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A. L. Millan, giving an interefting account of the Ile of France, where he now is.

Helvetius has been commented upon by the two greateft men of the prefent age, Voltaire and Rouffeau. The former published his Obfervations during his life-time; and a copy of the book De l'Esprit, with Rouffeau's marginal notes, has been lately difcovered.

The following men of letters and artifts, fome time fince, received THREE THOUSAND livres apiece, by way of encouragement, from the legislature:

Brunck, editor and tranflator of feveral of the Greek poets-Deparcieux, naturalift-Dotteville, tranflator of Tacitus and Salluft-Lebas, accoucheur, or man-midwife-Lemonnier, aftronomerMoitte, fculptor-Naigeon and Sedaine, men of letters-Parmentier, physicianVincent and Vien, painters-and Wailly,

grammarian.

N. B. Barthelemy, uncle of the navigator of the fame name, and author of Le Voyage du jeune Anacharfis, alfo received a prefent of 3000 livres in the name of the Republic, a little before his death.

The following have received Two THOUSAND livres each:

Schiveig Haeufer; Berenger; Caftillon (of Toulouse); Deforges; Fenouillet-Falbaire; Leclerk, men of letters-Gail, tranflator of Xenophon, Theocritus, &c. -Bridan, fculptor-Giraud-Kéraudon, mathematician-Le Blanc, poet-Millan, author of the Antiquities of FranceSylveftre-Sacy, on account of his proficiency in the oriental languages—and, Thuellier,, geometrician.

FIFTEEN HUNDRED livres have been prefented to each of the following:

Beffroi; Defaulnais; Imbert Laplatiere; Lieble; Soules, men of lettersDevoges; Ferlus, fchoolmasters-Brion and Robert Vaugondy, geographersDevoges; Renou; and Vanloo, painters Duvaure, a farmer-Louis Ribiere, engraver-Stouff, fculptor-Saverien, naturalift-Sejan and Miroir, organifts.

[To be continued.]

MATHEMATICAL CORRESPONDENCE.

To the Editor of the Monthly Magazine.

SIR,

ON perufing the laft number of your Mifcellany, I obferve, p. 394, an anfwer of Mr. Hickman, to queftion VI, from which I am inclined to think he has

rather mifapprehended the problem. The defideratum ftated is, to cut a given cone through a given point in its fide, by two planes, one parallel to the bafe of the cone, and the other obliquely cutting both fides, fo that the two fections may have equal areas. This problem is capable of being folved in every affignable cafe, whatever be the quantity of the vertical angle; but Mr. Hickman, by fuppofing, in addition to the conditions required, another, which is by no means fo, viz. that the tranfverse diameter of the ellipfe to be formed shall pass through the given point, has very much narrowed its application: it being only poffible to perform this with refpect to a cone whofe vertical angle does not exceed 23° 54′20′′. Our correfpondent's deduction of the equa463,2

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a-6262+6), giving the elliptical ones;-the two latter roots being alfo poffible only when b is not greater than (2-1) a. From hence it is evident, that no cone whofe vertical angle is greater than 23° 54′ 20′′ can be cut as required, if the given point be to form one of the extremeties of the tranfverfe; but that every one which is more accute may be thus cut by two different oblique planes, making with each other an angle at the point T, which is evanefcent when the angle of the cone is of the above value, and becomes a right angle when the latter is =o.

Mr. H's deduction, in his ift corollary, feems

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To the Editor of the Monthly Magazine.

I

SIR,

SEND anfwers to the Mathematical
Queftions in your laft, and alfo to

Question VII, which I do not find has
yet been answered. I before fent an
anfwer with Question XI, but the in-
clofed is fomewhat more concife.
Your's, &c.

20th June, 1796.

J. F --r.

QUESTION VII (No. II). Answered by
Mr. J. Fr.

