A. L. Millan, giving an interesting account of the Ifle of France, where he now is. Helvetius has been commented upon by the two greatest men of the present age, Voltaire and Rousseau. The former published his Observations 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 artists, some time since, received THREE THOUSAND livres apiece, by way of encouragement, from the legislature: Brunck, editor and tranflater of several of the Greek poets-Deparcieux, naturalist-Dotteville, tranflator of Tacitus and Sallust-Lebas, accoucheur, or man-midwife-Lemonnier, astronomer Moitte, 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, also received a present of 3000 livres in the name of the Republic, a little before his death. The following have received Two THOUSAND livres each: Schiveig-Haeuser; Berenger; Castillon (of Toulouse); Deforges; Fenouillet-Falbaire; Leclerk, men of letters-Gail, tranflator of Xenophon, Theocritus, &c. -Bridan, sculptor-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 presented to each of the following: Beffroi; Defaulnais; Imbert Lapla tiere; Lieble; Soules, men of letters Devoges; Ferlus, schoolmasters-Brion and Robert Vaugondy, geographersDevoges; Renou; and Vanloo, paintersDuvaure, a farmer-Louis Ribiere, engraver-Stouff, sculptor-Saverien, naturalist-Sejan and Miroir, organists. [To be continued.] MATHEMATICAL CORRESPONDENCE. To the Editor of the Monthly Magazine. SIR, ON perufing the last number of your Mifcellany, I observe, p. 394, an anfwer of Mr. Hickman, to question VI, from which I am inclined to think he has rather misapprehended the problem. The defideratum stated is, to cut a given cone through a given point in its fide, by two planes, one parallel to the base of the cone, and the other obliquely cutting both fides, so that the two sections may have equal areas. This problem is capable of being solved in every affignable cafe, whatever be the quantity of the vertical angle; but Mr. Hickman, by supposing, in addition to the conditions required, another, which is by no means fo, viz. that the tranfverfe 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 respect to a cone whofe vertical angle does not exceed 23° 54' 20". Ourcorrespondent's deduction of the equa b 2(2+2) x(b2-3r2+√b-2222-714) being so only when b is not less than √ 11+82.7, and the vertical angle consequently not greater than the above value. We shall get a somewhat simpler expression for the folution of this problem as put, by Mr. Hickman, by using the fides of the cone inftead of the perpendicular.-For putting a=LT or LH (vide diagram, P. 394, col. i.) b=TH and z=LV, we shall get 23- (2a--+ 62 a ±√a-6a2b2+64), giving the elliptical ones;-the two latter roots being also poffible only when b is not greater than (2-1) a. From hence it is evident, that no cone whose vertical angle is greater than 230 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 ist corollary, feems 1796.] Mathematical Correspondence. 475 leems a little mistaken. The roots of the To the Editor of the Monthly Magazine. b 3 3(6+2)×(26-21-14) Queftion VII, which I whose limits of possibility are when b:r I SEND answers to the Mathematical 20th June, 1796. gent of 15°; that is, when the angle of the cone is 30° or 150°. -By a fimilar pro- QUESTION VII (No. II). Anfwered by cefs, we shall get for a maximum or mini. I Mr. J. Fr. LET AB be z=3x2a2-b2±√a-4a22+64), the pole, the whose limits are when a:b::1: √2+√3, the chord of 30° or 150o.Mr. H. observes in his scholium, 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 30o, the plane TV in re C eye of the obfer- D F G are fimilar, as are also the triangles BFD and CFE; therefore, AD: DG :: CE: GE; or AD volving about T, from H towards L, CE (39): DG+GE (=30) :: CE (=13) forms a feries of ellipses whose areas constantly decrease to a certain minimum, which is indicated by the greater value of zor z as given above, and then again increase to a maximum, which is indicated by the leffer value of these 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, consequently, really the greatest possible ellipse, but less if the angle is between that value and 30o, in which cafe, there confequently will be greater ellipfes comprehended between the circle TH and the abovementioned :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 observer, and (CF+BF: CF, or) DE: FE::4:2:47619 inches, its breadth at the end farthest from him. The fame answered 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 correspondent, J. C. has faid, "these combinations are not to be ascertained by any known rule, but by experiment only." The most concise 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, Put, therefore, a, b, and the cotangents of the angles of elevation observed at A, B, and C, respectively. Find the line AD: AB::c:6, and the line CD BC a b. - About A and C with the radii AD and CD, describe arcs interfecting in D, and draw BD. Find AE: AB:: CD: BD, and CE: CB:: AD: BD, and with these as radii, defcribe arcs about A and C, whose 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 also in like manner proved of the triangles CBE and ABD.-But a:b:: CD:CB:: AE: BE, and c:6:: AD :AB:: CE: BE. Therefore, AE, BE, CE are as a, b, and c, respectively; and, consequently, the point E is rightly found From which the height is obtained either by construction, or by one analogy, viz. Cotangent angle elevation: distance 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 also be the point E required; which gives another method of conftruction, as easy as the former. In the cafe given, we shall have for calculation the following analogies; b:c:: AB: 892.708AD; and b:a:: BC: 1818-921 CD. 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 shutters) 1+1--1 for any number whatever. But as the positions of each single shutter is to be added, the rule will stand 28-1. 