Herfchel's obfervations relative to Venus and Saturn, and Mifs Caroline Herfchel's account of a new comet, being the fixth fhe has difcovered; this, however, was feen at Paris, by citizen Meffier, on the 17th of September..

P. Piazzi, an aftronomer of Palermo, in Sicily, has published the defcription of a fuperb inftrument, a circle of five feet diameter, constructed by Ramfden, and alfo an account of the obfervations made with it. Thefe a e contained in a volume in folio, entitled: " Della fpecola aftronomica de' regi fludi di Falermo, libri quatro de Giuseppe Piazzi, C. R. regio profeffore d'Aftronomia, &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 furprised to discover in this work, that, in a country lying fo far to the fouthward, the fky, fo ferene and agreeable to mankind, fhould be but little favourable to the aftronomer, during eight months in the year. Citizen Dangos, who has refided a long time at Malta, which is in latitude 36°, has made a fimilar remark refpecting it.

The aftronomers of Milan have finifhed the triangles for their grand meridian as far as Genoa, and measured the bafe; but they have not as yet received the large fector, with which they hope to be able to measure the celeftial

arc.

Doctor Slop, aftronomer at Pifa, has continued his obfervations from 1782' to 1786, with the calculations dependent on them; and M. Klügel, profeffor at Halle, has published in the Memoirs of the academy of Gottingen, fome 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 Herfchel.

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

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

labours of its obfervatory; Marcus Au-. guftus Pictet Turtin (born on the 23d July, 1752), who is entrusted with the fuperintendence, hopes to render it ferviceable to the caufe of aftronomy, and he has come to Paris, exprefly for the purpofe of procuring new inftruments.

The profeffors Tralles, of Berne, and Hafiler, have measured triangles and bafes 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 fteeple of Berne is 46° 56′ 55′′; and that it is fituated 20' 25" to the eaft of Paris.

M. Schroeder has conftructed a 25feet telescope at Lilienthal, near Bremen, which equals his expectations; and M. Schrader, of Kiel, in Holftein, 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 houfe, at Kemplin, near Hamburgh; the fmall 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.

this

M. Bode, the celebrated aftronomer of Berlin, who publishes an ephemeris annually, has juft added a fupplementary volume, which is faid to contain obferva tions of confiderable importance: circumftance has induced me to study German, and alfo to request, that a profeffor 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 "Acta Aftronomica Cre"mifanenfia," but we have loft the author. Placidus Fixlmillner was born May 29, 1721, at the caftle of Achleuthe, near the abbey of Benedictines, at Cremf munfter, in Auftria. He ftudied philofophy at Saltzburgh, in 1735, and became attached to mathematics; but on his entering into the order of Benedictines, he was diverted for feveral 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 aftronomy, on account of the tranfit of Venus acrofs 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 ob jects which fuftain courage, and excite

emulation,

1.796.]

emulation, this amiable man dedicated his, life to the moft abftrufe parts of the fcience. 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 alfo conftructed tables. The various orders of friars, hitherto ufelefs to mankind, have an opportunity, in thofe countries where they are ftill permitted to exift, to be of fome little fervice to the world, by following the noble example of the convent of Cremfmunfter.

Aftronomy has experienced other loffes, during the prefent year; in particular, we have reafon to regret Bailly, Du Sejour, and Saron. On the 14th Brumaire (14th Nov.) died the citizen Flecheux, author of an ingenious planifphere, and of a geocylic machine (Machine Géocy lique) for reprefenting the parallelifm of the earth's axis: he was 55 years old. On the 21 Brumaire (11th November) perished Silvain Bailly, hofe éloge I have already publifhed. On the 3d Ni.. vofe (23d December) Philip Lefne, my relation and pupil, died of a difeafe contracted while ferving his country, in the mares of La Vendée. His death is a great lofs to aftronomy. On the 8th Nivofe (28th December) P. M. T. Lebrun fuffered on a public fcaffold. He lived for fome time at the obfervatory, but he foon embarked in other purfuits, and rofe to the head of the Foreign department. He, however, brought up his younger brother, Achilles Tondu, to aftronomy; in confequence of which he accompanied the ambaffador, Choifeul Gouffier, to Conftantinople, and died there, in 1787, only 28 years of age, after having made a variety of remarks extremely useful to geographers, refpect 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 obfervations at Trebifonde, and Sinope; the English and Ruffians alfo oppofed this plan be fides this, we about the fame time loft the two beft-informed muffulmans be

longing to the whole empire One of them was the Vifir Halib Pacha, decollated at Tenedos. He had formed a fchool for the artillery and engineers, and caufed our elementary treatifes to be tranflated for their inftruction. The other was the Vice-Admiral CapitanaBey, whofe head was ftruck off in October 1787; he was in poffeffion of excellent inftruments, and had publifhed

