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43429 of the height



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From this expression of the time we learn that

30103 however great the velocity of projection, and which it will rise in the air is the beight to which this body will rise, may be, the time of its ascent is limited. It never can

to which it will rise in vacuo, or

in round exceed the time of falling from the height a in

numbers. vacuo in a greater proportion than that of a quadrantal arch to the radius, nearly the proportion

We thought it necessary to treat of the perof eight to five. A twenty-four pound iron ball pendicular ascents and descents of heavy bodies cannot continue rising above fourteen seconds, through the air thus particularly, that the reader ven if the resistance to quick motions did not may conceive distinctly the quantities which he increase faster than the square of the velocity. is thus combining in his algebraic operations, and It probably will attain its greatest height in less may see their connexion in nature with each than twelve seconds, let its velocity be ever so

other. We shall also find that, in the present great. In the preceding example of the whole state of our mathematical knowledge, this simple


state of the case contains almost all that we can ascent v = 0, and the time t =-X arc. tan. determine with any confidence.

Sect. VI.-OF THE OBLIQUE PROJECTION. or arc. 30° 48'. Now 30° 48' = 1848', and the 8

We would now proceed to the general problem

1848 radius 1 contains 3483: therefore the arch=;

to determine the motion of a body projected in 3438 any direction, and with any velocity. But our

readers will believe that this must be a difficult =0-5376; and 21":54. Therefore t=21":54 subject, when they see the simplest cases of rec

8 x 0-5376, + 11":58, or nearly 11} seconds. tilineal motion abundantly abstruse : it is indeed The body would have risen to the same height so difficult that Sir Isaac Newton has not given in a void in 10 seconds.

a solution of it, and has thought himself well emCor. 1. The time in which a body, projected ployed, in making several approximations, in in the air with any velocity V, will attain its which the fertility of his genius appears in great

lustre. In the tenth and subsequent propositions greatest height, is to that in which it would attain its greatest height in vacuo as the arch whose what state of density in the air will comport

of the second book of his Principia he shows tangent expresses the velocity is to the tangent; with the motion of a body in any curve whatfor the time of the ascent in the air is -x arch; ever; and then, by applying this discovery to


several curves which have some similarity to the V

V the time of the ascent in vacuo is

Now is path of a projectile, he finds one which is not

very different from what we may suppose to ob

tain in our atmosphere. But even this approx=tan. and V=uX tan.and

imation was involved in such intricate calculaFrom inspecting the diagram p. 153, it is evi- tions that it seemed impossible to make any use dent that the arch À I is to the tangent Á G as the of it. sector ICA to the triangle CGA; therefore the

In the second edition of the Principia, pubtime of attaining the greatest height in the air is

lished in 1713, Newton corrects some mistakes to that of attaining the greatest height in vacuo

in the first, and carries his approximations much (the velocities of projection being the same) as

farther, but still does not attempt a direct investhe circular sector to the corresponding triangle. ţigation of the path which a body will describe If therefore a body be projected upwards with in our atmosphere. This is somewhat surprising. the terminal velocity, the time of its ascent will In prop. 14, &c., he shows how a body, actube to the time of acquiring this velocity in vacuo ated by a centripetal force, in a medium of denas the area of a circle to the area of the circum- sity varying according to certain law, will described square.

scribe an eccentric spiral, of which he assigns 2. The height H to which a body will rise in a the properties, and the law of description. llad void is to the height h to which it would rise he supposed the density constant, and the differthrough the air when projected with the same

ence between the greatest and least distances u'+V?

from the ceritre of centripetal force exceedingly velocity V as M:V2 to u xa ; for the small in comparison with the distances them


selves, his spiral would have coincided with the height to which it will rise in vacuo is and path of a projectile in the air of uniform density,


and the steps of his investigation would have led

u? the height which it rises in the air is a him immediately to the complete solution of the


problem. For this is the real state of the case. "


A heavy body is not acted on by equal and pa; therefore H : h =

2g Mg

rallel gravity, but by a gravity inversely proporu* +? u?


tional to the square of the distance from the Χ 2 λν

V2: centre of the earth, and in lines tending to that u?

centre nearly; and it was with the view of simpliM:V?: u? X X u?+ V2

fying the investigation that mathematicians have M

adopted the other hypothesis. If the body, therefore, be projected with its . Not long after the publication of this second terminal velocity, so that V =u, the height to edition of the Principia, the dispute about the in

