The utility of the other little tract just mentioned, the Deletus Problematum, cannot be doubted. It is a work exactly of the kind that is most wanted as an elementary inftitution in this branch of fcience. The problems are in general well chofen, with ingenious and elegant folutions, laid down ftrictly according to the method of the ancient geometers.

Some remarks, that form a fcholium at the end of the Data, contain an encomium on the geometric analyfis, but tending too much to deprefs the algebraic. This fhould be carefully avoided; and, however fenfible we may be of the great beauty and elegance of the former, and of the valuable effects produced by the study of it on the powers of the mind, we should not forget, that in the most general and difficult fpeculations of the pure mathematics, and in all the most important branches of the mixt, it is the latter only that can be employed to advantage.. An accurate inquiry into the extent of their different provinces, and into the principles on which the difference between the two branches of analyfis depends, are objects that well deferve the attention of mathematicians. Dr Horfley has not touched on that fubject.

One of the tracts in this volume contains the re-invention of a fort of table, known by the name of the Sieve of Eratofthenes, which appears to be no other than a method of finding out the prime numbers. If the contrivance of the Greek geometer was the fame with Dr Horfley's, which we think extremely probable, it was very simple, and confifted in ranging all the numbers, 1, 2, 3, &c. in a table, and effacing from that table, in fucceffion, all the multiples of 2, of 3, of 5, 7, &c.; fo that what remained muft obviously be the prime numbers, or fuch as are not multiples of any other number. This device, though fomewhat ingenious, is fimple and obvious enough; fo that we cannot acquiefce in the very high encomium which Dr Horfley beftows on it. Cribrum igitur Eratofthenis, lector benevole, jam tibi ut fruaris eo, in manus traditum eft, non fictum aliquid aut adulterinum, fed quale ab auctore ipfo olim illud concinnatum effe omnino exiftimandum eft. Quin et illud te monitum effe velim, inter veterum mathematicorum inventa, vix in aliud quodvis te incidere poffe, quod vel magis artificiofe, vel magis ad utilitatem (in iis faltem que calculo indaganda funt): ufpiam excogitatum eft.'

Now, of the great ingenuity of this invention, we see no proof: Nothing is performed here, but what has been done, and that very nearly in the fame way, by every one who ever fet about forming a table of the divifors of numbers. The prime numbers have their places, in fuch a table, afcertained

almost

almost exactly in the fame manner as in the Sieve of Eratofthenes; and there feems hardly any arithmetical device more fimple or more obvious. Yet Dr Horfley holds it up, in this paf fage, as one of the most ingenious and fubtle inventions of the ancients in matters of arithmetic. To us it feems, on the other hand, that there is hardly a problem in all the thirteen arithmetical books of Diophantus, that does not difplay vastly more ingenuity and contrivance. The invention is ufeful, becaufe, in many refearches, it is of importance to diftinguish the prime numbers. This, however, is the fimpleft problem which can be propofed with refpect to thefe numbers, and throws no light at all on thofe that are more difficult. If a number, beyond the limits of the table of prime numbers, is given; to find whether it be a prime number, or not, is fometimes a work of much difficulty; and what is faid here, will not help us to the folution of it. Were it propofed, for inftance, to find whether 262657 be a prime number, we should find the investigation require fome thought, and would derive no benefit from the Sieve,

The tract on the Sieve of Eratofthenes was published in the Philofophical Tranfactions many years ago, and is now republifhed, having, as the author informs us, been abridged and tranilated into Latin by the Dean of Chrift-Church. He alfo expreffes his thankfulness to Dr Jackfon for aflifting him in draw ing up his prefaces; and adds, Particeps igitur laborum in laudis etiam partem veniat. Some will no doubt fay, that as the labour has been but fmall, the glory muft be little in proportion; but all will confefs, that the less a morfel is, there is the more merit in dividing it with another; and that, on the prefent occafion, it is highly edifying to fee these two great men fitting down contentedly to fo meagre a repaft.

The volume which we are now treating of, befides the tracts already enumerated, contains a book on Sphæricks, from the first and fecond of Theodofius, in which the propofitions demonftrated are very elementary, and the whole not very interefting, as keeping at a great diftance from any application to fphe rical trigonometry: Next comes the meafure of the circumfe rence of the circle, from Archimedes: And, laftly, Keil's differtation on Logarithms, as ufually annexed to his Euclid; a work of great merit, and which is here accompanied with notes by Dr Horfley, that are many of them very ufeful, and not a few which, though useful, appear ludicrous from the parade with which they are brought forward. At p. 134, Dr Horfley finds the logarithin of the cube root of a decimal fraction by a procefs a little different from the common, and, as he thinks, fome

what

a

b

what easier. He immediately ftops to admire the ingenuity of the proceeding; yet, the device which the learned Bifhop efteems fo much, is one for which a mafter might applaud a very young pupil who had difcovered it of himself, and, in doing fo, he would allow it its full measure of praife; for, in reality, it amounts to no more than that is equal to + Yet the commentator of Newton calls this a difcovery which he had made, Dís propitiis ufus. The rule, Nec Deus interfit nifi dignus vindice nodus acciderit, was probably never more violated in poetical fiction than it is here, amid the fobriety of an arithmetical calculation.

