then DO: CO≈m: n, and DC: CO—m—n:n, and Therefore any velo DC-2 CO, or te n AC and BD are as AC and BD'. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC: therefore the velocities at all the other instants x, g, D, are properly represented by the corresponding ordinates. Hence, 1. Since the abscissæ of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and AC xe is to Acgf as the space described in the time Cx to the space described in the time C g (1st 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 is e 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 ev E, or 2 a. 3. Draw the tangent A; then, by the hyperbola Cx-CO: now Cx is the time in which the resistance to the velocity AC would extinguish it; for the tangent coinciding with the elemental arc A a of the curve, the first impulse of the uniform action of the resistance is the same with the first impulse of its varied action. By this the velocity AC is reduced to a c. If this operated uniformly like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle ACx. This triangle, therefore, represents the height 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 BD 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 AC is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion, is to the extinguishing time alone: for AC: BD=OD (or OC+CD): OC; or V: vee+i. 6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles AFOC, BDOG are equal; and therefore AVGF and BVCD are equal, and VC: VA VG: VB; thereforete and ext v -V. 7. Any velocity is reduced in the proportion of m to n in the time e m-n For, let AC BD≈m : n; : n m-n n city is reduced to one half in the time in which the initial resistance would have extinguished it by its uniform action. mode of Thus may the chief circumstances of this motion be Anethe determined by means of the hyperbola, the ordinates determinand abscissæ exhibiting the relations of the times and; ing this velocities, and the areas exhibiting the relations of both motion. to the spaces described. But we may render the conception of these circumstances infinitely more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line ML, parallel to OD, for its abscissa, and of such a nature, that if the ordinates to the hyperbola AC, ex, fg, BD, &c. be produced till they cut this curve in L, p, n, K, &c. and the abscissa in L, s, h, d, &c. the ordinates s p, h, n, d K, &c. may be proportional to the hyperbolic areas e AC, ƒA C g, Ac K. Let us examine what kind of curve this will be. Make OC: 0 x=0x: 0g; then (Hamilton's Conics, IV. 14. Cor.), the areas AC x e, ex gfare equal: therefore drawing ps, n t perpendicular to OM, we shall have (by the assumed nature of the curve Lp K), Ms=st; and if the abscissa OD be divided into any number of small parts in geometrical progression (reekoning the commencement of them all from O), the a.is Viof this curve will be divided by its ordinates into the same number of equal parts; and this curve will have its ordinates LM, p s, nt, &c. in geometrical progression, and its abscisse in arithmetical progres sion. Also, let KN, MV touch the curve in K and L, and let OC be supposed to be to O c, as OD to O d, and therefore Ce to D d as OC to OD; and let these lines Cc, D d be indefinitely small; then (by the nature of the curve) Lo is equal to Kr: for the areas a AC c, b BD d are in this case equal. Also ko is to k r, as LM to KI, because c Cd D=CO: DO: Therefore IN: IK≈r Krk 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 Kas the rectangle BDOG or ACOF to the parabolic arex BDCA. For let fg hn be an ordinate very near to BDK; and let n cut the curve in a, and the ordinate Kl in 9; then we have Kq: qn KI: IN, or but BD: AC CO: DO; Therefore the sum of all the rectangles BD. Dg is to the sum of all the rectangles AC. qn, as CO to IN; but 56 The whole but the sum of the rectangles BD. Dg is the space space Hence it follows that QL expresses the area BVA, and in general, that the part of the line parallel to OM, which lies between the tangent KN and the curve LpK, expresses the corresponding area of the hyperbola which lies without the rectangle BDOG. And now, by the help of this curve, we have an easy way of convincing and computing the motion of a body through the air. For the subtangent of our curve now represents twice the height through which the ball must fall in vacuo, in order to acquire the terminal velocity; and therefore serves for a scale on which to measure all the other representatives of the motion. But it remains to make another observation on the reduced to curve L pK, which will save us all the trouble of graphical operations, and reduce the whole to a very cal compu- simple arithmetical computation. It is of such a na a simple arithmeti ation. ture, that when MI is considered as the abscissa, and is In constructing this curve we were limited to no par- our proportions would have still held good; because this Since then we have tables of logarithms calculated I x I 1 is properly represented by the fluxion of MI. In short the line MI is divided precisely as the line of numbers on a Gunter's scale, which is therefore a line of logarithms; and the numbers called logarithms are just the lengths of the different parts of this line measured on a scale of equal parts. Therefore, when we meet with such an expression as viz. the fluxion E, for a ball of cast-iron one inch diameter, and if it spaces, showing the motion during each successive se cond; the fourth column is the velocity at the end of the time t; and the last column is the differences of ve locity, showing its diminution in each successive second. We see that at the distance of 1000 yards the velocity is reduced to one half, and at the distance of less than a mile it is reduced to one-third. II. It may be required to determine the distance at which the initial velocity V is reduced to any other quantity v. This question is solved in the very same manner, by substituting the logarithms of V and u for those of e+t and e; for AC: BD=OD: OC, and AC OD V e+t OC' therefore log.log. or log. - log. e ย ย Thus it is required to determine the distance in which the velocity 