Wind

Fig. 86.

of the impulse in this direction, it must be diminished velocity of still farther in the proportion of radius to the cosine of the Wind angle LPO or sine of CPO. Hence the general pro-" position: The effective impulse is as the surface, as the square of the velocity of the wind, as the square of the sine of the angle of incidence, and as the sine of the obliquity jointly, which we may express by the symbol RSV sin.Isin. O; and as the impulse depends on the density of the impelling fluid, we may take in every circumstance by the equation R➡S D-V2 sin.2 I · sin. O. If the impulse be estimated in the direction of the stream, the angle of obliquity ACD is the same with the angle of incidence, and the impulse in this direction is as the surface, as the square of the velocity, and as the cube of the angle of incidence jointly.

Velocity of RESISTANCE of Fluids, and confine ourselves at present to what relates to the impulse and resistance of air alone; anticipating a few of the general propositions of that theory, but without demonstration, in order to understand the applications which may be made of it. Suppose then a plane surface, of which a C (fig. 86.) is the section, exposed to the action of a stream of wind blowing in the direction QC, perpendicular to a C. The motion of the wind will be obstructed, and the surface a C pressed forward. And as all impulse or pressure is exerted in a direction perpendicular to the surface, and is resisted in the opposite direction, the surface will be impelled in the direction CD, the continuation of QC. And as the mutual actions of bodies depend on their relative motions, the force acting on the surface a C will be the same, if we shall suppose the air at rest, and the surface moving equally swift in the opposite direction. The resistance of the air to the motion of the body will be equal to the impulse of the air in the former case. Thus resistance and impulse are equal and contrary.

315 Air mo

a double velocity will gene

If the air be moving twice as fast, its particles will ving with give a double impulse; but in this case a double number of particles will exert their impulse in the same time: the impulse will therefore be fourfold; and in general it will be as the square of the velocity: or if the air and body be both in motion, the impulse and resistance will be proportional to the square of the relative velocity.

rally impel

as the

square of

that velo

city.

This is the first proposition on the subject, and it appears very consonant to reason. There will therefore be some analogy between the force of the air's impulse or the resistance of a body, and the weight of a column of air incumbent on the surface; for it is a principle in the action of fluids, that the heights of the columns of fluid are as the squares of the velocities which their pressures produce. Accordingly the second proposition is, that the absolute impulse of a stream of air, blowing perpendicularly on any surface, is equal to the weight of a column of air which has that surface for its base, and for its height the space through which a body must fall in order to acquire the velocity of the air.

Thirdly, Suppose the surface AC equal to a C no longer to be perpendicular to the stream of air, but inclined to it in the angle ACD, which we shall call the angle of incidence; then, by the resolution of forces, it follows, that the action of each particle is diminished in the proportion of radius to the sine of the angle of incidence, or of AC to AL, AL being perpendicular to CD.

Again Draw AK parallel to CD. It is plain that no air lying farther from CD than KA is will strike the plane. The quantity of impulse therefore is diminished still farther in the proportion of a C to KC, or of AC to AL. Therefore, on the whole, the absolute impulse is diminished in the proportion of AC2 to AL' hence the proposition, that the impulse and resistance of a given surface are in the proportion of the square of the sine of the angle of incidence.

Fourthly, This impulse is in the direction PL, perpendicular to the impelled surface, and the surface tends te move in this direction: but suppose it moveable only in some other direction PO, or that it is in the direction PO that we wish to employ this impulse, its action is therefore oblique; and if we wish to know the intensity

It evidently follows from these premises, that if ACA' be a wedge, of which the base AA' is perpendicular to the wind, and the angle ACA' bisected by its direction, the direct or perpendicular impulse on the base is to the oblique impulse on the sides as radius to the square the sine of half the angle ACA'.

of

The same must be affirmed of a pyramid or cone ACA', of which the axis is in the direction of the wind.

If ACA' (fig. 87.) represent the section of a solid, Fig. 5% produced by the revolution of a curve line APC round the axis CD, which lies in the direction of the wind, the impulse on this body may be compared with the direct impulse on this base, or the resistance to the motion of this body through the air may be compared with the direct resistance of its base, by resolving its surface into elementary planes P p, which are coincident with a tangent plane PR, and comparing the impulse on Pp with the direct impulse on the corresponding part K k of the base.

