Some method of facilitatiog the back-observation in the use of Hadley's quadrant, is absolutely necessary to the perfection of this useful inftrument. In order to this, the back horizon-glass mult be carefully adjufted 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 construction, of which we have given a brief account in the preceding article. The proper ado justment of the line of fight, or axis of the telescope, is the subject of this article. If the quadrant be not fitted with a telescope, a director of the fight should by no means be omitted: but when a telescope is used, 'the exact position of it is a mat ter of great importance; and therefore Mr. M. has suggested several directions for this purpose. He recórómends an adjusting piece to be applied to the telescope, in order to make its axis parallel to the plane of the quadrant ; the flyering of the back horizon-glass; and the placing of two filver thick wires within the eye-tube in the focus of the eye glass, parallel to one another and to the plane of the quadrant. He then proposes twa methods for bringing the axis of the telescope into a position parallel to the plane of the quadrant. In the lequel of the paper there are many instructions 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 Maskelyne, F. R. S. Astronomer Royal, containing a Sketch of the Signs of the Zodiac, found in a Pagoda, near Cape Comorin in India, This letter is attended with a drawing, taken from the cielo ing of a Choultry or Pagoda at Verdapettah in the Madurah coun: try. The cieling is of a square figure, from the center of which is suspended by two hooks a throne on which the Deity or Swamy fits, when exhibited to the worlhippers. In the fides and at the angular points are delineated the figures of the 12 signs of the Zodiac : Aries and Taurus are to the East; Gea mini in the South East angle; Cancer and Leo to the South; Virgo in the South-West corner ; Libra and Scorpio to the West; Sagittarius to the North-Weft; Capricornus and Aqua; rius to the North, and Piscis to the NorthEast, Mr. Call in: forms us, that he has often met with detached pieces of this kind, but with only one so complete. And he conjectures, that the Signs of the Zodiac now in use among Europeans were ori, ginally derived from the Indians by Zoroaster and Pythagoras s and as these philosophers are still spoken of in India under the names of Zerdhurft and Pyttagore, he suggests the idea, that, chę worship of the cow, which still prevails in that country, was transplanted from thence into Egypt. He, thinks it may be safely pronounced that no part of the world has more marks of antiquity for arts, sciences, and cultivation, than the Peninfula 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 Horsley, F. R. S. The nature and distinction of prime and compofite numbers are generally understood; fo is likewise the method of determining, whether several numbers proposed be prime or compofite 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 solved ; and whereby a table may be constructed, including all the prime numbers to any given limit. Eratosthenes, who was so juftly celebrated among the sages of the Alexandrian school,' contrived an indiret method for constructing such a table, and for carrying it to a great length, in a short time, and with little labour. 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 Eiraywyn Ap. Spenloxn, and lived in the 3d or 4th century, and Boethius, whose treatise of numbers is only an abridgment of the wretched performance' of the former, are but little known. 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 arithmetic. He has not thought it necessary to confine himself in every particular to the account of Nicomachus, most of whose observations are either erroneous or foreign to the purpose ; and that the learned may judge how far he has done justice to this invention, he has subjoined extra&s both from the treatise of Nicomachus, and the Arithmetica of Boethius. Mr. H. observes, that the fieve of Eratosthenes Eratofthenes is a very different thing from that table, which has been falsely 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 Kooxivov Epatocers.' This, he apprehends, was copied from fome 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 inven. tion 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 prime number ; but, except 2, 'no even number is prime, because every even' number, except 2, is die visible by 2, and is therefore composite. Hence it follows, that all the prime numbers, except the Humber 2, ate 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. 35. &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, anią 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 compofite numbers, may be casily diftinguished and exterminated.' For between 3 and its first multiple in the series (9) two num. bers intervene. Between 'g 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 man. ner, between every pair of the multiples of 7, as they stand 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 n, as they stand in their natural order in the series, I 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 Steve. Count all the terms of the series following the number 3; by three, and expunge every third number. Thus all the mula tiples 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 s; by fives, and expurige every fifth number, if not expunged before. Thus all the multiples of s are expunged, which were not at first expunged, among the multiples of . The what The next uncancelled number to s is 7. Expunge the fquare of 7. Çount all the terms of the series following the square of 2by sevens, and expunge every seventh number, if not exfunged before. Thus all the multiples of 7 are expunged, which were not before expunged among the multiples of 3 or Si's Continue these expunctions till the first uncancelled number that appears, next to that whose multiples have been laft expunged, is such, that its square is greater than the last and greatest number to which the series is extended. The numbers wbich then remain uncancelled are all the prime num: bers, except the number 2, which occur in the natural progression of number from 1 to the limit of the series. By the şimit 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 Astronomical Problems, by the late Dr. Henry Pemberton, Communicated by Mathew Kaper, Esq; F. R. S. The first of these problems is to find in the Ecliptic the point of longest ascenfion; the second is to find when the arc of the Ecliptic differs most from its oblique afcenfion; and the third is to find the Tropic, by Dr. Halley's method, without any consideration 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, à Comedy ; as it is performed at the Theatre-Royal, in Drury-Lane, 8vo. I s. 6 d. Becket. 1774 THIS play (as usual 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 success: 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 sentence; and if he has been faved, they would not come prepossessed 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 persons of judgment and taste, we would advise him, in future editions, to let the + 3. 5. 7. g. 11. 13.15: 17-19. 41. 23: 75017. 29. 31. 33: 25. 37. 39. 41. 43. 45. 47. 4g. gr. 53. 83. 54. 59. 61. 63. 63. 69. 69. 71. 73. 45.77. 79. 81. 83. 85. 87. 89. 91. 93. 95. * Vide Philosophical Transactions, No. 215. preface R..s. preface be forgotten. At present, however, it thus ferves to speak of his opinions and purposes : • The Author's chief study has been to steer between the extremes of sentimental gloom, and the excesses of uninteresting levity; 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 Muse when a little grave, he must think that the degenerates into farce, where the grand business of instruction is neglected, and confider it as a heresy in criticism to fay that one of the most arduous tasks within the reach of literature, hould, when executed, be wholly without utility.' • The Author having been presumptuous enough to assert 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, Irithmen, 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 confisted in making dificulties for the purpose of removing them; in difresling 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, still in extenuation of his offence, he begs leave to observe, that having only men and women to introduce upon the Itage, he was obliged to compose his Dramatis Personæ of meer flesh and blood; if however he has thrown this felh and this blood into new situations ; 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 suppose, the Author means to prepossess 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 praile which he may really deserve, if he had not, in this manner, preferred his 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 men, is repaid in kind by the ladies, with an offensive and malculine hardiness; and all the soft and winning graces of the sex are almost lost to the world. The Author has also very happily exposed the folly and absurdity of duelling. The first Act is opened by two lovers privately engaged Captain Sayage, and Miss Wallingham; whole converiation principally turns on an intrigue of Belville's. This Belville is the husband who furniches the wife with subjects for her ler. D 2 sons. |