mn n n2 AC and BD are as AC and BD’. Now, because the then DO: CO=m:n, and DC: CO=m-:n, and resistance is measured by the momentary diminution of DC= co, or t=en Therefore any velovelocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and city is reduced to one half in the time in which the inithe velocities diminish by the same law; and the initial tial resistance would have extinguished it by its uniform velocity was represented by AC: therefore the veloci action. ties at all the other instants x,g, D, are properiy repre Thus may the chief circumstances of this motion be Anether sented by the corresponding ordinates. Ilence, determined by means of the hyperbola, the ordinates made ce determis 1. Since the abscissæ of the hyperbola are as the and abscissæ exhibiting the relations of the times and times, and the ordinates are as the velocities, the areas velocities, and the areas exhibiting the relations of botli botiga. will be as the spaces described, and ACxe is to A c gf to the spaces described. But we may render the conas the space described in the time C x to the space ception of these circumstances infinitely more easy and described in the time C g (1st Theorem on varied mo simple, by expressing them all by lines, instead of this tions). combination of lines and surfaces. We shall accom. 2. The rectan rle ACOF is to tbe area ACDB as plish this purpose by constructing another curve LKP, the space formerly expressed by 2 a, or E to the space having the line MLS, parallel to OD, for its abscissa, described in the resisting medium during the time CD: and of such a nature, that if the ordinates to the hyperfor AC being the velocity V, and OC the extinguishing bola AC, ex, fg, BD, &c. be produced till they cut time e, this rectangle is se V, or E, or 2 a, of our for this curve in L, P, n, K, &c. and the abscissa in L, s, mer disquisitions ; and because all the rectangles, such h, d, &c. the ordinates 6 p, h, n,d K, &c. may be proas ACOF, BDOG, &c. are equal, this corresponds portional to the hyperbolic areas é A CXfAĆg, with our former observation, that the space uniformly S Ac K. Let us examine what kind of curve this wil described with any velocity during the time in which it be. would be uniformly extinguished by the corresponding Make OC: 0,=0x: 0g; then (Hamilton's Coresistance is a constant quantity, viz. that in which we nics, IV. 14. Cor.), the areas AC xe, ex gf are equal : always had e v= E, or 2 a. therefore drawing ps, n t perpendicular to OM, we 3. Draw the tangent Ax; then, by the hyperbola shall have (by the assumed nature of the curve LpK), Cx=CO: now Cx is the time in which the resistance to M s=st; and if the abscissa OD be divided into any the velocity AC would extinguish it; for the tangent number of small parts in geometrical progression (reccoinciding with the elemental arc A a of the curve, the koning the commencement of them all from 0), the first impulse of the uniform action of the resistance is a.is V i of this curve will be divided by its ordinates the same with the first impulse of its varied action. By into the same number of equal parts; and this curre this the velocity AC is reduced to a c. If this opera will have its ordinates LM, ps, nt, &c. in geometrited uniformly like gravity, the velocities would diminish cal progression, and its abscissæ in arithmetical progresuniformly, and the space described would be represent- sion. ed by the triangle AC x. Also, let KN, MV touch the curve in K and L, and This triangle, therefore, represents the height through let OC be supposed to be to O c, as OD to 0 d, and which a heavy body must fall in vacuo, in order to ac therefore C c to D d as OC to OD; and let these lines quire the terminal velocity. C, D d be indefinitely small; then by the nature of 4. The motion of a body resisted in the duplicate ra the curve) Lois equal to Kr: for the areas a ACC, tio of the velocity will continue without end, and a b BD d are in this case equal. Also ko is to k r, as LM space will be described which is greater than any as to KI, because c Cid D=CO: DO: signable space, and the velocity will grow less than any Therefore IN : IK=r Kirk that can be assigned; for the hyperbola approaches IK : ML=rk ;ol ML : MV-01:0L and IN: MIN=rk:oL be continued till it exceeds any surface that can be That is, the subtangent IN, or MV, is of the same magassigned. nitude, or is a constant quantity in every part of the 5. The initial velocity AC is to the final velocity BD as the sum of the extinguishing time and the time Lastly, the subtangent IN, corresponding to the of the retarded motion, is to the extinguishing time point