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 fpot better inhabited, or filled with towns, temples, and villages, than this fpace is throughout, if China and fome parts of Europe are excepted.'

Mr. Call has tranfmitted to the Society the manufcripts of the late Mr. Robins, which he entrusted with him at his death; they have fince been examined by feveral of the members, who found, that they contain nothing material more than has been already printed; excepting a treatife on military difcipline; which may probably be inferted in the next edition of his works.

MATHEMATICS.

Article 22. ΚΟΣΚΙΝΟΝ ΕΡΑΤΟΣΘΕΝΟΥΣ : or, The Sieve of Eratofthenes. Being an Account of his Method of finding all the Prime Numbers. By the Rev. Samuel Horfley, F. R. S.

The nature and diftinction of prime and compofite numbers are generally understood; fo is likewife the method of determining, whether several numbers propofed be prime or compofite with refpect to one another: this is a problem, the folution of which Euclid has given in the three firft 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 propofed be abfolutely prime or compofite is much more difficult; nor does there feem to be any general method, whereby this problem may be directly folved; and whereby a table may be conftructed, including all the prime numbers to any given limit. Eratofthenes, who was fo juftly celebrated among the fages of the Alexandrian school,' contrived an indirect method for conftructing such a table, and for carrying it to a great length, in a fhort time, and with little labour. This curious invention has been described only by two very obfcure writers, and has therefore in a great measure efcaped notice. The names of Nicomachus Gerafinus, who, among other treatifes, wrote an Eiraywyn ApiSunloun, and lived in the 3d or 4th century, and Boethius, whofe treatife of numbers is only an abridgment of the wretched performance' of the former, are but little known.

Mr. Horfley prefents the Society with a particular account of this extraordinary invention: which he confiders as one of the most precious remnants of antient arithmetic.' He has not thought it neceffary to confine himself in every particular to the account of Nicomachus, moft of whofe obfervations are either erroneous or foreign to the purpose; and that the learned may judge how far he has done juftice to this invention, he has fubjoined extracts both from the treatife of Nicomachus, and the Arithmetica of Boethius. Mr. H. obferves, that the fieye of

Eratofthenes

Eratofthenes is a very different, thing from that table, which has been falfely afcribed to him, and which is printed at the end of the beautiful edition of Aretus published at Oxford in 1762, and adorned with the title of Kaσivov Epator Jevas. This, he apprehends, was copied from fome Greek comment upon the arithmetic of Nicomachus, and to have been the production of fome monk in a barbarous age, and not the whole of the invention of Eratofthenes.

We will transcribe this problem, with its folution, for the amufement of our mathematical Readers:

Problem. To, find all the prime numbers.

The number 2 is a prime number; but, except 2, no even number is prime, becaufe every even number, except 2, is divifible by 2, and is therefore compofite. Hence it follows, that all the prime numbers, except the number 2, are included in the feries of the odd numbers in their natural order, infinitely extended, that is, in the feries,

3.5.7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. &c. Every number, which is not prime, is a multiple of tome prime number, as Euclid hath demonftrated (Element. 7. prop. 33); therefore the foregoing feries confifts of the prime numbers, and of multiples of the primes. And the multiples of every number in the feries follow at regular diftances; by attending to which circumftance all the multiples, that is, all the compofite numbers, may be eafily diffinguifhed and exterminated."

For between 3 and its first multiple in the feries (9) two numbers intervene. Between 9 and the next multiple of 3-(15) two numbers likewife intervene, which are not multiples of 3.

Again, between 5 and its firft 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 feries, fix numbers intervene, which are not multiples of 7. Univerfally, between every two mul tiples of any number n, as they stand in their natural order in the feries, n-1 numbers intervene, which are not multiples of Hence may be derived an operation for exterminating the compofite numbers, which I take to have been the operation of the fieve, and is as follows:

n.

The Operation of the Sieve.

