MATHEMATIC S. A NEW METHOD OF CONSTRUCTING MAGIC SQUARES, COMMUNICATED BY MR. JOHN CLARKE, OF FARNHAM. TH HE properties of numbers disposed placing unity in that cell which is im in magic squares, though perhaps mediately under the center one, and of no real use, are, nevertheleis fome proceed diagonally down, towards the of them so surprising and curious, that right hand lower corner, in the order it is not wonderful so many ingenicus of the natural numbers. When you men have amused themselves in study- get out of the limits of the square ar ing them. Amongst those who have the bottom, go to the top cell of that tumed their thoughts to this fubject, column where you would kave written M. de la Hire and M. Sauveur, both the next figure, if there had been a cell of the Royal Academy of Sciences at within the limits of the table to reParis, certainly stand foremost; and ceive it; and proceed again diagonally indeed it was thought by fome that downwards towards the right hand. they had even exhausted the subject. When you happen to meet with the The following method, however, does limit of the square towards the right not appear to have suggested itself to hand side, go to that cell on the left them. It is chiefly, perhaps fulely, hand side of the square which is di. applicable to squares in which the num- rectly oppofite to that where you would ber of cells is odd: but no method, have written the number, if there had hitherto proposed, that I know of, is been a cell for it on the right hand; fo direct and expeditious, as it requires and proceed again diagonally from no primitive square or calculations of thence towards the lower right hand any kind; but the cells may be filled corner, as before. up as fast, and almost with as little If you meet with a cell which has trouble as the figures could be written a figure in it already, the figures which down in a ftraight line. you would have put there, if it had Let ABCD be a magic square, the been empty, must be placed in the next side of which is 9, and, of consequence, cell to it in a diagonal direction tothe number of its cells 81. Begin by wards the left hand lower corner; and А B thence you must proceed, diago nally, towards the right hand lower 137178|29|7e|2 7162 1315415 .corner again. 6138170130171|22|63|14/46 In that particular instance where you 471 739180.31172123 55115 go out at the right hand corner cell, 101481 8/40 8113216412456 go to the uppermost cell but one in the right hand column; and thence you must 571171491 914117.3|33|65|25 proceed as before. And in this man261581181501 1142174134166 ner you are to go on until all the cells 67.27159110151 243175135 are occupied, let them be ever so many. 3616811016011152' 3'44176 We shall, in some future number of our work, present our readers with -7,2846e|20161112153) 4145 some of the more remarkable proper ties of numbers disposed in such squares as these. OF THE ROOTS OF QUADRATIC EQUATIONS. BY ANALYTICUS. IF t?-ax+b=0, be an equation whose roots we would approximate hy the ine. thod delivered in Lord Stanhope's paper (Philos. Transact. Vol. LXXI. pr195) 2n+dm : 9+zem + dr. let + dmn to ema n2 b; where six of the q2 + df q + ef2 9. + dp9 + ep? eight =a, and eight quantities a, b, d, e, m, n, p, q, may be taken at pleasure, so that all be greater than 46: then will the two feries, requifite to carry on the approximation, m + nx be derived, by division, from þ + 9x and ; that is, any term of 1-dx + ex? i-dxt ex? met on &c. pet da the feries, &c. + x-'+ m + dm + n. x + dam 7dn-em.*?, &c. €2 m + 1x (derived from being divided by the correspondent term of the feries, 1-dx + x2 + 9 **+ P + dp +9.* + 2 p + dq - ep.x?, &c. (derived 82 from P dx fexa) the quotients successively taken both ways (as well of the defcending powers of x as of the afcending) will approximate to those roots as limits. Ex. 1. If a be = "5, b = 5, d=1,1= -1, m=1, n=2, p=1, and ? 1 53; the equation and series will be the same as in Lord Stanhope's paper, abovementioned. Ex. 2. If p be = 1, and the other seven quantities as in the preceding example; the roots of the same equation (11 x?-15x+s=o) will be approximated by the quotients of the correiponding terms of the series, arifing as above expressed; but the branches of the ferits which in that example approximated the lesser root, will in this example approximate the greater; the quotients of the coefficients of the increasing powers of x, in the series derived from it 2x and i + 3% I-X-X? 1-x-*2? approximating the greater root; and the quotients of the coefficients of the de. creasing powers of %, in the series so derived, approximating the lesser root. 11 II OBSERVATIONS OF SOME OCCULTATIONS OF FIXED STARS BY THE MOON; MADE AT DANTZIC, BY DR. WOLF, F. R. Ş. same Society. 3 1776. Occultation of wat by the Moon. 3 3 19 Apparent time go 27 7 Emersion by the Clock urh. 46' 43" Clock too fast 3 Equat. Time subt. 3 19 8 July 5th. Cecultation of it by the ). Clock too slow 4 13 13 48 3 A1-84 h. 43' mean time, the emersion was certainly over; the star being at too steats. distance from the moon's limbi Occultation 28 Occultation of 2d y en by the Moon. of July 5th. Immersion by the Clock 14 h. 39' 47" Apparent time 14 35 32 7 Clouds prevented the emersion from being seen. These observations were made with a telescope of Mr. Short's, which magnified the diameter of the object 200 times. In all three, the far disappeared at the distance of more than a quarter of an inch and nearly half of one from the moon's limb; which was extreinely weli defined. This fpace or diftance, I found, by re. peated trials, was run over by the moon in 7 or 8 seconds; and, therefore, at the mean distance of the moon from the earth, amounts to about 24,600 feet. Hence it appears that the moon has an atmosphere, in which vapours afcend to the same height as in our's. Perhaps the sudden disappearance of the star constantly observed by astronomers (being also seen with an irradiation or glare of light upon the disk) may be owing to the use of glasses with small magnifying powers, as most commodious for these observations. EXAMINATION OF THE CLOCK. 