nious contrivances formerly communicated to the Public. Dr. Hutton's theorem approximates, with great eafe, any indefinite poffible root whatfoever, whether its index be rational, irrational, fractional, fimple, compound, &c. N being the given number, n the given index, and a an affumed quantity nearly equal to the n + 1. N+ n — 1.an Xa=VN. For the

required root, then

n- 1. N+ n + 1. an

x

investigation and demonftration of this we must refer our Readers to the work, fince any abridgment of it would be unintelligible. The fifth tract fhews a method of finding in general and finite terms, the near values of the roots of high equations whofe figns alternately as - px”-1 + qx" are + and +9x"-2-rx"-3 &c.=0. Though we are already poffeffed of feveral methods for determining the roots of higher equations, yet in many respects this is better than the old ones. It elucidates the nature of equations in general, and difplays feveral properties of numbers, which, befide their curiofity, may be of fingular ufe in many particular cafes.

The fixth is a demonftration of the truth of the Newtonian binomial theorem in the general cafe of fractional exponents. The Author fhews the infufficiency of all former demonftrations of this theorem on account of the feries which is ufually affumed, viz. 1.+ ax + bx2 + cx2 &c. being of too limited a nature, where the form of it with refpect to the powers of x had been taken for granted. Doubting therefore the form of the terms themselves, increafing by the regular integral powers of x when the index is fractional, Dr. H. affumes 1+x=1+A+B+C&c. where A, B, C, &c. reprefent the whole of the 2d, 3d, 4th, &c. terms, both coefficients and powers of x, whatsoever they may be. In this tract our Author gives a hiftory of the various demonftrations which feveral mathematicians have given of this ufeful theorem, and fhews that Newton, as is generally fuppofed, was not the first inventor of it, but that it was well known to Briggs and others long before Newton's time. The curious reader will be much entertained with this treatife.

The feventh tract contains the demonftrations of fome curious properties of the fphere, deduced from contemplating the nature of the commcn fections of a fphere and cone. This paper is purely geometrical; and fince the feveral propofitions it contains are intimately connected with, and depend on each other, it cannot be abridged. Ever fince the time of Archimedes, the cone and sphere have been fruitful fources of geometrical fpeculation, and we doubt not but that these will be acceptable to the admirers of ancient geometry,

The eighth is a folution of a problem firft publicly proposed by Mr. Lawion at the end of his Diflertations in the year 1774,

• Το

To divide a circle into any number of parts, which shall be as well equal in area as in circumference.'

The laft, which is the moft important, tract, contained in this volume, is the defcription and refult of feveral experiments inftituted at Woolwich for the fake of improving the theory of gunnery. Any defcription of thefe experiments which we could give would be very imperfect without the plates explanatory of the extenfive apparatus and machinery used in making them. We shall therefore confine ourselves to the bare enumeration of the refults, and the conclufions made from them, only acquainting our Readers that the method for determining the velocity of the ball was twofold; 1ft, by obferving the vibration of a pendulum against which the ball was fired, according to Mr. Robins; and, 2dly, by obferving the recoil of the gun, which was for that purpofe fufpended, with its axis horizontal, on a pendulum. The calculations, for reducing the velocity of the ball to feet in a second of time, are intricate and of curious inveftigation.

From the refults of thefe experiments, it is concluded that the velocity of the ball is directly as the fquare root of the weight of powder, as far as to the charge of 8 oz. and this proportion would hold good were the gun of an indefinite length; but as the length of the charge bears a greater proportion to the length of the gun, the velocity is decreafed, and confequently falls fhort of the proportion above mentioned. It appears, that the velocity of the ball increases with the quantity of the charge, until that quantity arrive at a certain point peculiar to each gun, where the velocity will be the greateft; and if the charge be increased beyond this point, the velocity of the ball will be continually decreafed until the bore be quite full of powder.

The charge for producing the greatest velocity of the ball (the determination of which is moft undoubtedly a matter of the utmoft confequence in gunnery) is greater as the gun is longer, but not in the direct proportion of the length of the gun; for the part of the gun filled with powder bears a lefs proportion to the whole length of the bore in long guns than it does in shorter guns. The greatest velocity is given to the ball, when the length of the charge (i. e. the part of the bore filled with powder) is in the reciprocal fubduplicate ratio of the length of the empty

part.

An increase of the length of the gun increases the velocity of the ball; the velocities being in a lefs ratio than that of the fquare roots of the lengths, and greater than that of the cube roots of the lengths, Dr. H. therefore takes the mean ratio between the two.

With refpect to the range, it is found to increase in a much lefs ratio than the velocity, or, the gun and elevation being the

fame, nearly in the ratio of the fquare root of the velocity. When this is compared with the ratio of the velocity and length of the gun, it appears that very little is gained in the range by an increafe of the length of the gun, the charge remaining the fame. Dr. H. concludes the range to be nearly as the fifth root of the length of the bore, which is a very fmall increafe, for, according to this ratio, a double length of gun only increases the range about a feventh part of the whole. In making the observations on the ranges, it was found, that the gun and elevation being the fame, the time of flight of the ball was nearly as the range, fo that the time of flight is not much increased by the length of the gun.

The velocity and range were not found to be at all varied by changing 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. The common received opinions, concerning wads and ramming, are therefore to be exploded. We fhould have expected tight ramming would leffen the velocity of the ball, fince a confiderable force must be applied to overcome the refiftance of the wadding and the friction it muft confequently make on the infide of the bore.

The variation of the velocity arifing from the different degrees of windage is very confiderable. It appears that in a windage of sth of the caliber no lefs a quantity than ths of the charge is loft. And fince balls are in general smaller than the intended fize, it frequently happens that one half of the powder is loft by the unneceffary windage. This is furely a circumftance of vaft weight!

