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MATHEMATICS.

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On the Analysis of indeterminate Equations. By M. LE GENDRE.

This memoir is divided into four parts. The first treats of those indeterminate equations which admit of an infinite number of answers in whole numbers, and which may be represented by the general form Ay= ax + bx tox &c. M. LE GENDRE confines his enquiries, chiefly, to the particular equation Ay =*-B which affords many remarkable theorems.

The second part describes the uses of the analysis of indeterminate equations in finding the divisors to equations. This is a curious as well as an useful part of the present abftrufe memoir ; a direct method of finding the divisors has long been a dehideratum in Algebra ; but though M. Le Gendre's contrivances are truly ingenious, and will afford much fatisfa&tion to the speculative algebraift, it muft, nevertheless, be confessed, 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 series.

The third part contains a theorem for ascertaining the posfibility 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 extensive field where the human faculties may range at large in the pursuit of truth, these speculative enquiries have led the way to the noblest and moft useful discoveries. Logarithms might be produced to confirm the assertion: they were discovered by the speculative NAPIER, while he was investigating the curious properties of numbers in general, without any particular view for fhortening 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 singular success, has not left the demonftrations of those theorems which he bas given. Among these propositions M. LE GENDRE particularly remarks the two following; every number is composed of three triangular numbers at most; and, every prime number of the form 8n - 1, is of the form pa t q? + 2r?, or, what comes to the fame, its double is the Jum of three squares. He observes, with regard to the last, that

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it by no means characterizes those primes which are of the form 8n-1, for there is no uneven number, fimple or com pounded, which may not be of the form på +2° + 2r?, or which may not be, at the same time, of the two forms pé+qa to?, and p'+ q*+2r?, except only such numbers (whether primes or not) as are of the form on-1, which never can be of the form pa t qtir?, but which are always of the form p?tq + 2r2. The demonstration of these propofitions is ingenious, and it is, on account of the many dedu&ions that may be made from it, no less useful than curious.

The memoir closes with four tables, containing the different divisors of the number t + au, a being a prime, and of the form either 8n - 3, 8n+1, 8n +3 or 8-1. In a minute dea scription 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 specified; and the whole memoir, containing 94 pages, is replete with curious and interesting properties of different kinds of numbers.

ASTRONOMY, concluded. Observation on the Moon at the Time of its Pasage over the Me

ridian, about two Hours before the Occultation of several Stars in the Pleiades, December 13, 1785. By M. JEAURAT.

From this observation, which seems to have been accurately made with good inftruments, it appears that on Dec. 13, 1785, at 9' 59' 7" true time at the Paris observatory, 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 solar tables, M. JEAURAT has found that these last 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 also relates to the difference observable in the right ascenfion and declination of the different ftars in the Pleiades in 1748, 1755, and 1785: hence the results of this ob. servation are subservient (which is a circumstance that the aca. demician has not mentioned) to the determination of the obliquity of the ecliptic and of the preceflion of the equinoxes. Occultations of certain Stars of the Pleiades by the Moon, on

April 11, and December 13, 1785. By M. MessieR. The occultations on the Inih of April not being announced in the ephemerides, were in part paft, before M. MESSER applied himself to his telescope. He has given the true times of the immersions of seven of the stars.

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The Connoissance des Temps predicted, that on the 13th of December, the moon would eclipse 37 stars of the Pleiades. The greatest part of them were telescopic stars, and even these were invisible, at that time, through a large magnifier, because of the great light of the moon, which was then near the full. Twelve observations of immersions or emerfions are recorded, beside 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 cables. Observations 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 observatory at the time of his departure with M. DE LA PEYROUSE, on their voyage round the world. The valuable observations, published 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 observations did not commence before 1785; fo that the present memoir contains a series of observations which connects M. Cassini's with those of the preceding French astronomers.

