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ably greater than the weight of the column of air,, to 39 by being put into this form. Similar devi ations occur in the experiments of Chev. Borda; and it may be collected from both, that the refiftances diminish more nearly in proportion of the fines of incidence than in the proportion of the fquares of thofe fines.

whole height would produce the velocity in a falIng body. Mr Robins experiments in a fquare of 15 inches, defcribing 25'2 feet per fecond, in dicate the refiftance to be to this weight nearly as 4. Borda's experiments on the fame furface the difproportion ftill greater. The refiftare not in the proportion of the furfaces, tmcreafe confiderably 'fafter. Surfaces of 9 16, 36, and 84 inches, moving with one velocity; had resitances in proportion of 9, 174, 42, and Now as this deviation from the propor. tion of the furfaces increafes with great regularity, it is not probabile that it continues to increase in Larfaces of ftill greater extent; and these are the mat generally to be met with in practice in the ath of wind on ships and mills.

The irregularity in the resistance of curved furfaces is as great as in, plane furfaces. In general, the theory gives, the, oblique impulfes on plane furfaces much too small, and the impulfes on curved furfaces too great. The refiftance of a fphere does not exceed the 4th part of the refiftance of its great circle, instead of being its half; but the anomaly is fuch as to leave hardly any room for calculation. It would be very defirable to have the experiments on this fubject repeated in a greater variety of cafes, and on larger furfaces, fo that the errors of the experiments may be of lefs confequence. Till this matter be reduced to fome rule, the art of working ships must remain very imperfect, as muft alfo the conftruction of windmills, བསྟན་བྱ་ནན་གསྟན་ནི་ན

C. Box DA's experiments on 81 inches, fhow that the impuife of wind moving one foot per fe. cond, is about one 300th of a pound on a fquare fit. Therefore, to find the impulfe on a foot correfponding to any velocity, divide the fquare of the velocity by 500, and we obtain the impulfe pounds. Mr Rouse of Leicestershire made1 many experiments which are mentioned with. great approbation by Mr Smeaton. His great fagcity and experience in the erection of wind

oblige us to pay a confiderable deference to bus judgment. Thefe experiments confirm our C, that the impulfes increase fafter than the fus. The following table was calculated fr Mr Roufe's obfervations, and may be confi. dered as pretty accurate. Velocity

Impulse on a

in Feet. Foot in Pounds.

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The fquare of the velocity in feet, being multiplied by 16, the product will be the impulfe or reiflance on a fquare foot in grains, according to Roufe's numbers. The greateft deviation from theory occurs in the oblique impulfes. Mr Rabus compared the refiftance of a wedge, whofe gle was 90°, with the refiftance of its base; and tead of finding it lefs in the proportion of 1 to 1, as determined by the theory, he found It greater in the proportion of 55 to 68 nearly; and when he formed the body into a pyramid, of which the fides had the fame furface and the fame chration as the fides of the wedge, the refiftace of the bafe and face were now as 55 to 39 early: fo that here the fame furface with the fame actination, had its refiftance reduced from 68

7

SECT. IX, Of the RESISTANCE of the AIR, i
GUNNERY.

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in

The cafe in which we are most interested in the knowledge of the refiftance of the air is the motion of bullets and fhells. Writers on artillery have long been fenfible of the great effects of the air's refiftance. This confideration chiefly engaged Sir Maac Newton to confider the motions of bodies in a refifting medium. A propofition or two would have fufficed for fhewing the in compatibility of the planetary motion with the fuppofition, that the celeftial spaces were filled with a fluid matter; but he inveftigated the motion of a body projected on the furface of the earth, and its deviation from the parabolic track, affigned by Galileo. He bestowed more pains on? this problem than any other in his whole work; and his inveftigation has pointed out almoft all the improvements which have been made in the application of mathematical knowledge to the ftu.... dy of nature. Nowhere does this fagacity and fertility of refource appear in so strong a light as in the ad book of the Principia, which is almoft wholly occupied by this problem. The celebrated John Bernouilli engaged in it as the finest op<. portunity of difplaying his fuperiority. A mif. take committed by Newton in his attempt to a folution, was matter of triumph to him; and the whole of his performance, though a piece of ele gant and elaborate geometry, is tarnished by his continually bringing this trifling mistake into view. The difficulty of the fubject is fo great, that fub. fequent mathematicians, and many voluminous writers on military projectiles, have kept aloof from it. They have fpoken indeed of the refiftance of the air as affecting the flight of thot, but have faved themfelves from the task of inveftigating this effect, by fuppofing that it was not fo great as to render their theories and practical deductions very erroneous. Mr ROBINS was the first who feriously examined the fubject. He fhowed, that even the Newtonian theory (which had been corrected, but not improved or extended in its principles) was fufficient to fhow

