fula of India, from the Ganges to Cape Comorin ; nor is there in the world a finer climate, or face of the country, nor a spot better inhabited, or filled with towns, temples, and villages, than this space is throughout, if China and some parts of Europe are excepted.' Mr. Call has transmitted to the Society the manuscripts of the late Mr. Robins, which he entrusted with him at his death; they have since been examined by several of the members, who found, that they contain nothing material more than has been already printed; excepting a treatise on military discipline ; which may probably be inserted in the next edition of his works. MATHEMATICS. Article 22. KOEKINON EPATOEOENOTE: or, The Sieve of Eratosthenes. Being an Account of his Method of finding all the Prime Numbers. By the Rev. Samuel Horsey, F. R. S. The nature and distinction of prime and compofite numbers are generally understood ; so is likewise the method of determining, whether several numbers proposed be prime or composite with respect to one another : this is a problem, the solution of which Euclid has given in the three first propofitions of the 7th book of the Elements, and it is to be met with in the common treatises of arithmetic and algebra. But to determine whether any number proposed be absolutely prime or composite is much more difficult ; nor does there seem to be any general method, whereby this problem may be directly folved ; and whereby a cable may be constructed, including all the prime numbers to any given limit. Eratosthenes, who was fu juftly celebrated among the sages of the Alexandrian school,' contrived an indirect method for constructing such a table, and for carrying it to a great length, in a short time, and with little la. bour. This curious invention has been described only by two very obscure writers, and has therefore in a great measure escaped notice. The names of Nicomachus Gerafinus, who, among other treatises, wrote an Esrapwgn Apofuentoxn, and lived in the 3d or 4th century, and Boethius, whose treatife of numbers is only an abridgment of the • wretched performance of the former, are but little knuwn. Mr. Horsley presents the Society with a particular account of this extraordinary invention : which he confiders as one of the most precious remnants of antient arithmetici' He has not thought it necessary to confine himself in every particular to the account of Nicomachus, moft of whose observations are either erroneous or foreign to the purposes and that the learned may judge how far he has done justice to this invention, he has subjoined extraets both from the treatise of Nicomachus, and the Arithmetica of Boethius. Mr. H. obferves, that the fieye of Eratosthenes Eratofthenes is a very different thing from that table, which has been falsely ascribed to him, and which is printed at the end of the beautiful edition of Aretus publithed a: Oxford in 1762, and adorned with the title of Kapuivov Eputer Ievas. This, he apprehends, was copied from some Greek comment upon the arithmetic of Nicomachus, and to have been the production of some monk in a barbarous age, and not the whole of the invention of Eratosthenes. We will transcribe this problem, with its solution, for the amusement of our mathematical Readers : • Problem. To find all the prime numbers. The number 2 is a princ number; but, except 2, no even number is prime, because every even number, except 2, is divisible by 2, and is therefore composite. Hence it follows, that all the prime numbers, except the number 2, are included in the series of the odd numbers in their natural order, infinitely extended, that is, in the series, 3. 5.7. 9. 11. 13. 15. 17. 19.21.23 25. 27. 29. 31. 33. 33. &c. Every number; which is not prime, is a multiple of lome prime number, as Euclid hath demonstrated (Element. 7. prop. 33); therefore the foregoing series consists of the prime numbers, and of multiples of the primes. And the multiples of every number in the series follow at regular distances; by attending to, which circumstance all the multiples, that is, all the composite numbers, may be easily diftinguished and exterminated.' • For between 3 and its first multiple in the series (9) two numbers intervene. Between 9 and the next multiple of 3 (15) two numbers likewise intervene, which are not multiples of 3.' - Again, between 5 and its first multiple (15) four numbers intervene, which are not multiples of 5: In like manner, between every pair of the multiples of 7, as they ftand in their natural order in the series, fix numbers intervene, which are not multiples of 7 Universally, between every two mula tiples of any number in, as they stand in their natural order in the series, n-1 