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small degree of windage. Indeed with the usual established windage only, namely, about th of the calibre, no less than between and of the powder escapes and is lost. And as the balls are often smaller than that size, it frequently happens that the powder is lost by unnecessary windage.
Sthly. It appears that the resisting force of wood, to balls fired into it, is not constant. And that the depths penetrated by different velocities or charges, are nearly as the logarithms of the charges, instead of being as the charges themselves, or, which is the same thing as the square of the velocity.
9thly. These, and most other experiments, show, that balls are greatly deflected from the direction they are projected in; and that ⚫ so much as 300 or 400 yards in a range of a mile, or almost th of the range, which is nearly a deflection of an angle of 15 degrees.
10thly. Finally, these experiments furnish us with the following concomitant data, to a tolerable degree of accuracy; namely, the dimensions and elevation of the gun, the weight and dimensions of the powder and shot, with the range and time of flight, and first velocity of the ball; from which it is to be hoped, that the measure of the resistance of the air to projectiles may be determined, and thereby the foundation be laid for a true and practical system of gunnery, which may be as well useful in service as in theory; especially after a few more accurate ranges are determined, with better and larger balls than some of the last employed on the foregoing ranges.'
In the year 1789, experiments were made to afford a comparison between guns of different lengths. The results are tabulated, and the following inferences drawn from the whole.
First. By comparing the first-mentioned velocities by the short 3-pounder, with the velocities of the long one in this table, loaded with one pound of powder, at different distances, it may be perceived that the superior velocity with the long gun, which was found to be near greater, is reduced to an equality with the short one, during the flight of the ball through only 20 feet, or less than 77 yards; and as the length of these guns are very nearly in the proportion of 2 to 1, which is as great a difference as ever occurs in service between any two guns of equal calibre, it fully accounts for the small advan tage obtained in the ranges of shot, by any increasd length which the limits of practice will admit of: and also how very subject to error any decision must be, in determining the velocities corresponding with a certain length of gun, which is founded on the extent of their respective ranges; since the irregularities in the shots' flight, added to the last mentioned circumstance, must render them very uncertain criteria, in all cases where great velocities are concerned.
2dly. This table also affords a further confirmation of the small advantage, in point of range, which is obtained by increasing the charge, beyond what is necessary to communicate a certain velocity to the ball; since the increased resistance to great velocities operates so powerfully, that they are quickly reduced, and soon destroyed: for example, it appears by the table, that the velocity communicated by 16 ounces of powder, after the shot has passed through a space of
230 feet, is reduced to nearly the same with that of 12 ounces of powder, in a flight of 30 feet. It may also be observed, that the velocity with 24 07. of powder, at 180 feet distance, and that with 16 oz. at 30 feet distance, are nearly equal; differing only by 4 feet per second, though at their first discharge from the piece they differ by as much as 149 feet per second.
3dly. From the foregoing table, it is evident also, that the velocities communicated by different quantities of powder, are nearly in the proportion of the square roots of those charges.-Also, by a due computation from the quantity of velocity lost in the several distances, the resistance of the air to the ball of 2.78 inches diameter, moving with several velocities, will be nearly as expressed in the foregoing table, in p. 126, where the first column shows the velocity per second, with which the ball moves, and the other columus, show the corresponding resistances of the air, in ounces or pounds; that is, when a ball of that size moves with a velocity of suppose 1700 feet per se. cond, it is resisted by the air with a force which is equal to the weight or pressure of 2472 ounces, or 154 pounds; and so of the rest.
From this table of resistances it appears also, that there is a gra dual and regular increase of resistance, as the velocity is increased, from the least to the greatest, and without showing the appearance of such a very sudden or abrupt change in the nature and quantity of that resistance, as Mr. Robins suspected might obtain. But that the law of resistance gradually and slowly increases like the velocity itself, probably on account of the increasing partial vacuum behind the ball in its flight, from the slowest motion, when the resistance changes as the square of the velocity nearly, up to about the velocity of 1200 or 1400 feet, when, the vacuum being completed, the law of increase appears to have attained its highest pitch, being then nearly as the 2 power of the velocity; after which it gradually decreases again more and more, as the velocity increases higher, till it arrive at about the 22 power, and perhaps still lower; which, among several others, is a law that was unknown till it was discovered by means of these experiments.' Vol. III. pp. 129, 130.