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24

that the path of a cannon ball could not refemble a parabola. Even this theory fhowed that the refiftance was more than 8 times' the weight of the ball, and should produce a greater deviation from the parabola than the parabola deviated from a ftraight line.

He added, that the differences between the ranges by the Newtonian theory and by experiment, were fo great, that the refiftance of the air must be vaftly fuperior to what that theory fuppofed. This fuggefted to him the neceffity of experiments to afcertain this point. We have feen the refult of thefe experiments in moderate velocities; and that they were fufficient for calling the whole theory in question, or at least for rendering it useless. It became neceffary, therefore, to fettle every point by a direct experiment. Here was a great difficulty, how to measure either thefe great velocities which are obferved in the motions of cannon shot, or the refiftances which thefe Mr Robins did enormous velocities occafion. both. The method which he took for measuring the velocity of a mufket-ball, was quite original: and it was fufceptible of great accuracy. (See PROJECTILES.) Having gained this point, the other was not difficult. In the moderate velocities he had determined the refiftances by the forces which balanced them, the weights which kept ftate of uniform motion. the refifted body in In the great velocities, he proposed to determine the refiftances by their immediate effects, by the This was retardations which they occafioned. to be done by firft afcertaining the velocity of the ball, and then measuring its velocity after it had paffed through a certain quantity of air. The difference of thefe velocities is the retardation, and the proper measure of the refiftance; for, by the initial and final velocities of the ball, we learn the time which was employed in paffing through this air with the medium velocity. In this time the air's refiftance dimifhed the velocity by a certain quantity. Compare this with the velocity which a body projected directly upwards would lofe in the fame time by the refiftance of gravity. The two forces must be in the proportion of their effects. Thus we learn the proportion of the refiftance of the air to the weight of the ball. It is true, that the time of paffing through this space is not accurately had by taking the arithmetical medium of the initial and final velocities, nor does the refiftance deduced from this calculation accurately correfpond to this mean velocity; but both may be accurately found by a very troublefome computation, as is fhown in the sth and 6th propofitions of the 2d book of Newton's Principia. The difference between the quantities thus found, and thofe deduced from the fimple procefs, is quite trifling, and may there be difregarded.

་ ་་

Mr ROBINS made many experiments on this fubject; but unfortunately he has published only a very few, to afcertain the point he had in view. He intended a regular work on the sub. ject, in which the gradual variations of refiftance correfponding to different velocities fhould all be determined by experiment: but he was then newly engaged in an important and laborious employ ment, as chief engineer to the Eaft India Com

pany, in whofe fervice he went out to Indi where he died in less than two years. It is to b regretted that no perfon has profecuted thefe e periments. It would, add more to the improv ment of artillery than any thing that has bee done fince Mr Robins's death, if we except th profecution of his experiments on the initial ve locities of cannon-fhot by Dr Charles Hutton, roy, profeffor at the Woolwich Academy. As M Robins has not given us the mode of deduction we muft compare the refults with experimen He has indeed given a very extenfive compariso with the numerous experiments made both i Britain and on the continent; and the agreemen is very great.

The general refult of Robins's experiments o the retardation of mufket-fhot is, that althoug in moderate velocities the refiftance is fo near in the duplicate proportion of the velocities tha we cannot observe any deviation, yet in velocitie exceeding 200 feet per fecond, the retardations in creafe fafter, and the deviation from this rate in creafes rapidly with the velocity. He afcribe this to the condensation of the air before the ball and to the rarefaction behind, in confequence o the air not immediately occupying the space lef by the bullet. This increase is so great, that i the refiftance to a ball moving with the velocity of 1700 feet in a fecond be computed on the fup pofition that the refiftance obferved in moderat velocities is increased in the duplicate ratio of the velocity, it will be found hardly one 3d part of its real quantity. He found, for inftance, that a ball moving through 1670 feet in a fecond, loft about 125 feet per fecond of its velocity in paffing through 50 feet of air. This it must have done in the one 3ad of a fecond, in which time it would have loft one foot if projected directly upwards; from which it appears that the refiftance was about 125 times its weight, and more than three times greater than if it had increased from the refiftance in fmall velocities, in the duplicate ratio of the velocities. His other experiments fhew fimilar refults.