555

my Abridgement of Aftronomy in his vernacular tongue. Since the death of Tondu, M. Jumelin, a phyfician, M. Chevalier, and M. Racord, a pilot on board a French brig, have made a few obfervations at Conftantinople; but in order to fix, pretty nearly, the exact pofition of the eastern part of the Black Sea, at the fame time with the fouth of the Caspian, citizen Beauchamp has been fent into Perfia, at my folicitation, and he has been appointed conful at Malcate, in Arabia, which will enable him to furnish us with still more important materials.

On the 7th Ventofe (25th of February) was executed the ci-devant Baron, de. Marivetz, who had been employed in a work, called "La Phyfique 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 aftronomy.

Citizen Saron, in his 64th year, fell alfo a victim to that tribunal of blood, which fpared neither fcience nor virtue. His fole crime appears to have been, the poffeffion of a large fortune; in addition to this, he was formerly first prefident of the late parliament of Paris. He was received into the Academy, in 1779, and was extremely ufeful to us, more efpecially in the calculation of comets, all thofe obferved for several years, were calculated by him, and that, too, with a moft aftonifhing facility. He procured inftruments at a great expence, and lent them to men of fcience, with an exemplary generofity.

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

We have also to regret M. Niewland, of Leyden, who had compofed an interefting work on Nautical Aftronomy, which the Dutch ftood in great need of, as this branch of fcience is too much

neglected in their country. He had spent a whole fummer in the grand obfervatory belonging to M. Zach, at Gotha, and we expected great things from his zeal and kill.

The last mi fortune of this kind, in the course of 1794, was in the perfon of citizen Achilles Peter Dionis du Sejour, of the ci-devant Academy of Sciences, the Academies of London, Stockholm, 4 B 2

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 ftudied at the college of Jefuits, from 1743 to 1750. He was admitted in the Academy as an affociate in 1765, at which time, his brethren in the parliament pretended he could only fit as an honorary member; but he defpifed the fuggeftions 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 affiduity and fuccefs; this was the application of the algebraic analyfis to all the branches of aftronomy, and cfpecially to eclipfes. Aftronomers have always neglected analyfes too much; the obfervations and calculations neceffary to produce refults, demanding fo much time, that they would have little or no leifure for abftract fpeculations. 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 eclipfes of 1764`and 1769.

In confequence of a Memoir, written by me, refpecting the comets which had affrighted all France in 1773, he drew up a Treatife on this fubject. 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 difficult problems in aftronomy. 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 fo far as that. 1 know that fuch an affertion ought to be accompanied by reftrictions, but it was neceffary to difpel terror, and nothing could be more useful than a pubication 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 publifh a volume in 8vo, in 1776, 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 des Corps célefies."

It was in the adft of labours fuch as thefe, notwithstanding every appearance of a robust constitution, that he was at

tacked by a malignant fever, which his conftant uneafinefs fince the death of citizen Freteau, rendered more dangerous. He died on the 5th Fructidor (22d Aug.) in the 60th year of his age, at his country houfe, at Angerville, near Fontainbleau, which had formerly belonged to the famous Lord Bolingbroke,

His fimplicity was correspondent to his learning and virtues, for there was nothing in his drefs or ma ners, that announced the poffeffion of great knowledge, an exalted fituation, or a large fortune.

MATHEMATICAL CORRESPONDENCE. To the Editor of the Monthly Magazine.

SIR,

THE letters of your correfpondents

A. SEARCH and NO CONJURER, revived fome early impreffions 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 reafoning, by which fuch wonders are faid to be performed.

In the course of this purfuit, Mr. FREND'S Algebra was lately put into my hands; and I found myfelf in the fituation of those perfons, whom he deferibes in his preface, as having waded "through a few chapters of Maclaurin's Algebra; but frightened, and with

66

good reafon, at Cardan's Rule," and, confequently, unable to proceed farther in that part of my mathematical ftudies." There was no great difficulty, indeed, in comprehending Cardan's procefs: 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 uniels two impoffible roots were in the equation: how to make thefe impoffible roots, or to difcover whether they were in any propofed equation, I was totally at 8 lofs.

As the rule was demonftrated to me, was made equal to a+b and then I 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 fays, that 3ab can be equal to 9 only in particular cafes; and brings as a proof, the equa tion +27-280, in which x1, confequently, a+b=1; and, therefore,

Mr.

