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MATHEMATICS.

On the Analysis of indeterminate Equations. By M. LE GENDRE. This memoir is divided into four parts. The first treats of thofe indeterminate equations which admit of an infinite number of answers in whole numbers, and which may be represented

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by the general form Ay= ax + bx +cx &c. M. LE GENDRE confines his enquiries, chiefly, to the particular equation Ayx-B which affords many remarkable theorems.

The fecond part defcribes the ufes of the analysis of indeterminate equations in finding the divifors to equations. This is a curious as well as an useful part of the present abftruse memoir; a direct method of finding the divifors has long been a defideratum in Algebra; but though M. LE GENDRE's contrivances are truly ingenious, and will afford much fatisfaction to the fpeculative algebraift, it muft, nevertheless, be confeffed, that his direct method will in some cases be more laborious than the common one by trial, or even than the solution by approximation, by means of infinite feries.

The third part contains a theorem for afcertaining the poffibility of indeterminate equations of the second degree.

The fourth is a collection of theorems on the properties of prime numbers. The immediate utility of contemplations on the properties of numbers may not perhaps be apparent to the fuperficial enquirer: they are, however, the bafis of all arithmetical operations. They frequently afford concise methods for fimplifying complicated calculations; and, independently of the pleasure they afford, by opening an extenfive field where the human faculties may range at large in the pursuit of truth, these speculative enquiries have led the way to the nobleft and moft useful discoveries. Logarithms might be produced to confirm the affertion: they were discovered by the fpeculative NAPIER, while he was inveftigating the curious properties of numbers in general, without any particular view for thortening the labours of trigonometrical or other computations to which they may be applied.

M. LE GENDRE regrets that M. FERMAT, who has cultivated the theory of numbers with fingular fuccefs, has not left the demonftrations of those theorems which he has given. Among these propofitions M. LE GENDRE particularly remarks the two following; every number is compofed of three triangular numbers at moft; and, every prime number of the form 8n-1, is of the form p2+q2 + 212, or, what comes to the fame, its double is the fum of three fquares. He obferves, with regard to the last, that

it by no means characterizes thofe primes which are of the form 87-1, for there is no uneven number, fimple or compounded, which may not be of the form p+q+2r2, or which may not be, at the fame time, of the two forms p+g+r2, and p2+q2+2r2, except only fuch numbers (whether primes or not) as are of the form 8n-1, which never can be of the form p2+q2+r2, but which are always of the form p2+q2+2r2. The demonftration of these propofitions is ingenious, and it is, on account of the many deductions that may be made from it, no less useful than curious.

The memoir closes with four tables, containing the different divifors of the number + au2, a being a prime, and of the form either 8n 3, 8n+1, 8n +3 or 8n-1. In a minute description of these tables, M. LE GENDRE has introduced a variety of curious remarks on the properties of those prime numbers which come under the forms above fpecified; and the whole memoir, containing 94 pages, is replete with curious and interefting properties of different kinds of numbers.

ASTRONOMY, concluded.

Obfervation on the Moon at the Time of its Paffage over the Meridian, about two Hours before the Occultation of feveral Stars in the Pleiades, December 13, 1785. By M. JEAURat.

From this obfervation, which feems to have been accurately made with good inftruments, it appears that on Dec. 13, 1785, at 9h 597 true time at the Paris obfervatory, there is an error in Mayer's tables of +39" in the moon's longitude, and of-10" in her latitude. In comparing MAYER's with DE LA CAILLE's folar tables, M. JEAURAT has found that these laft give 3". 5 more than MAYER'S.

On the Conjunction of the Moon with the Pleiades, Dec. 13, 1785. By M. LE MONNIER.

This, like the preceding memoir, points out the errors in the tables: and it alfo relates to the difference obfervable in the right afcenfion and declination of the different ftars in the Pleiades in 1748, 1755, and 1785: hence the refults of this obfervation are fubfervient (which is a circumftance that the academician has not mentioned) to the determination of the obliquity of the ecliptic and of the preceffion of the equinoxes. Occultations of certain Stars of the Pleiades by the Moon, on

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April 11, and December 13, 1785. By M. MESSIER. The occultations on the 11th of April not being announced in the ephemerides, were in part paft, before M. MESSIER applied himself to his telefoope. He has given the true times of the immerfions of feven of the ftars.

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The Connoiffance des Temps predicted, that on the 13th of December, the moon would eclipfe 37 ftars of the Pleiades. The greatest part of them were telescopic ftars, and even thefe were invisible, at that time, through a large magnifier, because of the great light of the moon, which was then near the full. Twelve obfervations of immerfions or emerfions are recorded, befide others of the moon's diameter, as measured by the micrometer. No remarks are added to this memoir, relative to the errors in the lunar or other tables.

Obfervations on the Planets, made at the Military School, in 1783. By M. D'AGELET.

This memoir is compiled by the Academy, from the journal which M. D'AGELET left in the obfervatory at the time of his departure with M. DE LA PEYROUSE, on their voyage round the world. The valuable obfervations, publifhed by Meff. LE MONNIER, DARQUIER, and Dr. MASKELYNE, are the models which the academicians have followed in arranging M. D'AGELET's remarks. M. CASSINI'S obfervations did not commence before 1785; fo that the prefent memoir contains a series of obfervations which connects M. CASSINI's with thofe of the preceding French astronomers.