But he alfo mentions a fingular circumftance, that till the velocities exceed 1100 feet per fecond, the refiftances increase pretty regularly, in a ratio exceeding the duplicate ratio of the velocities; but that in greater velocities the refiftances become fuddenly triple of what they would have been, even according to this law of increafe. He thinks this explicable by the vacuum which is then left behind the ball, it being well known that air rushes into a vacuum with the velocity of 1130 feet per fecond nearly. Mr EULER controverts this conclufion, as inconfiftent with that gradation which is obferved in all the operations of nature; but Mr Robins's affertion has every argu ment for its truth that the nature of the thing will admit; and his experiments prove this dimi nution of refiftance. It clearly appears from them, that in a velocity of 1700 feet the refiftance is more than three times the refiftance determined by the theory which he fuppofes the common

one.

When the velocity was 1065 feet, the ac tual refiftance was of the theoretical; and when the velocity was 400 feet, the actual reliftance was about four 3ds of the theoretical.

M:

Mr ROBINS, in fumming up the results of his obfervations on this fubject, gives a rule very easily remembered for computing the refiftances to that very motions.

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B

D

Let AB reprefent the velocity of 1700 feet per feand, and AC any other velocity. Make BD to AD as the resistance given by the ordinary theory to the resistance actually obferved in the velocity 1700: then will CD be to AD as the resistance figned by the ordinary theory to the velocity AC is to that which really correfponds to it. To accommodate this to experiment, a sphere of the size of a 12 pound iron shot, moving 25 feet in a fecond, had a resistance of one 22d of a pound. Augment this in the ratio of 25 to 17002, and we obtain aro nearly for the theoretical reistance to this velocity; but by comparing its diameter of 4 inches with 4, the diameter of the leaden ball, which had a resistance of at leaft 11 with this velocity, we conclude that the 12 lb. hot would have had a resistance of 396 lb. therefore BD: AD=210: 369, and AB: AD=186: 396; and AB being 1700, AD will be 3613. Let AD-a, AC, and let Rbe the resistance to a 1 pound iron fhot moving one foot per fecond, dr the resistance (in pounds) wanted for the

r=R2 Mr Robins's ex

velocity; we have r=R

13750

periments give R
0163135, which is nearly one fourth. Thus
our formula becomes r
0*263235 x
or very
3613-*

very nearly. This gives

early

43613

falling fhort of the truth a

SECT. X. Of the UNDULATION of the Air. THERE is another motion of which air and other elastic fluids are fufceptible, viz. an UNDU LATION, or internal VIBRATION of their particles by which any extended portion of air is diftribut ed into alternate parcels of condensed and rarefied air, which are continually changing their condi tion without changing their places. By this change the condensation which is produced in one part of the air, is gradually transferred along the mass of air to the greateft diftances in all directions. It is of importance to have fome diftinct conception of this motion. It is by this that diftant bodies produce in us the fenfation of found. See ACOUSTICS, and SOUND. Sir Ifaac Newton treated this fubject with his accuftomed ingenuity, and gave a theory of it in the end of the 2d book of his Principia. This theory has been objected to with refpect to the conduct of the argu ment, and other explanations have been given by the moft eminent mathematicians. Though they appear to differ from Newton's, their refults are precifely the fame; but, on a close examination, they differ no more than John Bernouilli's theo rem of centripetal forces differs from Newton's, viz. the one being expreffed by geometry and the other by literal analysis.