Count all the terms of the feries following the number 3, by three, and expunge every third number. Thus all the thultiples of 3 are expunged. The firft uncancelled number that appears in the feries, after 3, is 5. Expunge the fquare of 5. Count all the terms of the feries, which follow the fquare of 5, by fives, and expunge every fifth number, if not expunged before. Thus all the multiples of 5 are expunged, which were not at first expunged, among the multiples of 3REV. Jan. 1774

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The

The next uncancelled number to 5 is 7. Expunge the fquare of 7. Count all the terms of the feries following the fquare of 7, by fevens, and expunge every feventh number, if not expunged before. Thus all the multiples of 7 are expunged, which were not before expunged among the multiples of 3 or 5. Continue thefe expunctions till the first uncancelled number that appears, next to that whofe multiples have been laft expunged, is fuch, that its fquare is greater than the last and greatest number to which the feries is extended. The numbers which then remain uncancelled are all the prime numbers, except the number 2, which occur in the natural progreffion of number from 1 to the limit of the feries. By the Fimit of the feries I mean the laft and greatest number, to which it is thought proper to extend it. Thus the prime numbers are found to any given limit +.'

Article 30. Geometrical Solutions of three celebrated Aftronomical Problems, by the late Dr. Henry Pemberton. Communicated by Mathew Raper, Efq; F. R. S.

The firft of thefe problems is to find in the Ecliptic the point of longeft afcenfion; the fecond is to find when the arc of the Ecliptic differs moft from its oblique afcenfion; 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 premifed; 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. I s. 6d. Becket. 1774.

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HIS play (as ufual fince the days of Dryden) is preceded by a preface; and it has occurred to us, in perusing it, that the author of a play, fhould write his preliminary difcourfe before he has known his fuccefs: if damned, his readers would not then, by his abufe and ill-nature, be put into an humour that might provoke them to repeat the fentence; and if he has been faved, they would not come prepoffeffed against him, as a coxcomb, from a vain parade of his aims and intentions, and his infipid 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 wishes it to be read with pleasure by perfons of judgment and tafte, we would advife him, in future editions, to let the

† 3.5.7. g. 11. 13. 13. 17. 19. 2x. 23. 25, 47. 29. 3·1. 33. 88. 37. 39. 41. 43. 45. 47. 49. 55. 53. 53. 57. 59. 61. 63. 0g. 67. 69. 71. 73. 78. 77. 79. 81. 83. 88. $7. 89. 91. 92. 95.

Vide Philofophical Tranfactions, No. 215.

preface

preface be forgotten. At prefent, however, it thus ferves to Speak of his opinions and purposes:

• The Author's chief study has been to fteer between the extremes of fentimental gloom, and the exceffes of uninteresting levity; he has fome laugh, yet he hopes he has alfo fome leffon; and fashionable as it has lately been for the wits, even with his friend Mr. Garrick at their head, to ridicule the Comic Mufe when a little grave, he must think that the degenerates into farce, where the grand bufinefs of inftruction is neglected, and confider it as a herefy in criticifm to fay that one of the most arduous tafks within the reach of literature, fhould, when executed, be wholly without utility.'

The Author having been prefumptuous enough to affert that he has not purloined a fingle fprig of bays from the brow of any other writer, he may perhaps be afked, if there are not feveral plays in the English language, which, before his, produced generals, lawyers, Irishmen, duels, mafquerades, and mistakes? He anfwers, Yes; and confeffes, moreover, that all the comedies before his, were compofed not only of men and women, but that before his, the great bufincfs of comedy confifted in making difficulties for the purpose of removing them; in diftreffing poor young lovers, and in rendering a happy marriage the object of every catastrophe.

Yet though the Author of the School for Wives pleads guilty to all thefe charges,. ftill in extenuation of his offence, he begs leave to observe, that having only men and women to introduce upon the ftage, he was obliged to compofe his Dramatis Perfonæ of meer flefl and blood; if however he has thrown this flesh and this blood into new fituations; if he has given a new 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 fome fmall pretenfions to originality.'