-=-X tan.
u u

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vention of the fluxionary calculus became very tigation, it is totally changed: and it is not tos violent, and the great promoters of that calculus much to say that with this complication it reupon the continent proposed difficult problems quires the knowledge and address of a Euler to to the mathematicians. "Challenges of this kind make even a partial and very limited approximafrequently passed between the British and tion to a solution. foreigners. Dr. Keill of Oxford had keenly Any law of the resistance, therefore, which is espoused the claim of Sir Isaac Newton to this more complicated than what Bernouilli has azinvention, and had engaged in a very acrimonious sumed, namely, that of a simple power of the altercation with the celebrated John Bernouilli of velocity, is abandoned by all the mathematicians, Basle. Bernouilli had published in the Acta as exceeding their data : and they have attempted Eruditorum Lipsiæ, an investigation of the law to avoid the error arising from the assumption of of forces, by which a body, moving in a resisting the duplicate ratio of the velocity either by supmedium, might describe any proposed curve, re- posing the resistance throughout the whole traducing the whole to the simplest geometry. This jectory to be greater than what it is in general, is perhaps the most elegant specimen which he or they have divided the trajectory into different has given of his great talents. Dr. Keill proposed portions, and assigned different resistances to to him the particular problem of the trajectory each, which yary, through the whole of that porand motion of a body moving through the air, as tion, in the duplicate ratio of the velocities. Thus one of the most difficult

. Bernouilli very soon they make up a trajectory and motion which corsolved the problem in a way much more general responds, in some tolerable degree, with what? than it had been proposed, viz. without any limit- With an accurate theory? No; but with a seation either of the law of resistance, the law of ries of experiments. For, in the first place, the centripetal force, or the law of density, pro- every theoretical computation which we make vided only that they were regular, and capable proceeds on a supposed initial velocity; and of being expressed algebraically. Dr. Brooke this cannot be ascertained with any thing apTaylor, the celebrated author of the Method of proaching to precision by any theory of the Increments, solved it at the same time, in the action of gunpowder that we are yet possessed limited form in which it was proposed. Other of. In the next place, our theories of the resistauthors since that time have given other solu- ing power of the air are entirely established on tions; but they are all (as indeed they must be) the experiments on the flight of shot and she is, the same in substance with Bernouilli's. Indeed and are corrected and amended till they tally they are all (Bernouilli's not excepted) the same with the most approved experiments we can with Newton's first approximations, modified by find. We do not learn the ranges of a gun by the steps introduced into the investigation of the theory, but the theory by the range of the gun. spiral motions mentioned above; and we still After all, therefore, the practical artillerist think it most strange that Sir Isaac did not per- must rely chiefty on the records of experiments ceive that the variation of curvature, which he contained in the books of practice at the acadeintroduced in that investigation, made the whole mies, or those made in a more public manner. difference between his approximations and the Even a perfect theory of the air's resistance can complete solution.

do him little service, unless the force of gunpowAll the solutions given of this problem depend der were uniform. But this is far from being the upon a particular law of resistance assumed, case. without proving that to be the law by which a The experiments of Mr. Robins and Dr. Hutbody is resisted in its motion through the air. ton show, in the most incontrovertible manner, This resistance is supposed to be in the duplicate that the resistance to a motion exceeding 1100 ratio of the velocity ; but even theory points out feet in a second is almost three times greater than many causes of deviation from this law, such as in the duplicate ratio to the resistance to modethe pressure and condensation of the air, in the rate velocities. Euler's translator, in his comcase of very swift motions: and Mr. Robins's parison of the author's trajectories with experiexperiments are sufficient to prove that the devi- ment, supposes it to be no greater. Yet the coations must be exceedingly great in such cases. incidence is very great. The same may be said Euler and all subsequent writers have allowed of the Chevalier de Borda's. Nay, the same that it may be three times greater, even in cases may be said of Mr. Robins's own practical rules; which frequently occur; and Euler gives a rule and yet his rules are confirmed by experience. for ascertaining with tolerable accuracy what this But we must not infer, from all this, that the increase and the whole resistance may amount physical theory is of no use to the practical arto. Let H be the height of a column of airtillerist. It plainly shows him the impropriety whose weight is equivalent to the resistance of giving the projectile an enormous velocity. taken in the duplicate ratio of the velocity. The This velocity is of no effect after 200 or 300 whole resistance will be expressed by H + yards at farthest, because it is so rapidly reduced H?

This number 28845 is the height in feet by the prodigious resistance of the air. Mr. Ro28845

bins bas deduced several practical maxims of the of a column of air whose weight balances its greatest importance from what we already know elasticity. We shall not at present call in ques- of this subject, and which could hardly have been tion his reasons for assigning this precise addi even conjectured without this knowledge. And tion. They are rather reasons of arithmetical we must still acknowledge that this branch of conveniency than of physical import. It is physical science is highly interesting to the phienough to observe thai, if this measure of the losopher; nor should we despair of carrying it resistance is introduced into the process of inves- to greater perfection.