1796.]

Mr. Frend fays, that, as a and b are both lefs than unity, 3ab 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 muft lead every one who depends upon it, into continual error, unless there is fome method pointed out by Algebraifts, which tel's 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, a13tro, and calls it abfurd: for, fays he, three numbers added together, cannot be equal to nothing. Doubtlefs, according to his pofition, which does not admit of negative numbers, the expreffion is abfurd: but I fhould be much obliged to fome one of your correfpondents to inform me, what is the real ufe of these negative numbers; and whether, if equations can be folved without them, the fuppofition should be admitted into a work of science? In Mr. Frend's book, various equations are solved, without admitting them: the true folution is brought out by one root, when, according 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 fcience, there feems to me to be fomething gained by fimplifying the principles; but, before I give up entirely the old mode, I fhould like to be well informed, what lofs will be fuftained in the higher parts of algebra, by reject ing the negative quantity for, to fay the truth, it frequently puzzled me fo 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,

EXOTERICUS.

July 20, 1796. QUESTION XIII (No. IV.).—Answered by J. Fr.

The difference between the true and apparent level, is the difference between the earth's femi-diameter and the fecant of an arc of its circumference, whofe length is the given distance. The verfed fines of circular arcs are as the fquares of their chords; the verfed fines of fmall

357

arcs are alfo nearly as the above-mentioned differences, and the arcs themfeives. nearly as the chords: therefore, the abovementioned differences, when the arcs are small, are as the fquares 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; whofe radius will, confequently, be 317579 chains. From hence, we have one chain.649499 of a fecond, the difference between whofe natural fecant 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 eafy.

The conftruction of a table from these data, is too obvious for explanation. It might be calculated for every 100 chains as far as neceffary; but, as the first dif. ferences 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 conftructed upon the common theorem for its interpolation; fo that, upon the whole, it feems better to calculate it for any particular cafe, from Mr. Waddington's rule, which will be fomewhat nearer the truth if we put 1247 for 124, and cut off fix places inftead 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 ufe logarithms, we fhall g get a rule which, I think, may be found fomewhat shorter in its application, viz.

From double the logarithm of the distance in chains, fubftract 2.904193, and the remainder will be the logarithm of the differ ence in inches between the apparent" and true levels.

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

tain this difference in a great diftance, as for inftance exceeding 20 miles; the beft method will be to find the fecant of the arc by the following analogy-Tabular radius: 251523000 tabular fecant of the arc fecant required, - ufing a table of natural fecants extending to 10 or 12 places, and fubtracting from the fecant thus found the before-mentioned radius 251523000, the remainder will be the difference of levels required in inches. In this latter cafe, the following table will be found of fome ufe for the reduction of chains into degrees, minutes, and feconds of a great circle: Degrees.

1 Chain

= .000180416

.000360833

.000541249

The fame anfwered by Mr. I. H. Not having ever feen "Waddington's Land Surveyor's Companion," I am not able to enter into his method of drawing the rule given in his question; but it appears to be obtained by fuppofing the earth's radius equal to 3968 miles; for 3968×2 64 which is Mr. Waddington's multiplier. But as the earth's diameter is now found to be 7958 miles, 7958 126, which

then

802

100

=64, and

64

will be a nearer multiplier than may be tried from what follows.

Let A reprefent the B earth's centre. A line equally diftant from A, is called the line of true

=124,

124, as

DE

H

level; but the line of fight BCDE is the apparent level; and the difference between them is evidently Á CF, DG, EH. By the 47th Euclid's 1ft, √AB2+BC2=AC, then AC-AF CF, the difference required. Suppofe BC2 miles, the earth's radius = 3979, then

ABBC-AF=.00050263882, which multiplied by 5280 (Feet in one Mile) 2.65393297 feet = 2 feet 7.847. inches. Whence, by the fame rule, may any difference of the true and apparent level be obtained.

The fame anfwered by Mr. Wm. Adam, of the Free School, Wooburn.

It is evident, that the pail in the coach wheel defcribes a cycloid. Hence, as 3.1415927: 47 miles: 8.91267

miles, the mean velocity of the nail re

quired. See the article CYCLOID in Doctor Hutton's Dictionary.

ERRATA. In No. III, page 214, instead of Cor. 3 to the problem, fubftitute the following: Cor. 3. the base vary, the locus of the pomt E will be If the equal fides be conftant, and a circle, whofe centre is C: alfo the folid under AE, BE, and CE, will be conftant.