Memoir concerning Trigonometrical Operations, the Refults of which depend on the Figure of the Earth. By M. LE GENDRE.

This memoir has an immediate reference to the series of triangles, in France, to be connected with those measured in England by General Roy: it contains the formulæ neceffary for reducing and calculating them, and for afcertaining the fituation of ftations on the furface of a fpheroid. M. LE GENDRE apprehends that, when the distances are confiderable, the ufual method of determining them, by perpendiculars to the meridian, may multiply errors; he would therefore calculate the pofition of each point by that of another in the fame triangle.

The first section of this paper contains the formula for reducing an angle to the horizon. Let A be the angle, a and B the depreffion of its legs beneath the plane of the horizon, which quantities, in cafes of elevation, will of courfe be negative; fuppofe R, the radius of the tables reduced to feconds, the logarithm of which is 5,314425; then if ap and 3=q,

2

the seconds, to be added to the angle A, will be

2

The next fection treats of the angle of depreffion or elevation of a point obferved, with refpect to the horizon of the place of obfervation. Let H be the height of the latter above the level of the fea, b that of the point obferved; D the distance between them; r the radius of a great circle, and a the depreffion H-h fought: then a R+R. For the logarithm of r • D M. LE GENTIL affumes 6,515439, which is a mean between the logarithm of the radius of the curvature of the earth, and that of the radius of the fection at right angles with the meridian.

D

21

With respect to refraction, the academician obferves that, by feveral experiments,, he is induced to allow for it a fourteenth part of the diftance of the place obferved, expreffed in degrees and minutes of a great circle. Thus, if the distance be 14000 toifes, the refraction will be th of a degree, or

In order to determine how much the fum of the angles of a triangle, reduced to the horizon, are greater than two right 'angles, the following formula is propofed. Let a be a fide, A its oppofite, and B and C its adjacent angles; the area of the a fine B fine C triangle will be and the excess fought will be

2 fine A

the conftant logarithm 1,982525 must be added to that of twice the area. Thus in an equilateral triangle, the fide of which is 200co toifes, the fum of the angles is greater than 180°, by 3". If then, a third part of this excefs be fubtracted from each angle of any triangle, the fphericity of which is infinitely small, it may be confidered and computed as rectilinear.

The next fection relates to the value of the degrees of the meridian on a fpheroid. M. LE GENDRE fupposes the axis to be 1 and 1+a: but, as the quantity a is very small, and not exactly known, its fquare may be neglected; let b represent half the lefs axis of the fpheroid, M the forty-fifth degree of the meridian, and the ratio of the circumference to the diameter; then b—M (1− a) and the value of a degree, in any given latitude L, will be M (1— a cof. 2 L); fo that, when the latitude increases one degree, the degree of the meri

180

These formulæ are followed by others for determining the relative pofition of places on a spheroid: but, as these cannot be made intelligible without diagrams, we muft refer our readers to the memoir, and fhall only give an abridged view of the author's manner of applying his corrections to the position of Dunkirk, Calais, and Dover.

The value of a degree of the meridian, in forty-five degrees of latitude, is estimated at 57028+10 w toifes: the correction wis, in all probability, rather affirmative than negative, and cannot exceed two units. The difference between the two

1+3

320

axes of the fperoid a=- ; there is the utmost reafon to fuppose that a does not exceed 20: but should it be 5, ß would yet be less than an unit. The bafe, from Dunkirk to Hondfcot, reduced to the horizon, is 8167,4+0 toifes: the error cannot be above a toise: but as all the fides of the triangles are affected by this error of the bafe, their logarithms must be increased by 5320; or by 530, if they contain only fix places of

decimals.

The latitude of Dunkirk fteeple is 51° 2′ 10′′: but M. LE GENDRE prefers the divifion of a minute into a thousand parts, and expreffes it 51° 2, 167+1000 x: this correction cannot be greater than four feconds: its longitude from the royal obfervatory of Paris, with a fimilar correction, is 2',367 +1000 y; and its azimuth, weftward from Calais, expreffed in the fame manner, is 102° 59′,867+1000 z: but this error cannot amount to more than 15′′.

Nn 2

Though

10%

1-50°

176°

11%

/57° / 16 %/

Though these elements are accompanied by indefinite cor rections, yet the author obferves that they may be computed by logarithms, as if they were known: the difference of the tables will shew what must be added to each logarithm, or to each corresponding number, for the indefinite part: thus the logarithm of the fine of 102° 59′,867.+1000 z. is 9,988728-29 z, the part 29 z being expreffed, like the logarithm, in decimal units of the fixth place: or, if the logarithm of an arc, expreffed in minutes, be 1,497427-1498-76w+530—292+157x, the arc itself will be 31,436-1085w+40-2z+11x.

These corrections, applied to the church of Notre Dame at Calais, and to the northern tower of Dover Castle, are as follows:

Lon. of the church 29,217-108,3-5+49-22+11x-10008 557,523 + 53+ w—8—6z+1000x

Latitude,
Azimuth, eastward
of Dunkirk,

Longitude of the

}:35,743+ 848+44—38—998z—15×

northern tower +0,928-2183-11+80+32+22x-1000y

of Dover Castle,

Latitude,

Azimuth, eaftwd
of Calais,

}

7,813-48--12z+1000x

+653′,334+1698+8-60-30x- 1002Z

The remainder of the memoir contains practical directions for measuring with accuracy; for which purpose, iron bars are recommended in preference to chains; and for determining the direction of a coaft with refpect to the meridian, by means of a spherical triangle, formed by the zenith and pole with the place obferved. For thefe, and the corrections of the observations of the polar ftar, we must refer to the work itself.