In this way it follows that the impulse on a sphere is one half of the impulse on its great circle, or on the base of a cylinder of equal diameter.

We shall conclude this sketch of the doctrine with a very important proposition to determine the most advantageous position of a plane surface, when required to move in one direction while it is impelled by the wind blowing in a different direction. Thus,

316

doctrine

Let AB (fig. 88.) be the sail of a ship, CA the di-Important rection in which the wind blows, and AD the line of inference the ship's course. It is required to place the yard AC from this in such a position that the impulse of the wind upon the sail may have the greatest effect possible in impelling the ship along AD.

Let AB, A b, be two positions of the sail very near Fig. §§. the best position, but on opposite sides of it. Draw BE be, perpendicular to CA, and BF, bf, perpendicular to AD, calling AB radius; it is evident that BE, BF, are the sines of impulse and obliquity, and that the ef fective impulse is BE1× BF, or be xbf. This must be a maximum.

Let the points B, b, continually approach and ultimately coincide; the chord 6 B will utimately coincide with a straight line CBD touching the circle in B; the triangles CBE, c be are similar, as also the triangles DBF, Dbf: therefore BE': b e2 — BC3 : bc2, and BF:bf=BD:6D; and BE1× BF ; be × bf=CB*X BD cbxbD. Therefore when AB is in the dest position, so that BE1× BF is greater than be × bf, we shallhave CBX BD greater than CbxbD, ore BX BD

:

Velocity of is also a maximum. This we know to be the case when Wind CB 2BD: therefore the sail must be so placed that the tangent of the angle of incidence shall be double of the tangent of the angle of the sail and keel.

In a common windmill the angle CAD is necessarily a right angle; for the sail moves in a circle to which the wind is perpendicular: therefore the best angle of the sail and axle will be 54°.44 nearly.

Such is the theory of the resistance and impulse of the air. It is extremely simple and of easy application. In all physical theories there are assumptions which depend on other principles, and those on the judgment of the naturalist; so that it is always proper to confront the theory with experiment. There are even circumstances in the present case which have not been attended to in the theory. When a stream of air is obstructed by a solid body, or when a solid body moves along in air, the air is condensed before it and rarefied behind. There is therefore a pressure on the anterior parts arising from this want of equilibrium in the elasticity of the air. This must be superadded to the force arising from the impetus or inertia of the air. We cannot tell with precision what may be the amount of this condensation ; it depends on the velocity with which any condensation diffuses itself.

Also, if the motion be so rapid that the pressure of the atmosphere cannot make the air immediately occupy the place quitted by the body, it will sustain this pressure on its fore part to be added to the other forces. Experiments on this subject are by no means numethe princi- rous; at least such experiments as can be depended on pal experi- for the foundation of any practical application. The ments on first that have this character are those published by Mr

317

Account of

this subject.

Robins in 1742 in his treatise on Gunnery. They were repeated with some additions by the Chevalier Borda, and some account of them published in the Memoirs of the Academy of Sciences in 1763. In the Philosophical Transactions of the Royal Society of London, vol. lxxiii. there are some experiments of the same kind on a larger scale by Mr Edgeworth. These were all made in the way described in our account of Mr Robins's improvements in gunnery. Bodies were made to move with determined velocities, and the resistances were measured by weights.

In all these experiments the resistances were found very exactly in the proportion of the squares of the velocities; but they were found considerably greater than the weight of the column of air, whose height would produce the velocity in a falling body. Mr Robins's experiments on a square of 16 inches, describing 25.2 feet per second, indicate the resistance to be to this weight nearly as 4 to 3. Borda's experiments on the same surface state the disproportion still greater.

The resistances are found not to be in the proportion of the surfaces, but increase considerably faster. Surfaces of 9, 16, 36, and 81 inches, moving with one velocity, had resistances in the proportion of 9, 171, 424, and 104.

Now as this deviation from the proportion of the surfaces increases with great regularity, it is most probable that it continues to increase in surfaces of still greater extent; and these are the most generally to be met with in practice in the action of wind on ships and mills. Porda's experiments on 81 inches show that the im

Wind.

