this motion, in the same time, that is, in the time of describing four-thirds md; but the resistance of the air would do this in the time of describing eignt-thirds md; that is, in twice the time. The resistance therefore is equal to half the weight of the ball, or to half the weight of the column of air whose height is the height producing the velocity. But the resistance to different velocities are as the squares of the velocities; and therefore as their producing heights, and, in general, the resistance of the air to a sphere moving with any velocity, is equal to the half weight of a column of air of equal section, and whose altitude is the height producing the velocity. The result of this investigation has been acquiesced in by all Sir Isaac Newton's commentators. Many faults have indeed been found with his reasoning, and even with his principles; and it must be acknowledged that although this investigation is by far the most ingenious of any in the Principia, and sets his acuteness and address in the most conspicuous light, his reasoning is liable to serious objections, which his most ingenious commentators have not completely removed. Yet the conclusion has been acquiesced in, but as if derived from other principles, or by more logical reasoning. The reasonings or as sumptions, however, of these mathematicians are no better than Newton's; and all the causes of deviation from the duplicate ratio of the velocities, and the causes of increased resistance, which the latter authors have valued themselves for discovering and introducing into their investigations, were actually pointed out by Sir Isaac Newton, but purposely omitted by him to facilitate the discussion in re difficillima (See Schol. prop. 37. b. 2). The weight of a cubic foot of water is 624 lbs. and the medium density of the air is of water; therefore let a be the height producing the velocity (in feet), and d the diameter of the ball (in inches), and the periphery of a circle whose diameter is 1; the resistance of the air will be: = pounds, ° d° 315417 Example.-A ball of cast iron weighing twelve pounds is four inches and a half in diameter. Suppose this ball to move at the rate of 25 feet in a second. The height which will produce this velocity in a falling body is 97 feet. The area of its great circle is 0-11044 feet, or of one foot. Suppose water to be 840 times heavier than air, the weight of the air incumbent on this great circle, and 9 feet high, is 0-081151 lbs. half of this is 0 0405755 or or nearly of a pound. This should be the resistance of the air to this motion of the ball. It is proper, in all matters of physical discussion, to confront every theoretical conclusion with experiment. This is particularly necessary in the present instance, because the theory on which this proposition is founded is extremely uncertain. Newton speaks of it with the most cautious diffidence, and secures the justness of the conclusions by the conditions which he as sumes in his investigation. He describes witn the greatest precision the state of the fluid in which the body must move, so as that the demonstrations may be strict, and leaves it to others to pronounce whether this is the real constitution of our atmosphere It must be granted that it is not; and that many other suppositions have been introduced by his commentators and foilowers to suit his investigation (for little or nothing has been added to it) to the circumstances of the case. Sir Isaac Newton himself, therefore, attempted to compare his proportions with experiment. Some were made by dropping balls from the dome of St. Paul's cathedral; and all these showed as great a coincidence with his theory as they did with each other: but the irregularities were too great to allow him to say with precision what was the resistance. It appeared to follow the proportion of the squares of the velocities with sufficient exactness; and, though he could not say that the resistance was equal to the weight of the column of air having the height necessary for communicating the velocity, it was always equal to a determinate part of it; and might be stated na, n being a number to be fixed by numerous experiments. One great source of uncertainty in his experiments seems to have escaped his observation: the air in that dome is almost always in a state of motion. In summer there is a very sensible current of air downwards, and frequently in winter it is upwards: and this current bears a very great proportion to the velocity of the descents. Sir Isaac takes no notice of this. He made another set of experiments with pendulums; and pointed out some very curious and unexpected circumstances of their motions in a resisting medium. There is hardly any part of his noble work in which his address, his patience, and his astonishing penetration, appear