Certainly the most complete set of experi 6. The ranges and times of flight of balls, ments made with a view of obtaining a rational with the velocities, by striking the pendulum at theory of projectiles are those of Dr. Hutton, various distances, to compare them with initial which were carried on at Woolwich during the velocities, for determining the resistance of the years 1775, 1783, 1784, 1785, 1787, 1788, 1789, medium. and 1791, the objects of which were very vari 7. The effects of wads, of ramming, of windous, and some of the results highly important. age, &c.' The latter are thus enumerated by the author in the second volume of his Tracts :

We shall now quote this author's expression *1. It is made evident, by these experiments, for the resistance of the air, deduced from these ihat powder fires almost instantaneously, seeing experiments, and thence determine the ranges, hat nearly the whole of the charge fires, though the times of fight, &c., of projectiles according to time be much diminished.

that hypothesis. 2. The velocities communicated to the shot THEOREM.—The resistance of the ait, to a ball of the same weight, with different quantities of projected into it with any considerable velocity, powder, are nearly in the subduplicate ratio of is expressed by the formula r = ('000007565 v2 those quantities ; a very small variation, in de 00175 v) d?. But, for the smaller velocities, fect, taking place when the quantities of powder r=0000044 do ve will be a sufficiently near apbecome great.

proximation, where r represents the resistance in 3. And when shot of different weights are avoirdupois pounds, d the diameter of the ball fired with the same quantities of powder the ve in inches, v the velocity in English feet. See locities communicated to them are nearly in the Hutton's Tracts, vol. iii. p. 232. reciprocal subduplicate ratio of their weights. PROB. I.-To determine the height to which 4. So that, universally, shots which are of ball

, projected perpendicularly upwards, will different weights, and impelled by the firing of ascend, being resisted by the atmosphere. different quantities of powder, acquire velocities Putting x to denote any variable and increaswhich are directly as the square roots of the ing height ascended by the ball; v its variable quantity of powder, and inversely as the square and decreasing velocity there ; d'the diameter of roots of the weight of the shot, nearly.

the ball, its weight being w; m = .000007565, *5. It would therefore be a great improve- and n= .00175, the co-efficients of the two terms ment in artillery to make use of shot of a long in the above theorem. Then (m v2 — nv) [? form, or of heavier matter ; for thus the momen- will be the resistance of the air against the ball i tum of the shot, when fired with the same weight avoirdupois pounds, to which, if the weight of the of powder, would be increased in the ratio of ball be added, then (m v2 n v) d? + w will be the square root of the weight of the shot. the whole resistance to the ball's motion, and 66. It would also be an improvement to di

(m vi —nv) d? + w (m v— nv) minish the windage; for, by so doing, one-third con

consequently or more of the quantity of powder might be d? +1=f, the retarding force. Hence the gesaved.' (This, however, must be understood

neral formula vů = 2gf i becomes vu=2 only to be true within certain limits.)

(mv2 — nv) d? + w *7. When the improvements mentioned in the

8 IX

making v negative, betwo last cases are considered as both taking place it is evident that about half the quantity of cause the velocity is decreasing, whereg

1612 powder might be saved, which is a very conside feet, or sixteen feet, the descent of a budv in one able object. But, important as this saving may

second by gravity. be, it seems still to be exceeded by that of the

Hence =guns : for thus a small gun may be made to

29 (m vi — nv)d? + w have the effect of one of two or three times its

vo Х

The fluent of size, in the present way, by discharging a long 2 g mda shot of two or three times the weight of its na

q2 tural ball, or round shot: and thus a small ship which being taken, and corrected for the instant might discharge sbot as heavy as those of the of the first velocity V, when x = 0, gives r= greatest now made use of.' The objects of the latter courses of experiments

are thus detailed : viz. to ascertain,

2 gm ď

m d? *1. The velocities with which balls are pro- log.

P jected by equal charges of powder, from pieces

(arc. tan. (V-p)

m d? of the same weight and calibre, but of different lengths.

p to rad. 9) where p and pe + q? 62. The greatest velocities due to the different charges of powder, the weight and length of the gun being the same. '3. The greatest velocity due to the different


But as part of this fluent, denoted by lengths of guns ; to be obtained by increasing the charge as far as the resistance of the piece is difference of the two arcs to tan. (V—p) and capable of sustaining.

p, is always very small in comparison with the 64. The effect of varying the weight of the other preceding terms, it

may be omitted without piece; every thing else being the same.

any material error in practical cases; in which 65. The penetration of balls into blocks of case we have, wood.