725 pulse of wind moving one foot per second is about 0 Velocity of of a pound on a square foot. Therefore to find the impulse on a foot corresponding to any velocity, divide the square of the velocity by 500, and we obtain the impulse in pounds. Mr Rouse of Leicestershire made many experiments, which are mentioned with great approbation by Mr Smeaton. His great sagacity and experience in the erection of windmills oblige us to pay a considerable deference to his judgment. These experiments confirm our opinion, that the impulses increase faster than the surfaces. The following table was calculated from Mr Rouse's observations, and may be considered as pretty near the truth.

If we multiply the square of the velocity in feet by 16, the product will be the impulse or resistance in a square foot in grains, according to Mr Rouse's numbers.

The greatest deviation from the theory occurs in the oblique impulses. Mr Robins compared the resistance of a wedge, whose angle was 90°, with the resistance of its base; and instead of finding it less in the proportion of/2 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 sides had the same surface and the same inclination as the sides of the wedge, the resistance of the base and face were now as 55 to 39 nearly so that here the same surface with the same inclination had its resistance reduced from 68 to 39 by being put into this form. Similar deviations occur in the experiments of the Chevalier Borda; and it may be collected from both, that the resistances diminish more nearly in the proportion of the sines of incidence than in the proportion of the squares of those sines.

The irregularity in the resistance of curved surfaces is as great as in plane surfaces. In general, the theory gives the oblique impulses on plane surfaces much too small, and the impulses on curved surfaces too great. The resistance of a sphere does not exceed the fourth part of the resistance of its great circle, instead of being its half; but the anomaly is such as to leave hardly any room for calculation. It would be very desirable to have the experiments on this subject repeated in a greater variety of cases, and on larger surfaces, so that the errors of the experiments may be of less consequence.

Till

Resistance Till this matter be reduced to some rule, the art of of Air in working ships must remain very imperfect, as must also Gunnery. the construction of windmills.

318 It is of

know the

resistance

&c.

The case in which we are most interested in the knowledge of the resistance of the air is the motion of great con- bullets and shells. Writers on artillery have long been sequence to sensible of the great effect of the air's resistance. It seems to have been this consideration that chiefly engaof air in ged Sir Isaac Newton to consider the motions of bodies the motion in a resisting medium. A proposition or two would of bullets, have sufficed for showing the incompatibility of the planetary motions with the supposition that the celestial spaces were filled with a fluid matter; but he has with great solicitude considered the motion of a body projected on the surface of the earth, and its deviation from the parabolic track assigned by Galileo. He has bestowed more pains on this problem than any other in his whole work; and his investigation has pointed out almost all the improvements which have been made in the application of mathematical knowledge to the study of nature. Nowhere does his sagacity and fertility of resource appear in so strong a light as in the second book of the Principia, which is almost wholly occupied by this problem. The celebrated mathematician John Bernouilli engaged in it as the finest opportunity of displaying his superiority. A mistake committed by Newton in his attempt to a solution was matter of triumph to him; and the whole of his performance, though a piece of elegant and elaborate geometry, is greatly hurt by his continually bringing this mistake (which is a mere trifle) into view. The difficulty of the subject is so great, that subsequent mathematicians seem to have kept aloof from it; and it has been entirely overlooked by the many voluminous writers who have treated professedly on military projectiles. They have spoken indeed of the resistance of the air as affecting the flight of shot, but have saved themselves from the task of investigating this effect (a task to which they were unequal), by supposing that it was not so great as to render their theories and practical deductions very erroneous. Mr Robins was the first who seriously examined the subject. He showed, that even the Newtonian theory (which had been corrected, but not in the smallest degree improved or extended in its principles) was sufficient to show that the path of a cannon ball could not resemble a parabola. Even this theory showed that the resistance was more than eight times the weight of the ball, and should produce a greater deviation from the parabola than the parabola deviated from a straight line.

in this re

319 The igno This simple but singular observation was a strong rance of proof how faulty the professed writers on artillery had the writers been, in rather amusing themselves with elegant but use. on artillery less applications of easy geometry, than in endeavouring spect. to give their readers any useful information. He added, that the difference between the ranges by the Newtonian theory and by experiment was so great, that so great, that the resistance of the air must be vastly superior to what that theory supposed. It was this which suggested to him the necessity of experiments to ascertain this point. We have seen the result of these experiments in moderate velocities; and that they were sufficient for calling the whole theory in question, or at least for rendering it useless. It became necessary, therefore, to settle every point by means of a direct experiment. Here was a great difficulty. How shall we measure eithe these