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 whatever. Second, to find the number of changes or alterations which any number of quantities and from any one of these, can undergo, when combined in all poffible ways, with themselves and each other, both as to things themselves, and the order or position of them, which some authors call the compo-fition of things; the general rule, then, is, Then, by plain trigonometry, we get the angles BAD=33° 7' 3", and BCD=15° 33′ 15′′; and hence BD=549.077. Alfo, As BD: { CD::AB: 3313023-ΑΕ. Cot. 15°: ÀE :: radius: 887-722, &c. yards, quired. 1796.] New Mathematical Questions. put m=AB-1000, n=BC=1500, s=m-t-n =AC=2500, atang. angle. AOD or 75°, b=tang. BOD or 72°, ==tang. COD or 70°, and x=the common perp. OD. Then, by trigonometry, as I:a::x:ax=AD; in like manner is BD bx, and CD=ax. Again, as AC (s): AD+CD (ax+cx) :: AD-CD (ax-cx): =AF-CF; and in like manner as BC: BD+CD :: BD-CD: 62x22x2 71 a2x2-2x2 S ✓ BF-CF; the difference of these NEW MATHEMATICAL QUESTIONS. To be answered in No. VIII, the Mag. for September. QUESTION XVI.-By Mr. O. G. Gregory, of Yuxley. THE dimensions of a cylindrical tube are such, that, having a plate of tin at one end, with an aperrure 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 hemisphere: The young perusers of the mathematical part of the Monthly Magazine are requested 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, situated in a given latitude, and on a given day; to determine the height of the mountain? ANECDOTES 477 QUESTION XIX.-By Philalethes. Of seven numbers, in a continued geometrical progreffion, having given the sum of the two leaft=90, and the fum of the two greateft = 281,250: to find the seven 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. Johnson's, Bookseller, St. Paul's Church Yard, London. 2h2 2+h2 2 = 2h2 Ib. line 13 from the bottom, for 37 Cd read 37 Ca. Pa. 396, col. i, 1. 7, for ABFX, read ABF+. Pa. 213, col. i, 1. 24 from the bottom, for xx and a read xa and x-b. Pa. 213, col. ii, 1. 2, for proper read proposed. Ib. col. ii, 1. 3, for quadratures read quadraties. Pa. 304, col i, 1. 11 and 12, read, " that is Nx2,"&c. Pa. 305, col. i, 1.2, N-nNx+1.2 EMINENT PERSONS. [This article is devoted to the reception of Biographical Anecdotes, Papers, Letters, &c. and we request the Communication of fuch of our Readers as can afsist us in these objects.] ANECDOTES OF PERSONSCONNECTED Bossuet, and minifter of the marine, on WITH THE FRENCH REVOLUTION. [Continued from our last.] plodding, man; totally incapacitated, by nature and education, to act the important part affigned to him by friendship, on one hand, and the want of able and patriotic competitors, on the other for all those appertaining to the ancient marine-royal, from the minister of the department down to the enseigne, which answers to our midshipman, was, at this period, notorioufly counter-revolutionary. Monge had folved several difficult problems while a boy, before the Academy of Sciences, a circumstance which had captivated the regard of the fecretary. As the inspector of a seminary for ship-building, this might have been a sufficient qualification; but when, in ftead 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 almost annihilated, during the administration of a minifter, an adept indeed in geometry, but an ignoramus in respect to mankind. BUZOT Was one of the Girondists, and his attachment to a federative republic, fuch as those of Greece, America, and Switzerland, instead of a republic, one and indivine, coft him his life. How much mait the idea of royalty have been dreaded in France, when his enemies could undermine his reputation, and ruin his character, by the opprobrious nick name of le roi Buzol! But this was at a period, and the custom is not yet abolished, when naughty children were whipped by their parents for being les petits aristocrats! BAILLY, 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 composed that Memoir, than been prefident of the States-General; " for," added he," there are, assuredly, 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 such a differtation as that; it will, therefore, of course, become a more certain passport to the notice of posterity." Jean Silvain-Bailly exhibited a rare instance of modesty, zeal, affiduity, and talents, united in one and the same perfon; it was a great misfortune, both for himself and his country, that he should have quitted the retreats of science, and embarked on the ftormy fea of politics. During his mayoralty, he was induced, by Lafayette, to hoist the red flag, the symbol of infurrection, on the top of the Hotel de Ville, and thus countenance the mafficre, as it was called, of the Champ de Mars, which ensued. He was tried for this upwards of two years afterwards, before a tribunal, stained with blood; and executed, by the unsparing guillotine, on the 21st Brumaire, (11th November) 1793, in the 57th year of his age. CHAMPAGNEUX Was the editor of one of the three-score newspapers, that imparted the revolutionary stimulous 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 speculation! He was selected by Roland on account of his industry and talents; and was put by him at the head of the principal division 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 Rolands élèves, and does great credit to his difcernment. Soon after the refignation of his friend, he quitted the home department, was elected a member of the Convention, and is now Archevift to the present legiflature. He was one of the deputies delivered over by Dumouriez to, and confined by, the Prince de Cobourg. From an Austrian 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, |