M. D'AGELET had been long engaged in a fet of obfervations on the fixed ftars, with a view to give a complete catalogue of thofe in the northern hemifphere; thefe obfervations. are omitted in this collection, and the reafon affigned for the omiffion is, that M. D'AGELET purposes to renew his observa

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* They fet fail in July 1785. In March 1786, they were Chili; and in January 1787, at Macao. Thus far we have the authority of the book before us. According to the public prints, we learn, that M. de la Peyrouse left Macao in February, 1787, and proceeded to Manilla; where he made a fhort stay, in order to take in provifions and water. From Manilla he fet fail on the 9th of April, and paffing eastward of Formofa, he directed his course between Japan and the peninfula of Corea, advancing as high as 52 degrees by a channel unknown to European navigators; it is formed by the coafts of eaftern Tartary on one fide, and by two large iflands on the other. M. de la Peyroufe found the northern extremity of this channel so obftructed by shelves and fands, as to render the paffage through it impracticable; he therefore refumed his courfe fouthward; and in 46 degrees of latitude, found a ftrait that brought him into the fea weft of the Kurile Ilands; and he failed thence to Avatska, on the fouth fide of the peninfula of Kamfchatka: where he arrived on the 6th of September. This account was brought by M. de Leffeps, Vice Conful of Cronstadt, and interpreter of the Ruffian language on board M. de la Peyroufe's fhip. He left the officers and crews of the two fhips in perfect health, on September the 30th, 1787.

tions on his return, and to give the refult of his labours in a feparate work. This indefatigable aftronomer has often taken the places of upwards of 80 ftars in one night; and in his journal, which is yet in a very imperfect ftate, the places of ábove 4 millions are recorded.

Any abridgment of the obfervations would be impoffible. They chiefly confift of the time of the planets' paffage over the meridian, and their zenith diftance at that time. Many obfervations on the fun feem to have been made with a view to enlarge or at least verify what was advanced, concerning the duration of the year, in the memoirs of the academy for 1782*. On the Oppofition of the new or feventh Planet. By M. Lɛ MONNIER.

From the obfervations recorded in this memoir, it is deduced that the new planet was, on Jan. 7th, 1785, 11 43′ 4′′ in longitude 104° 13′ 25, and north lat. 0° 25′ 56′′. and on O&. 15th, 1784, at 5 48′ 11′′ in the morning, its right ascenfion was 107° 51′ 0′′ or 2′′, and its declination north 25° 51 27.

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Occultation of Venus by the Moon, April 12, 1785. By M,

MESSIER.

As thefe obfervations are useful for correcting the errors of the planetary tables, we fhall transcribe the fix following: Immerfion of the enlightened edge of Venus oh 1′ 50′′ true time of the center of the difc, computed

of the laft horn of Venus

Emerfion of the enlightened edge of Venus

of the center of the difc, computed

of the last horn of Venus

236
3 8

50 28

51 18

52 5

Other Obfervations of the Conjunction of the Moon with Venus, April 12, 1785. By M. LE MONNIER.

Thefe fubfequent obfervations were made on the paffage of the moon and Venus over the meridian foon after their conjunction, and they afford feveral useful deductions respecting the motions of these two planets. At 2° 53′ 4′′, when the moon paffed the meridian, M. LE MONNIER makes the Moon's long. I 7° 33′ 59′′ lat. north 4° 46′ 43′′ Venus's long. I 6 30 II lat. 4 33 29 Obfervations on the Comet of January 1785, made at the Marine Obfervatory. By M. MESSIER.

After making fome obfervations on Herfchel's planet, on January 7th, M. MESSIER looking at fome fixed ftars, difcovered the comet of which he bere treats. M. MECHAIN had alfo

* See Review, vol. lxxv. p. 503.
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feen

seen the comet nearly at the fame time, from the royal obfer vatory at Paris. The fhort appearance of the comet, viz. from Jan. 7, to Jan. 16, did not afford much time for many obfervations. During this short interval, however, M. MESSIER made 32 obfervations of its places, by comparing it with fixed ftars of the conftellation of the Whale, through which it paffed. A chart of the comet's path, and the fixed ftars with which it was compared, is annexed to the memoir, with two tables, one containing 32 apparent places of the comet, and the other, the right afcenfions and declinations of 25 fixed ftars used in ascertaining these places.

This makes the twenty-third comet which the vigilant M. MESSIER has obferved at the marine obfervatory, and the 71ft whofe orbit has been calculated. Its elements are as follow: Place of the ascending node 8s 24° 12′ 15′′ Inclination of the orbit

Place of the perihelion

Log. of the perihelian diftance

2 10 14 12

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19 51 56 27d7 58′4′′ 0.0581975

Mean time of the perihelion at Paris, 1785, Jan.
Motion direct.

Obfervations on the Comet of March and April 1785, made at the
Marine Objervatory. By the fame.

We have here 59 places of this comet; a chart of the comet's path; and the right afcenfions and declinations of 19 fixed ftars, with which it was compared. The elements are:

Place of the afcending node

Inclination of the orbit.

Place of the perihelion

Log. of the perihelian distance

Motion retrograde.

2° 4° 44′ 40′′

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2 17

7

9 27 34 30

9.631024.

Mean time of the perihelion at Paris, 1785, Apr. 8d 11h 29' 0"

POLITICA L.

Continuation of the Effay for afcertaining the Population of France. By M. DU SEJOUR, le Marquis DE CONDORCET, and M, DE LA PLACE.

This is a long series of tables and calculations, which relate to the year 1784, and of which the refults are, that, comparing this with the preceding year, the births have increased 17,707, the marriages have increased 1,196; and the deaths diminished 65,050. The general refult for the year 1784 ftands thus:

Births, 965,648; marriages, 229,827; deaths, 887,155. The numbers for 1783 will be found in our laft Appendix, p. 618. Having now gone through thofe memoirs which we had not before noticed, we shall refume the confideration of thofe which we had barely enumerated in our former article,

The

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