great velocities which are observed in the motions of Resiste cannon-shot, or the resistances which these enormous of Air in velocities occasion? Mr Robins had the ingenuity to do GEY both. The method which he took for measuring the ve locity of a musket-ball was quite original; and it was susceptible of great accuracy. We have already given an account of it under the article GUNNERY. Having gained this point, the other was not difficult. In the moderate velocities he had determined the resistances by the forces which balanced them, the weights which kept the resisted body in a state of uniform motion. In the great velocities, he proposed to determine the resistances by their immediate effects, by the retardations which they occasioned. This was to be done by first ascertaining the velocity of the ball, and then measuring its velocity after it had passed through a certain quantity of air. The difference of these velocities is the retardation, and the proper measure of the resistance; for, by the initial and final velocities of the ball, we learn the time which was employed in passing through this air with the medium velocity. In this time the air's resistance diminished the velocity by a certain quantity. Compare this with the velocity which a body projected directly upwards would lose in the same time by the resistance of gravity. The two forces must be in the proportion of their effects. Thus we learn the proportion of the resistance of the air to the weight of the ball. It is indeed true, that the time of passing through this space is not accurately had by taking the arithmetical medium of the initial and final velocities, nor does the resistance deduced from this calculation accurately correspond to this mean velocity; but both may be accurately found by the experiment by a very troublesome computation, as is shown in the 5th and 6th propositions of the second book of Newton's Principia. The difference between the quantities thus found and those deduced from the simple process is quite trifling, and far within the limits of accuracy attain. able in experiments of this kind; it may, therefore, be safely neglected.

ments on

Mr Robins made many experiments on this subject; Mr Rober but unfortunately he has published only a very few, such made my as were sufficient for ascertaining the point he had in view. He intended a regular work on the subject, in this sub which the gradual variations of resistance corresponding ject. to different velocities should all be determined by experiment: but he was then newly engaged in an important and laborious employment, as chief engineer to the East India Company, in whose service he went out to India, where he died in less than two years. It is to be regretted that no person has prosecuted these experiments. It would be neither laborious nor difficult, and would add more to the improvement of artillery than any thing that has been done since Mr Robins's death, if we except the prosecution of his experiments on the initial veloci ties of cannon-shot by Dr Charles Hutton royal professor at the Woolwich Academy. It is to be hoped that this gentleman, after having with such effect and success extended Mr Robins's experiments on the initial velocities of musket-shot to cannon, will take up this other subject, and thus give the art of artillery all the scientific foundation which it can receive in the present state of our mathematical knowledge. Till then we must content ourselves with the practical rules which Robins has deduced from his own experiments. As he has not given us the mode of deduction, we must compare the results

Resistance with experiment. He has indeed given a very extensive of Air in comparison with the numerous experiments made both in Gannery. Britain and on the continent; and the agreement is very great. His learned commentator Euler has been at no pains to investigate these rules, and has employed himself chiefly in detecting errors, most of which are supposed, because he takes for a finished work what Mr Robins only gives to the public as a hasty but useful sketch of a new and very difficult branch of science.

321

General result of them, &c.

The general result of Robins's experiments on the retardation of musket-shot is, that although in moderate velocities the resistance is so nearly in the duplicate proportion of the velocities that we cannot observe any deviation, yet in velocities exceeding 200 feet per second the retardations increase faster, and the deviation from this rate increases rapidly with the velocity. He ascribes this to the causes already mentioned, viz. the condensation of the air before the ball and to the rarefaction behind, in consequence of the air not immediately occupying the space left by the bullet. This increase is so great, that if the resistance to a ball moving with the velocity of 1700 feet in a second be computed on the supposition that the resistance observed in moderate velocities is increased in the duplicate ratio of the velocity, it will be found hardly one-third part of its real quantity. He found, for instance, that a ball moving through 1670 feet in a second lost about 125 feet per second of its velocity in passing through 50 feet of air. This it must have done in the of a second, in which time it would have lost one foot if projected directly upwards; from which it appears that the resistance was about 125 times its weight, and more than three times greater than if it had increased from the resistance in small velocities in the duplicate ratio of the velocities. He relates other experiments which show similar results. But he also mentions a singular circumstance, that till the velocities exceed 1100 feet per second, the resistances increase pretty regularly, in a ratio exceeding the duplicate ratio of the velocities; but that in greater velocities the resistances become suddenly triple of what they would have been, even according to this law of increase. 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 1132 feet Partly con- per second nearly. Mr Euler controverts this conclutroverted sion, as inconsistent with that gradation which is observed by Euler, in all the operations of nature; and says, that although the vacuum is not produced in smaller velocities than this, the air behind the ball must be so rare (the space grounds. being but imperfectly filled), that the pressure on the anterior part of the ball must gradually approximate to that pressure which an absolute vacuum would produce; but this is like his other criticisms. Robins does nowhere assert that this sudden change of resistance happens in the transition of the velocity from 1132 feet to that of 1131 feet 11 inches or the like, but only that it is very sudden and very great. It may be strictly demonstrated, that such a change must happen in a narrow enough limit of velocities to justify the appellation of sudden a similar fact may be observed in the motion of a solid through water. If it be gradually accelerated, the water will be found nearly to fill up its place, till the velocity arrives at a certain magnitude, corresponding to the immersion of the body in the water; and then the smallest augmentation of its motion imme

