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theory of its motions with nearly the fame exactness that the motions of the other planets are fettled.

The appearance of this planet, when viewed with the naked eye, or a fmall telescope, is not greatly different from that of a fixed ftar of the fifth or fixth magnitude, being fomething lefs bright than N. 132 of Taurus in Flamited's catalogue: but when examined with a good telefcope which magnifies zoo fimes or upwards, it is far otherwife; as it then appears under a fenfible diameter, and its light is more diluted than that of the fixed ftars.

With respect to its diameter, we are told that Dr. Mafkelyne eftimates it at 3 or 4 feconds: the obfervations of Mr. Herfchel, printed in the Philofophical Tranfactions for 1781, vary from about 3 to 5". M. de la Lande has calculated that if its apparent diameter be 3, its real diameter will be about 28,000 miles, or 3 times that of the earth: we may, therefore, conclude from the obfervations of Dr. Maikelyne and Mr. Herfchel, that its real diameter is not lefs than 4 times that of the earth, and its real diftance near 1800,000,000 miles.

I cannot conclude this paper without remarking, and rather with concern, that foreign aftronomers feem to fet their faces against the name which the ingenious difcoverer of this planet has thought proper to give it; though, at the fame time, they are not agreed amongst themselves in this matter. M. de la Lande will call it the PLANET OF HERSCHEL; M.Bode, of Berlin,proposes

to call itURANUS; M. de Sivry, CYBELE; and M. Profperin, of Upfal, NEPTUNE. M. de la Lande acknowledges that the three latter gentlemen have no reason. for what they propofe; and perhaps their propofitions might be made before they knew that Mr. Herfchel had aligned any name to it. M. de la Lande has not that excufe; but he alledges that gratitude to the author of fuch a rare difcovery, and the ardour which immortalizing his name, by calling the planet after it, will give to other learned men to pufue his fteps, in advancing the fciences, are his motives for it. I must confefs I have no idea of that gratitude which leads us to op pofe, in the moft direct manner, the wifhes of the perfon that we pretend to exprefs it for: and I conceive fei ence will be moft effentially encouraged if we can excite other monarchs to fol low the example of our most gracious fovereign, in rendering the lives of thofe eafy and happy here, whofe labours and difcoveries are of themselves fufficient to perpetuate their names hereafter, and in enabling them, in the moft liberal manner, to purfue their ftudies for the advancement of science. Mr. Herschel's name will not want the aids, M. de la Lande propofes, to perpe tuate it. The names of Galileo and Caffini would have been in no more danger of perifhing than they now are if the Satellites of Jupiter and Saturn had ftill retained the names of Medicean and Lodovicean ftars, as they wished them to do.

Perhaps M. de la Lande may be mistaken. A very facetious, if not a very pious divine, of our own country, has repeatedly affured us that the Chriftian religion, according to his cal culations, founded on the rate at which it has decayed in the courfe of the last 50 years, cannot poffibly last above 50 years longer. In confequence, I fuppofe, when he was at Rome, he made a very reverend bow to the statue of Jupiter, which still remains in the Pantheon; at the fame time defiring the dormant thunderer would take notice he had paid him that piece of refpect when his fortunes were at a very low ebb, and therefore hoped it would be remem bered to him for good" if ever his godhip came into play again. Is it not poffible thefe ingenious gentlemen may entertain fufpicions of a like kind, and theretore are paying their court to thefe gentry, that they may be received into their kingdom" at their restoration ?

DEMONSTRATIONS OF SOME PROPERTIES RELATING. TO RIGHT-ANGLED PLANE TRIANGLES. BY MR. JOSEPH KEECH.

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PROPOSITION I.

ET ABC be a plane triangle, right angled at B; and let the fquares ABLK, BCDE be defcribed on the two legs AB, BC; alfo let the ftraight lines AD, CK be drawn from the two acute angles to the oppofite angles D and K of the

the two fquares, cutting the legs of the triangle in F and H: I fay that BF fhall be equal to BH and each of them to the fide of a fquare HBFZ, inscribed in the triangle ABC.

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BF. Draw FZ parallel to AB, and, confequently, to CD, alfo join HZ. Then because the triangles AZF and ACD are fimilar, DE: FB:: AD: AF: CD: FZ. Hence as CD= DE, FZ FB HB; confequently HZ is equal and parallel to F, and the figure HBFZ is equilateral. More-K over the angles at B and F being

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right angles by conftruction, the oppofite ones at Z and H are right angles alfo, and the figure HBFZ is a fquare.

PROPOSITION II.

Q. E. D.

The fame things remaining as in the laft propofition: I fay that BF (BH) is a mean proportional between the other fegments AH and FC of the legs of the triangle ABC..

DEMONSTRATION.

The triangles KAH, CBH are fimilar, as well as the triangles ABF, DCF; therefore AK (AB): BC:: AH: BH:: BF: FC; and therefore, BH being equal to BF by propofition I. AH: BH:: BH (or BF): FC.

PROPOSITION III.