For many other highly important and curious results, we must refer the philosophical reader to the work itself.
The thirty-fifth Tract contains the description and use of a new gunpowder eprouvette. The principle of this machine is remarkably simple, being nothing more than a small gun or mortar suspended on an axis, which, being charged with a small quantity of powder, without ball, and fired, the quality or strength of the powder is inferred from the length of the arch through which the gun recoils. The idea of such an instrument was originally suggested by Mr. Robins; but his principle, simple as it is, does not seem to have been correctly acted upon till Dr. Hutton produced the eprouvette, of which the following is the description.
This machine may be described as consisting of either a small brass mortar, or gun, suspended by a metallic stem or rod, turning by an axis on a firm and strong frame, by means of which the piece
oscillates in a circular arch. A little below the axis, the stem divides. into two branches, reaching down to the gun or mortar, to which the lower ends of the branches are firmly fixed, the one near the muzzle, and the other near the breech of the piece. The upper end of the stem is firmly attached to the axis, which turns very freely by its extremities, in the sockets of the supporting frame; by which means the gun and stem vibrate together in a vertical plane, with a very small degree of friction. The piece is charged with a small, but proper quantity, of the powder to be proved, without any ball, and then fired; by the force of which the piece is made to recoil or vibrate, describing an arch or angle, which will be greater or less, according to the quantity or strength of the powder.
To measure the quantity of this recoil or vibration, and consequently the strength of the powder, a circular brazen or rather silvered arch, of a convenient extent, and of a radius equal to its distance below the axis, is fixed against the descending two branches of the stem, and graduated into divisions, according to the purpose required to be answered by the machine, viz. into equal parts, if we would know only the angle of vibration, as measured by the simple equal degrees of a circle; or into unequal parts according to the chords, or to the versed sines of the arcs, to measure either the velocity of the vibration, or the force and strength of the powder: the arch in my instrument had all those three scales of divisions on it. The divisions in these scales, answering to the angle of any recoil, are pointed out by a concentric index, fitted on the axis of vibration, by means of a round hole or socket, with which it embraces pretty closely the round part of the axis of the stem, but capable of being turned easily about it by the hand. By means of a spring, the round end or socket of this instrument is pressed sideways, along the direction of the axis, always moderately tight against the socket of the stem, which is firmly brazed to the same axis; thus connecting the index and the stem slightly together; by which means, these two always vibrate in conjunction with the arch, unless when the index is stopped by some obstacle. When the machine is at rest, and the, index brought to point to the beginning of the divisions on the arch, an additional piece fixed on the index bears agains a stop-bar, fixed across the frame of the machine; so that, when the powder is fired, the gun and arch together vibrate backwards, leaving the index at rest, bearing still against the stop, and the divisions of the arch passing by it, till the gun has recoiled to the utmost extent that the force of the explosion can impel it: then, returning again, it brings the index along with it (because of their friction in consequence of the pressure of the spring) still pointing to the proper recoil division on the arch, showing the extent of the vibration; which, on gently stopping the vibrations, is easily read off, and noted down.
The circumstances which are peculiar to this eprouvette, and by which it differs from all others, as far as known to me, are as follow. 1st. The convenient manner of placing the arch, which measures the recoil, below the axis of the machine. 2nd. The divisions on this arch being made, not only according to equal degress, but also according to the chords and versed sines of the recoil, by which the
true proportions of the velocities of balls, or the strength of powder, is shown. 3rd. The manner of applying the index, making it bear with a gentle pressure against the side of the socket of the stem, by means of a spring, and fixing it by a stop, while the gun and arch make the first or greatest vibration backwards.'
