Some method of facilitating the back-observation in the use of Hadley's quadrant, is abfolutely neceffary to the perfection of this ufeful inftrument. In order to this, the back horizon-glass must be carefully adjusted and the fight must be directed parallel to the plane of the quadrant. Mr. Dollond has contrived to obviate the first difficulty by a new conftruction, of which we have given a brief account in the preceding article. The proper ad juftment of the line of fight, or axis of the telescope, is the fubject of this article. If the quadrant be not fitted with a telescope, a director of the fight fhould by no means be omitted a but when a telescope is used, the exact pofition of it is a mat ter of great importance; and therefore Mr. M. has fuggefted feveral directions for this purpofe. He recommends an adjusting piece to be applied to the telescope, in order to make its axis parallel to the plane of the quadrant; the filvering of the back horizon-glass; and the placing of two filver thick wires within the eye-tube in the focus of the eye glafs, parallel to one another and to the plane of the quadrant. He then proposes two methods for bringing the axis of the telescope into a pofition parallel to the plane of the quadrant. In the fequel of the paper there are many inftructions and remarks, that may be of great use, both to those who make and to those who use this inftrument. " Article 24. A Letter from John Call, Efq; to Nevil Mafkelyne, This letter is attended with a drawing, taken from the cieling of a Choultry or Pagoda at Verdapettah in the Madurah country. The cieling is of a fquare figure, from the center of which is fufpended by two hooks a throne on which the Deity or Swamy fits, when exhibited to the worshippers. In the fides and at the angular points are delineated the figures of the 12 figns of the Zodiac: Aries and Taurus are to the East; Gemini in the South-Eaft angle; Cancer and Leo to the South; Virgo in the South-Weft corner; Libra and Scorpio to the Weft; Sagittarius to the North-Weft; Capricornus and Aquar rius to the North, and Pifcis to the North-East, Mr. Call in: forms us, that he has often met with detached pieces of this kind, but with only one fo complete. And he conjectures, that the Signs of the Zodiac now in ufe among Europeans were ori ginally derived from the Indians by Zoroafter and Pythagoras; and as these philofophers are ftill fpoken of in India under the names of Zerdhurst and Pyttagore, he fuggefts the idea, that the worship of the cow, which ftill prevails in that country, way transplanted from thence into Egypt. He thinks it may be fafely pronounced that no part of the world; has more marks of antiquity for arts, fciences, and cultivation, than the Penin 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 fome parts of Europe are excepted.' Mr. Call has tranfmitted to the Society the manuscripts of the late Mr. Robins, which he entrusted with him at his death; they have fince been examined by several 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. MATHEMATIC S. Article 22. KOEKINON EPATOZOENOTE: 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 distinction of prime and compofite numbers are generally understood; fo is likewife the method of determining, whether several numbers propofed be prime or compo→ fite with respect 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 fuch a table, and for carrying it to a great length, in a fhort time, and with little labour. This curious invention has been defcribed 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 ApiJμnlixn, 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. Horley 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 necessary to confine himself in every particular to the account of Nicomachus, most 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 fieve 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 publifhed at Oxford in 1762, and adorned with the title of Κοσκινον Ερατοσθενες. 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 inven tion of Eratofthenes. We will transcribe this problem, with its folution; for the amusement 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, ate 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. 3. &c. Every number, which is not prime, is a multiple of fome 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 cafily diftinguished and exterminated." For between 3 and its first multiple in the feries (9) two num bers 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 manher, 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 multiples of any number n, as they ftand 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 aumbers, which I take to have been the operation of the fieve, and is as follows: 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 mul tiples of 3 are expunged. The first 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 expurge 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 3REY. Jan. 1774 17 * 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 exFunged 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 firft uncancelled number that appears, next to that whofe multiples have been laft expunged, is fuch, that its fquare is greater than the laft The and greatest number to which the feries is extended. 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 Jimit 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 Preblems, 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. R..s [To be continued.] ART. VII. The School for Wives, a Comedy; as it is performed at Becket. I s. 6 d. the Theatre-Royal, in Drury-Lane. 8vo. 1774. HIS play (as ufual fince the days of Dryden) is preceded by a preface; and it has occurred to us, in perufing it, that the author of a play, should write his preliminary discourse before he has known his fuccefs: 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 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 fhould not trouble ourselves about the Author's preface; but if he wishes it to be read with pleasure by perfons of judgment and taste, we would advise him, in future editions, to let the 3.5.7.9.11. 13. 15. 17. 19. x. 23. 75, 77. 29. 31. 33. 28. 37. 39. 41. 43. 45. 47. 49. 8X. 53.85. 87. 59. 61. 63. 85. 67. 6. 71. 73.73. 77. 79. 81. 83. 85. 87. 89. 91. 93. 98. Vide Philofophical Tranfactions, No. 215. preface preface be forgotten. At prefent, however, it thus ferves to ípeak of his opinions and purposes: The Author's chief study has been to fleer between the extremes of fentimental gloom, and the exceffes of uninterefting 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 business of inftruction is neglected, and confider it as a herefy in criticism to fay that one of the most arduous talks within the reach of literature, should, 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 several plays in the English language, which, before his, produced generals, lawyers, Irishmen, duels, masquerades, and miftakes? 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 bufinefs 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 these charges, ftill in extenuation of his offence, he begs leave to obferve, that having only men and women to introduce upon the flage, he was obliged to compofe his Dramatis Perfonæ of meer flesh 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 some small 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 deserve, 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. The general moral of this play is, in itself, excellent, and peculiarly seasonable, at a time, when conjugal infidelity in the men, is repaid in kind by the ladies, with an offenfive and mafculine hardinefs; and all the foft and winning graces of the fex are almoft loft to the world. The Author has alfo very happily expofed the folly and abfurdity of duelling. The firft Act is opened by two lovers privately engagedCaptain Savage, and Mifs Walfingham; whofe converfation principally turns on an intrigue of Belville's. This Belville is the hufband who furnishes the wife with subjects for her lef D 2 fons. |