But since NEWTON published this theory of aerial undulations, and of their propagation along the air, and since the theory has been fo corrected and improved as to be received by the most accurate philofophers as a branch of natural philo fophy fufceptible of rigid demonstration, it has been freely reforted to by many writers on other parts of natural science, who did not profess to be mathematicians, but made ufe of it for explaining phenomena in their own line, on the authority of the mathematicians. Learning from them that this vibration, and the quaqua verfum propagation of the pulfes, were the neceffary proper ties of an elaftic fluid, and that the rapidity of this propagation had a certain affignable proportion to the elafticity and density of the fluid, they freely made ufe of these conceffions, introduced elaftic vibrating fluids into many facts, where others would fufpect no fuch thing, and attempted to explain by their means many abftrufe pheno

bout one 20th part. The simplicity of the for. mula recommends it, and when we increafe its refalt one 20th, it is amazingly near to the true refult of theory as corrected by Mr Robins. The resistance to other balls will be made by taking them in the duplicate ratio of the diameters. The firft mathematicians of Europe have lately employed themselves in improving this theory of the motion of bodies in a resisting medium; but their difcuffions are such as few artillerifts can underftand. mena of nature. Ethers are every where introproximation, and this by the quadrature of very duced, endued with great elafticity and tenuity. complicated curves. They have not been able Vibrations and pulfes are fuppofed in this æther, therefore to deduce from them any practical rules and thefe are offered as explanations. The doc of caly application, and have been obliged to trines of animal fpirits and nervous fluids, and the Compute tables fuited to different cafes. Of thefe whole mechanical fyftem of HARTLEY, by which performances, that of Chev. Borda, in the Mem. the operations of the SOUL are faid to be explain of the Acad. of Sciences, in 1769, feems the beft ed, have their foundation in this theory of aerial adapted to military readers, and the tables are undulations. If thefe fancied fluids, and their inrules of Mr Robins are of as much fervice, and are afcribed to them, any explanation that can be more easily remembered: besides, the nature of given of the phenomena from this principle, muft military fervice does not give room for the appli- be nothing elfe than fhowing, that the legitimate tation of any that can be derived from a perfect theory, the phenomena; or, if we are be an improvement in the conftruction of this laft ftep in the cafe of

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no more able to fee pieces of ordnance, and a more judicious appro- know to be one confequence of the aerial undulafound, (which we priation of certain velocities to certain purposes. tions, although we cannot tell how,) we must be The fervice of a gun or mortar must always bere able to point out, as in the case of found, certain

galated by the eye.

YOL. XVIII. PART I.

conftant relations between the general laws of D thefe

thefe undulations. and the general laws of the phenomena. It is only in this way that we are entitled to say that the aerial undulations are caufes of found; and it is because there is no fuch relation, but a total diffimilarity between the laws of elaftic undulations, and the laws of the propagation of light, that, we affert with confidence, that ethereal undulations are not the caufes of vision. See OPTICS, 153–156.

Explanations of this kind fuppofe, 1. That the philofopher who propofes them understands precifely the natures of thefe undulations; 2. That he makes his reader fensible of thofe circumstancs of them which are concerned in the effect to be explained; and 3. that he makes him understand how this circumftance of the vibrating Blaid is connected with the phenomenon.. But, with the exception of Euler's unfuccefsful attempt to explain the optical phenomena by the undulations of ether, we have met with no explanation of na tural phenomena, by means of elattic and vibrating fluids, where the author has fo much as at tempted any one of these three things, fo indifpenfably requisite in a logical explanation. They have talked of vibrations without defcribing them, or giving the reader the leaft notion of what kind they are; and in no inftance have they fhowed how fuch vibrations could have any influence in the phenomenon. Indeed, by not defcribing with precision the undulations, they were freed from ,the talk of fhewing them to be mechanical caufes of the phenomenon; and when any of them show any analogy between the general laws of elaftic undulations and the general laws of the phenomenon, the analogy is fo vague, indiftinct, or partial, that no perfou of common fenfe 'would receive it as argument.

We think it our duty to remonftrate against this flovenly way of writing: we would even hold it up to reprobation. It has been chiefly on this faithlefs foundation, that the blind vanity of men has railed that degrading fyftem of opinions called MATERIALISM, by which the affections and faculties of the foul of man have been refolved into vibrations and pulfes of ether.

