the Indians had no general method for all equations of these two powers, but only depended on their own ingenuity for artfully managing some particular cases of them; for at the conclusion of the above process the author emphatically adds, "the solution of such questions as these depends on correct judgment, together with the assistance of God."

The Beej Gunnit contains several curious specimens of problems in the application of algebra to geometry, from the solutions to which it is evident that the Indians were well acquainted with the chief properties relating to plane geometry in Euclid's Elements. The 47th problem of the first book is cited under the designation of the figure of the bride's chair,' in reference to the similarity of the diagram employed in the Indian mode of demonstration to a palanquin; and in one of the solutions the author of the Beej Gunnit observes, that 'the sum of the sides is always greater than the bypothenuse, by the ass's proposition,' from which it would seem, that the Indians as well as the Europeans have their pons asinorum.

This interesting account of the Indian algebra is followed by a description of the Arabian algebra, abridged principally from Mr. Davis's account of the Khulasat-ul-Hisab, written by Baha-ul-din, who died at Isfahan in the year 1653. We have here some curious particulars respecting the Arabian notation, for which, however, we must refer to the work before us. It is obvious from the whole, that the knowledge of mathematics, and of algebra especially, among the Arabians, was much inferior to that possessed by the Indians: they had no algebraic notation, no abbreviating symbols, no acquaintance with the indeterminate or Diophantine analysis, nor with any thing more than the easiest and elementary parts of the science.

Dr. Hutton next traces the history of algebra among the Italians, beginning with Leonard Bonacci, of Pisa, who about 1229 solved quadratics by completing the square, deriving his rules, and even the double values in the possible case of the equation r+nax, from geometrical considerations. The history is carried on with great research, and so as to furnish an excellent treatise on the science, to the end of the 17th century. As far as it goes it may be characterised as elaborate and satisfactory, and we have only to hope that the same masterly hand will, by selecting and classifying the additions and improvements made by Clairaut, Euler, Landen, Bezout, Waring, Lagrange, Lacroix, &c. bring it down to the termination of the 18th century.

We have now arrived at those tracts which, however interesting and important many of the preceding papers may be, will tend principally to stamp upon this work, in the estimation of the scientific world, its peculiar character of value and excellence; namely, those which relate to the theory and practice of gunnery

and

and the resistance of fluids. They occupy about 400 pages in the 2d and 3d volumes, and have in part appeared before in the Philosophical Transactions, and in the Doctor's quarto tracts, though the greater portion is original. Such of our readers as are at all conversant with the history of mixed mathematics, and especially that branch of it which relates to projectiles, know that the parabolic theory is of no farther use than as it furnishes a set of very elegant constructions and examples for young geometricians; and that, before the time of Robins, no progress, in effect, had been made in the true theory of military projectiles. And even after his valuable work, The New Principles of Gunnery,' had been published, and translated with the addition of a profound and elaborate commentary, by Euler, there still remained much to do in order to bring us acquainted with the real nature of the expansive force of gunpowder, the actual velocities of shot at the commencement of their motion or in different points of their path, the laws of the resistance experienced by balls and shells in their motion, and the true nature of the curve they describe. Borda and others had greatly extended the theory, but principally by means of gratuitous, and as is now known, inaccurate assumptions respecting the resistance of the air. In order, therefore, that this important and intricate department of philosophy might receive some essential improvement, it became desirable that a person possessing an active and ardent mind, with habits of regularity and perseverance, should be so circumstanced as to have both the inclination to enter upon this peculiar investigation, and the means of pursuing it: and this, by a happy coincidence, occurred by the late Duke of Richmond (a man of science and of great public spirit) being master-general of the ordnance just at the period when Dr. Hutton was, with all the zeal and activity of the meridian of life, discharging the duties of the mathematical professorship at the Royal Military Academy.

The mathematical sciences are taught at this institution with a view to their application to military purposes, and particularly to the practice of artillery: and Dr. Hutton was not likely to rest satisfied with affecting to teach, what, in truth, there were no data for teaching properly. He knew that if the doctrine of projectiles were ever to be so exalted as to become an integral part of mathematical science, it must rest upon the basis of well conducted experiment. He therefore began a series so early as the year 1775; and afterward carried on a far more extensive one, under the auspices of the Duke of Richmond (and officially under the direction of General Sir Thomas Blomfield) during the summers of 1783 and of many succeeding years.

The 34th tract contains a minute account of the experiments of every day, with a register of the weather, wind, thermometer, &c. For this we must refer to the tract itself, as well as for a description of the ballistic pendulum and other machinery employed in these experiments. Our limits will barely allow us to quote a few of the most important deductions.

And first, it is made evident by the experiments in 1775, that powder fires almost instantaneously, seeing that nearly the whole of the charge fires, though the time be much diminished.

(2.) The velocities communicated to shot of the same weight, with different quantities of powder, are nearly in the subduplicate ratio of those quantities. A very small variation, in defect, taking place when the quantities of powder become great.

(3.) And when shot of different weights are fired with the same quantity of powder, the velocities communicated to them, are nearly in the reciprocal subduplicate ratio of their weights.

(4.) So that, universally, shot which are of different weights, and impelled by the firing of different quantities of powder, acquire velocities which are directly as the square roots of the quantities of powder, and inversely as the square roots of the weights of the shot, nearly.

