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eight quantities a, b, d, e, m, n, p, q, may be taken at pleasure, fo that d2 be greater than 4: then will the two feries, requifite to carry on the approximation, that is, any term of

p+qx
1-dx+ex2

1+m+dm + n.x + dam+dn—em .x3,

be derived, by divifion, from

„ + nx
I-dx+ ex2

and

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Ex. 1.

&c.

being divided by the correfpondent term of the feries,

+ p + dp +q.x + d2 p + dq — ep.x2, &c. (derived

the quotients fucceffively taken both ways (as well of the de

a

be=

15
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fcending powers of x as of the afcending) will approximate to thofe roots as limits. and If g 5, d= 1, e = −1, m=1, n=2, p=2, =3; the equation and feries will be the fame 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 fame equation (11 x2—15x+5=0) will be approximated by the quotients of the corresponding terms of the feries, arifing as above expreffed; but the branches of the feries which in that example approximated the lesser root, will in this example approximate the greater; the quotients of the coefficients of 1 + 2x 1+ 3x the increafing powers of x, in the feries derived from I-x-x2 approximating the greater root; and the quotients of the coefficients of the decreafing powers of x, in the feries fo derived, approximating the leffer root.

and

OBSERVATIONS OF SOME OCCULTATIONS OF FIXED STARS
BY THE MOON; MADE AT DANTZIC, BY DR. WOLF, F. R. S.
WHICH SEEM TO POINT OUT A VÉRY PARTICULAR PHENOMENON.
Tranflated from a Latin paper, communicated by Mr. MAGELLAN, Fellow of the
fame Society.

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Immersion by the Clock

Clock too flow

Equat. of Time fubt.

Apparent time

"At 14 h. 43', mean time, the emerfion was certainly over; the star being at too

great distance from the moon's limb.

Occultation

Occultation of 2d by the Moon.

July 5th. Immersion by the Clock

Apparent time

14 h. 39' 47"

14 35 32 7

Clouds prevented the emerfion from being seen. Thefe obfervations were made with a telescope of Mr. Short's, which magnified the diameter of the object 200 times. In all three, the ftar disappeared at the distance of more than a quarter of an inch and nearly half of one from the moon's limb; which was extremely well defined. This fpace or diftance, I found, by repeated trials, was run over by the moon in 7 or 8 feconds; 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 fame height as in our's. Perhaps the fudden difappearance of the ftar conftantly observed by aftronomers (being alfo feen with an irradiation or glare of light upon the disk) may be owing to the ufe of glaffes with small magnifying powers, as most commodious for thefe obfervations.

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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 afcenfions of two ftars being given; it is required to find their diftance from the meridian when their difference of azimuth is the greatest or least poffible.

11. QUESTION III. by NUMERICUS.

Three school-boys laid out equal fums of money in fruit-apples and oranges: they all paid the fame 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 firft boy was but 9; whilft the fecond had 18, and the third no fewer than 24 moreover, the difference between the number of apples and the number of oranges, purchafed by the firft boy, was the leaft that the queftion will admit of. What was the number of apples bought by each boy, and what did each fort of fruit coft them?

12. QUESTION IV. by J. P.

Given the vertical angle of a plane triangle, the line bifecting it, and the fides of a rectangle infcribed in the triangle, to conftru&t it.

13. QUESTION V. by Mr. R. ROBBINS.

Given, in a plane triangle, the vertical angle, the greater fide adjoining to it, and the fum of the greater fegment of the bafe and the perpendicular, to conftruct the triangle.

14. QUESTION VI. by Mr. JER. AINSWORTH.

If ABC, MNH be two circles, the centers of which
are O and E, and radii OK, EH: when the circle ---
MNH is wholly within the circle ABC, if EO2=
OKXOK-NH; or when it is not wholly with the

B

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circle ABC, if OE2=OK × OK-†NH; then if two fides, AC, AB, of a triangle ABC, infcribed in the circle ABC touch the circle MNH, the other fide BC will touch it likewife. It is required to demonstrate this property geometrically.

15. QUESTION VII. by Mr. BENJ. LEGEND.

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At 8, A. M. in lat. 44° 10' N. and the wind at N. N. W. we saw a fail, due north of us, with her starboard tacks on board, to which we gave chafe by running within fix points of the wind, on the larboard tack, until the chafe was on our beam: we then tacked, and ran on the ftarboard tack, clofe hauled as before, until the chase was again on the beam; when we tacked, and fo 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 run, we obferved that we always ran twice as far on the ftarboard tack as we did on the larboard. Our rate of failing, that of the prize, the number of boards we made to fetch her, and the diftance run on each are required.

The anfwers to thefe queftions must be fent to Mr. Baldwin's, in Paternofter-row, poft paid, before the 1ft of November, 1783.

AN ESSAY ON THE ORIGIN AND PROGRESS OF NATURAL PHILOSOPHY. (Concluded from our laft.)

