triangle (fig. 53.); and the rectilinear angle $ACB$ is the measure of the spherical angle $ACB$ (fig. 53.).

PROPOSITION XIX. PROBLEM XI.

The three angles of a spherical triangle being given, to project it, and to find the measures of the sides.

Let $ABC$ (fig. 55.) be the spherical triangle of which the angles are given: construct another spherical triangle $EFG$, whose sides are the supplements of the given angles of the triangle $ABC$; and with the sides of this supplemental triangle describe the gnomonic projection, &c. as before.

It may be observed, that the supplemental triangle $EFG$ has also a supplemental part $EFg$; and when the sides $GE$, $GF$, which are substituted in place of the angles $A$, $B$, are obtuse, their supplements $gE$, $gF$ are to be used in the gnomonic projection of the triangle.

PROPOSITION XX. PROBLEM XII.

Given two sides, and the included angle of a spherical triangle, to describe the gnomonic projection of that triangle, and to find the measures of the other parts.

Let the sides $AC$, $CB$, and the angle $ACB$ (fig. 53.), be given: make the angles $CDA$, $CDG$ (fig. 56.) equal respectively to the sides $AC$, $CB$ (fig. 53.); also make the angle $ACB$ (fig. 56.) equal to the spherical angle $ACB$ (fig. 53.), and $CB$ equal to $CG$, and $ABC$ will be the projection of the spherical triangle.

To find the measure of the side $AB$: from $C$ draw $CL$ perpendicular to $AB$, and $CM$ parallel thereto, meeting the circumference of the primitive in $M$; make $LN$ equal to $LM$; join $AN$, $BN$, and the angle $ANB$ will be the measure of the side $AB$.

To find the measure of either of the spherical angles, as $BAC$: from $D$ draw $DK$ perpendicular to $AD$, and make $KH$ equal to $KD$: from $K$ draw $KI$ perpendicular to $CK$, and let $AB$ produced meet $KI$ in $I$, and join $HI$: then the rectilinear angle $KHI$ is the measure of the spherical angle $BAC$. By proceeding in a similar manner, the measure of the other angle will be found.

PROPOSITION XXI. PROBLEM XIII.

Two angles and the intermediate side given, to describe the gnomonic projection of the triangle; and to find the measures of the remaining parts.

Let the angles $CAB$, $ACB$, and the side $AC$ of the spherical triangle $ABC$ (fig. 53.), be given: make the angle $CDA$ (fig. 56.) equal to the measure of the given side $AC$ (fig. 53.); and the angle $ACB$ (fig. 56.) equal to the angle $ACB$ (fig. 53.); produce $AC$ to $H$, draw $DK$ perpendicular to $AD$, and make $KH$ equal to $KD$; draw $KI$ perpendicular to $CK$, and make the angle $KHI$ equal to the spherical angle $CAB$: from $I$, the intersection of $KI$, $HI$, to $A$ draw $IA$, and let it intersect $CB$ in $B$, and $ACB$ will be the gnomonic projection of the spherical triangle $ABC$ (fig. 53.). The unknown parts of this triangle may be measured by last problem.

PROPOSITION XXII. PROBLEM XIV.

Two sides of a spherical triangle, and an angle opposite to one of them given, to describe the projection of the triangle; and to find the measure of the remaining parts.

Let the sides $AC$, $CB$, and the angle $BAC$ of the spherical triangle $ABC$ (fig. 53.) be given: make the angles $CDA$, $CDG$ (fig. 56.) equal respectively to the measures of the given sides $AC$, $BC$: draw $DK$ perpendicular to $AD$, make $KH$ equal to $DK$, and the angle $KHI$ equal to the given spherical angle $BAC$: draw the perpendicular $KI$, meeting $HI$ in $I$; join $AI$; and from the centre $C$, with the distance $CG$, describe the arch $GB$, meeting $AI$ in $B$, join $CB$, and $ABC$ will be the rectilinear projection of the spherical triangle $ABC$ (fig. 53.) and the measures of the unknown parts of the triangle may be found as before.

PROPOSITION XXIII. PROBLEM XV.

Given two angles, and a side opposite to one of them, to describe the gnomonic projection of the triangle, and to find the measures of the other parts.