Q. E. D.

If the fame conftruction remain, and if the fquare HBFZ be circumfcribed by the circle HBFZ, meeting the fide AC of the triangle again in G; and if GB, GH, and GF be drawn: I fay that the angles FGC, FGB, BGH, and HGA are each of them equal to half a right angle.

DEMONSTRATION.

The angles AGB, and BGC are each of them right angles by Euc. III. 31. Now the angles, HFB and BHF are each of them half a right angle, becaufe the angle HBF is a right angle, and BH-BF. Hence the angles HGB and BGF, which ftand on the fame arcs with them, are each of them half a right angle: and if thefe be taken from the two right angles AGB, BGC, there will remain the two half right angles AGH and FGC.

PROPOSITION IV.

Q. E. D,

The fame conftruction remaining; if BG and DC be produced out until they meet in R: I fay that FC and FB, BH and HA, FG and GH, CG and GB, alfo RC and CD are all in the ratio of the given legs of the triangle BC, AB.

DEMONSTRATION.

Because of the parallel lines AB, ZF; BC, HZ, the triangles ABC, AHZ, and ZFC are fimilar; and the triangles ABC, BGC, and BCR, are fimilar by Euclid VI. 8. Moreover, because the angles HGF and BGC, are right, and the

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angles GHF and GBC ftand on the fame arc, GF, the triangle HGF is alfo fimilar to the triangle BGC, &c. Hence BC: BA:: FC : FB (=FZ) :: BH (=HZ): HA :: FG: GH:: CG: GB:: RC: CB, or CD. Q. E. D.

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The fame conftruction ftill remaining; I fay that the lines AD, CK interfect each other in the perpendicular, BG, Ict fall from the right angle, B, upon the fide, AC.

DEMONSTRATION.

The alternate angles PAB, and PDR being equal and alfo the vertical ones APB and DPR, the triangles APB, DPR are fimilar; and, by Prop. IV. RC: CD:: BH: HA; confequently CH paffes through the point P. Q. E. D.

**A line from H to F is omitted in the figure.

MATHEMATICAL QUESTIONS.
QUESTION I. by SLOKE.

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From the equation x5+ · x3 + rx2 + sx + t=0, in which r, s, and s are fuppofed given, it is required to find the value of x?

QUESTION II. by ASTRONOMICUS.

Suppofing the right afcenfion and declination of a ftar to be given, as alfo the right afcenfion of another star; it is required to determine the declination of this last, fo that the difference of their velocities in azimuth may be the greateft poflible when they are upon the fame vertical circle, in a given latitude.

QUESTION III. by Mr. WILLIAM KAY.

To determine a point in a given hyperbola which is neareft to any given point in the oppofite hyperbola.

QUESTION IV. by RUSTICUS.

Given the area, one of the angles, and the difference of the including fides of a plane triangle, to conftruct it.

QUESTION V. by CAPUT MORTUUM.

To furround a fifh-pond of a given area, and in the form of a given trapezium, with a walk of a given area, and of the fame breadth every where, by a geometrical conftruction.

N. B. This is Prob. IX. Newton's Universal Arithmetic, edit. 1720.

QUESTION VI. by Mr. J. WALSON.

Two numbers (47 and 59) prime to each other, being given; to find the least multiple of each of them, exceeding by unity a multiple of the other.

QUESTION VII. by Mr. JAMES WEBB.

What is the declination of that ftar which has the greatest altitude poffible 3h 37' after it has paffed the meridian in latitude 51° 31' N.

QUESTION VIII. by N. T.

Sailing N. N. W. I came in fight of two iflands, the one bearing N. and the other W. After running 8 miles, I found myfelf equally diftant from them, and when I had run 3 miles farther I was in a right line with them: it is required to find my distance from these two iflands at each time of fetting them.

The anfwers to thefe queftions are requefted to be fent (poft paid) to Mr. Baldwin in Paternofter-row, London, before the 1st of October, 1783; as none can be inferted that come to hand after that time.

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ON THE FUNERALS OF THE ANCIENT BRITONS..

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Hæc noffe, et dulce et utile.

HE ancient Britons and Gauls, we are well affured, burned the bodies of their dead, and after this ceremony, interred the remains in urns, a cuftom, which, in all probability, they borrowed from the Romans.

In many of the barrows, which are to be found in almost every part of this kingdom, thefe urns are frequently difcovered. Thofe of our ancestors are eafily to be diftinguished from thofe of the Romans, as the former are of a rude make, and formed of coarse materials, while the latter are remarkable for the elegance of their fhape, the neatness with which they are made, and the ornaments with which they are de

corated.

The ancients fometimes compofed thefe urns of very coftly materials, as Homer informs us that Patroclus's was made of gold. Thofe of filver, brass, marble, glass, and pottery ware, however, were the more common. They were tricked out with ribbands, flowers, and filk. Lycurgus, however, confined thofe of Sparta, to the fober drefs of olives and myrtles.

