felves, at the end of two years, on a valuable confideration, to deliver up, if the king fhould require it, Pietro "Santa to the Gencefe, provided the city of Genoa fhould at that time be under his majefty's command. That the ambassadown the 10,000 ducats agreed upon in the 'dors fhould pay capitulation of Florence, and the republic fhould have jewels in pledge for the reftitution of the caftles; which should be forfeited, if under any pretext whatsoever, they were not • restored that at the time they were given up they should lend the king 7000 ducats; for the payment of which four principal officers of the kingdom of France fhould become bound; that they fhould fend thefe 7000 ducats into the kingdom of Naples, and, according to inftructions, distribute them among the king's forces: that, provided they were not engaged in a war in Tuscany, they fhould fend two hundred and fifty men at arms into the kingdom of Naples, who fhould not be obliged to stay any longer than the end of the • month of October; that they fhould have a general indemnity and an immediate restoration of their effects: and that, for a fecurity of the performance of these articles, they should fend fix hoftages, of the principal citizens of Florence, at the king's choice, who fhould remain for a certain time at • his court.'

Novara was in this time reduced to the utmost distress: at length a way was opened for an accommodation, to which neither party were in reality averfe; the allies knowing the money received of the Florentines had been fent into Swifferland to raise new levies. Commiffioners were appointed on both fides; after many meetings, thofe on the part of France brought the final refolution of the confederates, as to the terms on which they would agree; the principal articles of which were, that there fhould be peace and friendship between the king and the duke of Milan, but without any prejudice to the duke's other alliances; that his majefty fhould give orders to the citizens of Novara, to deliver up the town to the duke of Milan, and his troops fhould evacuate the citadel; that all places taken in the war fhould be restored: that the king might equip at Genoa, his fief, what veffels he pleafed, and make ufe of any military ftores, provided they were not employed in favour of the enemies of that state; and for the fecurity of this article, the Genoefe fhould give hoftages: the duke of Milan fhould procure him the reftitution of the veffels taken at Rapalle, and the twelve gallies. detained at Genoa, and at his own expence fit out two large Genoa hips, which, with four of his own, were to be fent

to

to the fuccour of the kingdom of Naples, and the year following be obliged to have three more in readiness : that he fhould give a free pass to any troops the king fhould fend, ⚫ through his dutchy, on condition, that no more than two hundred lances at a time fhould pafs; and in cafe the king "returned himself, the duke fhould then accompany him with ' a certain number of men: that the Venetians, for two months, might have the liberty of acceding to this peace; and if they did, fhould be obliged to withdraw from Naples, ' and give no affiftance to Ferdinando; if afterwards they vio'lated their engagements, and the king on that account declared war, the duke then fhould affift him, and might keep poffeffion of whatever part of their territory he could make himself mafter of: that Lodovico, within the month of March enfuing, fhould pay 50,000 ducats to the duke of Orleans, for the expences he had incurred during the fiege of Novara, remit 80,000 of the money he had lent the king when he first paffed through his dominions, and allow a farther time for the payment of the remainder: that all prifoners fhould be fet at liberty: that no obftruction should be made to the Florentines in taking poffeffion of their forts, nor any disturbance given to them afterwards, in the pofleffion of them: that the caftilletto of Genoa fhould for two years be put into the hands of the duke of Ferrara, who 'fhould take his oath to deliver it to the king of France any time within that term, in cafe the duke of Milan did not 'perform the articles of this treaty: Ludovico, on figning the peace, fhould give hoftages to remain with the king, till the 'caftilletto was delivered to the duke of Ferrara.

These articles were strongly objected to by feveral of the French officers, but, after a long and warm debate, were accepted, and ratified by Charles; who, about the end of October 1495, returned over the mountains to France, more like a vanquished than a victorious prince.

Our author, whofe knowledge of mankind is abundantly confpicuous in the many judicious reflections he has interspersed in the courfe of his hiftory, finifhes his fecond book; which alfo concludes the first volume of this tranflation, with taking notice, that the earlieft appearance of the venereal difeafe happened during this period.

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T.L.

ART.

ART. 111. A new and comprehenfive method of investigating the parallactic angle, without regard to the nonagefimal degree: with fome few obfervations on the lunar theory. In a letter to the Rt. Hon. the Earl of Macclesfield, prefident of the royal fociety. 4to. Is. Sandby.

T

HE fubject of this fmall, but ingenious performance,

is of the greatest importance in aftronomy, as the method of finding the parallactic angle, by inveftigating the longitude and altitude of the nonagefimal degree, is a very intricate and laborious task.

It was the opinion of the great Dr. Halley, and which experience has fince confirmed, that the only probable means of, difcovering the longitude, was the having a correct theory of the lunar motions; and in order to this he has given us a complete feries of obfervations, compared with calculations from the tables, by which means the numbers may generally be corrected with fuch precilion, as not to err a minute in longitude: fo that the lunar aftronomy feems to want nothing to make it the defired means of obtaining the longitude, but fhorter methods of calculation. The inveftigation of the parallactic angle is one of the moft difficult and tedious of the whole; but by the method laid down by this learned author, it may now be obtained with very little trouble, tho' with the fame degree of exactnefs as before. The method of obtaining this angle, we fhall give in the author's own words.