at o
3

3 3

The two volumes which we have now been confidering, were preceded by another publifhed in 1801, the whole being intended to make one entire courfe of elementary geometry. That volume, as well as the other two, contains many things ufeful to a beginner, and particular in what regards the application of arithmetic to geometry. Yet the three together will form a courfe of which the parts are not very accurately proportioned, nor very happily arranged; and he who would use it as his text, muft fupply many things, retrench feveral, and tranfpofe not a few. But the work, whatever may be its defects, manifefts a degree of knowledge and talent which would deferve praife, if it came forward with lefs oftentation, and a lefs marked contempt for others. It is a proof of no common activity of mind, and tafte for fcience, in a ftation which has fometimes been thought too high, or too facred for the exercife of thefe fublunary virtues: And, to the credit of the learned Prelate, it fhould also be obferved, that his love of science has not turned him afide from the duties of his profeffion; that his investigations take a very extenfive range; and that, while he finds leifure to comment on Euclid and Eratofthenes, he demonftrates, beyond all contradiction, that France is not a country with wings, and that geographic maps were unknown to the prophet Ifaiah.

ÁRT.

ART. II. The Life and Poflbumous Writings of William Cowper, Efq. with an Introductory Letter to the Right Honourable Earl Cowper. By William. Hayley, Efq. Vol. III. 4to. pp. 416. Johnfon,

London. 1804.

THIS is the continuation of a work of which we formerly fubmitted a very ample account and a very full character to our readers: on that occafion, we took the liberty of obferving, that two quarto volumes feemed to be almoft as much as the biography of a fecluded scholar was entitled to occupy; and with a little judicious compreffion, we are ftill of opinion that the life and correfpondence of Cowper might be advantageoufly included in fomewhat narrower limits. We are by no means difpofed, however, to quarrel with this third volume, which is more interesting, if poffible, than either of the two former, and will be read, we have no doubt, with general admiration and delight.

Though it bears the title of the life of Cowper, this volume contains no farther particular, of his hiftory, but is entirely made up of a collection of his letters, introduced by a long, rambling fort of differtation on letter-writing in general, from the pen of his biographer. This prologue, we think, poffeffes no peculiar merit. The writer has no vigour, and very little vivacity; his mind feems to be cultivated, but not at all fertile; and, while he always keeps at a fafe diftance from extravagance or abfurdity, he does not feem to be uniformly capable of diftinguishing affectation from elegance, or dulnefs from good judgment. This difcourfe upon letter-writing, in short, contains nothing that might not have been omitted with confiderable advantage to the publication; and we are rather inclined to think, that those who are ambitious of being introduced to the prefence of Cowper, will do well not to linger very long in the antichamber with Mr Hayley.

Of the letters themfelves, we may fafely affert, that we have rarely met with any fimilar collection, of fuperior interest or beauty. Though the incidents to which they relate be of no public magnitude or moment, and the remarks which they contain be not uniformly profound or original, yet there is fomething in the fweetnefs and facility of the diction, and more perhaps in the glimpfes they afford of a pure and benevolent mind, that diffufes a charm over the whole collection, and communicates an interest that cannot always be commanded by performances of greater dignity and pretenfion. This intereft was promoted and affifted, no doubt, in a confiderable degree, by that curiofity which always feeks to penetrate into the privacy of celebrated men, and which had been almoft entirely fruftrated in the inftance of Cowper, till the appearance of this publication. Though his writings had

YOL. IV. No. 8.

S

Vol. II. p. 64, &c.

long

long been extremely popular, the author was fcarcely known to the public; and having lived in a state of entire feclufion from the world, there were no anecdotes of his converfation, his habits or opinions, in circulation among his admirers. The publication of his correfpondence has in a great measure fupplied this deficiency; and we now know almoft as much of Cowper as we do of thofe authors who have fpent their days in the centre and glare of literary or fashionable notoriety. Thefe letters, however, will continue to be read long after the curiofity is gratified to which perhaps they owed their first celebrity: for the character with which they make us acquainted, will always attract by its rarity, and engage by its elegance. The feminine delicacy and purity of Cowper's manners and difpofition, the romantic and unbroken retirement in which his life was paffed, and the fingular gentleness and modefty of his whole character, difarm him of those terrors that fo often fhed an atmosphere of repulfion around the perfons of celebrated writers, and make us more indulgent to his weakneffes, and more delighted with his excellences, than if he had been the centre of a circle of wits, or the oracle of a literary confederacy. The intereft of this picture is still farther heightened by the recol lection of that tremendous malady, to the vifitations of which he was fubject, and by the fpectacle of that perpetual conflict which was maintained, through the greater part of his life, between the depreflion of thofe conftitutional horrors, and the gayety that refalted from a playful imagination, and a heart aninated by the mildeft affections.

In the letters now before us, Cowper difplays a great deal of all thofe peculiaritics by which his character was adorned or diftinguished; he is frequently the fubject of his own obfervations, and often delineates the finer features of his understanding with all the induftry and impartiality of a ftranger. But the moft interesting traits are thofe which are unintentionally discovered, and which the reader collects from expreffions that were employed for very different purpofes. Among the most obvious, perhaps, as well as the most important of thefe, is that extraordinary combination of fhyness and amibition, to which we are probably indebted for the very exiftence of his poetry. Being difqualified, by the former, from vindicating his proper place in the ordinary fcenes either of bufinefs or of fociety, he was excited, by the latter, to attempt the only other avenue to reputation that appeared to be open, and to affert the real dignity of the talents with which he felt that he was gifted. If Cowper had acquired courage enough to read the jour nals of the Houfe of Lords, or been able to get over the diffidence which fettered his utterance in general fociety, his genius would probably have evaporated in converfation, or been contented with

the