Thefe urns are generally found in the middle of the barrow, and even near the edge, as Dr. Williams has informed us, in the Philofophical Tranfactions, for the year 1740. This circumftance is fuppofed to have been occafioned by a fecond interment; when the skirt of the barrow alone was opened, that the remains, first intombed, might not be difturbed. Sometimes, however, it should feem, that a whole family was buried in the fame barrow, as feveral urns have been found placed near one another."

Thefe urns are moft commonly inclofed in little ceils, formed of flone, in order to defend them againft all preffure.

The bones, however, before they were depofited, were burned, almost to afhes, and particularly the larger ones. By these means, they were, in fome meafure, freed from the filth and pol

VARRO.

lution which follows our mode of burial. When the bones were thus reduced, the urn was filled with them, and whatever could not be crouded into it, was placed round, and covered by the barrow.

There are many inftances of bones confiderably larger than thofe of the human body, being found in thefe heaps of ftones. Let not thefe be fuppofed to be the remains of giants, but rather of horfes, as those animals, as well as the arms of foldiers, were laid on the funeral pile: an honourable diftinction, which could only be claimed by the Equites, as the foot-foldiers were not permitted fo great an indulgence. At the funeral of Patroclus, we are told that

"Four fprightly courfers, with a deadly groan, "Pour forth their lives, and on the pyre are thrown."

Pope's Homer. Iliad xxiii. 209.

The bones were clofely confined in the urns, by earth placed over them, and fometimes they were cemented with mortar, to prevent the admiffion of the air, or any impure mixture. Achilles, in Homer, orders the bones of his friend Patroclus to be covered with a double coat of fat:

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Sometimes the bones are found not above half confumed, which may, perhaps, enable us to diftinguish the barrows of the rich and virtuous, from thofe of the poor and profligate. For we are informed by Suetonius, in his life of Tiberius, that the body of that tyrant was ordered to be half burned in the amphitheatre. In all probability then, where the bones are found in any quantity, unconfumed, the barrow was erected over fome perfon of low condition, or whofe vices had rendered him odious. On thefe accounts, the fune ral was carelessly attended, and the remains gathered haftily together. This treatment of the dead, indeed, might be occafioned by the hurry and confufion of war, as well as by the difrefpect which arifes from vice and tyranny.

On the contrary, however, where there are evidences, that the fire was ftrong, and of long continuance, fo that not only the bones, but even the armour and all the various trappings which decorated the pile, and fet off

"The laft fad honours that await the dead," are confumed, we may infer that the deceased were either of high quality, or fuch, as by their virtues had rendered themselves beloved and refpected. For the funeral obfequies were performed in these cafes with all poffible care, and the fires watched, till all the

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smaller bones were entirely confumed, and the larger were not put into the urn, until blanched quite white. But this could be produced only by a long and fierce flame, which every method was taken to raife and preserve. On this account, Achilles intreats the affiftance of the deities, when he finds the funeral pile of Patroclus burn flow ly, as Homer tells us in the following beautiful allegory:

Smokes, nor as yet the fullen Hames arife;
"Nor yet the pile, where dead Patroclus lies,
But, fait befide, Achilles flood in pray`r,
Invok'd the gods whole fpirit moves the air.
And victims promiled, and libations caft
To gentle Zephyr and the Boreal Llaft:
He called th' aerial pow'rs along the kies
To breathe, and whisper to the fires to rife.
The winged Iris heard the hero's call.
And infiant haften'd to their airy hall,
Where, in old Zephyr's open courts on high,
Sat all the blutt ring brethren of the iky.

She thone amidst them, on her painted bow
The rocky pavement glittered with the fhow.
All from the banquet rife, and each invites
The various goddess to partake the rites.
"Not fo (the dame reply'd) I hate to go
To facred Ocean and the floods below:
E'en now our folemn hecatombs attend,
And heav'n is feafting on the world's green end,
With righteous Ethiops (uncorrupted train!)

Far on th' extremeft limits of the main."
But Peleus' fon intreats, with facrifice,
The Western fpirit, and the North to rife;

Let on Patroclus' pile your blast be driven, And bear the blazing honours high to heav'n.' Pope. Il. xxiii. 236.

JUVENAL.

ON EDUCATION. Quibus artibus, et quibus hunc tu Moribus inftituas. 'N the tablature of Cebes, Life is painted under the form of a fpacious manfion, of which infancy forms the entrance. Fancies and opinions, as infinite in their number, as they are various in their purfuits, are defcribed attending the gate of this dwelling, in order to engage the notice, and attract the affections of every franger who approaches; while a good genius teaches them to difcriminate between truth and faifehood, and points out the appearances which are fallacious, and thofe on which they may depend.

In our infant ftate, it is the duty of our parents to perform the part of this LOND. MAG. July 1783

genius. The great difficulty, however, is to adopt a proper method for conveying this advice. Aufterity and rigour fhould not be equally exerted against the good and the bad, or the generous and the froward. As a want of method and measure in punishment, very frequently when the fuppreffion of vice has been intended, have inculcated a diftate for virtue. For of virtue, the inherent attractions are in themfelves without meretricious ornaments, or fecondary motives, fufficient to lead the hearts of youth to noble actions, and to incite them to pursue with ardour the paths of learning.

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