LET AB be

×(2a2--b2±√aa—4a2b2+6), the pole, C the

whofe limits are when ab :: 1 :

2+3, the chord of 30° or 150°. Mr. H. obferves in his fcholium, that the roots of the last-mentioned equations do not always indicate the greatest and leaft fections of which the cone is capable. The truth is, that when the angle of the cone is under 30°, the plane TV in revolving about T, from H towards L, forms a series of ellipfes whofe areas conftantly decrease to a certain minimum, which is indicated by the greater value of x or z as given above, and then again increase to a maximum, which is indicated by the leffer value of thefe quantities, again diminishing from thence to the vertex of the cone.-This maximum, when the vertical angle does not exceed 23° 54' 20", is greater than the area of the circle TH, and, confequently, really the greateft poffible ellipfe, but lefs if the angle is between that value and 30°, in which cafe, there confequently will be greater ellipfes comprehended between the circle TH and the abovementioned

minimum.

eye of the obfer-
ver, DE the fur-
face of the water,
and FG the ex-
tremities of the B
image of the pole,
appearing by re- D
flection.-By the

G

laws of optics, the triangles AGD and CGE are fimilar, as are alfo the triangles BFD and +CE (=39): DG+GE (=30) :: CE (=13) CFE; therefore, AD: DG :: CE: GE; or AD : GE=10; and, in like manner, BD+CE (=21): DF+FE (=30) : : CE (=13): FE= 18-5714285. And hence FG, the length of the image, is FE-GE-8.5714285 feet, or 8 feet 6 inches nearly.

We shall also have (CG+AG: CG or) DE GE:: 4 : 1 inch, the breadth of the image at the end nearest the obferver, and (CF+BF: CF, or) DE: FE:: 4:2:47619 inches, its breadth at the end farthest from him.

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QUESTION XI (No. III). Anfwered by
J. Fr.

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PUT the number of boards compofing the telegraph, then the whole number of fignals which can be made by it will be 2"-1 the fum ofn terms of the following series n+ A+ B+ C, &c. A, B, C, &c. being refpectively the ift, 2d, 3d, &c. terms, or the value of the terms immediately preceding thofe in which they appear.

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3

The general folution of the problem, in the terms in which it was ftated, I before fent you. The queftion giving fome room for fpeculation with respect to fome important properties of the cone, you will, perhaps, not deem the above remarks intrufive. Mr. Hickman will not, I am If be put equal any number of boards lefs fure, be difpleafed at the liberty which I than the whole, and it be required to find how have taken with his anfwer; he appears many fignals may be made in each of which a man of science, and is probably there-boards fhall be difplayed; the pth terms of the foregoing feries will give the anfwer, fore not deficient in candour.

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The fame anfwered by Mr. J. H.

In answer to Question XI, in No. III, of your Monthly Magazine, a variety of authors have given general rules for the doctrine of Combinations, permutations, &c. notwithstanding your ingenious correfpondent, J. C. has faid, thefe combinations are not to be afcertained by 86 any known rule, but by experiment only." The moft concife and plain method of treating that of combinations, is that of Dr Hutton's, in his valuable Mathematical Dictionary, which he comprises under two heads: First,

Having given any number of things, with the number in each combination, to find the number of combinations. This comes under the changes in the Telegraph; and the general rule is, (if n be the number of fhutters) -1 for any number whatever. But as the pofitions of each fingle shutter is to be added, the rule will stand 2-1. For instance, if n=6, then 26—1=63, the whole number of 6 things; if n=9, then 29-1=511, the whole number of 9 things, &c. for any number what

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ever.

Second, to find the number of changes or alterations which any number of quantities can undergo, when combined in all poffible ways, with themselves and each other, both as to things themselves, and the order or pofition of them, which fome authors call the compofition of things; the general rule, then, is,

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Put, therefore, a, b, and cthe cotangents of the angles of elevation obferved at A, B, and C, refpectively. Find the line AD: AB::c:b, and the line CD: BC:: a: b.-About A and C with the radii AD and CD, defcribe arcs interfecting in D, and draw BD.-Find AE : AB:: CD: BD, and CE: CB:: AD: BD, and with thefe as radii, defcribe arcs about A and C, whofe interfection E will be the point perpendicularly under the balloon.