in greater lustre. It requires the utmost intenseness of thought to follow him in these disquisitions. Their results were much more uniform, and confirmed his general theory; and it has been acquiesced in by the first mathematicians of Europe. But the deductions from this theory were so inconsistent with the observed motions of military projectiles, when the velocities are prodigious, that no application could be made which could be of any service for determining the path and motion of cannon shot and bombs; and although John Bernouilli gave, in 1718, a most elegant determination of the trajectory and motion of a body projected in a fluid which resists in the duplicate ratio of the velocities (a problem which even Newton did not attempt), it has remained a dead letter. Mr. Benjamin Robing was the first who suspected the true cause of the imperfection of the usually received theories and in 1737 he published a small tract, in which he showed clearly that even the Newtonia theory of resistance must cause a cannon ball discharged with a full allotment of powder, t deviate farther from the parabola, in which i would move in vacuo, than the parabola deviate from a straight line. But he farther asserted from good reasoning, that in such great velociti the resistance must be much greater than th theory assigns; because, besides the resistance arising from the inertia of the air which is put in motion by the ball, there must be a resistance arising from a condensation of the air on the anterior surface of the ball, and a rarefaction behind it and there must be a third resistance, arising from the statical pressure of the air on its anterior part, when the motion is so swift that there is a vacuum behind. Even these causes of disagreement with the theory had been foreseen and mentioned by Newton (see the Scholium to prop. 37, Book II. Princip.); but the subject seems to have been little attended to. Some authors, however, such as St. Remy, Antonini, and Le Blond, have given most valuable collections of experiments, ready for the use of the profound mathematician. SECT. IV. 10000000 resulting from Mr. Robins's experiments nearly in the proportion of seven to ten. Chev. de Borda made experiments similar to those of Mr. Robins, and his results exceeded those of Robins in the proportion of five to six. We must content ourselves, however, at present with the experimental measure mentioned above. To apply to our formulæ, therefore, we reduce this experiment, which was made on a ball of four inches and a half diameter, moving with the velocity of twenty-five feet and one-fifth per second, to what would be the resistance to a ball of one inch, having the velocity a foot. 0.04919 This will give R = being dimin4.52 x 25.2" ished in the duplicate ratio of the diameter and velocity. This gives R=0·00000381973 pound, 3.81973 of a pound. The ogarithm is, 1000000 от 4.58204. The resistance here determined is the inch in diameter Log. of W E Log. of E or terminal velocity Log, u a, or producing height. OBSERVATIONS BY MR. ROBINS, ON VELOCITY AND RESISTANCE. Two or three years after the appearance of his first publication, Mr. Robins discovered that ingenious method of measuring the velocities of military projectiles which has handed down his name to posterity with great honor: and, having ascertained these velocities, he discovered the prodigious resistance of the air, by observing the diminution of velocity which it occasioned. This made him anxious to examine what was the real resistance to any velocity whatever, in order to ascertain what was the law of its variation; and he was equally fortunate in this attempt likewise. From his Mathematical Works, vol. i. p. 205, it appears that a sphere of four inches and a half in diameter, moving at the rate of twenty-five feet one-fifth in a second, sustained a resistance of 0-04914 lb. or of a pound. This is a greater resistance than that of the Newtonian theory, which gave 305755 in the proportion, of 1000 to 1211, or very nearly in the proportion of five to six in small numbers. And we may adopt as a rule, in all moderate velocities, that the resistance to a sphere is equal to of the weight of a column of air having the great circle of the sphere for its base, and for its altitude the height through which a heavy body must fall in vacuo to acquire the velocity of projection. The importance of this experiment is great, because the ball is precisely the size of a twelve pound shot of cast iron; and its accuracy may be depended on. There is but one source of error. The whirling motion must have occasioned some whirl in the air, which would continue till the ba'l again passed through the same point of its revolution. The resistance observed is therefore probably somewhat less than the true resistance to the velocity