It appears that the refiftance of wood, againft which balls are fired, is not always conftant. The depths penetrated by the balls are nearly proportional to the logarithms of the charges, that is to the fquares of the velocities. This is perfectly agreeable to the refiftance of bodies moving in any medium.

Laftly, from the obfervations made on the ranges, it appears that the balls were greatly deflected from the direction in which they were projected, fome of them nearly to an angle of about 15 degrees. The reafon of this is not given.

Such is the general outline of the conclufions made by Dr. Hutton. The experiments have determined many circumstances which may ferve as a bafis for a practical fyftem of gunnery, a work which will require for its true foundation, befide the facts here obtained, a few others that are to form the fubject of a future volume.

Though the fubject of this tract is wholly adapted to the principles of gunnery and projeAiles, yet the nicety of the experiments, and the reafonings on the refults of them, together with the Author's remarks on the various appearances that pre

fented

fented themselves, will be acceptable to the natural philofopher, who will meet with feveral circumftances, that occur in this treatife, which tend to elucidate and explain many of the firft principles of phyfics.

R. m.

ART. IX. Gulielmi Bellendeni, Magiftri Supplicum Libellorum Augufti Regis Magna Britanniæ, &c. De Statu Libri tres. Editio fecunda, longè emendatior. Londini. 8vo. 12s. Boards. Deighton. 1787.

THE

HE fcarcity of books, intrinfically excellent, and once eminently known, is to be afcribed to various causes. The nobleft monuments of human induftry and genius, have fometimes been fuddenly deftroyed by the fury of the elements, by the mean and narrow jealoufy of priefts and politicians, or the blind rage of illiterate and brutal conquerors. They have perifhed under enemies, more ignoble, indeed, but not lets fatal, than the Caliph Omar, and the Emperor Leo Ifauricus. They have filently funk into oblivion, through the negligence of heirs, the prejudices of party, the fickleness of public tatte, the Janguor of literary curiofity, or the petty vanity and felfifhnefs of collectors. While, however, we lament the lofs, which learning may have fometimes fuftained, we muft reflect with fatisfaction on the general good sense of mankind, through the operation of which, many frivolous and pernicious books are forgotten, only because they do not deferve to be remembered. But we mean not to extend the invidious part of this obfervation to the work which is now before us. We were interested by its contents, and we rejoice in its republication.

Bellendenus was mafter of the pleas to James I. He was a man of extenfive reading and refined tafte. He had collected great stores of political knowledge from the writings of Cicero ; and during his refidence at Paris, where, through the bounty of his royal patron, he seems to have enjoyed the otium cum dignitate, he prefented the rich and abundant fruits of his ftudies, in various forms, and at various periods, to the literary world. The Cicero Princeps appeared firft. It was fucceeded by the Cicero Conful. Thefe tracts were afterwards republished by their writer, and the value of the collection was increased by the addition of an elaborate and masterly enquiry into the religion and philofophy of the Prifci Orbis.

The pofthumous work of Bellenden us, De tribus Luminibus Romanorum, is in the hands of fome few fcholars, and we are glad to find, from the infpection of catalogues, that the copies of it have been, of late years, multiplied in this country: but his three books De Statu are feldom * mentioned in the converfation

*There is no complete copy of the work, now republished, either in the Bodleian, or in the King's Library, or in the British Museum. REV. June, 1787.

LI

or

or writings of learned men. Within the circle of our own acquaintance, we have never met with any person, who profeffes to have read them.

The prefent edition is printed with a very neat type, and is faid, by the Editor, to be far more correct than the edition which Bellendenus himself fuperintended in its progrefs through the prefs. It is adorned with beautiful engravings of Mr. Burke, Lord North, and Mr. Fox, to whom the three treatifes are now refpectively dedicated. The collectors of rare or fine books will be eager to purchase this work: the politician, and the man of tafte, will read it with pleafure; and our brethren of North Britain will exult in feeing the precious remains of their illuftrious countryman rescued from obfcurity,-we had almoft faid, from oblivion.

In the little work which Bellendenus himfelf prefixed to his three treatifes, and which he calls "Tractatus de proceffu et fcriptoribus rei politice," there is a rich vein of mafculine tenfe and fervent piety. The origin of our errors in religion, and of our defects in policy and morals, is traced out with confiderable accuracy and learning. But while the Author condemns the monstrous tenets of ancient idolatry, and the grofs corruptions of philofophy, he beftows many juft encomiums on the wifdom and the patriotifm of fome ancient legiflators. He informs us, that, among the Greek theorifts, there is no fyftematic work on the fcience of politics at once comprehenfive in its principles, and applicable to real life. He acknowledges, that much ufeful information may be gathered from the writings of Xenophon, and the fragments of Solon, Charondas, and Zaleucus. On the authority of Cicero, he reprefents Demetrius Phalereus as the firft perfon, who united the practice of politics with a correct and profound knowledge of the art. He allows, however, very great merit to Plato, to Ariftotle, to Theophraftus, and other imitators of Hippodamus, who, it feems, was the first writer on the subject of government without being perfonally concerned in the administration of it. He then speaks with becoming and warm admiration of Cicero, and enumerates the political works of that writer which are come down to us,-thofe that were publifhed by him, but are now loft, and those which he intended to draw up, at the requeft of Atticus.

To preferve the memory of every compofition which flowed' from the pen of Bellendenus, the Editor has inferted an Epitha

*Bellendenus feems to affert this on the teftimony of Cicero, which we fhall confirm by that of Ariftotle:wtos Twv pen) TOXITEDO ενεχείρησε τι περὶ πολιτείας ἐιπῖιν της άριστης. De Rep. 1. 2. cap. 8. In this chapter the political opinions of Hippodamus are stated and difcuffed.