an expreffion, and make its fluxion equal to nothing, in order to determine the maximum, or minimum, without fhewing that it contains one in fact. The author himself, has fhewn feveral exceptions in his treatife upon fluxions, and yet he takes not the leaft notice of that, in this paper; on the contrary all his fuppofitions are vague, and by no means decifive.

Article 103. The invention of a general Method for determining the Sum of every 2d, 3d, 4th or 5th, &c. term of a Series, taken in order; the Sum of the whole Series being known. By Thomas Simpfon, F. R.'S.

The author fays, As the doctrine of feries is of very C great ufe in the higher branches of the mathematics, and their application to nature, every attempt tending to extend that doctrine, may justly merit fome degree of regard. The fubject of the paper, which I have now the honour to lay before the fociety, will be found an improvement of fome confequence in that part of fcience. And how far the bufinefs of finding fluents may, in fome cases, be facilitated thereby, will appear from the examples fubjoined, in illuftration of the general method here delivered.'

He then proceeds to fhew by examples, how the method he proposes may be refolved; but we fhall only obferve, that the author's intention seems to have been, to fhew a different way of demonftrating Mr. Cote's theorems, in his Harmonia Menfurarum. It feems to be a favourite topic of the author's; for he has treated this fubject in a different manner, in feveral of his works; and in fome of them, extended it a great length: it has likewife been treated by many others (but by none fo elegantly as by the inventor himself), and continued by Dr. Smith, the publisher of this work. It is true, the general problem was given without a demonftration; because it depends on the divifion of the circle, and is therefore eafily performed as Mr. Demoivre has fhewn in his Mifcellanea Analytica.

Article 110. Of the irregularities in the Motion of a Satellite, arif ing from the fpheroidical figure ofits primary Planet: in a letter to the Rev. James Bradley, D.D. Aftronomer Royal; F.R.S. and Member of the Royal Academy of Sciences at Paris by Mr. Charles Walmefly, F. R. S. and Member of the Royal Academy of Sciences at Berlin, &c.

Since the time, (fays the author) that aftronomers have been enabled by the perfection of their inftruments to determine with great accuracy, the motions of the celestial • bodies,

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bodies, they have been folicitous to feparate and diftinguish the feveral inequalities difcovered in these motions; and to know their caufe, quantity, and the laws according to which they are generated. This feems to furnish a fufficient motive to mathematicians, wherever there appears a cause 'capable of producing an alteration in those motions, to examine by theory what the refult may amount to, though • it comes out never fo fmall: for, as one can feldom depend fecurely upon mere guefs, for the quantity of any effect, it must be a blameable neglect entirely to overlook it, without being previously certain of its not being worth our notice.

Finding therefore, it had not been confidered what effect the figure of a planet, differing from that of a sphere, might produce in the motion of a fatellite revolving about it, and as it is the cafe of the bodies of the Earth and Jupiter, which have fatellites about them, not to be spherical, but ‹ spheroidical, I thought it worth while to enter upon the • examination of fuch a problem. When the primary planet is an exact globe, it is well known that the force by which the revolving fatellite is retained in its orbit, tends to the center of the planet, and varies in the inverse ratio of the fquare of the diftance from it; but when the primary planet is of a spheroidical figure, the fame rule then no longer holds the gravity of the fatellite is no more directed to the center of the planet, nor does it vary in the proportion above-mentioned; and if the plane of the fatellite's orbit, be not the fame with the plane of the planet's equator, the • protuberant matter about the equator, will by a conftant 'effort of its attraction, endeavour to make the two planes coincide. Hence the regularity of the fatellite's motion is neceffarily disturbed, and though upon examination, this effect is found to be but fmall in the moon, the figure of the earth differing fo little from that of a sphere, yet in fome cafes it might be thought worth notice; if not, it will be at least a fatisfaction to fee, that what is neglected can be of no confequence. But, however inconfiderable the change may be with regard to the moon, it becomes very fenfible in the motions of the fatellites of Jupiter, both on ' account of their nearer diftances to that planet, when com'pared it its femi-diameter, as alfo, because the figure of Jupite far recedes from that of a sphere. This, I have fhewn, and exemplified, in the fourth fatellite; in which cafe, indeed, the computation is more exact than it would be for the other fatellites; for as my firft defign, was to kamine only how far the moon'smotion could be affected

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by this caufe, I fuppofed the fatellite to revolve at a distance, fomewhat remote from the primary planet, and the difference of equatorial diameter, and the axis of the planet not to be very confiderable. There likewife arifes this other advantage from the prefent theory, that it furnishes means to fettle more accurately the proportion of the different forces which difturb the celeftial motions, by affigning the particular fhare of influence, which is to be afcribed to the figure of the central bodies round which thofe motions are performed.