(5.) It would therefore be a great improvement in artillery, to make use of shot of a long form, or of heavier matter; for thus the momentum of a shot, when fired with the same weight of powder, would be increased in the ratio of the square root of the weight of the shot.

'(6.) It would also be an improvement, to diminish the windage: for, by so doing, one third or more of the quantity of powder might be saved.

'(7.) When the improvements mentioned in the last two articles are considered as both taking place, it is evident that about half the quantity of powder might be saved, which is a very considerable object. But, important as this saving may be, it seems to be still exceeded by that of the guns: for thus a small gun may be made to have the effect and execution of one of two or three times the weight of its natural ball, or round shot: and thus a small ship might discharge shot as heavy as those of the greatest now made use of.'

Such were the information, and the probable advantages, derivable from the experiments in 1775: they led to the invention of carronades, a species of ordnance which, by means of large balls, and very small windage, produce considerable effects with small charges of powder.

In the description of his second course of experiments, which is carried on after the manner of a journal, occurs one of those touches of goodness and simplicity which we have had frequent occasions to admire in the course of our proceeding.

August 31, 1785. I took out with me, and employed the first class of gentlemen cadets belonging to the Royal Military Academy,

namely,

namely, Messrs. Bartlett, Rowley, De Butts, Bryce, W. Fenwick, Pilkington, Edridge, and Watkins, who have gone through the science of fluxions, and have applied it to several important considerations in natural philosophy. Those gentlemen I have voluntarily offered and undertaken to introduce to the practice of these experiments, with the application of the theory of them, which they have before studied under my care. For, though it be not my academy duty, I am desirous of doing this for their benefit, and as much as possible to assist the eager and diligent studies of so learned and amiable a class of young gentlemen; who, as well as the whole body of students now in the upper academy, form the best set of young men I ever knew in my life; nay, I did not think it even possible, in our state of society in this country, for such a number of gentlemen to exist together in the constant daily habits of so much regularity and good manners; their behaviour being indeed perfectly exemplary, such as would do honour to the purest and most perfect state of society that ever existed in the world: and I have no hesitation in predicting the great honour and future services, which will doubtless be rendered to the state by such eminent instances of virtue and abilities.'

Many of the results of this extensive series of experiments, are extremely important: but we must content ourselves with a very concise summary. After observing that they confirm the deductions from the former course, Dr. Hutton proceeds

'It farther appears also, that the velocity of the ball increases with the increase of charge only to a certain point, which is peculiar to each gun, where it is greatest; and that by farther 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 yet not greater 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 bore in the long guns, than it does in the shorter ones; the part which is filled being indeed nearly in the inverse ratio of the square root of the empty part.

'It appears too, that the velocity, with equal charges, always increases as the gun is longer; though the increase in velocity is but very small in comparison with 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 greater than that of the cube roots of the same, and is indeed nearly in the middle ratio between the two.

'It appears, again, from the table of ranges, that the range increases in a much lower ratio than the velocity, the gun and elevation being the same. And when this is compared with the proportion of the velocity and length of the gun in the last paragraph, it is evident that we gain extremely little in the range by a great increase in the length of the gun, with the same charge of powder. In fact, 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 a 7th part more range for a double length of DD 4 gun.

gun.

From the same table it also appears, that the time of the ball's flight is nearly as the range; the gun and elevation being the same.

It has been found, by these experiments, that no difference is caused in the velocity, or the 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. But that a very great difference in the velocity arises from a small degree in the windage indeed with the usual established windage only, viz. 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 the regulated size, it fre quently happens that half the powder is lost by unnecessary windage.

It appears too, that the resisting force of wood, to balls fired into it, is not constant: and that the depths penetrated by balls, with 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.-Lastly, these and most other experiments, show, that balls are greatly deflected from the direction in which they are projected: and that frequently as much as 300 or 400 yards in a range of a mile, or almost 4th of the range.'

Tract 36th describes a series of extensive and well-conducted experiments upon Robins's whirling machine, to determine the resistance of the air. These, together with those made by firing balls from artillery, constitute a complete and connected series of resistances to balls, from the slow velocities of 5 or 10 feet per second, to the rapid velocities of 1900 and 2000 feet. It ap pears from an examination of the results, that though the resistances are nearly as the squares of the velocities in very slow motions, they are never exactly so. The exponent of the velocity indicating the resistance always exceeds 2. At 200 feet per second that exponent is 2028: at 500 feet it is 2.042: at 1000 feet it is 2.115: from thence it keeps gradually increasing up to the velocity of 1500 or 1600 feet per second, where the exponent is 2153: and from this velocity the exponent gradually diminishes, being 2156 at the velocity of 2000 feet, the limit of the experiments.

He

That the resistance should not be accurately as the square of the velocity, must be evident to every one who attentively reflects upon the subject. But Dr. Hutton has gone farther, and at pp. 221, 222 of the third volume, has very satisfactorily developed the causes of the variable exponent in the ratio of the resistance. has also investigated three or four theorems for the resistance of balls; of which the following appears to be both accurate and convenient in use. Let v be the velocity in feet with which a ball, whose diameter is d feet, moves in air near the earth's surface, then will the resistance in avoirdupois pounds be expressed by the formula (000007565 v2-00175 v) d'.

Dr.