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1735 to 1774, when the operations at Mount Shichallin were commenced. These were conducted with great judgment and precision, and the result was that the plumb-line, at two stations on opposite sides of the mountain, experienced a deflection of 11.6, the half of which, or 5.8, is the mean effect of the attraction of the mountain. Thus were philosophers furnished with the finishing step of the analysis which firmly established the doctrine of universal gravitation. But the observations at Shichallin were calculated to furnish another result, nearly as important in its consequences, and much more difficult in the attainment,---namely, the mean density of the earth. Dr. Maskelyne terminated his paper on Mount Shichallin, with a few gross conjectures on this subject; but even that able astronomer and mathematician, with all the acuteness, ingenuity, and perseverance which he was known to possess, shrunk from the task. He had surrounded the mountain in all directions; but he dare not venture to "cut through it" and anatomize it, in the way · that was requisite to obtain this important result. After a most curious investigation of rules, and the performance of some thousand computations, Dr. Hutton has determined, as shown in this tract, that the mean density of the earth is 23 or almost 5 times that of water and it must not be forgotten in the history of sciences (though some men, especially foreigners, have already ascribed this honour to another) that Dr. H. was the first philosopher who exhibited this curious result.
To what useful purposes the knowledge of the mean density of the earth, as above found, may be applied, it is not necessary here to show. I shall therefore conclude this tract with a reflection or two on the premises that have been delivered. Sir Isaac Newton thought it probable, that the mean density of the earth might be 5 or 6 times as great as the density of water; and we have now found, by experiment, that it is very little less than what he had thought it to be: so much justness was there even in the surmises of this wonderful man! Since then the mean density of the whole earth is about double that of the general matter near the surface, and within our reach, it follows, that there must be somewhere within the earth, toward the more central parts, great quantities of metals, or such like dense matter, to counterbalance the lighter materials, and produce such a considerable mean density on the whole. If we suppose, for instance, the density of metal be 10, which is about a mean among the various kinds of it, the density of water being 1, it would require 16 parts out of 27, or considerably more than one-half of the matter in the whole earth, to be metal of this density, in order to compose a mass of such mean density as we have found the earth to possess by the experiment: or, or between and of the whole inagnitude will be metal; and consequently, or nearly of the diameter of the earth, is the central or metalline part, But if the metalline matter be chiefly iron, which as far as we know is by much the most predo
minant metal, then the half of the whole terrestrial magnitude would be the bulk of the ferruginous matter.
Another inference that readily occurs, is this: viz. that thus knowing the mean density of the earth in comparison with water, and the densities of all the planets relatively to the earth, we can now assign the proportions of the densities of them all, as compared to water, after the manner of a common table of specific gravities. And the numbers expressing their relative densities, in respect of water, will be as here annexed, supposing the densities of the planets, as compared to each other, to be as laid down in Mr. De la Lande's astronomy. Water 1; the sun 11; Mercury 101; Venus 6; the earth 5; Mars 3; the moon 3; Jupiter 1; Saturn Vol. II. pp. 65, 66.
The twenty-seventh tract contains investigations to determine at what point in the side of a hill its attraction will be the greatest. They constitute the substance of a paper read before the Royal Society in November, 1779. The general result of the inquiry is,
That at of the altitude, or very little more, is the best place for observation, to have the greatest attraction, from a hill in the form of a triangular prism, of an indefinite length. But when its length is limited, the point of greatest attraction will descend a little lower; and the shorter the hill is, the lower will that point descend. For the same reason, all pyramidal hills have their place of greatest attraction a little below that above determined. But if the hill have a considerable space flat at the top, after the manner of a frustum, then the said point will be a little higher than as above found. Commonly, however, 4 of the altitude may be used for the best place of observation, as the point of greatest attraction will seldom differ sensibly from that place.' Vol. II. p. 67.
A very valuable tract follows, on the subjects of cubic equations and infinite series. It contains many excellent remarks on the nature of cubic equations, and many ingenious rules for facilitating the solution in certain cases. Methods are also described of obtaining good approximations to the roots, by means of infinite series; and then, conversely, the author shows how to sum a great variety of such series, by means of the roots of certain cubic equations. We recommend to particular notice the observations at pages 91, 92, tending to explain the reason why Cardan's rule (as it is usually called) always gives the equation in an imaginary form when the equation has no imaginary roots, but in the form of a real quantity when it has imaginary roots; a kind of paradox with which many students are perplexed on their entrance upon this department of algebra.
Tract the twenty-ninth contains a project for a new division of the quadrantal arc of a circle, with a view to the construction of tables of sines, tangents, and secants of arcs, to equal parts of the radius of the circle. Several curious formula for the calculations are here given, as in the Philosophical Transactions
for 1784, where the paper was first published. The author, it appears, had proceeded a good way in the computation of tables. according to this plan, when the attempts of the French to introduce tables according to the centesimal division of the quadrantal arc, prevented the completion of his purpose.
The thirtieth tract is on the sections of spheroids and conoids, including demonstrations of these propositions: 1st, that all such plane sections are the same as conic sections; 2dly, that all the parallel sections in every such solid, are like and similar figures. The thirty-first relates to the comparison of curves of the same species, and shows their mutual relations; and is succeeded by one containing a useful theorem for the cube root of an algebraic binomial, one of the terms being a quadratic radical.
