Herschel's obfervations relative to Venus and Saturn, and Miss Caroline Herschel's account of a new comet, being the fixth she has discovered; this, however, was seen at Paris, by citizen Metfier, on the 17th of September..

P. Piazzi, an astronomer of Palermo, in Sicily, has published the description of a fuperb instrument, a circle of five feet diameter, constructed by Ramsden, and alfo an account of the observations made with it. These a e contained in a volume in folio, entitled: " Della Specola astronomica de regi ftudi di Falermo, libri quatro de Giuseppe Piazzi, C. R. regio profeffore d' Astronomia, &c." In this important work, we find the exact latitude of the Obfervatory of Palermo to be 38° 6'44"; and its longitude 44' 3" caft of Paris. One is surprised to discover in this work, that, in a country lying fo far to the fouthward, the sky, fo ferene and agreeable to mankind, should be but little. favourable to the astronomer, during eight months in the year. Citizen Dangos, who has refided a long time at Malta, which is in latitude 36°, has made fimilar remark respecting it.

a

The astronomers of Milan have finished the triangles for their grand meridian as far as Genoa, and measured the base; but they have not as yet received the large sector, with which they hope to be able to measure the celestial

arc.

Doctor Slop, astronomer at Pifa, has continued his observations from 1782' to 1786, with the calculations dependent on them; and M. Kltigel, profeffor at Halle, has published in the Memoirs of the academy of Gottingen, some enquiries relative to the perturbations of planets. Mr. Wurm, of Nurtingen, in Wirtemburg, has entered into a laborious examination of the diameters of planets, a fubject relative to which there is much uncertainty: for example, the diameter of Saturn is 11" according to M. Bugges, 13" according to M. Zach, and 20" according to Herschel.

M. Barry, aftronomer, at Manheim, has, until of late, carried on his observa tions with zeal and affiduity; but the bombs and bullets of the French army have nine different times struck the ob fervatory, which is one of the most remarkable and elevated objects belonging to that city: on this account the instru ments were all difmounted, and fent beyond the mountains of Suabia.

The revolution at Geneva, in 1794, has not interfered, in the least, with the

labours of its obfervatory; Marcus Augustus Pictet Turtin (born on the 23d July, 1752), who is entrusted with the superintendence, hopes to render it ferviceab'e to the cause of aftronomy, and he has come to Paris, exprefly for the purpose of procuring new instruments.

The professors Tralles, of Berne, and Haffler, have measured triangles and bases at Arau, in the canton of Soleure, in order to connect the chart of the cantons of Berne, Bafle, and Soleure, with that of France. From their obfervations, we learn, that the latitude of the steeple of Berne is 46° 56' 55"; and that it is firuated 20′ 25′′ to the east of Paris.

M. Schroeder has constructed a 25feet telescope at Lilienthal, wear Bremen, which equals his expectations; and M. Schrader, of Kiel, in Holstein, another of 26 feet. M. de Hahn, an opulent individual of Mecklenburg, has received a 20 feet telescope from Mr. Herschel, of a fuperior quality, which he has fixed at his house, at Kemplin, near Hamburgh; the small mirror is omitted, according to Herschel's method; this was proposed in France, fo early as 1728, by Lemaire. See, "Recueil des Machines approuvées par l'Académie." TOM. VI.

M. Bode, the celebrated astronomer of Berlin, who publishes an ephemeris annually, has just added a fupplementary volume, which is faid to contain observations of confiderable importance: this circumstance has induced me to study German, and also to request, that a professor of that language may be added to the establishment of the college of Prance.

The obfervations of P. Fixlmillner, from 1776 to 1791, have lately appeared, under the title of " Alta Aftronomica Cre "mifanenfia," but we have lost the author. Placidus Fixlmillner was born May 29, 1721, at the castle of Achleuthe, near the abbey of Benedictines, at Cremfmunster, in Austria. He studied philofophy at Saltzburgh, in 1735, and became attached to mathematics; but on his entering into the order of Benedictines, he was diverted for several years from his favourite pursuits, by theology and canon law Luckily, however, in 1761, being then in his 40th year, he was permitted to cultivate astronomy, on account of the tranfit of Venus across the fun: alone, and buried in the folitude of a remote province, at a diftance from cities, academies, and the learned; or, in other words, from all objeets which sustain courage, and excite emulation, 1796.]

emulation, this amiable man dedicated his life to the most abstruse parts of the science. He was very ferviceable to me when I published my tables of Mercury, and was one of the first who calculated the orbit of the planet Herschel, for which he also constructed tables. The various orders of friars, hitherto useless to mankind, have an opportunity, in those countries where they are still permitted to exist, to be of some little service to the world, by following the noble example of the convent of Cremfmunster.