322

but with

out suflicient

diately produces a void behind it, into which the water Resistance rushes in a violent manner and is dashed into froth. A of Air in gentleman, who has had many opportunities for such Gunnery. observations, assures us, that when standing near the line of direction of a cannon discharging a ball with a large allotment of powder, so that the initial velocity certainly exceeded 1100 feet per second, he always observed a very sudden diminution of the noise which the bullet made during its passage. Although the ball was coming towards him, and therefore its noise, if equable, would be continually increasing, he observed that it was loudest at first. That this continued for a second or two, and suddenly diminished, changing to a sound which was not only weaker, but differed in kind, and gradually increased as the bullet approached him. He said, that the first noise was like the hissing of red-hot iron in water, and that the subsequent noise rather resembled a hazy whistling. Such a change of sound is a necessary consequence of the different agitation of the air in the two cases. We know also, that air rushing

into a void, as when we break an exhausted bottle, makes a report like a musket.

Mr Robins's assertion therefore has every argument for its truth that the nature of the thing will admit. But we are not left to this vague reasoning: his experiments show us this diminution of resistance. It clearly appears from them, that in a velocity of 1700 feet the resistance is more than three times the resistance determined by the theory which he supposes the common one. When the velocity was 1065 feet, the actual resistance was of the theoretical; and when the velocity was 400 feet, the actual resistance was about 4 of the theoretical. That he assumed a theory of resistance which gave them all too small, is of no consequence in the present argument.

323

Robins for

Mr Robins, in summing up the results of his obser- Rule by vations on this subject, gives a rule very easily remem- computing bered for computing the resistances to those very rapid resistances motions. It has been already mentioned in the article and very GUNNERY, but we repeat it here, in order to accommo- rapid modate it to the quantities which have been determined in tions. some degree by experiment.

A

B

C D Let AB represent the velocity of 1700 feet per second, and AC any other velocity. Make BD to AD as the resistance given by the ordinary theory to the resistance actually observed in the velocity 1700: then will CD be to AD as the resistance assigned by the ordinary theory to the velocity AC is to that which really corresponds to it.

To accommodate this to experiment, recollect that a* See Gun sphere of the size of a 12 pound iron shot, moving 25 feet nery, No in a second, had a resistance of of a pound. Augment 19. &c, this in the ratio of 25 to 17003, and we obtain 210 nearly for the theoretical resistance to this velocity; but by comparing its diameter of 4 inches with, the diameter of the leaden ball, which had a resistance of at least 11 pounds with this velocity, we conclude that the 12 pound shot would have had a resistance of 396 pounds: therefore BD: AD=2ro: 396, and AB : AD=186: 396; and AB being 1700, AD will be 3613.

Let AD-a, AC=x, and let R be the resistance to a 12 pound iron shot moving one foot per second, and the resistance (in pounds) wanted for the velocity x;

Undulation of Air.

324

The dicus

13750

which is nearly one-fourth. Thus our formula becomes
0.263235
or very nearly
falling
3613-x
4(3613-x)'
short of the truth about th part. The simplicity of
the formula recommends it to our use, and when we in-
crease its result, it is incomparably nearer to the
true result of the theory as corrected by Mr Robins than
we can hope that the theory is to the actual resistance.
We can easily see that Mr Robins's correction is only a
sagacious approximation. If we suppose the velocity 3613
feet, a very possible thing, the resistance by this formula
is infinite, which cannot be. We may even suppose that
the resistance given by the formula is near the truth only
in such velocities as do not greatly exceed 1700 feet
per second.
No military projectile exceeds 2200,
and it is great folly to make it so great, because it is
reduced to 1700 almost in an instant, by the enormous
resistance.