M. D'Agelet had been long engaged in a set of observations on the fixed stars, with a view to give a complete catalogue of those in the northern hemisphere; these observations are omitted in this collection, and the reason assigned for the omiffion is, that M. D'AGELEt purposes to renew his observa

* They set sail in July 1785. In March 1786, they were at Chili; and in January 1787, at Macao. Thus far we have the aothority of the book before us. According to the public prints, we learn, that M. de la Peyrou se left Macao in February, 1787, and proceeded to Manilla ; where he made a short stay, in order to take in provisions and water. From Manilla he set sail on the 9th of April, and paffing eastward of Formosa, he directed his course between Japan and the peninsula of Corea, advancing as high as 52 degrees by a channel unknown to European navigators ; it is formed by the coafts of eastern Tartary on one side, and by two large islands on the other. M. de la Peyrouse found the northern extremity of this channel fo ob. structed by shelves and fands, as to render the passage through it impracticable; he therefore resumed his course fouthward; and in 46 degrees of latitude, found a strait that brought him into the sea weft of the Kurile Isands; and he sailed thence to Avatska, on the south side of the peninsula of Kamschatka: where he arrived on the 6th of September. This account was brought by M. de Lesseps, Vice Conful of Cronstadt, and interpreter of the Ruffian language on board M. de la Peyrouse's ship. He left the officers an crews of the two thips in perfect health, on September the 30th, 1787.

tions on his return, and to give the result 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 above 4 millions are recorded.

Any abridgment of the observations would be impoffible. They chiefly confift of the time of the planets' passage over the meridian, and their zenith distance at that time. Many observations on tbe fun feem to have been made with a view to enJarge or at least verify what was advanced, concerning the duration of the year, in the memoirs of the academy for 1782*. On the Opposition of the new or feventh Planet. By M. LE

MONNIER. From the observations recorded in this memoir, it is deduced that the new planet was, on Jan. 7th, 1785, 116 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"1. Occultation of Venus by the Moon, April 12, 1785. By M,

MESSIER.
As these observations are useful for correcting the errors of
the planetary tables, we fhall transcribe the fix following:
Immersion of the enlightened edge of Venus of 1' 50'true time

of the center of the disc, computed 2 36
of the last horn of Venus

3 8 Emersion of the enlightened edge of Venus 50 28

of the center of the disc, computed 51 18
of the last horn of Venus

52 5 Other Observations of the Conjunction of the Moon with Venus,

April 12, 1785. By M. LE MONNIER. These subsequent observations were made on the passage 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 passed the meridian, M. LE MONNIER makes the Moon's long.

1' 7° 33' 59" lat, north 4° 46' 43" Venus's long. 1

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lat.

4 33 29 Observations on the Comet of January 1785, made at the Marine

Observatory. By M. MESSIER. After making some observations on Herschel's planet, on January 7th, M. Messier looking at some fixed stars, discovered the comet of which he here treats. M. MECHAIN had also

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

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seen the comet nearly at the same time, from the royal obfer: vatory at Paris. The Ahort appearance of the comet, viz. from Jan. 7, to Jan. 16, did not afford much time for many observacions. During this fort interval, however, M, MESSIER made 32 observations of its places, by comparing it with fixed stars of the conftellation of the Whale, through which it passed. 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 ascensions and declinations of 25 fixed stars used in ascertaining these places.

This makes the twenty-third comet which the vigilant M. Messier has observed at the marine observatory, and the 71st whose orbit has been calculated. Its elements are as follow: Place of the ascending node

85 24 12 15 Inclination of the orbit

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12 Place of the perihelion

3 19 51 56 Log, of the perihelian distance

0.0581975 Mean time of the perihelion at Paris, 1785, Jan. 2747 58'4" Motion direct, Observations 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 ascensions and declinations of 19 fixed fars, with which it was compared. The elements are: Place of the ascending node

2' 4° 44' 40" Inclination of the orbit

7 Place of the perihelion

9 27 34 30 Log. of the perihelian distance

9.631024 Mean time of the perihelion at Paris, 1785, Apr. 84 119 29 0 Motion retrograde.

POLITICAL. Continuation of the Elay for ascertaining 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 results are, that, comparing this with the preceding year, the births have increased 17,707, the marriages have increased 1,196; and the deaths diminilhed 65,050. The general result for the year 1784 stands thus :

Birchs, 965,648; marriages, 229,827; deaths, 887,155. The numbers for 1783 will be found in our last Appendix, p. 618.

Having now gone through those memoirs which we had not before noticed, we all resume the confideration of those which we had barely enumerated in our former article,

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