Astronomy has experienced other losses, during the present year; in particular, we have reason to regret Bailly, Du Sejour, and Saron. On the 14th Brumaire (14th Nov.) died the citizen Flecheux, author of an ingenious planisphere, and of a geocylic machine (Machine Géocylique) for representing the parallelism of the earth's axis: he was 55 years old. On the 21st Brumaire (11th November) perished Silvain Bailly, hose éloge I have already published. On the 3d Ni.. vose (23d December) Philip Lesne, my relation and pupil, died of a disease contracted while serving his country, in the marshes of La Vendée. His death is a great loss to astronomy. On the 8th Nivofe (28th December) P. M. T. Lebrun fuffered on a public scaffold. He lived for some time at the observatory, but he foon embarked in other pursuits, and rose to the head of the Foreign department. He, however, brought up his younger brother, Achilles Tondu, to astronomy; in consequence of which he accompanied the ambassador, Choiseul Gouffier, to Constantinople, and died there, in 1787, only 28 years of age, after having made a variety of remarks extremely useful to geographers, respect ing the country as far as the mouth of the canal that communicates with the Black Sea. The Turks would not permit the French to make observations at Trebisonde, and Sinope; the English and Ruffians also opposed this plan: befides this, we about the same time loft the two best-informed mussulmans belonging to the whole empire One of them was the Visir Halib-Pacha, decollated at Tenedos. He had formed a school for the artillery and engineers, and-caused our elementary treatises to be tranflated for their instruction. The

other was the Vice-Admiral CapitanaBey, whose head was struck off in October 1787; he was in poffeffion of excellent instruments, and had published

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my Abridgement of Astronomy in his vernacular tongue. Since the death of Tondu, M. Jumelin, a physician, M. Chevalier, and M. Racord, a pilot on board a French brig, have made a few observations at Constantinople; but in order to fix, pretty nearly, the exact position of the eastern part of the Black Sea, at the same time with the fouth of the Caspian, citizen Beauchamp has been sent into Perfia, at my folicitation, and he has been appointed conful at Mascate, in Arabia, which will enable him to furnith us with still more important materials.

On the 7th Ventose (25th of February) was executed the ci-devant Baron de Marivetz, who had been employed in a work, called " La Physique, du Monde," published between 1780 and 1787, in 7 vols. 4to. His youth was spent amidst the diffipations of a court, and he had not applied himself to literature until a period of life, when old habits are renounced with great difficulty. Vols. II. and III. are dedicated to astronomy.

Citizen Saron, in his 64th year, fell also a victim to that tribunal of blood, which spared neither science nor virtue. His sole crime appears to have been, the poffeffion of a large fortune; in addition to this, he was formerly first president of the late parliament of Paris. He was received into the Academy, in 1779, and was extremely useful to us, more especially in the calculation of comets, all those observed for several years, were calculated by him, and that, too, with a most astonishing facility. He procured instruments at a great expence, and lent them to men of science, with an exemplary generofity.

To the other losses sustained during a tyranny of nine months, I may fairly add that of Lavoisier, who perished on the 19th Floreal (May 8th), and Walle who fell on the ninth Thermidor (Jay 27th).

We have also to regret M. Niewland, of Leyden, who had composed an interesting work on Nautical Aftronomy, which the Dutch stood in great need of, as this branch of fcience is too much neglected in their country. He had spent a whole summer in the grand observatory belonging to M. Zach, at Gotha, and we expected great things from his zeal and skill.

The last misfortune of this kind, in the course of 794, was in the person of citizen Achilles Peter Dionis du Sejour, of the ci-devant Academy of Sciences, the Academies of London, Stockholm, 4 B2 and

and Gottingen, and councellor of the great chamber of the late parliament of Paris. He was born in this capital, January 11th, 1734, and studied at the college of Jesuits, from 1743 to 1750. He was admitted in the Academy as an associate in 1765, at which time, his brethren in the parliament pretended he could only fit as an honorary member; but he despised the suggestions of vanity, and deemed himself honoured by belong. ing to a body of learned men, under any denomination

On this occafion, he undertook a feries of labours, which he followed up during thirty years, with equal affiduiry and success; this was the application of the algebraic analysis to all the branches of astronomy, and especially to eclipses. Aftronomers have always neglected analyses too much; the obfervations and calculations necessary to produce results, demanding so much time, that they would have little or no leifure for ab

stract speculations. Du Sejour is the first who addicted himself entirely to this branch of science, and he made an important application of it, in determining the longitudes of a great number of towns by means of the eclipses of 1764 and 1769.

In confequence of a Memoir, written by me, respecting the comets which had affrighted alt France in 1773, he drew up a Treatise on this subject. He published it in 1775, and exhibited a mode of calculating the orbit of a comet, by means of three obfervations; this is one of the most difficul problems in aftro-' nomy. In this work, he demonstrated, how difficult it was to conceive the encounter of a comet with the earth, in the order of probabilities, or even in poffibilities for he went so far as that. 1 know that such an affertion ought to be accompanied by restrictions, but it was necessary to dispel terror, and nothing could be more useful than a publication of this kind, in order to comfort the public.

The difappearance of Saturn's ring, which ha pens once every fifteen years, induced D Sejour to publish a volume in 8vo, in 1726, on this fubject. In 1786 and 1789, he completed two large 4to volumes of his works, under the title of "Traité Analytique des Mouvemens apparens ues Corps célestes."