1776. o June 2d. 41 15{1313 8 39 II 12 II Alurude Times by the Clock. 1) 23 P. M. 39 5}Equat. to equal altitudes. Six others agreed with this. 15 P. M. 52 23 9 29 3 52 + 4 Equation to equal altitudes. 11 24 July 4. 45 15? 2 2 II 22 45 301 9 54 26 1 MATHEMATICAL QUESTIONS. 9. QUESTION I. by Mr. J. Walson. A parabola being given, and a point without it; it is required to determine the shortest distance between that point and the curve. 10. QUESTION II. by ASTRONOMICUS. The declinations and right ascensions of two stars being given; it is required to find their distance from the meridian when their difference of azimuth is the greatest or least possible. u. QUESTION III. by NUMERICUS. Three school-boys laid out equal sums of money in fruit-apples and oranges: they all paid the same price, a-piece, for their apples, as well as for their oranges; and yet the whole number of apples and oranges (together) purchased by the first boy was but 9; whilft the second had 18, and the third no fewer than 24: moreover, the difference between the number of apples and the number of oranges, purchased by the first boy, was the least that the question will admit of. What was the number of apples bought by cach boy, and what did cach sort of fruit cost them? 12. QUESTION IV. by J. P. Given the vertical angle of a plane triangle, the line bisecting it, and the fides of a rectangle inscribed in the triangle, to construct it. 13. QUESTION V. by Mr. R. ROBBINS. Given, in a plane triangle, the vertical angle, the greater fide adjoining to it, and the sum of the greater segment of the base and the perpendicular, to conftruct the triangle. 14. QUESTION VI. by Mr. Jer. AINSWORTH, K If ABC, MNH be two circles, the centers of which are O and E, and radii OK, EH: when the eireles MNH is wholly within the circle ABC, if EO2= OKOR-NH; or when it is not wholly with the B M circle ABC, if OE-=OK OK-HNH; then if two B fides, AC, AB, of a triangle ABC, inscribed in the circle ABC touch the circle MNH, the other side BC will touch it likewise. It is required to demonstrate this property geometrically. |НГ c K 15. QUESTION VII. by Mr. Bens. LECEND. M E N board tacks on board, to which we gave chase by running within fix points of the wind, on the larboard tack, until the chase was on our beam: we then tacked, and ran on the starboard tack, clofe hauled as before, until the chase was again on the beam; when we tacked, and so continued to do 'till ten the next morning; at which time we came up with and took her in latitude 44° 40' N. We then found that her course had been due W. the whole time; and, though we kept no account of the distance ron, we observed that we always ran twice as far on the starboard tack as we did on the larboard. Our rate of sailing, that of the prize, the number of boards we made to fetch her, and the distance run on each are required. + The answers to these questions must be sent to Mr. Baldwin's, in Pa. ternoster-row, poft paid, before the ift of November, 1783. AN ESSAY ON THE ORIGIN AND PROGRESS OF NATURAL PHILOSOPHY. (Concluded from our laft.) IT , our admiration of this philosopher, Modest and unassuming, he continued that he was deficient in no one requi- his enquiries with the same cautious at. site calculated to form the great man. tention with which he began, and free It is a thing too often confirmed by from that avarice for reputation which daily experience, that a personal ac- induces philosophers to pursue in foliquaintance with men of acknowledged tude the path to discovery, he publishmerit scarcely ever fails to diminish ex- ed his thoughts, and pointed out the ceedingly that respect, which their wri- way to foliow his footsteps. But all tings have procured them. In books his successors have not been entirely we see the author, but in private life the able to remove the difficulties of which man alone appears, uninfluenced by an explication is demanded of them. that awe which the venerable presence Sir Isaac Newton may justly be styled of the public inspires. The near view the father of physical astronomy, but presents bigotry, arrogance, affected the prodigious cilat of his discoveries fingularity, and many other imperfec- seems to have occafioned a temporary tions, to which the state of mortality ftand in some other parts of natural is subject. What an infinite addition philosophy. The action of electricity muit it be to the fame of Sir Isaac began about this time to be particularNewton that when in the entire pof- ly attended to, and chemistry, which sestion of the enthusiastic applause of from immemorial time had existed not mankind, who regarded him even in as a science but as the basis of various his life-time as a being of a fuperior trades, had a century before been introorder, what an addition must it be to duced into medicine by the active and his fame, that at this inebriating height extravagant Paracelsus, and since culti |