For because AE and AB are as CD and BD, and the angle ABE DBC, the triangles ABE and CBD are fimilar; as is alfo in like manner proved of the triangles CBE and ABD.-But a:b:: CD: CB:: AE: BE, and c:b:: AD :AB::CE: BE. Therefore, AE, BE, CE are as a, b, and c, refpectively; and, confequently, -From which the point E is rightly found the height is obtained either by conftruction, or by one analogy, viz. Cotangent angle elevation: diftance of perpendicular from station: : radius: height required.

Cor. If a circle be described about ADC, and DB be produced to cut its circumference on the fide E of AC, the point of interfection will alfo be the point E required; which gives another method of conftruction, as eafy as the former.

In the cafe given, we fhall have for calculation the following analogies; b:c:: AB: 892708=AD; and b: a :: BC: 1818.921—

CD.

Then, by plain trigonometry, we get the angles BAD=33° 7' 3", and BCD=15° 33′ 15′′; and hence BD 549 077. Alfo, CD: AB: 3313023=AE. As BD: AB: BC 2732.127=BE. AD: BC:2438-994-CE.. and from any one of thefe,

Cot. 15°: AE): : radius: 887-722, &c. yards,
Cot. 180 BE the height of the balloon re-
Cot. 20°: CE
quired.

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1796.]

New Mathematical Questions.

put m AB-1000, BC=1500, sm-n AC 2500, a tang. angle. AOD or 75°, btang, BOD or 72°, tang. COD or 70°, and the common perp. OD. Then, by trigonometry, as Ia::x:ax=AD; in like manner is BDbx, and CDax. Again, as AC (5) AD+CD (ax+cx) :: AD-CD a2x2-c2x2 (ax-cx): =AF-CF; and in like manner as BC: BD+CD : : BD-CD:

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NEW MATHEMATICAL QUESTIONS. To be answered in No. VIII, the Mag. for September. QUESTION XVI.-By Mr. O. G. Gregory, of Yaxley.

THE dimenfions of a cylindrical tube are fuch, that, having a plate of tin at one end, with an aperture in its centre, the other end being open and turned towards the heavens, the field of view it takes in, is one-twentieth of the hemifphere:-The young perufers of the mathematical part of the Monthly Magazine are requefted to find the internal diameter of this tube, the length being one foot eight inches.

QUESTION XVIII.—By Mr. E. Warren. Given the difference of the times of fun-rifing at the top and bottom of a mountain, fituated in a given latitude, and on a given day; to determine the height of the mountain?

ANECDOTES

QUESTION XIX.-By Philalethes.

477

Of feven numbers, in a continued geometrical progreffion, having given the fum of the two leaft 90, and the fum of the two greatest 281,250: to find the feven numbers?

The folutions of the above questions must be fent, at the latest, in the first week of September; but the fooner the better. And all Communications must be poft paid, and directed, For the Monthly Magazine, at Mr. Johnfon's, Bookfeller, St. Paul's Church Yard, London.

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EMINENT PERSONS.

[This article is devoted to the reception of Biographical Anecdotes, Papers, Letters, &c. and we requeft the Communication of fuch of our Readers as can assist us in these objects.]

ANECDOTES OF PERSONS CONNECTED
WITH THE FRENCH REVOLUTION.

[Continued from our laft.]
MONGE,

RIGINALLY a ftone-cutter at
O
Mezieres, in Champagne, became
a mathematician of fome celebrity, in
confequence of the liberality of the Abbé

Boffuet, and minifter of the marine, on the recommendation of Condorcet. He was an honest and virtuous, but dull and plodding,

*«On ne favoit qui mettre à la marine; Condorcet parla de Monge, parce qu'il l'avoit ou réfoudre des problèmes de géométrie à l'Académie

des

plodding, man; totally incapacitated, by nature and education, to act the important part affigned to him by friendhip, on one hand, and the want of able and patriotic competitors, on the other-for all thofe appertaining to the ancient marine-royal, from the minifter of the department down to the enfeigne, which answers to our midshipman, was, at this period, notorioufly counter-revolutionary.