Let the angles $A$, $B$, and the side $BC$ of the triangle

angle ABC (fig. 55.), be given: let the supplemental triangle EFE be formed, in which the angles E, F, G, are the supplements of the sides BC, CA, AB respectively, and the sides EF, FG, GE, the supplements of the angles C, A, B. Now, at the centre C (fig. 56.) make the angles CDA, CDK equal to the measures of the sides GE, GF respectively, being the supplements of the angles B and A; and let the lines DA, DK intersect the diameter of the primitive EF, in the points A and K: draw DG perpendicular to AD, make GH equal to DG, and at the point H make the angle GHI equal to the angle E, or to its supplement; and let EI, perpendicular to CH, meet HI in I, and join AI: then from the centre C, with the distance CG, describe an arch intersecting AI in B; join CB, and ABC will be the gnomonic projection of the given triangle ABC (fig. 55.): the supplement of the angle ACB (fig. 56.) is the measure of the side AB, (fig. 55.); the measures of the other parts are found as before.

It has already been observed, that this method of projection has, for the most part, been applied to dialling only. However, from the preceding propositions, it appears that all the common problems of the sphere may be more easily resolved by this than by either of the preceding methods of projection; and the facility with which these problems are resolved by this method has given it the preference in dialling. It may not perhaps be amiss, in this place, to give a brief illustration of it in this particular branch of science.

In an horizontal dial, the centre of projection Z (fig. 57.) represents the zenith of the place for which the dial is to be constructed; ZA the perpendicular height of the style: the angle ZPA, equal to the given latitude, determines the distance ZP of the zenith from the pole; and AP the edge of the style, which by its shadow gives the hour: the angle ZAP, equal also to the latitude, gives the distance of the equator EQ from the zenith: let E a be equal to EA, and a will be the dividing point of the equator. Hence if the angles E a I, E a II, &c. E a XI, E a X, &c. be made equal to 15°, 30°, &c. the equator will be divided into hours; and lines drawn from P to these points of division will be hour lines.

If the dial is either vertical, or inclined to the horizon, then the point Z will be the zenith of that place whose horizon is parallel to the plane of the dial: ZE

will be the latitude of that place; and the hours on the Gnomonic former dial will now be changed into others, by a quantity equal to the difference of longitude between the given place and that for which the dial is to be constructed. Thus if it is noon when the shadow of the style falls on the line PX, then the difference of meridians is the angle E a X, or 30°. Hence, when a dial is to be constructed upon a given plane, either perpendicular or inclined to the horizon, the declination and inclination of that plane must be previously found.

In an erect direct south dial, its zenith Z is the south point of the horizon, ZP is the distance of this point from the pole, and ZE its distance from the equator. If the dial is directed to the north, Z represents the north point of the horizon; PZ the distance of Z from the pole under the horizon; and ZE the elevation of the equator above the horizon.

If the dial is an erect east or west dial, the zenith Z is the east or west points of the horizon accordingly, and the pole P is at an infinite distance, for the angle ZAP is a right angle; and therefore the line AP will not meet the meridian PZ. The line ZA produced is the equator, and is divided into hours by lines perpendicular to it.

If the plane of the dial is parallel to the equator, its zenith Z coincides with one of the poles of the equator P; and hence the hour lines of this dial are formed by drawing lines from the point Z, containing angles equal to 15°.

In the preceding methods of projection of the sphere, equal portions of a great circle on the sphere are represented by unequal portions in the plane of projection, and this inequality increases with the distance from the centre of projection. Hence, in projections of the earth, those places towards the circumference of the projection are very much distorted. In order to avoid this inconveniency, M. de la Hire* proposed, that the eye should be placed in the axis produced at the distance of the sine of 45° beyond the pole: In this case arches of the sphere and their projections are very nearly proportional to each other. Hence in a map of the earth agreeable to this construction, the axis, instead of being divided into a line of semitangents, is divided equally, in like manner as the circumference. The map of the world, Plate CCXIV. is constructed agreeable to this method of projection.

PROJECTION, in perspective, denotes the appearance, or representation of an object on the perspective plane.

The projection, e. gr. of a point, as B (Fig. 1. Plate CCCLXXXIII.) is a point b, through which the optic ray BE passes from the objective point through the plane to the eye; or it is the point wherein the plane cuts the optic ray.

And hence is easily conceived what is meant by the projection of a line, a plane, or a solid.

PROJECTION in Alchemy, the casting of a certain imaginary powder, called powder of projection, into a crucible, or other vessel, full of some prepared metal, or

other matter; which is to be hereby presently transmuted into gold.

POWDER OF PROJECTION, or of the philosophers stone, is a powder supposed to have the virtue of changing any quantity of an imperfect metal, as copper or lead, into a more perfect one, as silver or gold, by the admixture of a little quantity thereof.

The mark to which alchemists direct all their endeavours, is to find the powder of projection; which every one of them has been within an age of a hundred times. See PHILOSOPHER'S STONE.

PROJECTURE, in architecture, the outjetting and prominence, or embossing, which the mouldings