K of the curve, is to the ordinate Kd as tho alone: for AC: BD-OD (or OC+CD): 0C; or rectangle BDOG or ACOF to the parabolic area V:v=e:e tot. BDCA. 6. The extinguishing time is to the time of the re For let fg h n be an ordinate very near to BDIK; tarded motion as the final velocity is to tbe velocity and let h n cut the curve in n, and the ordinate Kl in lost during the retarded motion : for the rectangles q; then we have AFOC, B DOG are equal; and therefore AVGF and BVCD are equal, and VC: VA=VG:VB; there. Kq:qn=KI : IN, or Dgn=DO: IN; and e="vame but BD : AC=CO :DO; therefore B D.Dg: AC.qn=c0: IN. 7. Any velocity is reduced in the proportion of m Therefore the sum of all the rectangles BD. Dg is to to n in the time e' For, let AC: BD m:n; the sum of all the rectangles Ac.gn, as CO to IN; curve. foret n but I but the sum of the rectangles BD.Dg is the space our proportions would have still held good ; because this derable trouble to suit it to every case. There are two Hence it follows that QL expresses the area BVA, and sets of logarithmic tables in common use. One is callin general, that the part of the line parallel to OM, ed a table of hyperbolic or natural logarithms. It is which lies between the tangent KN and the curve Lpk, suited to such a curve as is drawn in the figure, where expresses the corresponding area of the hyperbola which the subtangent is equal to that ordinate su which corlies without the rectangle BDOG. responds to the sider 0 of the square #02 O inserted beAnd now, by the help of this curve, we have an easy tween the hyperbola and its assymptotes. This square way of convincing and computing the motion of a body is the unit of surface, by which the hyperbolic areas through the air. For the subtangent of our curve now are expressed ; its side is the unit of length, by which represents twice the height through which the ball must the lines belonging to the hyperbola are expressed ; T.! fall in vacuo, in order to acquire the terminal velocity; is =1, or the unit of numbers to which the logarithms and therefore serves for a scale on which to measure all are suited, and then IN is also 1. Now the square 56 the other representatives of the motion. 070, being unity, the area BACD will be some number; The whole But it remains to make another observation on the “O being also unity, OD is some number: Call it x. reduced to curve L p K, which will save us all the trouble of Then, by the nature of the byperbola, OB : 0r= a simple urithmeti. graphical operations, and reduce the whole to a very : DB: That is, x:1=I: so that DB is al compu- simple arithmetical computation. It is of such a naation. ture, that when MI is considered as the abscissa, and is now calling D dě, the area BD db, which is the divided into a number of equal parts, and ordinates are fluxion (ultimately) of the hyperbolic area, is Nowy is properly represented by the flusion of MI. In of many problems in the most simple and expeditious short the line MI is divided precisely as the line of manner, on which the genius of the ancient mathema numbers on a Gunter's scale, which is therefore a ticians had been exercised in vain. Few of our readers line of logarithms; and the numbers called logarithms are ignorant, that the numbers called logarithms are of are just the lengths of the diflerent parts of this line measured on a scale of equal parts. Therefore, when we meet with such an expression as viz. the fluxion it is therefore strictly true that the fluent of is the hy- Certain reasons of convenience have given rise to ano- ther set of logarithms; these are suited to a logistic numbers. curve whose subtangent is only *** of the ordinate In constructing this curve we were limited to no par I v, which is equal to the side of the lyperbolic square, ticular length of the line LR, which represented the and which is assumed for the unit of number. We shall 'space ACDB; and all that we had to take care of was, suit our applications of the preceding investigation to both these, and sball first use the common logarithms The whole subject will be best illustrated by taking Illustrated Lave been twice as long, as is shown in the dotted curve an example of the different questions which may be pro- by examwhich is drawn tbrough L. All the lines which serve to posed. ples. u measure the hyperbolic spaces would then have been Recollect that the rectangle ACOF is=2 a, or doubled. But NI would also bave been doubled, and E, , ог or Ed, e e ett e E, for a bail of cast-iron one inch diameter, and if it spaces, showing the motion during each successive se. uod cond ; the fourth column is the velocity at the end