The resistance to other balls will be made by taking them in the duplicate ratio of the diameters.

It has been already observed, that the first mathemasions of ma-ticians of Europe have lately employed themselves in improving this theory of the motion of bodies in a resistsily applied, ing medium; but their discussions are such as few ar

thematici ans not ea

325

Robins's apparently the best.

tillerists can understand. The problem can only be solved by approximation, and this by the quadrature of very complicated curves. They have not been able therefore to deduce from them any practical rules of easy application, and have been obliged to compute Borda's and tables suited to different cases. Of these performances, that of the Chevalier Borda, in the Memoirs of the Academy of Sciences for 1769, seems the best adapted to military readers, and the tables are undoubtedly of considerably use; but it is not too much to say, that the simple rules of Mr Robins are of as much service, and are more easily remembered: besides, it must be observed, that the nature of military service does not give room for the application of any very precise rule. The only advantage that we can derive from a perfect theory would be an improvement in the construction of pieces of ordnance, and a more judicious appropriation of certain velocities to certain purposes. The service of a gun or mortar must always be regulated by the eye.

326 Undulation ,of air.

There is another motion of which air and other elastic fluids are susceptible, viz. an internal vibration of their particles, or undulation, by which any extended portion of air is distributed into alternate parcels of condensed and rarefied air, which are continually changing their condition 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 greatest distances in all directions. It is of importance to have some distinct conception of this motion. It is found to be by this means that distant bodies produce in us the sensation of sound. See AcOUSTICS. Sir Isaac Newton treated this subject with his accustomed ingenuity, and has given us a theory of it in the end of the second book of his Principia. This theory has been objected to with respect to the conduct of the argument, and other explanations have been given by the most eminent mathematicians. Though they appear to differ from Newton's,

317

nety of na

But since Newton published this theory of aerial un- Has been dulations, and of their propagation along the air, and used to ex since the theory has been so corrected and improved as plain a vato be received by the most accurate philosophers as a tural phe branch of natural philosophy susceptible of rigid demonstration, it has been freely resorted to by many writers on other parts of natural science, who did not profess to be mathematicians, but made use of it for explaining phenomena in their own line on the autho rity of the mathematicians themselves. Learning from them that this vibration, and the quaquaversum propagation of the pulses, were the necessary properties of an elastic fluid, and that the rapidity of this propagation had a certain assignable proportion to the elasticity and density of the fluid, they freely made use of these concessions, and have introduced elastic vibrating fluids into many facts, where others would suspect no such thing, and have attempted to explain by their means many abstruse phenomena of nature. Æthers are everywhere introduced, endued with great elasticity and tenuity. Vibrations and pulses are supposed in this æther, and these are offered as explanations. The doctrines of animal spirits and nervous fluids, and the whole mechanical system of Hartley, by which the operations of the soul are said to be explained, have their foundation in this theory of aerial undulations. If these fancied fluids, and their internal vibrations, really operate in the phenomena ascribed to them, any explanation that can be given of the phenomena from this principle must be nothing else than showing that the legitimate consequences of these undulations are similar to the phenomena; or, if we are no more able to see this last step than in the case of sound (which we know to be one consequence of the aerial undulations, although we cannot tell how), we must be able to point out, as in the case of sound, certain constant relations between the general laws of these undulations and the general laws of the phenomena. It is only in this way that we think ourselves entitled to say that the aerial undulations are causes, though not the only causes, of sound; an it is because there is no such relation, but, on the contrary, a total dissimilarity, to be observed between the laws of elastic undulation and the laws of the propagation of light, that we assert with confidence that ethereal undulations are not the causes of vision.

328

sufficient

Explanations of this kind suppose, therefore, in the But the ap first place, that the philosopher who proposes them un- pacation derstands precisely the nature of these undulations; in not being the next place, that he makes his reader sensible of made with those circumstances of them which are concerned in the effect to be explained; and, in the third place, that he makes the reader understand how this circumstance of the vibrating fluid is connected with the phenomenon, either by showing it to be its mechanical cause,

precision