By this method of expatiating, we fuppofe, the Author means to prepoffefs people in favour of his play; but in our apprehenfion he is mistaken. We imagine that his Readers - would have more readily yielded him the praise which he may really deferve, if he had not, in this manner, preferred his claim to it. Reviewers, however, are grave, difpaffionate men; and ever difpofed to overlook the little infirmities and foibles of deferving Authors. They will therefore forgive the faults of the preface; and proceed to confider the work which it introduces to our notice.

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The general moral of this play is, in itself, excellent, and peculiarly feasonable, at a time, when conjugal infidelity in the men, is repaid in kind by the ladies, with an offensive and mafculine hardinefs; and all the foft and winning graces of the sex are almost lost to the world.-The Author has also very happily exposed the folly and abfurdity of duelling.

The firft Act is opened by two lovers privately engaged Captain Savage, and Mifs Walfingham; whofe converfation principally turns on an intrigue of Belville's. This Belville is the husband who furnishes the wife with fubjects for her lef

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fons.

fons. He had got acquainted with and deluded Mifs Leefon, niece of Mrs. Tempeft, the miftrefs of General Savage, who is the Captain's father. Belville had effected this under pretence of being an Irifh manager, and had engaged the Lady for the Dublin ftage. Mrs. Tempeft procured fome knowledge of his defign, and had upbraided him with it in the hearing of Mrs. Belville; but in fo outrageous a manner, that Belville éafily perfuaded his good wife that the woman was mad. Mr. and Mrs. Belville join Captain Savage and Mifs Walfingham; and a few words pafs on this fubject, when Lady Rachel Mildew fends her compliments and fays the will wait on Mr. and Mrs. Belville. Some witty hints are given of a love-affair between this Lady, who is a poet and a wit, and Torrington, an old lawyer; and Mifs Walfingham tells us, that Lady Rachel puts her charms into fuch repair, whenever fhe expects to meet him, that her cheeks look for all the world 1ke a rafberry ice upon a ground of custard.'-This piece of wit has been applauded, but we apprehend it to be defective in many eflential requifites of a fimile. It is not at all to be understood, but by thofe who are admitted to the tables of the great; and it gives extraordinary trouble to a Reviewer, who mult of neceffity, be at a lofs to judge of the propriety of fuch dainty allufions. However, as the Author may, in this inftance at leaft, object to the competency of the court, we fhall drop the point, and proceed.

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The fcene changes to Leefon's chambers in the Temple. Leefon is brother to the girl who is deluded by Belville. And Conolly is a faithful and affectionate Irifh fervant. Leefon is in difficulties, which are to be removed by his running away with a girl of large fortune. In the mean time he fends a challenge to Belville for the injury done to his fifter.-The fcenc removes us to an apartment at Belville's; and opens with one of the beft leffons in the School for Wives.

Mrs. Bel. How ftrangely this affair of Mrs. Tempeft hangs upon my fpirits! though I have every reafon from the tenderness, the politeness, fand the generofity of Mr. Belville, as well as from the woman's behaviour, to believe the whole charge the refult of a difturbed imagination-Yet fuppofe it fhould be actually true :-heigho! well, fuppofe it fhould;-I would endeavour-I think I would endeavour to keep my temper:-a frowning face never recovered a heart that was not to be fixed with a fmiling one:-but women in general, forget this grand article of the matrimonial creed entirely; the dignity of infulted virtue obliges them to play the fool, whenever their Corydons play the libertine ;-and, poh! they must pull down the house about the traitor's ears, though they are themfelves to be crushed in pieces by the ruins.'

This excellent foliloquy is interrupted by the introduction of Lady Rachel Mildew, and the converfation turns on love, on poetry, and on Mifs Leefon, as a candidate for the ftage.

They