numbers intervene, which are not multiples of n. Hence may be derived an operation for exterminating the composite numbers, which I take to have been the operation of the fieve, and is as follows: The Operation of the Sievé. Count all the terms of the series following the number 3, by three, and expunge every third number. Thus all the inultiples of 3 are expunged. The first uncancelled' number that appears in the series, after 3, is 5. Expunge the square of s. Count all the terms of the series, which follow the square of 5, by fives, and expunge every fifth number, if noc expunged before. Thus all the multiples of 5 are expunged, which were not at first expunged, among the multiples of 3. Rev. Jan. 1774. D The The next uncancelled number to ş is 7. Expunge the square of 7. Count all the terms of the series following the square of 7by sevens, and expunge every feventh number, if not exfunged before. Thus all the multiples of 7 are expunged, which were not before expunged among the multiples of 3 or 5' Continue these expun&tions till the firft uncancelled number that appears, next to that whose multiples have been last expunged, is such, that its square is greater than the last and greatest number to which the series is extended. The numbers which then remain uncancelled are al} the prime numbers, except the number 2, which occur in the natural progression of number from 1 to the limit of the series. By the fimit of the series I mean the last and greatest number, to which it is thought proper to extend it. Thus the prime numbers are found to any given limit t.' Article 30. Geometrical Solutions of three celebrated Apronomical Problems, by the late Dr. Henry Pemberton. Communicated by Mathew Raper, Eja; F. R.S. The first of these problems is to find in the Ecliptic the point of longest afcenfion; the fecond is to find when the arc of the Ecliptic differs moft from its oblique ascension ; and the third is to find the Tropic, by Dr. Halley's method", without any confideration of the parabola. To these three problems a lemma is premised; but as they are purely geometrical, they admit of no extract or abridgment. [To be continued.] ART. VIII. The School for Wives, a Comedy ; as it is performed at the Theatre-Royal, in Drury. Lane, 8vo. Is. 6 d. Becket. 1774. HIS play (as usual fince the days of Dryden) is preceded that the author of a play, should write his preliminary discourse before he has known his fuccess: if damned, his readers would not then, by his abuse and ill-nature, be put into an humour that might provoke them to repeat the fentence'; and if he has been saved, they would not come prepofTeffed against him, as a coxcomb, from a vain parade of his aims and intentions, and his insipid compliments to the actors. If we did not think the School for Wives a comedy of merit, we should not trouble ourselves about the Author's preface; but if he wilhes it to be read with pleasure by perfons of judgment and taste, we would advise him, in future editions, to let the 3. 5. 7. g. 11, 13. 13. 17. 19.4*. 23. 18,77. 29. 36. 37. 38. -37, 39, 41. 43. 45. 4.7. 49. BL. 53. 85. 87. 59. 61. 63. 07. 07. 8g. 71. 73. 78.7779. 81. 83. 85. 87. 89. 91. 92. 95. • Vide Philosophical Transactions, No. 21j. preface preface be forgotten. At presene, however, it-thus ferves to speak of his opinions and purposes : · The Author's chief ftudy has been to feer between the extremes of sentimental gloom, and the excesfes of uninteresting tevity; he has fome laugh, yet he hopes he has also some leffon; and fashionable as it has lately been for the wits, even with his friend Mr. Garrick at their head, to ridicule the Comic Mule when a little grave, he must think that she degenerates into farce, where the grand business of instruction is neglected, and consider it as a herely in criticism to say that one of the moft arduous talks within the reach of literature, pould, when executed, be wholly without utility' • The Author having been presumptuous enough to affert that he has not purloined a single sprig of bays from the brow of any other writer, he may perhaps be asked, if there are not several plays in the English language, which, before his. produced generals, lawyers, Irishmen, duels, masquerades, and mistakes? He answers, Yes; and confesses, moreover, that all the comedies before his, were composed not only of men and women, but that before his, the great business of comedy consisted in making difficulties for the purpose of removing them; in diftrefling poor young lovers, and in rendering a happy marriage the objeět of every cataftrophe. - Yet though the Author of the School for Wives pleads guilty to all these charges, till in extenuation of