I have added at the end a propofition, concerning the di⚫urnal motion of the earth. This motion has been generally esteemed to be exactly uniform; but as there is a caufe that muft neceffarily fomewhat alter it, I was glad to examine what that alteration could amount to. If we first fuppofe the globe of the earth to be exactly fpherical, revolving about its axis in a given time, and afterwards conceive that by the force of the fun, or moon, raifing the waters, its figure be changed into that of a fpheroid, then according as the axis of revolution becomes a different diameter of the fpheroid, the velocity of the revolution • mult increase or diminish: for, fince fome parts of the ter raqueous globe, are removed from the axis of revolution, and others depreffed towards it, and that in a different proportion as the fun, or moon, approaches to, or recedes from the equator, when the whole quantity of motion, which always remains the fame, is diftributed through the fpheroid, the velocity of the diurnal rotation cannot be conftantly the fame. This variation, however, will scarce be obfervable, but as it is real, it may not be thought amifs to determine, what its precife quantity is.'

After this, the ingenious and learned author, proceeds to demonftrate his propofition, in the moft clear and eligible. manner, which the fubject will admit. We recommend this paper as a mafter-piece of the kind,

The other Papers in this publication of the Philofophical Tranfactions, will be confidered in a future article.

M

The Safety and Perpetuity of the British State, under the influence of Political and religious Zeal. Being the fubftance of feveral Difcourfes preached before the University of Cambridge during the late Rebellion and prefent War. By W. Wefton, B. D. Fellow of St. John's College, Cambridge. 8vo. 3s. 6d. Whifton, &c.

THE

HE author's defign in this work, is to rouze the attention of his countrymen to their true intereft and fafety; to infpire them with zeal for their religion and their liberties. How he has conducted himself in the execution of his defign, will appear from a fhort view of what he has advanced, His performance is divided into thirteen chapters; in the two first of which he fhews, from a detail of many eminent examples in hiftory, the exceeding great difficulty of conquering any nation, which is zealous for its government and religion. But because it is not enough to thew what qualities we ought to have, unlefs the means of attaining them are likewife pointed out, he proceeds to this part of his design in the third chapter, wherein he treats of the exceffes of ridicule in this kingdom: and,

The first advice he gives us is, to refolve to be SERIOUS; for fimple as this remedy may feem, he fays, it will in the end effectually root out, one of the most dangerous maladies that has infected the state, viz. that profufion of wanton and indifcriminate banter, which has taken poffeffion of the appetites, the reafon, and the heart. The affections of men chained down to earth, and devoted to fenfe, are not more averfe, we are told, to heavenly things, than the prefent age, abandoned to laughter and ridicule, is abhorrent of fedate and fober relection. That this is not paradox but fact, Mr. WESTON fays, will appear inconteftably to those who view with difcerning eyes, the manners and cuftoms, the diverfions and employments, the books and converfation, of those who file themselves the more liberal part of the ENGLISH people: a people, that were once dignified with the manly epithets of fedate, open and fincere; but are now become light, artificial, and disguised. He goes on:

If fedatenefs and fobriety fhould in any places, or, on any occafion, fhine forth with diftinguished luftre, one would think it should be at the tribunals of law, and in all things. - relating to the church and the state. Yet, in the first of these, what wantonnefs and levity, nay, what farce and buffoonery are often feen! How do the advocates of the respective parties frolic it with each other, and make it in a manner their principal

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principal contention who fhall moft cruelly fport with property, reputation, and, in fome particular occafions, with even life itfelf! This is too often verefied.

In regard to the affairs of ftate, no conduct of the ftatef man, we are told, tho' ever fo confummately wife, and no zeal of the patriot, tho' ever fo exuberantly warm, can escape the strokes of obloquy and fcorn. If a Proteftant king is courted, Proteftantifm is made a jeft; if a Catholic, we laugh at all true religion.-If a good treaty is made, it was made only to be broken; if a bad, it is just what ought to be expected from BRITISH politicians.--If foreigners are called in, the diftrefs is vifionary, and their aid fuperfluous; if our own ftrength is relied on, the distress is dreadful, and never to be furmounted by the filken fons of luxury and floth.-If our foldiery are kept at home, the defign is to comply with their original inftitution, for oftentation, and parade; if they are fent abroad, it is wantonly to employ them where honour will be our difgrace, and where victory will be our ruin.If an invafion is feared from an inveterate enemy, the fear arifes from imaginary fleets and armies in the clouds; if we invade them, it is only a piece of the moft extravagant Quixotifm against walls of brafs.

As to religion-How, fays he, is the SACRED inftitution of TYTHES become fubject to banter, and the TENTH of the INCREASE made the fong of the drunkard, and the jest of the profane! The doctrine of the NEW BIRTH, is a point as clear from the Scriptures, as that the Scriptures themselves are true; and yet how cautiously do we tread on this obnoxious ground! How is the RENOVATION of the CARNAL MAN unfelt, and ridiculed by this degenerate age! and how are fuch as infift on it, ignominioufly claffed with thofe that cant in the conventicle, or bawl out in the field?-The terrors of HELL, and the tortures of the DAMNED, are vilified, and fet at nought; and the FIRE and BRIMSTONE which were once fo terrific, are toffed about like the arrows of SOLOMON's fool, and with the fame reflection, Are we not in port?

Our author concludes his third chapter with obferving, that this train of light and ludicrous reflections, does not only proceed from thofe who are both Infidels in principle, and trangers to the wife and generous fentiments of the ancient lawgivers, who held thofe in execration who infulted the eftablished religion, but likewife from numbers who feem to be in earneft in the profeffion of their faith, and would take it ill not to be thought fo; nay, from those too who would

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