We shall therefore give fuch an account of this motion of elaftic fluids, as fhall be understood by thofe who are not mathematicians, because thofe only are in danger of being milled by the improper application of them. Mathematical difcuflion is, however, unavoidable in a fubject purely mathematical; but we fhall introduce nothing that may not be easily understood or confided in. The first thing incumbent on us, is to show how elaftic fluids differ from the unelaftic in the propagation of any agitation of their parts. When a long tube is filled with water, and any one part of it pushed out of its place, the whole is inftantly moved like a folid mals. But this is not the cafe with air. If a door be fuddenly fhut, the window at the farther end of a long and clofe room will rattle; but fome time will elapfe between the shutting of the door and the motion of the window. If fome light luft be lying on a braced drum, and another be violently beat at a little distance from it, an obferver will fee the duft dance up from the parchment; but this will be at the inftant he hears the found of the ftroke on the other drum, a.! a fensible

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time after the ftroke. Many fuch familiar fac how that the agitation is gradually communicate along the air; and therefore that when one partic is agitated by any fensible motion, a finite tim however fmall, muft elapfe before the adjoinin particle is agitated in the fame manner. Th would not be the cafe in water, if water be perfec ly incompreffible. We think that this may b made intelligible with very little trouble; thus: D

A a

· .

Въ

с

Let A, a, B, b, C, D, &c. bé a row of aeri particles, at fuch diftances that their elafticity ju balances the preffure of the atmosphere; and l us fuppofe (as is deducible from the obferved der sity of air being proportional to the compreffir force) that the clafticity of the particles, by whic they keep each other at a diftance, is as their di tances inverfely. Let us farther fuppofe that th particle A has been carried, with an uniform mo tion, to a by fome external force. It is eviden that B cannot remain in its prefent ftate; for be ing now nearer to a than to C, it is propelled to wards C by the excess of the elafticity of A abov the natural elafticity of C. Let E be the natura elafticity of the particles, or the force correfpond ing to the diftance BC or BA, and let F be th force which impels B towards C, and let ƒ be th We have force exerted by A when at a.

E:ƒ=Ba: BC, =Ba: BA; and E:-E-Ba: BA-Ba=Ba: Aa; or E: F Ba: Aa.

Now in fig. 60. Pl. 281. let ABC be the lin joining 3 particles, to which draw FG, PH paralle and IAF, HBG perpendicular. Take IF or HO to reprefent the elasticity correfponding to the di tance AB. Let the particle A be fuppofed to hav been carried with an uniform motion to a by fom external force, and draw R a M perpendicular to RG, and make FI : RM Ba: BA. We ha then have FI: PM-Ba: Aa; and PM will re prefent the force with which the particle B is ur ged towards C. Suppofe this conftruction to bo made for every part of the line AB, and that point M is thus determined for each of them, ma thematicians know that all thefe points M lie in the curve of hyperbola, of which FG and GH are the affymptotes. It is alfo known by the ele ments of mechanics, that since the motion of A along AB is uniform, A a or IP may be taken to reprefent the time of defcribing Aa; and that the area IPM reprefents the whole velocity which has acquired in its motion towards C when A bas come to a, the force urging B being always as the portion PM of the ordinate. Take GX of any length in HG produced, and let GX reprefent the velocity which the uniform action of the ra tural elafticity IF could communicate to the particle B during the time that A would uniformly defcribe AB. Make GX to GY as the rectangle IFGH to the hyperbolic fpace IFRM, and draw YS cutting MR produced in S, and draw FX cut ting MR in T. It is known to the mathemati cians that the point S is in a curve fine FS, called the logarithmic curve; of which the leading pro perty is, that any line RS parallel to GX is to GA

A

as the reangle FGH is to the hyperbolic space
IFRM, and that FX touches the curve in F.
This being the cafe, it is plain, that because
RT increafes in the fame proportion with FR, or
with the rectangle IFRP, and RS increases in the
proportion of the fpace IFRM, TS increafes in the
proportion of the fpace IPM. Therefore TS is
proportional to the velocity of B when A has
rached, and RT is proportional to the velocity
which the uniform action of the natural elafticity
would communicate to B in the fame time. Then
face FT is as the time, and TS is as the velocity,
the area FTS will be as the space defcribed by B
urged by the variable force PM), while A, urged
by the external force, defcribes Aa, and the
tangle FRT will reprefent the fpace which the
orm action of the natural elafticity would caufe
Bto defcribe in the fame time. And thus it is
plain that these three motions can be compared
together the uniform motion of the agitated
particle A, the uniformly accelerated motion which
the natural elafticity would communicate to B by
its conftant action, and the motion produced in B
by the agitation of A. But this comparifon, re-
quiring the quadrature of the hyperbola and loga-
rithmic curve, would lead us into moft intricate
and tedious computations. Of thefe we need only
e the refult, and make fome other comparifons
which are palpable.

therefore, fuppofing it accurate from the very first particle, it follows, from the equable motion of A, that each fucceeding particle moves through an equal space in acquiring the motion of A.