It was in the audit of labours such as these, notwithstanding every appearance of a robust constitution, that he was at

SIR,

THE letters of your correfpondents A. SEARCH and No CONJURER, revived some early impeffions made on my mind, in the course of my youthful studies; and I was excited to re-examine the difficulties, which I had encountered in a fcience, in the endeavour to obtain the comprehenfion of a mode of reasoning, by which such wonders are faid to be performed.

In the course of this pursuit, Mr. FREND'S Algebra was lately put into my hands; and I found myself in the situation of those perfons, whom he deferibes in his preface, as having waded " through a tew chapters of Moclaurin's "Algebra; but frightened, and with "good reason, at Cardan's Rule," and, confequently, unable to proceed farther in that part of my mathematical studies. There was no great difficulty, indeed, in comprehending Cardan's process: but when I came to the application of it to practice, I do not know whether it fucceeded once in the equations which I formed at random; and I was told by the initiated, that it would not do unless two impoffible roots were in the equation: how to make these impoffible roots, or to discover whether they were in any proposed equation, I was totally at 2 lofs.

As the rule was demonstrated to me, x was made equal to afb and then was told, that as only one fuppofition had been, another might be made, namely, that 3ab might be equal to q Mr. Frend denies this, and says, that zab can be equal to only in particular cafes; and brings as a proof, the equa. tion x2+27-28-0, in which x=1, consequently, a+b=1; and, therefore, Mr.

1796.)

Mr. Frend says, that, as a and b are both less than unity, zab cannot be equal to 27. If this is really the cafe, and I fee no means of contradicting it, the adoption of Cardan's Rule must lead every one who depends upon it, into continual error, unless there is some method pointed out by Algebraifts, which tells him, when he may apply the rule to a particular cafe, or when it fails. I have heard, indeed, that there must be two impoffible roots in an equation, to bring it under Cardan's method: but the procefs of finding them out, must make the rule very tedious and difficult of appli. cation

Again, Mr. Frend objects to the equation ufed in explaining Cardan's Rule, a+b+r=o, and calls it absurd: for, says he, three numbers added together, cannot be equal to nothing. Doubtless, according to his position, which does not admit of negative numbers, the expreffion is absurd: but I should be much obliged to some one of your correspondents to inform me, what is the real use of these negative numbers; and whether, if equations can be folved without them, the supposition should be admitted into a work of science? In Mr. Frend's book, various equations are solved, with out admitting them: the true solution is brought out by one root, when, accord ing to the common mode, two roots appear; and the learner is to try which of them is the true one. If this method may be pursued throughout the whole of the science, there seems to me to be fomething gained by simplifying the principles; but, before I give up entirely the old mode, I should like to be well informed, what loss will be sustained in the higher parts of algebra, by rejecting the negative quantity? for, to say the truth, it frequently puzzled me so much, that, though I can get through a quadratic equation, all beyond feems to me to be enveloped in impenetrable dark nefs and mystery.

I remain, fir, yours,

July 20, 1796.

ExOTERICUS.

QUESTION XIII (No. IV.).-Answered by J. F-r.

The difference between the true and apparent level, is the difference between the earth's femi-diameter and the secant of an arc of its circumference, whose length is the given distance. The versed fines of circular arcs are as the squares of their chords; the versed fines of small

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arcs are also nearly as the above-mentioned differences, and the arcs themferves nearly as the chords: therefore, the abovementioned differences, when the arcs are small, are as the squares of the arcs, quam proximè.

A mean of the principal measures of a degree of latitude, taken fince 1736, by Maupertuis, Caffini, Boscowich, Mafon, and Dixon; Bouguer and de la Condamine, de la Caille, &c. in different parallels, gives 69.076947 English miles, or 5526.15576 chains, which multiplied by 1.003, being about half the ratio of the equatorial diameter to the axis, gives 5542.7 chains, for a mean degree of a great circle; whose radius will, confequently, be 317579 chains. From hence, we have one chain. 649499 of a fecond, the difference between whose natural secant and radius is = (11)4957156; and this multiplied by 317579 gives .00000157429 of a chain, or .00124683 of an inch; from whence the derivation of the rule is easy.

The construction of a table from these data, is too obvious for explanation. It might be calculated for every 100 chains as far as necessary; but, as the first differences of the terms would not be equal, it would be necessary, if confiderable accuracy were required, to be prepared with a table of equations of second difference constructed upon the common theorem for its interpolation; so that, upon the whole, it seems better to calculate it for any particular case, from Mr. Waddington's rule, which will be somewhat nearer the truth if we put 1247 for 124, and cut off fix places instead of five. Or it will be the easiest way of any, by using the number 125, which is nearer than Mr. Waddington's, and being = one-eight of 1000, there. fore only cut off two figures, and divide by 8, or take the 800th part.

If we use logarithms, we shall get a rule which, I think, may be found somewhat shorter in its application, viz.

From double the logarithm of the distance in chains, substract 2.904193, and the remainder will be the logarithm of the difference in inches between the apparent and

true levels.

Either of the above rules, the last of which is nearest the truth, will do till the arc becomes so large as to render the error of the first hypothesis confiderable, which will not be the cafe within the limits of any ordinary operation of this kind. Should it be necessary to afcer