One of the principal advantages derived from the method of constructing folar eclipfes or ocultations geometrically, is, that we have the phænomenon transferred from the heavens to the earth, and we fee at one view the progrefs of the lunar path over the difk, the parallels it approaches to, or interiects; the points of interfection, and the diftance of any particular place on the globe from thofe points, as alfo the longitudes and latitudes of the places where the lunar fhadow enters or leaves the earth's difk; all which may be ge-⚫ometrically determined from the projection. We have also in this projection a scheme of the earth's afpect in refpect of the fun or ftar for the time given; and can fee represented to us, how every place on the hemifphere before us defcribes its own peculiar path in its paffage over the disk, and in refpect of which the fun or ftar, on the plane of whofe univerlan Horizon the projection is made, is feen to rife, culminate, or fet, according to the different pofitions of the place given on the plane. The angle too, which the poles of the earth, and the ecliptic, make upon the plane of the

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+ Spencer lowper, Year of Durham

horizon given, is reprefented as well as that which the axis of the ecliptic makes with a vertical circle drawn thro' the ⚫ center of any particular place. This angle is the complement of that, which, in fpherical trigonometry, is called the parallactic angle: and the fides, which the triangle fubtends by ftreight lines, drawn from the center of a given place on the diurnal path, one to the center of the disk, and the other to the axis of the ecliptic, cutting it at right angles, together with the intercepted portion of the axis of the ecliptic; thefe form the parallaxes in altitude, longitude, and latitude, and bear fuch proportion, as is well known to ⚫ the whole disk, as the parallaxes themselves bear to the hori⚫zontal parallax of the moon in the heavens.

The having fome time ago, this comprehenfive view before me in a conftruction of the late folar eclipfe; and obferving that the parallactic angle here, tho' fo intricate and perplexed in the trigonometrical calculation, was fo clearly and fimply laid down in the geometrical construction, being but a part of a plain right-angled triangle; of which, one fide, as well as the right-angle were already known; I could not help forming fome hopes, that this angle might be folved, and by it the whole triangle, without the operofe method of inveftigating the altitude and longitude of the 90th degree, thro' fo many spherical theorems as were neceflary for that purpose. Some opportunities that offered of reconfidering this point fully confirmed them; for I found, on examining carefully the projection before me, that the angle fought for was always either the fum or dif⚫ference of those angles which are formed by the vertic line ' of the place at the center of the difk with the prime meridian, and the angle of the prime meridian or pole of the earth with the axis of the ecliptic. Having thus far fucC ceeded, I foon difcovered that the angle at the center might be obtained in the fame method of calculation the parallactic angle is, when the altitude and place of the 90th degree are given, with the diftance of the moon from it; only affuming the compliment of the moon's or fun's declination, inftead of their diftance from the pole of the ecliptic: and that to, or from the angle found, adding or fubtracting the angular distance of the poles, as the cafe required, would give the angle whofe complement fhould be the parallactic ⚫ angle fought.

The very conftruction will fhew, that when the northpole is projected on the east-fide of the axis of the ecliptic, if the vertical line, paffing thro' the center of the place given,

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given, is weftward of the prime meridian; that then the angle fought is the difference of these angles; that is, to fpeak more intelligibly, when the moon or ftar is eastward of the prime meridian; but when the planet is weftward, or when (which is the fame thing) the meridian of the place lies to the eastward of the prime meridian, then the fum of the two angles is the angle fought. The contrary is to be obferved, when the axis of the earth lies to the weft'ward.'

This compendious method the author has illuftrated by three examples, both in and out of the fyzigies, to shew its certainty and comprehenfiveness; from whence it evidently appears to be fufficiently exact for all the purposes relating to that important problem, the finding the longitude at fea.

After fhewing the method of finding the parallactic angle, the author proceeds to make fome obfervations on the lunar tables, in order to render the calculus more perfect and fhorten the method of performing it; in which he has shewn that all the equations given us by Sir Ifaac are well founded.

Few fubjects have given rife to more difputes than the famous theory of the moon, delivered by the immortal Sir Ifaac Newton; but it has been found that most of those who pretended to difcover errors in it have been unequal to the task. And as truth receives an additional brightnefs from a rigid infpection, fo Sir Ifaac's theory, from a close examination, has appeared with a double luftre. Some time fince M. Clairaut pretended that the Newtonian law of attraction was inconfiftent with the motion of the moon's apogee; but this was prefently contuted by the learned M. de Buffon. And M. Clairaut, on re-examining his calculations publickly retracted his opinion; for he found it owed its rife to an error of his own. The great M. Euler owns that he was formerly of the fame opinion with M. Clairaut, that the theory did not agree with the motion of the apogee of the moon; but after making the most tedious calculations, he found, to his fatisfaction, that the theory was entirely fufficient to account for that motion. We would not however be understood to mean that this theory is abfolutely perfect; what we contend for is, that all the equations given by Sir Ifaac are truly founded, and confequently that to omit any of them, as fome have of late pretended to be neceflary, is to abandon truth; and that whatever corrections may be neceffary they must be expected from enlarging it in what is ftill wanting to complete it, and not by mutilating or taking away any part of that which this great author has deduced from the unerring laws of nature.