of twenty-five feet one-fifth, because it was exerted in a relative velocity which was less than this, and is, in fact, the resistance mpetent to this relative and smaller velocity. Accordingly, Mr. Smeaton places great confidence in the observations of Mr. Rouse of Leicestershire, who measured the resistance by the effct of the wind on a plane properly exposed to it. He does not tell us how the velocity of the wind was ascertained; but our opinion of kas penetration and experience leads us to beLeve that this point was well determined The resistance observed by Mr. Rouse exceeds that . lbs. 0.13648 0.21533 These numbers are of frequent use in all questions on this subject. Mr. Robins gives an expeditious rule for readily finding a, which he calls F, by which it is made 900 feet for a castiron ball of an inch diameter. But no theory of resistance which he professes to use will make this height necessary for producing the terminal velocity. His F, therefore, is an empirical quantity, analogous indeed to the producing height, but accommodated to his theory of the trajectory of cannon-shot, which he promised to publish, but did not live to execute. We need not be very anxious about this; for all our quantities change in the same proportion with R, and need only a correction by a multiplier or divisor, when R shall be accurately established. The use of these formulae may be illustrated by an example or two. Ex. 1. To find the resistance to a twenty-four pouna ball moving with the velocity of 1670 feet in a second, which is nearly the velocity communicated by sixteen pounds of powder. The diameter is 503 inches. Log, R Log.d Log. 16702 Log. 3344 lbs.r + 4.58204 + 1·49674 + 6·44548 2-52426 But it is found, by unequivocal experiments on the retardation of such a motion, that it is 504 lbs. This is owing to the above causes, the additional resistance to great velocities, arising from the condensation of the air, and from its pressure into the vacuum left by the ball. Ex. 2. Required the terminal velocity of this ball? 6.07878 a 5.30143 We proceed to consider these motions through their whole course: and we shall first consider them as affected by the resistance only; then we shall consider the perpendicular ascents and descents of heavy bodies through the air; and, lastly, their motion in a curvilineal trajectory, when projected obliquely. This must be done by the help of the abstruser parts of fluxionary mathematics. To make it more perspicuous, we shall consider the simply resisted rectilineal mo tions geometrically, in the manner of Sir Isaac Newton. As we advance, we shall quit this track, and prosecute it algebraically, having by this time acquired distinct ideas of the algebraic quantities. We must remember the fundamental theorems of varied motions. 1. The momentary variation of the velocity is proportional to the force and the moment of time jointly, and may therefore be represented by v=ft, where v is the momentary increment or decrement of the velocity v,f the accelerating or retarding force, and the moment or increment of the time t. 2. The momentary variation of the square of the velocity is as the force, and as the increment or decrement of the space jointly; and may be represented by vv=fs. The first proposition is familiarly known. The second is the 39th of Newton's Principia, B. I. It is demonstrated in the article OPTICS, and is the most extensively useful proposition in mechanics. Having premised these things, let the straight line AC (fig. 2) represent the initial velocity V, and let C O, perpendicular to AC, be the time Fig. 2. in which this velocity would be extinguished by the uniform action of the resistance. Draw through the point A an equilateral hyperbola A e B having OF, OCD, for its assymptotes; then let the time of the resisted motion be represented by the line C B, C being the first instant of the motion. If there be drawn perpendicular ordinates ke, fg, DB, &c., to the hyperbola, they will be proportional to the velocities of the body at the instant; x,g, D, &c., and the hyperbolic areas AC xe, AC, fg, ACD B, &c., will be proportional to the spaces described during the times C, Cg, CB, &c. For suppose the time divided into an indefinite number of small and equal moments, C c, Dd, &c., draw the ordinates ac, bd, and the perpendiculars bß, aa. Then, by the nature of the hyperbola, A C: uc =0c: 0C. and A C—a c : ac=0c-OC OC, that is, A a: ac=Cc: OC, and A a: Cc ac: OC, AC ac: A COC; in like manner, BB: D d = BDb D: BDO D. Now Dd=Cc, because the moments of time were Mi t li taken equal, and the rectangles AC CO, BD DO, are equal by the nature of the hyperbola; therefore Aa: Bß⇒ AC ac: BDbd: but as the points c, d, continually approach, and ultimately coincide with C, D, the ultimate ratio of ACac to BD bd is that of A C2 to B D2; therefore the momentary decrements of A C and BD are as A C2 and BD. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC; therefore the velocities at all the other instants x,g, D, are properly represented by the corresponding ordinates. Hence, 1. As the abscissa of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and AC ke is to A c gf as the space described in the time C to the space described in the time Cg (first theorem on varied motions). 2. The rectangle ACOF is to the area ACDB as the space formerly expressed by 2 a, or E to the space described in the resisting medium during the time CD; for AC being the velocity V, and OC the extinguishing time e, this rectangle ise V, or E, or 2 a, of our former disquisitions; and because all the rectangles such as ACOF, BDOG, &c., are equal, this corresponds with our former observation, that the space uniformly described with any velocity during the time in which it would be uniformly extinguished by the corresponding resistance is a constant quantity, viz. that in which we always had v E, or 2 a. 3. Draw the tangent Ax; then, by the hyperbola CCO: now C x is the time in which the resistance to the velocity AC would extinguish it; for the tangent coinciding with the elemental arc A a of the curve, the first impulse of the uniform action of the resistance is the same with its first impulse of its varied action. By this the velocity AC is reduced to ac. If this operated uniformly, like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle AC. This triangle, therefore, represents the height through which a heavy body must fall in vacuo, in order to acquire the terminal velocity. 4. The motion of a body resisted in the duplicate ratio of the velocity will continue without end, and a space will be described which is greater than any assignable space, and the velocity will grow less than any that can be assigned; for the hyperbola approaches continually to the assymptote, but never coincides with it. There is no velocity B D so small, but a smaller ZP will be found beyond it; and the hyperbolic space may be continued till it exceeds any surface that can be assigned. 5. The initial velocity A C is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion is to the extinguishing time alone; for AC: BD0D (or OC x CD): OC: or V: v=e: ext. 6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles AFOC, BDOG, are equal; and therefore AVG F and BVCD are equal and e V-v VC: VA VG: VB; therefore t= times and velocities, and the areas exhibiting the relations of both to the spaces described., But we may render the conception of these circum stances much more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line ML8, parallel to OD for its abscissa, and of such a nature that if the ordinates to the hyperbola A Cex, fg, BD, &c. be produced till they cut this curve in L, p, n, K, &c., and the abscissa in L, e, h, 8, &c., the ordinates e, p, h, n, d, K, &c., may be proportional to the hyperbolic areas e Ack, ƒ A cg, d A c K. Let us examine what kind of curve this will be. Make OC: 0x0 x : 0 g; then (Hamilton's Conics, IV. 14. Cor.) the areas AC re, exgf are equal: therefore drawing ps, nt, perpendicular to OM, we shall have (by the assumed nature of the curve Lp K), Ms=st; and if the abscissa O D be divided into any number of small parts in geometrical progression, (reckoning the commencement of them all from O), the axis V i of this curve will be divided by its ordinates into the same number of equal parts; and this curve will have its ordinates LM, ps, nt, &c., in geometrical progression, and its abscissæ in geometrical progression. Also, let K N, M V, touch the curve in K and L, and let OC be supposed to be to Oc, as OD to Od, and therefore Cc to D d as OC to OD; and let these lines Cc, Dd, be indefinitely small; then (by the nature of the curve) Lo is equal to Kr; for the areas a AC c, b BDd are in this case equal. Also lo is to kr, as L M to KI, because cC: dD=CO: DO: Therefore IN: IK=rK:rk IK: ML=r =rk: ol ML: MVol: 0 L and IN: MN=rK: 0 L. That is the subtangent IN, or MV, is of the same magnitude, or is a constant quantity in every part of the curve. Lastly, the subtangent IN, corresponding to the point K of the curve, is to the ordinate K8 as the rectangle BDOG or ACOF to the parabolic area BDCA. For let fghn be an ordinate very near to BD 8 K; and let hn cut the curve in n, and the ordinate KI in g; then we have Kqqn=KI: IN, or Dg: qn=DO:IN; but BĎ: AC=CO: DO; therefore BD. Dg: AC.qn=CO: IN: Therefore the sum of all the rectangles B D.Dg is to the sum of all the rectangles A C. qn, as CO to IN; but the sum of the rectangles AC is given, the sum of the rectangles AC⋅ q n BD Dg is the space ACDB; and, because is the rectangle of A C, and the sum of all the lines on; that is, the rectangle