of has the diameter d, it is or 2 a d, or Ed. the time t; and the last column is the differences of ve. 8 I. It may be required to determine what will be the locity, showing its diminution in each successive second. space described in a given time t by a ball setting out We see that at the distance of 1000 yards the velocity with a given velocity V, and what will be its velocity v is reduced to one half, and at the distance of less than at the end of that time. a mile it is reduced to one-third. Here we Irave NI : MI=ACOF: BDCA; now II. It may be required to determine the distance at NI is the subtangent of the logistic curve ; MI is the which the initial velocity V is reduced to any otber difference between the logarithms of OD and OC; that quantity v. This question is solved in the very same is, the difference betweeen the logarithms of ett and e; manner, by substituting the logarithms of V and v for those of ett and e; for AC : BD-OD : OC, and uod ACOF is 2 a d, or AC OD V BD or log. OC Therefore by common logarithmg 0,43429 : log. Thus it is required to determine the distance in wbich ett-log. e=2ad: S, = space described, the velocity 1780 of a 24 pound ball (which is the meett dium velocity of such a ball discharged with 16 pounds or 0,43429 : log. = 2ad: S, of powder) will be reduced to 1500. Here d is 5.68, and therefore the loga- +3.78671 0,43429 V Log. -=0.07433, of which the log.is +8.87116 by hyperbolic logarithms S=2ad x log. Let the ball be a 12 pounder, and the initial velocity Log. 0.43429 -9.63778 be 1600 feet, and the time 20 seconds. We must first 2 ad Log. 1047-3 feet, or 349 yards 3.0 2009 find e, which is This reduction will be produced in about z of a seV cond. Therefore, log. 2 a +3.03236 III. Another question may be to determine the time log. d (4,5) +0.65321 which a ball, beginning to move with a certain velocilog. V. (1600) -3.20415 ty, employs in passing over a given space, and the dimi nution of velocity which it sustains from the resistance Log. of 3",03,=e And ett is 23"03, of which the log.is 1.36229 We may proceed thus : from which take the logo of e 0.48145 et Then to log. 0.88084 ett add log.e, and we obtain log. e+t, ande +t; from This must be considered as a common number by which if we take e we have t. Then to find v, say 2ad which we are to multiply ett:e=V: v. 0.43429 We sball conclude these examples by applying this application Therefore add the logarithms of 2 ad +3.68557 last rule to Mr Robins's experiment on a musket bullet of an expert ett + 9.94490 of of an inch in diameter, which had its velocity re duced from 1670 to 1425 by passing through 100 feet see Rolog. 0.43429 - 9.63778 of air. This we do in order to discover the resistance bina's Neth which it sustained, and compare it with the resistance to Works , vol. ip Log. S. 9833 feet 3.99269 a velocity of 1 foot per second. For the final velocity, We must first ascertain the first term of our analogy. "35: OD: OC=AC: BD, or ett :e=V: v. The ball was of lead, and therefore 2a must be multi23",03 : 3",03=1600 : 2101, =v. plied by d and by m, which expresses the ratio of the The ball has therefore gone 3278 yards, and its ve density of lead to that of cast iron. dis 0.75, and m is locity is reduced from 1600 to 210. 11.37 It may be agreeable to the reader to see the gradual = 1.577. Therefore log. 20 3.03236 7.21 d 9.87506 m 0.19782 Log. 2adm 3.10524 and 2adm=1274.2 ett Now 1274.2 : 100=0.43429 : 0.03408=log. 744 114 4" 4080 690 645 86 604 240m=0.763, and its logarithm = 9.88252, 537 The first column is the time of the motion, the second which, added to 0.03408, gives 9.91660, which is the is the space described, the third is the differences of the log. of e+t, =0.825, from which take e, and there 0.48145 of the air. 2ad: S=0,43429 : log. med remains the log. ofe+ e e log. riment of Mr Robin e 964 160 804 2456 3336 3" But e= 6" 5294 remains V or 62 followed the proportion of the hyperbolic areas, we remains t=, 05.062, or of a second, for the time shewed the nature of another curve, where lines could be found which increase in the very same manner as the path of the projectile increases ; so that a point describ- ing the abscissa MI of this curve moves precisely as city was only 1425, the ball having lost 245 ; whereas the projectile does. Then, discovering that this line is by this computation it should have lost only 126. It the same with the line of logarithms on a Gunter's appears, therefore, that the resistance is double of what scale, we shewed how the logarithm of a number really it irould have been if the resistance increased in the represents the path or space described by the projectile. duplicate proportion of the velocity. Mr Robins says Having thus, we hope, enabled the reader to con- the geometrical method, and prosecute the rest of the 60 The time e, in which the resistance of the air would We are, in the next place, to consider the perpendi- Of the perextinguish the velocity is o".763. Gravity, or the cular ascents and descents of heavy projectiles, where pendicular the resistance of the air is combined with the action of rascents of 1670 heavy proweight of the bullet, would have done it in or 52"; gravity: and we shall begin with the descents. 