MEDICINE.

Obfervations on the Effects of Mephitic Vapours on Man, fe cond Memoir. By M. PORTAL.

This memoir is intended to explain the immediate causes of the death which these vapours occafion. It is written in a very pompous manner: but the information which it contains is either very trite, or very doubtful. Of thofe pages which are employed in proving that the mephitic air affects animals br means of their lungs, and produces a kind of apoplexy, we ca only fay that we have often been told fo before: but when the author, because he finds that laudanum, poured on the heart of a frog, ftopped its palpitation, tells us that the mephitic vapour is carried into the blood by the pulmonary veflels, and on being tranfmitted to the heart, acts on it in a fimilar man

ner, we must observe that his premises do not warrant his conclufion.

Account of a Work concerning Hofpitals. By M. LE ROY.

This article contains a detail of the defects of the old Hotel Dieu at Paris, and a plan of a new hospital, in which each ward is a separate building. We cannot but applaud the humane and judicious attention which the author beftows on every circumftance that may contribute to alleviate the diftrefs, to promote the comfort, and to restore the health, of those, for whole accommodation the building is intended.

The claffes of Mathematics and Political Oeconomy contain each only one memoir. The former of these is by M. LE GENDRE, on the Integration of Equations with partial differences. Of this memoir we cannot give a fatisfactory account, without greatly tranfgreffing the limits of this article; and shall therefore only obferve that the ingenious academician ascertains the general cafes, in which partial differential equations are integrable, either by means of an ordinary differential of the first order with two variable quantities, or by two equations of three

variables.

The other Memoir is, A Continuation of the Effay to afcertain the Population of France; by Meffrs. DU SEJOUR, DE CONDORCET, and DE LA PLACE: which does not admit of any abridgment.

The volumes closes with a memoir, fent by the Royal Academy of Sciences at Montpellier, entitled, Obfervations on the Oxygenated Muriatic Acid. By M. CHAPTAL. The utility of this acid in whitening linens, cotton, and wax, has already been pointed out by M. Berthollet. M. CHAPTAL fhews that it is equally ferviceable in whitening paper. An hundred weight of pulp, defigned for blotting paper, being thus whitened, the expence of which was not more than seven per cent. was found to be increafed in value twenty-five per cent. He also recommends it for cleaning old books and prints, and fays, it renders them as fresh as if they were just come from the prefs. The muria tic acid may be oxygenated in a very cheap and eafy manner, by mixing it, when diluted, with manganese, in a very strong glafs bottle, which must not be quite filled; air bubbles will then be formed on its furface, whence a greenish vapour will rife after standing a few hours, the acid may be diluted with water, and ufed.

This article has been protracted to a greater length than we at firft intended. The only apology which we can make, is our wish to gratify our readers with all the information that can tend to promote the interefts and extend the limits of fci

Nn3

ence;

ence; and, in this refpect, we trust that the variety and importance of matter, in the volume which we have reviewed, will sufficiently plead in our behalf.

Sow.

ART. III. L'Education de Henri IV. &c. ie. The Education of Henry IV. By M. D. of Béarn. With fix Copper-plates, defigned by Marillier, and engraved by Duflos the Younger Two Volumes. 8vo. About 230 pages in each. Paris. 1791.

THE

'HESE Volumes, which are neatly printed and adorned with very elegant engravings, were written in humble imitation of Xenophon's Cyropædia: but, though not deftitute of merit, they admit of no comparifon with this moft excellent model. The narrative part, which is compiled from various memoirs and hiftories of the period to which it carries us back, commences with the birth of the hero whom it celebrates. His mother, Jeanne D'Albret, when pregnant, left her husband in Picardy, and travelled to Pau in Béarn, the refidence of her father the King of Navarre; where the was delivered of Henry. The good old man, left the fhould bring forth a crying peevith child, infifted on her finging a fong while in labour. With this ridiculous requeft the complied, and the king took on him the care of having the infant properly educated. On the birth of Jeanne D'Albret, the Spaniards, in allufion to the arms of Béarn, which are two cows, faid the king's cow had brought forth an ewe but on the birth of Henry, he gave them the retort courteous, by obferving that his ewe had brought forth a lion. On the death of his grandfather, the prince was brought to the court of Henry II. and left there by the queen under the care of La Gaucherie his tutor; this excellent man, for as fuch he is reprefented, died when Henry had attained his thirteenth year. After this event, his mother carried him down to Béarn, and committed his education to Florent Chrétien; and, on his taking poffeffion of the government of Guienne, appointed Beauvais to fuperintend his conduct. The latter was murdered in the horrid maffacre of St.Bartholomew. With this event, and with the feigned converfion of Henry and the Prince of Condé to the Roman catholic religion, the author rather abruptly concludes. That part of the work which relates to the education of Henry, confifts of conferences between him and his inftructors; these turn chiefly on well known hiftorical anecdotes ; and the reflections, to which they give occafion, however juft, are by no means remarkable either for their depth, or for their novelty.

The author profeffes an abhorrence of perfecution, which we believe to be fincere; and, with great liberality, does justice