The conclufion which we must draw from all this is, that when the agitation of A has been fully communicated to a particle at a fenfible diftance, the intervening particles, all moving forward with a common velocity, are equally compreffed as to fenfe, except a very few of the first particles; and that this communication, or this propagation of the original agitation, goes on with an uniform velocity. Thefe computations need not be attended to by fuch as do not wish for an accurate knowledge of the precife agitation of each particle. It is enough for fuch readers to fee clearly that time mu efcape between the agitation of A and that of a diftant particle; and this is abundantly manifeft from the incomparability of the nafcent rectangle IFRP with the nafcent triangle FRT, and the incomparability of FRT with FTS.

What has now been fhown of the communication of any fenfible motion A a mult hold equally with refpect to any change of this motion. Therefore if a tremulous motion of a body, fuch as a foring or bell, fhould agitate the adjoining particle A by pufhing it forward in the direction AB, and then allowing it to come back again in the Let Aa be fuppofed indefinitely small in com- direction BA, an agitation fimilar to this will take prion of AB. The space defcribed by A is there- place in all the particles of the row one after the are indefinitely fmall; but in this cafe we know other. Now if this body vibrate according to that the ratio of the space FRT to the rectangle the law of motion of a pendulum vibrating in a FRP is indefinitely fmall. There is therefore no cycloid, the neighbouring particle of air will of necomparifon between the agitation of A by the city vibrate in the fame manner; and then NEWexternal force, and the agitation which natural TON'S demonftration in the article ACOUSTICS aticity would produce on a fingle particle in the needs no apology. Its only deficiency was, that fame time, the laft being incomparably finaller than it feemed to prove that this would be the way in the fit. And this fpace FRT is incomparably which every particle would of neceffity vibrate; greater than FTS, and therefore the space which which is not true, for the fucceffive parcels of air would defcribe by the uniform action of the will be differently agitated according to the ori atural ciafticity is incomparably greater than what ginal agitation. Newton only wants to prove the it would defcribe in confequence of the agitation uniform propagation of the agitations, and he feof A. From this realoning we fee evidently that lects that form which renders the proof cafiéft. Amait be fenfibly moved, or a finite or meafurable He proves, in the moft unexceptionable manner, line must elapfe before B acquires a meafurable that if the particles of a pulfe of air are really mo tuation. In like manner B must move during a ving like a cycloidal pendulum, the forces acting meslusable time before C acquires a measurable on each particle, in confequence of the comprefstion, &c.; and therefore the agitation of A is fion and dilatation of the different parts of the Communicated to the diftant particles in gradual pulfe, are precisely such as are necessary for continu

fucceffion.

C

ing this motion, and therefore no other forces are

By a farther comparison of thefe fpaces we required. Then fince, each particle is in a certain a the time in which each fucceeding particle part of its path, is moving in a certain direction, equires the very agitation of A. If the particles and with a certain velocity, and urged by a deBand Conly are confidered, and the motion of termined force, it must move in that very manner. Begledded, it will be found that B bas acquired The objection started by John Bernouilli againft the motion of A a little before it has defcribed + Newton's demonftration (in a fingle line) of the of the space defcribed by A; but if the motion of elliptical motion of a body urged by a force in C be confidered, the acceleration of B must be in- the inverfe duplicate ratio of the distance from

created

A. By computation it appears, that when both of aerial undulations, and is equally futile.
B and C have acquired the velocity of A, B has

pace in proportion to that defcribed by againft Newton's demonftration of the progrefs

It muft, however, be obferved, that Newton's

defcribed nearly of A's motion, and C more demonftration proceeds on the fuppofition that nearly fd. Extending this to D, we fhall find that the linear agitations of a particle are incomparaAnd from the nature of the computation it ap- This is not strictly the cafe in any refiftance, and in Dhat defcribed ftill more nearly 4th of A's motion. bly fmaller than the extent of an undulation. that this

D 2

twang

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