of AC and RL; therefore the space ACDB: AC.RL=CO : IN, and AC DB ×IN=AC.CO.RL: and therefore IN: RLAC.CO: ACDB. Hence it follows that QL expresses the area BV A, and, in general, that the part of the line parallel to O M, which lies between the tangent KN and the curve LpK, expresses the corre sponding area of the hyperbola which lies with. out the rectangle BDOG. And now, by the help of this curve, we have an easy way of conceiving and computing the motion of a body through the air. For the subtangent of our curve now presents twice the height through which the ball must fall in vacuo, in order to acquire the terminal velocity; and therefore serves for a scale on which to measure all the other representatives of the motion. It remains to make another observation on the curve Lp K, which will save us all the trouble of geographical operations, and reduce the whole to a very simple arithmetical computation. In constructing this curve we were limited to no particular length of the line LR, which represented the space A CDB; and all that we had to take care of was, that when O C, OK, O g, were taken in geometrical progression, Ms, M t, should be in arithmetical progression. The abscissæ having ordinates equal to ps, nt, &c., might have been twice as long as is shown in the dotted curve which is drawn through L. All the lines which serve to measure the hyperbolic spaces would then have been doubled. But NI would also have been doubled, and our proportions would have still held good; because this sub-tangent is the scale of measurement of our figure, as E or 2 a is the scale of measurement for the motions. called logarithms are just the lengths of the different parts of this line measured on a scale of equal parts. 100000 Reasons of convenience nave given rise to another set of logarithms: these are suited to a logistic curve whose subtangent is only of the ordinate rv, which is equal to the side of the hyperbolic square, and which is assumed for the unit of number. We shall suit our applications of the preceding investigation to both these, and shall first use the common logarithms whose subtangent is 0.43429. The whole subject will be best illustrated by taking an example of the different questions which may be proposed: Recolu lect that the rectangle ACO F is 2 a, or or E, for a ball of cast iron one inch diameter, u'd and, if it has the diameter d, it is or 2 a d, or E d. H g I. It may be required to determine what will be the space described in a given time t by a ball setting out with a given velocity V, and what will be its velocity v at the end of that time. Here we have NI: MIACOF: BDCA; now NI is the subtangent of the logistic curve; MI is the difference between the logarithms of OD and OC; that is, the difference between the logarithms of e+t and e; A COF is 2 ad, or or Ed. Therefore by common logag rithms 0-43429: log. e+t-log.e=2ad: S,= space described, u'd et! or 0-43429 log. =2ad: S, 2 ad 0.43429 e by hyperbolic logarithms S2 ad x log. e + t R Let the ball be a twelve pounder; the initial velocity 1600 feet, and the time twenty seconds. 2 ad We must first find e, which is Therefore, log. 2 a Since then we have tables of logarithms calculated for every number, we may make use of them instead of this geometrical figure, which till requires considerable trouble to suit it to every case. There are two sets of logarithmic tables in common use. One is called a table of hyperbolic or natural logarithms. It is suited to such a curve as is drawn in the figure, where the subtangent is equal to that ordinate rv which and S. corresponds to the side #O of the square O inserted between the hyperbola and its assymptotes. This square is the unit of surface, by which the hyperbolic areas are expressed; its side is the unit of length, by which the lines belonging to the hyperbola are expressed; r v is 1, or the unit of numbers to which the logarithms are suited, and then IN is also 1. Now the square Ox being unity, the area BACD will be some number; # O being also unity, OD is some number: call it x. Then, by the nature of the hyperbola, OB: 0 = 0: DB; that is, r: 11:, so that D B is Now, calling D d the area, B D d b, which is the fluxion (ultimately) of the hyperbolic area, is --Now in the curve Lp K, MI has the same ratio to NI that BACD has to @XO. Therefore, if there be a scale of which N I is the unit, the number on this scale corresponding to MI has the same ratio to 1 which the number measuring BA CD has to 1; and I i, which corresponds to BD db, is the fluxion (ultimately) of MI ; Therefore, if MI be called the logarithm of r, is properly represented by the fluxion of MI. In short, the line M I is divided precisely as the line of numbers on a Gunter's scale, which is therefore a line of logarithms; and the numbers 1 Log. of 3",03, e |