32 jectiles. Let u, as before, be the terminal velocity, and g the 52 therefore the resistance is times, or nearly 68 times accelerating power of gravity: When the body moves 0.763 with the velocity u, the resistance is equal to g; and its weight, by this theory, or 5-97 pounds. If we cal in cnlate from Ár Robins's experiment, we must say log. every other velocity v, we must have uo: vi = g : =r, for the resistance to that velocity. In the 1:0.43429=100: e V, which will be 630.23, and descent the body is urged by gravity g, and opposed 52 va : therefore the remaining acce0.3774 proportion of the resistance to the weight, and makes the resistance 12.07 pounds, fully double of the other. lerating force, which we shall call f, is g. 35 , or It is to be observed that with this velocity, which greatly exceeds that with which the air can rush into u_g va g(u—00) a void, there must be a statical pressure of the atmosphere equal to 6 pounds. This will make up the dif Now the fundamental theorem for varied motions is ference, and allows us to conclude that the resistance arising solely from the motion communicated to the air fizui, and s= Х follows very nearly the duplicate proportion of the ve 8 u- -X -0% 1-09 gives a resistance equal to 157 times the weight of the =- hyperb. log. of Jua_va. For the fluxion of bullet, and this bears a much greater proportion to the former than 1690* does to 1670', which shows, that W? is and this divided by the although these experiments clearly demonstrate a pro Tu -0% digious augmentation of resistance, yet they are by no quantity Vue—v?, of which it is the fluxion, gives means susceptible of the precision wbich is necessary for discovering the law of this augmentation, or for a precisely which is therefore the fluxion of good foundation of practical rules; and it is still great u_0% ly to be wished that a more accurate mode of investi its hyperbolic logarithm. Therefore S= Х gation could be discovered. Thus we have explained, in great detail, the princi- Lu-v*+C. Where L means the hyperbolic loples and the process of calculation for the simple case of garithm of the quantity annexed to it, and a may be the motion of projectiles through the air. The learn used to express its common logarithm. (See article ed reader will think that we have been unreasonably Fluxions). prolix, and that the whole might have been comprised The constant quantity C for completing the fluent in less room by taking the algebraic method. We is determined from this consideration, that the space knowledge that it might have been done even in a few described is , when the velocity is o: therefore Clines. But we have observed, and our observation has u been confirmed by persons well versed in such subjects, -XL Vu= , and C = XL Vio and the that in all cases where the fluxionary process introduces 8 us the fluxion of a logarithm, there is a great want of di- complete fluent S L Vest - L Vu - v*, stinct ideas to accompany the hand and eye. 8 lution comes out by a sort of magic or legerdemain, we u ua cannot tell either how or why. We therefore thought XL Χλ 8 0.43429 8 it our duty to furnish the reader with distinct conceptions or (putting M for 0.43429, the modulus or subtangent of the things and quantities treated of. For this reason, after showing, in Sir Isaac Newton's manner, how the u* of the common logistic curve)= spaces described in the retarded motion of a projectile Vol. XVII. Part II, + This VV VV and s = vv+c. Now the fluent of VU -UV uv e ac Х The so 3 F is 14850, and 323,62 8 2gs n 61 and 13,897 ry u This equation establishes the relation between the 1. Let the height h be sought which will produce the space fallen through, and the velocity acquired by the velocity 323,62 the terminal velocity of the ball being fall. We obtain by it =L and 689,34 =L or, which is still more conveni- 0,46947, which is the sine of 28o. The logarithmic ! MX2g S secant of this arch is 0,05407. Now M or 0,43429 : ent for us, that is, equal to 0,05407=14850 : 1848, the height wanted. us u*_02 the logarithm of a certain number: therefore having falling 1848 feet. Say. 14850 : 1848 = 0,43429 2. Required the velocity acquired by the body by found the natural number corresponding to the fraction 0,05407. Look for this number among the logarithMx 26S, consider it as a logarithm, and take out the mic secants. It will be found at 28°, of which the logarithmic sine is 9.67161 number corresponding to it: call this n. Then, since Add to this the log. of u 2.83844 us n is equal to we have nunv=u', 2.51005 and nu?