out the rectangle BDO G. 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 CD B; and all that we had to take care of was, that when O C, Or, Og, were taken in geometrical progression, Ms, M t, should be in arithmetical progression. The abscissae having ordinates equal to ps, n t, &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 N I would also have been doubled, and our proportions would Thave 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. Since then we have tables of logarithms calculated for every number, we may make use of them instead of this geometrical figure, which still 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 corresponds to the side or O of the square trox 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 vis1, or the unit of numbers to which the logarithms are suited, and then I N is also 1. Now the square 0 it OA being unity, the area B A CD will be some number; it O being also unity, OD is some number: call it r. Then, by the nature of the hyperbola, O B : O T = T 0: D B ; that calling D di the area, B D d b, which is the fluxion (ultimately) of the hyperbolic area, is +. Now in the curve Lp K, M I has the same ratio to N I that B A C D has to 0X () or . Therefore, if there be a scale of which NI is the unit, the number on this scale corresponding to M I has the same ratio to 1 which the number measuring B A CD has to 1; and I i, which corresponds to BD d b, is the fluxion (ultimately) of M I ; Therefore, if M I be called the logarithm of 1,– ar is properly represented by the fluxion of M.I. 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 called logarithms are just the lengths of the dif- ferent questions which may be proposed: Recol2. lect that the rectangle ACOF is - 2a, or +, or E, for a ball of cast iron one inch diameter, 2 and, if it has the diameter d, it is *d. or 2 a d, & 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 : MI = AC OF : B DC A; Row NI is the subtangent of the logistic curve; MI is the difference between the logarithms of Q D and O C ; that is, the difference between the logarithms ofe + t and e : A COF is 2 ad, Let the ball be a twelve pounder; the initial velocity 1600 feet, and the time twenty seconds. We must first find e, which is 2 a.d. This must be considered as a common number Therefore + 3.68557 + 9-94.490 – 9:63778 =. 3.99.269 The first column shows the time of the motion; the second the space described; the third the differences of the spaces, showing the motion during each successive seconds; the fourth the velocity at the end of the time t; and the last the differences of velocity, showing its diminution in each successive second. 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. 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 v for those of e-H4 and e ; for A C : This reduction will be produced in about seveneighths of a second. III. To determine the time which a ball, beginning to move with a certain velocity, employs in passing over a given space, and the diminution of velocity which it sustains from the resistance of the air; proceed thus:– 2 ad : S= 0:43429 : log e-Ft =t. Then to log. e *: add log. e, and we obtain log. e-H t, and e-H t , from which if we take e we have t. Then, to find v, say e--t: e = V : v. These examples may be concluded by applying this last rule to Mr. Robins's experiments on a musket bullet of three-fourths of an inch in diameter, which had its velocity reduced from 1670 to 1425 by passing through 100 feet of air. This we do to discover the resistance which it sustained, and compare it with the resistance to a velocity of one foot per second. We must first ascertain the first term of our analogy. The ball was of lead, and therefore 2 a must be multiplied by d and by m, which expresses the ratio f the density of lead to that of cast iron, d is = 0.763, and its logarithm = 9.88252, which, added to 0-03408, gives 9-91660, which is the log, of e-H t, = 0.825, from which take e, and - -- 62 there remains t = 0"-062, or 1000 for the time of passage. Now, to find the remaining velocity, say 825: 763 = 1670: 1544, = y. But in Mr. Robins's experiment the remaining velocity was only 1425, the ball having lost 245; whereas by this computation it should have lost only 126. It appears, therefore, that the resistance is double of what it would have been if the resistance increased in the duplicate proportion of the velocity. Mr. Robins says it is nearly triple. But he supposes the resistance to slow motions much smaller than his own experiment, so often mentioned, fully warrants. The time e in which the resistance of the air would extinguish the velocity is 0".763. Gravity, or the weight of the bullet, would have done it in “or so therefore the resistance is “ or 52 ; therefore the resistance is 0-763 times, or nearly sixty-eight times its weight, by this theory, or 5.97 pounds. If we calculate from Mr. Robins's experiment, we must say log. 630-23 *. 52 and e = 1670 T 0°-3774, and 0.3774 gives 138 for the proportion of the resistance to the weight, and makes the resistance 12:07 pounds, fully double of the other. With this velocity, which greatly exceeds that with which the air can rush into a void, there must be a statical pressure of the atmosphere equal to six pounds and a half. This will make up the difference; and allows us to conclude that the resistance, arising solely from the motion communicated to the air, follows very nearly the duplicate proportion of the velocity. The next experiment, with a velocity of 1690 feet, gives a resistance equal to 157 times the weight of the bullet, and this bears a much greater proportion to the former than 1690 does to 1670°; which shows that, although these experiments clearly demonstrate a prodigious augmentation of resistance, yet they are by no means susceptible of the precision which is necessary for discovering the law of this augmentation, or for a good foundation of practical rules; and it is still greatly to be wished that a more accurate mode of investigation could be discovered. We have thus explained, in 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 of a second, observed, and our observation has been confirmed by persons well versed in such subjects, that in all cases where the fluxionary process introduces the fluxion of a logarithm, there is a great want of distinct ideas to accompany the hand and eye. The solution comes out by a sort of magic or legerdemain, we cannot tell either how or why. We therefore thought it necessary to furnish the reader with distinct conceptions of the things and quantities treated of. For this reason, after showing, in Sir Isaac Newton's manner, how the spaces described in the retarded motion of a projectile followed the proportion of the hyperbolic areas, we showed 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 describing the abscissa M I of this curve moves precisely as the projec tile does. Then, discovering that this line is the . same with the line of logarithms on a Gunter's scale, we showed how the logarithm of a number really represents the path or space described by the projectile. SECT. V.-Of the PERPEND1cu LAR Ascents AND DEscENTs of HEAvy PRojectiles. Having thus enabled the reader to conceive distinctly the quantities employed, we shall leave the geometrical method, and prosecute the rest of the subject in a more compendious manner. We are next to consider the perpendicular ascents and descents of heavy projectiles, where the resistance of the air is combined with the action of gravity: and we shall begin with the descents. Let u, as before, be the terminal velocity, and g the accelerating power of gravity: when the body moves with the velocity u, the resistance is equal to g; and in every other velocity v, we g vo the hyperbolic logarithm of the quantity annexed to it, and A may be used as to express its common logarithm. See article Fluxions. The constant quantity C for completing the fluent is determined from this consideration, that the space described is 0, when the velocity is o : therefore, considering the above fraction as a logarithmic secant, look for it in the tables, and then take the sine of the arc of which this is the secant, and multiply it by u ; the product is the velocity required. An example may be given of a ball whose terminal velocity is 689 feet, and ascertain its velocity after a fall of 1848 feet. Here, is the logarithm of 323:62, the velocity required. From these solutions we see that the acquired velocity continually approaches to, but never equals, the terminal velocity. For it is always expressed by the sine of an arch of which the terminal velocity is the radius. The motion of a body projected downwards next merits consideration. While the velocity of projection is less than the terminal velocity, the motion is determined by what we have already said; for we must compute the height necessary for acquiring this velocity in the air, and |. the motion to have begun there. But, if the velocity of projection be greater, this method fails. We pass it over (though not in the least more difficult than what has gone before) because it is of mere curiosity, and never occurs in any interesting case. We may just observe that, since the motion is swifter than the terminal velocity, the resistance must be greater than the weight, and the motion will be retarded. The very same process will give us for the space the logarithmic sines, and take the logarithmic cotangent of the half complement of the corresponding arch. This, considered as a common number, will be the second term of our proportion. This is a shorter process than the former. |