In tract the thirty-second we are presented with a very elabo rate, curious, and instructive history of algebra; being much enlarged and improved from Dr. Hutton's valuable article ALGEBRA, in his Mathematical and Philosophical Dictionary. Here, the history of this interesting and useful branch of mathematical science is traced, from its probable origin, through its practice and successive improvements among the ancient Greeks, the Indians, Persians, and Arabians; after which the discoveries and improvements of the principal European authors are successively developed. The whole is carried through with great perspicuity, and, which is also no small recommendation, with much impartiality. Among other very interesting particulars in this tract, the scitentific student will be much entertained and instructed by the account of Diophantus's Algebra, of the Bija Ganita, a work translated from the Hindu into the Persian language about 1634, and of the Lilawati, another Indian work on algebra, translated into the Persian about 1585. The controversy between Tartalea and Cardon on the subject of cubic equations, is also highly amusing and instructive.
The thirty-fourth tract occupying 78 pages in Vol. II. and 152 pages in Vol. III, contains an account of the author's valuable experiments carried on at Woolwich in the years 1775, 1783, 1784, 1785, 1787, 1788, 1789, and 1791, for the purpose of deducing new rules tending to improve the theory and practice of gunnery Some of these have been already published in the Philosophical Transactions, and the Doctor's quarto Tracts; but above 70 pages, detailing the experiments for the four last years above specified, and the deductions from them, have not been printed before.
The objects of this course have been various. But the principal articles of it are as follow: (1.) The velocities with which balls are projected by equal charges of powder, from pieces of the same weight and calibre, but of different lengths. (2.) The velocities with dif
ferent charges of powder, the weight and length of the gun being the same. (3.) The greatest velocity due to the different lengths of guns; to be obtained by increasing the charge as far as the resistance of the piece is capable of sustaining. (4) The effect of varying the weight of the piece; every thing else being the same. (5.) The penetration of balls into blocks of wood. (6.) The ranges and times of flight of balls, with the velocities by striking the pendulum at various distances: to compare them with their initial velocities, for determining the resistance of the medium. (7.) The effect of wads; of different degrees of ramming, or compressing the charge: of different degrees of windage; of different positions of the vent; of chambers, and trunnions, and every other circumstance necessary to be known for the improvement of artillery.' Vol. II pp. 311, 312.
In these experiments the two principal things to determine were, the actual velocity with which a ball was projected from the mouth of the piece, and the velocity which it possessed at any given distance from it. The first of these particulars it was thought might be ascertained by the recoil of the gun; but, after many accurate experiments, it was found that this method was not to be relied upon, that is, that the effect of the inflamed powder on the recoil, was not exactly the same when it was fired without a ball, as when it was fired with one; other means, therefore, were employed to ascertain this point. The second particular was determined by means of the vibrations of the ballistic pendulum, on being struck at different distances from the muzzle of the piece whence the respective balls were fired. To give the requisite precision to these experiments with the ballistic pendulum, several preliminary investigations were necessary: the Doctor has conducted them with great skill and precision; and finds, that no sensible error is occasioned, either by the friction on the axis of the pendulum, or by the resistance of the air to the back of the pendulum itself, or by the circumstance of the ball's penetration into the vibrating block. The investigation shows clearly, and it is a curious circumstance, that the velocity of the pendulum is the same, whatever be the resisting force of the wood, and therefore to whatever depth the ball penetrates, and the same as if the wood were perfectly hard, or the ball made no penetration at all.'
We are not able to quote so largely from this most interesting part of the Tracts, as the importance of the subjects may seem to demand; yet we cannot forbear presenting some of Dr. H.'s most important results. At the end of the experiments of 1786 we have the following:
1st. It may be remarked, that the former law [i. e. that deduced from the experiments of 1775] between the charge and velocity of ball, is again confirmed, namely, that the velocity is directly as the square root of the weight of powder, as far as to about the charge of 8
ounces and so it would continue for all charges, were the guns of an indefinite length. But as the length of the charge is increased, and bears a more considerable proportion to the length of the bore, the velocity falls the more short of that proportion.
2nd. That the velocity of the ball increases with the charge, to a certain point, which is peculiar to each gun, where it is greatest; and that by further increasing the charge, the velocity gradually diminishes, till the bore is quite full of powder. That this charge for the greatest velocity is greater as the gun is longer, but not greater however, in so high a proportion as the length of the gun is; so that the part of the bore filled with powder bears a less proportion to the whole in the long guns, than it does in the shorter ones; the part of the whole which is filled being indeed nearly in the reciprocal subduplicate ratio of the length of the empty part. And the other circum. stances are as in this tablet.
3dly. It appears that the velocity continually increases as the gun is longer, though the increase in velocity is but very small in respect to the increase in length, the velocities being in a ratio somewhat less than that of the square roots of the length of the bore, but somewhat greater than that of the cube roots of the length, and is indeed nearly in the middle ratio between the two.
4thly. It appears from the table of ranges in Art. 121, p. 76, that the range increases in a much less ratio than the velocity, and indeed is nearly as the square root of the velocity, the gun and elevation being the same. And when this is compared with the property of the velocity and length of gun in the foregoing paragraph, it appears that we gain extremely little in the range by a great increase in the length of the gun, the charge being the same. And indeed the range is nearly as the 5th root of the length of the bore; which is so small an increase, as to amount only to about 4th part more range for a double length of gun.
5thly. From the same table in Art. 121, it also appears, that the time of flight is nearly as the range; the gun and elevation being the
6thly. It appears that there is no difference caused in the velocity or range, by varying the weight of the gun, nor by the use of wads, nor by different degrees of ramming, nor by firing the charge of powder in different parts of it.
7thly. But a very great difference in the velociy arises from a