1780 of a 24 pound ball (which is the medium velocity of such a ball discharged with 16 pounds of powder) will be reduced to 1500. Here d is 5.68, and therefore the logarithm of 2 ad is by hyperbolic logarithms S=2ad X log. e+t e +3.78671 +8.87116 v Let the ball be a 12 pounder, and the initial velocity Log. 0.43429 be 1600 feet, and the time 20 seconds. We must first This reduction will be produced in about of a se- +3.03236 We may proceed thus: 1.36229 0.48145 =t. Then to log. add log. e, and we obtain log. e+t, and e+t; from which if we take e we have t. Then to find v, say e+t:e= =V : V. riment Mr We shall conclude these examples by applying this app +3.68557 last rule to Mr Robins's experiment on a musket bullet of of an inch in diameter, which had its velocity re +9.9449° duced from 1670 to 1425 by passing through 100 feet see 9.63778 of air. This we do in order to discover the resistance which it sustained, and compare it with the resistance to r a velocity of 1 foot per second. OC=AC: BD, or e+t: eV: v. 23",03: 3′′,03=1600: 2101, v. 3.99269 The ball has therefore gone 3278 yards, and its velocity is reduced from 1600 to 210. It may be agreeable to the reader to see the gradual progress of the ball during some seconds of its motion. T. S. Diff. V. Diff. We must first ascertain the first term of our analogy." The ball was of lead, and therefore 2a must be multiplied by d and by m, which expresses the ratio of the density of lead to that of cast iron. dis 0.75, and m is and 2adm=1274.2 3" 3336 744 4" 4080 690 645 804 114 86 5" 4725 604 569 67 537 6" 5294 The first column is the time of the motion, the second is the space described, the third is the differences of the 59 ecapita stion. 62 1000 of passage. Now, to find the remaining velocity, say But in Mr Robins's experiment the remaining velo- The time e, in which the resistance of the air would or 52"; its weight, by this theory, or 5.97 pounds. If we cal- V v e= 630.23 The next experiment, with a velocity of 1690 feet, Thus we have explained, in great detail, the principles and the process of calculation for the simple case of the motion of projectiles through the air. The learned reader will think that we have been unreasonably prolix, and that the whole might have been comprised in less room by taking the algebraic method. We acknowledge that it might have been done even in a few lines. followed the proportion of the hyperbolic areas, we Having thus, we hope, enabled the reader to con- 60 We are, in the next place, to consider the perpendi- Of the percular ascents and descents of heavy projectiles, where pendicular the resistance of the air is combined with the action of ascents of heavy projectiles. gravity and we shall begin with the descents. Let u, as before, be the terminal velocity, and g the 9 g v : its hyperbolic logarithm. Therefore S- The constant quantity C for completing the fluent g g u2 u— But we have observed, and our observation has = ·x L g Χλ 0.43429 g This equation establishes the relation between the space fallen through, and the velocity acquired by the gS fall. 2g S u We obtain by it =L u2 =L. v and is 14850, and 323,62 or, which is still more conveni- 0,46947, which is the sine of 28°. The logarithmic u2 -* ent for us, 2. Required the velocity acquired by the body by falling 1848 feet. Say, 14850 1848 = 0,43429: 0,05407. Look for this number among the logarithIt will be found at 28°, of which the lo n is equal to u1 u2-va number corresponding to it: call this n. Then, since we have nu—n v3 — u3, and nu2—u3— n v3, or n v2 = ua × n—v, and v3 u2 xn―I = n is the logarithm of 323,62, the velocity required. We may observe, from these solutions, that the acquired velocity continually approaches to, but never equals, the terminal velocity. For it is always expres sed by the sine of an arch of which the terminal velocity is the radius. We cannot help taking notice here Erronect of a very strange assertion of Mr Muller, late professor assertion of of mathematics and director of the royal academy at Mr Maler, Woolwich. He maintains, in his Treatise on Gunnery, his Treatise of Fluxions, and in many of his numerous works, that a body cannot possibly move through the air with a greater velocity than this; and he makes this a fundamental principle, on which he establishes a theory of motion in a resisting medium, which he asserts with great confidence to be the only just theory; saying, that all the investigations of Bernoulli, Euler, Robins, Simpson, and others, are erroneous. We use this strong expression, because, in his criticisms on the works of those celebrated mathematicians, he lays aside good manners, and taxes them not only with ignorance, but with dishonesty, saying, for instance, that it required no small dexterity in Robins to confirm by his experiments a theory founded on false principles; and that 5.67688 Thomas Simpson, in attempting to conceal his obliga2.83844 tions to him for some valuable propositions, by chan1.50515 ging their form, had ignorantly fallen into gross errors. 3.26670 Nothing can be more palpably absurd than this asserA blown bladder will have but a +1.44396 tion of Mr Muller. +5.26670 small terminal velocity; and when moving with this 5.67688 velocity, or one very near it, there can be no doubt that it will be made to move much swifter by a smart stroke. Were the assertion true, it would be impossible for a portion of air to be put into motion through the rest, for its terminal velocity is nothing. Yet this author makes this assertion a principle of argument, saying, that it is impossible that a ball can issue from the mouth Robins and others are grossly mistaken, when they give of a cannon with a greater velocity than this; and that them velocities three or four times greater, and resistances which are 10 or 20 times greater than is possible; The process of this solution suggests a very perspicu- and by thus compensating his small velocities by still perspicu-maller resistances, he confirms his theory by many ex We shall take an example of a ball whose terminal velocity is 689 feet, and ascertain its velocity after a fall of 1848 feet. Here, |