-H= n v*, or n vi = u x n - v, and ve is the logarithm of 323,62, the velocity required. observe, from these solutions, that the acu? X – quired velocity continually approaches to, but never equals, the terminal velocity. For it is always expresTo expedite all the computations on this subject, it sed by the sine of an arch of which the terminal velociwill be convenient to have multipliers ready computed ty is the radius. We cannot help taking notice here Errones for MX2g, and its half, of a very strange assertion of Mr Muller, late professor assertion of viz. 27,794, whose log.is 1.44396 of mathematics and director of the royal academy at Mr Male. 1.14293 Woolwich. He maintains, in his Treatise on Gunnery, But v may be found much more expeditiously by his Treatise of Fluxions, and in many of his numerous works, that a body cannot possibly move through the observing that is the secant of an arch ulja air with a greater relocity than this; and he makes this of a circle whose radius is u, and whose sine is v, or a fundamental principle, on which he establishes a theo of motion in a resisting medium, which he asserts whose radius is unity and sine = : therefore, consi- with great confidence to be the only just theory; say. ing, that all the investigations of Bernoulli, Euler, Rodering the above fraction as a logarithmic secant, look bins, Simpson, and others, are erroneous. We use this for it in the tables, and then take the sine of the arc of strong expression, because, in his criticisms on the works wbich this is the secant, and multiply it by u; the pro- of those celebrated mathematicians, he lays aside good duct is the velocity required. manpers, and taxes them not only with ignorance, but We shall take an example of a ball whose terminal with dishonesty; saying, for instance, that it required velocity is 689} feet, and ascertain its velocity after a no small dexterity in Robins to confirm by bis experifall of 1848 feet. Here, ments a theory founded on false principles; and that u’=475200 and its log. = 5.67688 Thomas Simpson, in attempting to conceal bis obligau =6891 2.83844 tions to him for some valuable propositions, by chang=32 1.50515 ging their form, had ignorantly fallen into gross errors. S =1848 3.26670 Nothing can be more palpably absurd than this asserThen log. 27,794 + 1.44396 tion of Mr Muller. A blown bladder will bave but a + 5.26670 small terminal velocity; and when moving with this log. u* 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. Log. of 0,10809= log. n 9.03378 Were the assertion true, it would be impossible for a 0,10809 is the logarithm of 1,2826= n, and 'n-I= portion of air to be put into motion through the rest, tư X năI for its terminal velocity is nothing. Yet this author 0,2826, and = 323,6", = U, and v= makes this agsertion a principle of argument, saying, that it is impossible that a ball can issue from the mouth In like manner, 0,054045 (which is half of 0,10809) Robins and others are grossly mistaken, when they give of a cannon with a greater velocity than this; and that will be found to be the logarithmic secant of 28°, whose them velocities three or four times greater, and resistsine 0,46947 multiplied by 6891 gives 324 for the ve ances wbich are 10 or 20 times greater than is possible; The process of this solution suggests a very perspicu. smaller resistances, be confirms his theory by many exo and by thus compensating bis small velocities by still ous manner of conceiving the law of descent; and it may be thus expressed : periments adduced in support of the others. No reaM is to the logarithin of the secant of an arch whose son whatever can be given for the assertion. Newton, or perhaps Huygens, was the first who observed that sine is 7, and radius 1, as aa is to the height through communicate to a body; and this limit was found by there was a limit to the velocity which gravity could which the body must fall in order to acquire the velo. his commentators to be a term to which it was vastly city v. Thus, to take the same example. convenient to refer all itother motions. It therefore log. s n 323,6. locity. became |