- W


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arc. tan.


2 m


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- + the

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4 gmda

m d?



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V? _"_V +
the latter arc as inconsiderable ; P


q reprem de senting the same as before. x byp. log.

Erample 1.- Let it be proposed to find the time in which an iron ball, two inches in diame

ter, will acquire its greatest height, when projectfor the greatest height to which the ball will as

ed with a velocity of 2000 feet per second. cend in air; supposing its density uniformly the same as at the earth's surface. Now for the nu Here = 115] =p, and =p2 +

2 m

m d? merical value of the general coefficient

4 gm d' 9°, gires 9

= 737153—p* = 154;: whencet=

V and the term -; because the mass of the

Х arc. tan.

= 11.81 se

2 ball to the diameter d is '5236 d', if its specific conds. gravily be s, its weight will be ·5236 s = w; If we take the second example above to find

the time, we shall have p= 115} as before, and therefore

= -5236 s d, and

= 69259 sd;
m da
9 = 299.4 ; therefore t

X arc. tan.

2 g mq da this divided by 4 g or 64, gives



= 16.89 seconds, 1082 s d for the value of the general coefficient, 9 to any diameter d, and specific gravity s. And After the investigation of these problems, and if we farther suppose the ball to be cast iron, the some others of a similar nature, Dr. Hutton specific gravity of which, or the weight of a cubic proceeds to the investigation of his principal one, inch, is 26855 lbs., it becomes 290: s d for that viz. to determine the circumstances of ranges at co-efficient; also 69259 s d=18600 d=

different degrees of elevation; which we tranm 2' scribe.

*Rules for the general solution of this problem and = 231.5.

would be best derived from experiments; and Hence the preceding fluent becomes

these should be made at all elevations, and with V? — 231.5 V + 18600 d all charges, and with various sizes of balls, obx = 290.6 x hyp. log.

serving both the ranges and times of flight in 18600 d

Such experiments would V? — 231.5 V + 18600 d every experiment. x= 669 d x com. log.

give us the relations existing, in all cases, 18600 d

amongst these four terms, viz. the ranges, the which is in general expression for the altitude in times of flight, the velocity or charges, and the feet ascended by an iron bullet whose diameter size of the balls. Numerous and various as are is d, and projectile velocity V:

our experiments, as before related, and fruitfulas Example 1.-Suppose a ball of cast iron,

they are in useful consequences, we have obtainwhose diameter is two inches, and, therefore, its

ed but a small portion of those alluded to; nor weight 1{ lb., to be projected upwards with a ve- do I know of any proper set of such experiments locity of 2000 feet per second, to find the greatest any where to be found. Such must, therefore, height to which it will ascend.

still remain a valuable desideratum; the few that Here, substituting for d, w, and V their respec- we have been able to make afford us but very tive values, we have

few and imperfect rules, being chiefly as follows: V? — 231.5 V + 18600d -1. That the ranges with the one-pound balls, =669d X com. log.

18600 d

at an elevation of 15°, are nearly proportional to = 2653 feet.

the times of Alight. 2. That the ranges with the Erample. 2.-Again, let the ball weigh twenty three-pound balls, at 45° of elevation, are nearly four pounds, and, therefore, its diameter 5:6, and

as the times of flight, and also as the projectile velocity 2000 feet per second, as before ; then

velocities. Besides these inferences, it does not V? 231.5 V + 18600 d appear that the experiments are extensive enough x = 669 d x com. log.

to afford any more useful conclusion.

18600 d = 5782 feet, the height required.

By trials, however, amongst many of the nu

numIn the first of these examples, where the height at an elevation between 45° and 30°, the time of

bers in art. 24, it appears that in most of them is found to be only about half a mile, the ball would ascend to nearly twelve miles in a non

flight is nearly equal to one-fourth of the square resisting medium; and hence we may see the

root of the range in feet, in which respect it immense effect of atmospheric resistance to the

nearly agrees with the similar rule for the time motion of projectiles.

of flight in the parabolic theory, at the angle of PROB. II.-To determine the time in which a

45° for the greatest range, which time, it is well ball will have acquired its greatest height, using

known, is equal to one-fourth of the square root the same formula of resistance as in the last case.

of the said range in feet. Whence it is probable Here the general value of t, determined on

that, with the help of a few other ranges at seveprinciples similar to those above employed, gives

ral elevations, some general relations might be

evinced between the ranges and the times of х

fight, with the tangents of the elevation. 9


* But such experiments and enquiries as these, X arc. tan.

unfortunately, it is no longer in my power either

rejecting 2 g mod


to procure, or by any means to promote; and

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arc. tan.

arc. tan.