IT is a circumftance which increases our admiration of this philofopher, that he was deficient in no one requifite calculated to form the great man. It is a thing too often confirmed by daily experience, that a perfonal acquaintance with men of acknowledged merit fcarcely ever fails to diminish exceedingly that refpect, which their writings have procured them. In books we see the author, but in private life the man alone appears, uninfluenced by that awe which the venerable prefence of the public inspires. The near view prefents bigotry, arrogance, affected fingularity, and many other imperfections, to which the ftate of mortality is fubject. What an infinite addition muft it be to the fame of Sir Ifaac Newton that when in the entire poffeffion of the enthufiaftic applaufe of mankind, who regarded him even in his life-time as a being of a fuperior order, what an addition muft it be to his fame, that at this inebriating height

he continued to poffefs every virtue. Modeft and unaffuming, he continued his enquiries with the fame cautious attention with which he began, and free from that avarice for reputation which induces philofophers to purfue in folitude the path to difcovery, he published his thoughts, and pointed out the way to follow his footsteps. But all his fucceffors have not been entirely able to remove the difficulties of which an explication is demanded of them.

Sir Ifaac Newton may juftly be ftyled the father of phyfical aftronomy, but the prodigious eclat of his difcoveries feems to have occafioned a temporary ftand in fome other parts of natural philofophy. The action of electricity began about this time to be particularly attended to, and chemistry, which from immemorial time had existed not as a fcience but as the bafis of various trades, had a century before been introduced into medicine by the active and extravagant Paracelfus, and fince culti

vated by various practicioners, chiefly. Ifaac's niece, relates the affair in the German. The well-known Robert following words: (Elemens de NewBoyle was one of the firft who pursued ton.) it from philofophical motives; but the foundation of our prefent theory is due to Tachenius, Becker, and Stahl. Among experimenters in this fcience, Profeffor Boerhaave of Leyden, ftands

first.

The latter end of the feventeenth, and the beginning of the eighteenth centuries form an era of difcoveries. The weight of the atmosphere difcovered by Torricellius, but not allowed by all the philofophical world, was deeifively fhewn by Boyle, by the help of the air pump, which he had greatly improved from an invention of Otto Guericke. The various refrangibility of the rays of light difcovered by New ton put an end to the attentpts of grinding lenfes of the figure of the conic fections; but the reflecting inftrument first perfected by him remained unattended for near fifty years. The velocity of light obferved by Romer, was confirmed and established by Bradley, who first observed the aberration of the fixed ftars. And the difcoveries of Hawkfbee and Stephen Gray are the foundation of our present electrical knowledge.

After the quotations concerning gravitation, which were pointed out in our last number, it will feem ftrange that the knowledge of this univerfal property of matter was really a difcovery of Newton, though fufpected and even known to fo many before him. In fact, the men who are bufied in accumulating a flock of ideas from the investigations of others are not often thofe who themfelves add the moft to the general mafs of knowledge by their difcoveries. Dr. Pemberton affures us, that Newton was not intimately acquainted with the works of his contemporaries, even when far advanced in life, and in the earlier part he might well be fuppofed to have overlooked thofe paffages of the ancients which might have led him to the knowledge of gravity. However, we are indebted to an accidental circumftance for this grand difcovery. Voltaire, who received his information from Mrs, Conduit, Sir

"Un jour en l'année, &c." One day in the year 1666, Newton being retired into the country, was led, by obferving fruit falling from a tree, into a profound meditation on the caufe which thus dratvs all bodies in lines, that if prolonged would pafs through the center of the earth. What, faid he to himself, is this force, which cannot poflibly arife from the falfely imagined vortices of Defcartes? It acts upon bodies in proportion to their maffes, and not after the ratio of their furfaces: it would act upon thefe fruits that have fallen, even if elevated thoufands of fathoms above the furface of the earth. If this be true, this force must act from the regions in which the moon is placed even to the center of the earth; and confequently this power, whatever it is, may be the fame which caufes the planets to bend towards the fun, and occafions the fatellites of Jupiter to gravitate towards his body. Now, it is demonftrated by inductions drawn from the laws of Kepler, that all thefe fecondary planets gravitate towards the centers of their orbits with a greater degree of force at a lefs diftance; that is to fay, with a force which is as the fquare of the diftance inverfely. A body placed in the orbit of the moon, and another at the furface of the earth, ought both to gravitate according to this law if the fuppofition be true.

In order to be affured that the force

which retains the planets in their orbits is the fame as that which caufes heavy bodies to fall to the earth, nothing more is required than proper admeafurement. It is only required to determine the space which a heavy body falls through in a given time near the furface of the earth, and what space a heavy body would fail through in the fame time at the region of the moon. The moon itfelf is this heavy body, and may be confidered as really falling during the whole time the revolves in her orbit. But the prefent bufinefs is not an hypothefis, the work of fancy, which may be adapted at pleasure to

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