A more particular account of the construction and use of this ingenious contrivance, illustrated by engravings, may be seen in the Tract from which the above is quoted.
The tract next in order contains an account of a most valuable series of experiments made with the whirling machine, in the years 1786, 1787, and 1788, to determine the resistance of the air. In the experiments with the ballistic pendulum, the resistance of the air to balls moving through it, is determined, with considerable accuracy, for all velocities from 2000 down to about 300 feet per second. Lower than this latter limit, experiments of that kind could not be carried; because with such velocities it was found that the ball could scarcely ever be made to lodge in the block, but rebounded from it, and defeated the experiment. To obtain the resistance to lower velocities, this indefatigable experimenter had recourse to the whirling machine; and had the satisfaction to find that the resistances deduced from the two distinct classes of experiments formed one regular, orderly, and unbroken series; as much so, as they could possibly have done, had all the experiments from the highest to the lowest velocities been performed by means of one and the same contrivance. This is a remarkable proof of the accuracy of the whole. It appears from the experiments generally, that if d be the diameter of a ball in inches, and the velocity with which it moves in feet, then will the resistance it experiences from the air, in avoirdupois pounds, be expressed by (000007565 v2·00175 v) d2.
But our author did not confine his attention to the resistance experienced by globes merely he also employed flat surfaces, cylinders, cones, and bodies of other figures; and thus ascertained the resistances of the air to bodies of different kinds moving through it. Many of his results are exceedingly cu-rious; but we can only here present the general formula for the resistance to a rectangular plane, with area a, moving with a velocity v, in a path whose angle of inclination to the plane has for its sine s and cosine c. The formula is 03- 2.04 av feet. For water, or any other fluid different from air, the theorem will be varied in the relation cf the density of the fluid to that of air.
The thirty-seventh Tract is on the Theory and Practice of Gunnery, as dependant on the resistance of the air;' and is greatly
improved from the 3d Vol. of the Woolwich Course of Mathematics, where it was originally published.
The thirty-eighth or last tract contains a variety of mathematical problems, serving for the illustration and practice of the principles which have been established or developed in the former parts of these volumes. Among these there are two amusing problems relative to the division of the circle into any number of parts; one by means of concentric circles, the other also by means of circles, but so as to cause the several parts to be eq both in surface and in perimeter. The history of these problems is terminated with the following curious specimen of the manner in which a celebrated Northern Professortreats his friends.
Finding the two constructions introduced, by my friend Mr. Leslie, the ingenious and learned mathematical professor in the university of Edinburgh, into the first edition of his Geometry, published in 1809, both together in pages 222 and 223: as these blems were rather of an uncommon nature, I did think some mention might have been made of their origin, or the circumstances that have attended them; and I hinted as much to my ingenious friend. In consequence of which, probably, I find that the learned author has, in the 2nd edition of his work, separated those two constructions, placing one among the elements at p. 181, and the other among the notes at p. 432. accompanied with the note, than it was the result of a principle briefly suggested by Mr. Lawson, and afterwards explained and demonstrated in Dr. Hutton's Mathematical Tracts.' This change and announce seemed to make the matter rather worse than before, as it appeared less unfriendly, or less uncivil, to omit noticing a fact entirely, than to mis-state it. For, certain it is, that Mr. Lawson never suggested any principle or extension, nor any mode of solution whatever; the discovery having been made and published by myself alone.'
We have no inclination to make any remarks upon such a story as this. Mr. Leslie alone can furnish the proper explanation; and we trust he will think his character with the public of sufficient moment to offer it speedily.
We feel rather inclined to apologize for the shortness, than for the length of this article. Three volumes written by so distinguished a mathematician and philosopher as Dr. Hutton, and in every way worthy of him, can never be despatched in haste In truth, we have read them with much deliberation; and, we need not be ashamed to say, with great pleasure and improvement. The venerable author says, as this is, in all probability, the last original work that I may ever be able to offer to the notice of the public, I am the more anxious that it should be found worthy of their acceptance and regard.' His desire, as far as our judgement goes, is fully realized; and we conclude