his offence, he begs leave to observe, that having only men and women to introduce upon the Lage, he was obliged in compose bis Dramatis Perfonæ of meer flell. and blood ; if hoxever he has thro:vo this flesh and chis blood into new situations; if he has given a nezu fable, and placed his characters in a point of light hitherto unexhibited :-he flatters himself that he may call his play, a new play; and though it did not exist before the creation of the world, like the famous Welch pedigree, that he may have some small pretentions to originality.' By this method of expatiating, we suppose, the Author means to prepoffefs people in favour of his play ; but in our apprehenGon he is mistaken. We imagine ibat bis Readers would have more readily yielded him the praise which he may really delerve, if he had noty in this manner, preferred bis claim to it. - Reviewers, however, are grave, dispassionate men; and ever disposed to overlook the little infirmities and foibles of deserving Authors. They will therefore forgive the faults of the preface, and proceed to consider the work which it introduces to our notice. The general moral of this play is, in itself, excellent, and peculiarly seasonable, at a time, when conjugal infidelity in the mes, is repaid in kind by the ladies, with an offenGye and mar. culine hardiness; and all the soft and winning graces of the sex are almoft loft to the world. The Author has also very happily exposed the folly and abfurdity of duelling. The first Act is opened by two lovers privately engaged Captain Savage, and Miss Walfingham; whose conversation principally turns on an intrigue of Belville's. This Belville is the tulband who furnishes the wife with subjects for her les. fons. sons. He had got acquainted with and deluded Miss Leefon, niece of Mrs. Tempeft, the mistress of General Savage, who is the Captain's father. Bel ille had effcéted this i'nder pretence of being an Irish manager, and had engaged the Lady for the Dublin fiage. Mrs. Tempest procureu Tome knowledge of his design, and had upbraided him with it in the hearing of Mrs. Belville ;. but in so outragcous a manner, that Belville easily persuaded his good wife that the woman was mad. Mr. and Niis. Belville join Captain Savage and Miss Wallingham ; and a few words pals on this fubject, when Lady Rachel Mildew fends her compliments and lays the wiil wait on Mr. and Mrs. Belville. Some witty hints are given of a love-affair bet vcen this Lady, who is a pott and a wit, and Torrington, an old Jawyer; and Miss Wallingham tells us, that Lady Rachel puts her charins into such repair, whenever she expeéis to meet him, that her checks look for all the world like a rasb:rry ice upon a ground of custard.'— This piece of wit has been applauded, but we apprehend it to be defective in many effential requisites of a fimile. It is not at all to be underitood, but by chole who are adınitted to the tables of the great; and it gives extraordinary trouble to a Reviewer, who mult of neceffity, be at a loss to judge of the propriety of such dainty allusions.—However, as the Author may, in this instance at least, object to the competency of the court, we shall drop the point, and proceed. The scene changes to Leeson's chambers in the Temple. Leeson is brother to the girl who is deluded by Belville. And Conolly is a faithful and affectionate Irith servant. Leeson is in difficulties, which are to be removed by his running away with a girl of large fortune. In the mean time he sends a chala lenge to Belville for the injury done to his fiiter.--The scene removes us to an apartment at Belville's ; and opens with one of the best lessons in the School for Wives. • Mrs. Bel How strangely this affair of Mrs. Tempel hangs upon my spirits ! though I have every reason from the tenderness, the politeness, and the generosity of Mr. Belville, as well as from the woman's behaviour, to believe the whole charge the result of a difturbed imagination-Yet suppose it should be actually true :-heigho! well, fuppose it should ;-I would er.deavour-I think I would endeavour to keep my temper.:-a frowning face never recovered a heart that was not to be fixed with a smiling one :--but women in general, forget this grand article of the matrimonial creed entirely; the dignity of insulted virtue obliges them to play the fool, whenever their Corydons play the libertine ;-and, poh! they must pull down the houle about the traitor's ears, though they are themselves to be crushed in pieces by the ruins.' This excellent soliloquy is interrupted by the introduction of Lady Rachel Mildew, and the conversation turns on love, on poetry, and on Miss Leeson, as a candidate for the stage. They |