Monge had folved feveral difficult problems while a boy, before the Academy of Sciences, a circumftance which had captivated the regard of the fecretary. As the infpector of a feminary for fhip-building, this might have been a fufficient qualification; but when, inftead of contending with the paffive figns of triangles and parallelograms, the mathematician was to enter upon active life, and regulate men and fleets, he was quite bewildered. The refult was, accordingly, what might have been expected-the French marine became almoft annihilated, during the adminiftration of a minifter, an adept indeed in geometry, but an ignoramus in refpect to mankind.

Buzor

Was one of the Girondifts, and his attachment to a federative republic, fuch as thofe of Greece, America, and Switzerland, instead of a republic, one and indroje, coft him his life. How much mut the idea of royalty have been dreaded in France, when his enemies could undermine his reputation, and ruin his character, by the opprobrious nickname of le roi Buzo!! But this was at a period, and the cuffom is not yet abolihed, when naughty children were whipped by their parents for being les petits ariftocrats!

BAILLY,

The first mayor of Paris, was a man of fcience rather than a politician. He diftinguished himself by his Hiftory of Aftronomy, in 5 vols. 4to; by his Theory of the Satellites of Jupiter, which had engaged his attention ever fince 1763; and by feveral learned Memoirs, inferted in the proceedings of the Academy of Sciences. Jerome Lalande, one of the first aftronomers of the prefent day, and who, at this moment, prefides over the National Obfervatory, was fo much pleafed with the paper on the des Sciences, Monge fut élu. C'est un effèce d'original qui feroit bien des fingeries à la manière des ours que j'ai vus jouer dans les foffes de la Ville de Berne, &c. Appel de Mad. Roland.

C

light emitted from the Satellites of Jupiter, published in 1771, that he told the author, then in the height of his glory, that he would rather have compofed that Memoir, than been prefident of the States - General; " for," added he," there are, affuredly, more citizens, worthy of being mayor of Paris, or of filling the chair of the National Affembly, but there are not ten men in all Europe, capable of writing fuch a differtation as that; it will, therefore, of course, become a more certain paffport to the notice of pofterity."

Jean Silvain-Bailly exhibited a rare inftance of modefty, zeal, affiduity, and talents, united in one and the fame perfon; it was a great misfortune, both for himself and his country, that he should have quitted the retreats of fcience, and embarked on the ftormy fea of politics.

During his mayoralty, he was induced, by Lafayette, to hoift the red flag, the fymbol of infurrection, on the top of the Hotel de Ville, and thus countenance the macre, as it was called, of the Champ de Mars, which enfued.

He was tried for this upwards of two years afterwards, before a tribunal, stained with blood; and executed, by the unfparing guillotine, on the 21ft Brumaire, (11th November) 1793, in the 57th year of his age.

CHAMPAGNEUX

Was the editor of one of the three-fcore newspapers, that imparted the revolutionary ftimulous to France. He is the father of a numerous family; a man of unimpeached morals, and was attached to liberty from principle, at a time, and in a country, when it was not unusual to be fo from mere fpeculation! He was felected by Roland on account of his induftry and talents; and was put by him at the head of the principal divifion of the home department. In short, during his administration, he became, what is termed in England, under-fecretary of state.

CAMUS.

This is another of Roland's élèves, and after the refignation of his friend, he quitdoes great credit to his difcernment. Soon ted the home department, was elected a member of the Convention, and is now Arcbevift to the prefent legiflature. He was one of the deputies delivered over by Dumouriez to, and confined by, the Prince de Cobourg. From an Auftrian prifon he has been restored to the exercise of his legislative functions (for he is one of the two-thirds) and, on the first vacancy,

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