2gmq di *(arc

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or t =


52 ;

we can, therefore, only endeavour to render, The next was by Wm. Bourne, in 1643, in without them, what service we can to the state, his Art of Shooting in great Ordnance. His and to philosophy, by such means as are in our elevations were not regulated by the points of power.

the gunner's quadrant, but by degrees; and be • There are some few theoretical principles gives the proportions between the ranges at difwhich it may be useful to notice here, as first ferent elevations and the extent of the point-blanc mentioned by professor Robison. Thus balls shot, thus : if the extent of the point-blanc shot of equal density, discharged at the same elevation be represented by one, then the proportions of with velocities which are proportional to the the ranges at several elevations will be as below, square roots of their diameters, will describe si- viz.-milar curves; because then the resistances will

Bourne's PROPORTION OF RANGES. be in proportion as the momentum or quantity of motion. For the resistance r is v2 nearly;



d being the diameter, and v the velocity. But v

5 being as v d, v2 will be as d; consequently da


10 vi will be as.d; that is, is as w. But the mo


15 mentum is as the magnitude or mass, which is as


20 d also, the cube of the diameter. Therefore the resistance is proportional to the momentum,

and the greatest random when the velocity is as v d, or the square root of which greatest random, he says, in a calm day is the diameter of the ball. In this case, then, the at 42° elevation; but according to the strength horizontal velocity at the vertex of the curve will of the wind, and as it favors or opposes the be proportional to the terminal velocity; also the flight of the shot, the elevation may be from 43° ranges, and heights, and all other similar lines to 360. He does not say with what piece he in the curve, will be proportional to d, the dia- made his trials, though from his proportion it meter of the ball. And this principle may be of seems to have been a small one. This however considerable use; for thus, by means of a proper ought to have been mentioned, as the relation series of experiments on one ball, projected with between the extent of different ranges varies exdifferent velocities and elevations, tables may be tremely according to the velocity and density of constructed, by which may be ascertained the the bullet. motions in all similar cases.'

After him Eldred and Anderson, both EnglishWe shall have occasion to advert again to these men, also published treatises on this subject. valuable contributions of Dr. Hutton.

The former of these was many years gunner of

Dover Castle, where most of his experiments PART II.

were made, the earliest of which are dated 1611, OF THE PRACTICE OF GUNNERY, OR

though his book was not published till 1646,

and was entitled The Gunner's Glass. His prinMILITARY PROJECTILES.

ciples were simple, and within certain limits very Having laid before our readers the substance near the truth, though they were not rigorously of the latest and most inproved theories of

He has given the actual pro


of different jectiles, we proceed to give them a brief sketch pieces of artillery at small elevations, all under of the most improved modern practice.

10°. His experiments are numerous, and appear Mr. Robins, in his preface to his New Princi- to be made with great care and caution; and he ples of Gunnery, states that he had met with has honestly set down some which were not reonly four authors who had treated experimental- concileable to his method : upon the whole he ly on this subject. The first of these is Collado seems to have taken more pains, and to have in 1642, who has given the ranges of a falconet, had a juster knowledge of his business than is to carrying a three-pound shot, to every point of be found in most of his practical brethren. the gunner's quadrant, each point being the

Galileo printed his Dialogues on Motion in twelfth part, or 7° 30'. But from his numbers the year 1646. In these he pointed out the geit is manifest that the piece was not charged with neral laws observed by nature in the production its usual allotment of powder. The result of his and composition of motion, and was the first who trials shows the ranges at the point-blanc, and described the actions and effects of gravity on the several points of elevations as below. falling bodies : on these principles he determined

that the flight of a cannon-shot, or of any other Elevation at

Range in

projectile, would be in the curve of a parabola, • Points. Deg.


unless so far as it should be diverted from that 0


track by the resistance of the air. He also pro1 7) 594

posed the means of examining the inequalities 2 15 794

which arise thence, and of discovering what 3 22 954

sensible effects that resistance would produce in 4 30 1010

the motion of a bullet at some given distance 5 371 1040

from the piece. Notwithstanding these determi6 45 1053

nations and hints it seems, however, that those 7

52 between the 3d and 4th who came after Galileo never imagined that it 8 60

between the 2d and 3d was necessary to consider how far the operations 9

671 . between the 1st and 2d of gunnery were affected by this resistance. In10 75

between the () and 1st stead of this, they boldly asserted, without mak11

82} . fell very near the piece. ing the experiment, that no great variation could

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