now is. A. L. Millan, giving an interesting ac- rather misapprehended the problem. The count of the Me of France, where he desideratum stated is, to cut a given cone through a given point in its fide, by two Helvetius has been commented upon planes, one parallel to the base of the by the two greatest men of the present cone, and the other obliquely cutting age, Voltaire and Rousseau. The for. both sides, so that the two sections may mer published his Observations during have equal areas, This problem is cahis life-time ; and a copy of the book pable of being solved in every allignable De l’Esprit, with Rousseau's marginal care, whatever be the quantity of the notes, has been lately discovered. vertical angle ; but Mr. Hickman, by The following men of letters and supposing, in addition to the conditions artists, some time since, received THREE required, another, which is by no means THOUSAND livres apiece, by way of en. so, viz. that the transverse diameter of the couragement, from the legislature : ellipse to be formed ball pass brougb the Brunck, editor and translator of se- given point, has very much narrowed its veral of the Greek poets-Deparcieux, application : it being only possible to pernaturalist- Dotteville, translator of Ta: form this with respect to a cone' whofe citus and Salluft-Lebas, accoucheur, or vertical angle does not exceed 23° 54' 20". man-midwifc-Lemonnier, astronomer - Our correspondent’s deduction of the equaMoitre, sculptor-Naigeon and Sedaine, r22 45372 men of letters-Parmentier, physician tion #+26x +2+62 127.2 Vincent and Vien, painters and Wailly, is perfectly mathematical and correct, but grammarian. N. B. Barthelemy, uncle of the navi- he would have found, that the only root had he proceeded farther in his analysis, gator of the same name, and author of which is always possible, is x=b, giving Le Voyage du jeune Anacharsis, also re the circular fcation, the two others x= ceived a present of 3000 livres in the name of the Republic, a little before his b x (62—3r2+vbt-22r2b2—714) death. 2(1+2) The following have received two THOUSAND livres each : being so only when h is not less than Schiveig Haeuser; Berenger; Caftillon Vu+812.7, and the vertical angle (of Toulouse); Deforges; Fenouillet-Fal- consequently not greater than the above baire ; Leclerk, men of letters-Gail, value. We shall get a somewhat simpler translator of Xenophon, Theocritus, &c. expression for the folution of this problem - Bridan, sculptor-Giraud-Kéraudon, as put, by Mr. Hickman, by using the mathematician- Le Blanc, poet-Mil- sides of the cone instead of the perpendilan, author of the Antiquities of France- cular.–For putring a=LT or LH (vide Sylvestre-Sacy, on account of his profi- diagram, P. 394; col. i.) b=TH and ciency in the oriental languages-and, b2 Thuellier, geometrician. z=LV, we shall get 23- (2am 20ーーっす · FIFTEEN HUNDRED livres have been presented to each of the following: a z=ab?; whose roots are z=a, giving Beffroi ; Defaulnais ; Imbert Lapla- the circular fe&tion, and z=-x(22—62 tiere ; Lieble ; Soules, men of lettersDevoges; Ferlus, schoolmasters-Brion #vat-bu2b2+64), giving the elliptical and Robert Vaugondy, geographers~ ones ;--the two latter roots being also Devoges; Renou; and Vanloo, painters, pofsible only when b is not greater than Duvaure, a farmer-Louis Ribiere, en (2--1) a. From hence it is evident, graver — Stouff, sculptor --Saverien, natu cone whose vertical angle is ralift-Sejan and Miroir, organists. greater than 23° 54' 20" can be cut as re[To be continued.] quired, if the given point be to form one of the entiemeties of the transverse ; but that MATHEMATICAL CORRESPONDENCE. every one which is more accute may be To the Editor of the Monthly Magazine. thus cut by two different oblique planes, making with each other an angle at the SIR, point T, which is evanescent when the ON perusing the last number of your angle of the cone is of the above value, Mifcellany, i observe, p. 394, an and becomes a right angle when the answer of Mr. Hickman, to question VI, latter is =o. from which I am inclined to think he has Mr. H's deduction, in his eft corollary, seems =, I 2a that no 4 3 3 ::1: face of the water, 1796.] 475 feems a little miftaken. The roots of the To tbe Editor of the Montbly Magazine. grah2 bz SIR, equation 42+35x et=0, are z 72 +12 I SEND answers to the Mathematical b 3(644x2)x(25—27£/5–19643+ri) Question VII, which I do'not find has yet been answered. I before fen: an whose limits of posibility are when bir answer with Question XI, but the in closed is somewhat more concise. the tangent or cotan Your's, &c. 7+413 2016 June, 1796. J.F---r. gent of 15°; that is, when the angle of che cone is 30° or 150°.-By a fimilar pro- QUESTION VII (No. Il). Answered by cess, we shall get for a maximum or mini. Mr. J. F LET AB be mum =-x(za?--b11a-4225276*), the pole, the 3а eye of the obserwhose limits are when a : 6::1: ver, DE the furWatu 3, the chord of 30° or 150°.- and FG the exMr. H. observes in his scholium, that the tremities of the B! roots of the last-mentioned equations do image of the pole, not always indicate the greatest and least appearing by re- dl sections of which the cone is capable laws of optics, the triangles AGD and CGE The truth is, that when the angle of the cone is under 30°, the plane TV in re are similar, as are also the triangles BFD and volving about T, from towards L, CFE; therefore, AD: DG :: CE: GE; or AD forms å series of ellipses whose areas con +CE (=39): DG+GE (=30) :: CE (=13) : GE=10; and, in like manner, BD+CE stantly decrease to a certain minimum, (=21): DF+FE (=30):: CE (=13),: FE= which is indicated by the greater value 18.5714285. 'And hence FG, the length of the of x or z as given above, and then again image, is =FE-GE=8.5714285 feet, or 8 increase to a maximum, which is indicated feet 6inches nearly. by the lesser value of these quantities, We shall also have (CG+AG: CG or) DE again'diminishing from thence to the ver- : GE :: 4:1 inch, the breadth of the image at tex of the cone. --This maximum, when the end neareft the observer, and (CF+BF: the vertical angle does not exceed 23o CF, or) DE : FE :: 4 : 2:47619 inches, its 54' 20"; is greater than the area of the breadth at the end farthest from him. circle TH, and, consequently, really the greatest possible elliple, but less if the QUESTION XI (No. III).- Anfwered by angle is between that value and 30°, in 7. F-t. which case, there consequently will be greater ellipses comprehended between PUT n=the number of boards composing the the circle TH and the above mentioned telegraph, then the whole number of signals minimum. which can be made by it will be =2N-> The general solution of the problem, the sum of n terms of the following series nt in the terms in which it was stated, I be. A+"2B+ C, &c. A, B, C, &c. fore sent you. The question giving fome 3 room for speculation with respect to some being respectively the ift, 2d, 3d, &c. terms, important properties of the cone, you or the value of the terms immediately preceding will, perhaps, not deem the above remarks those in which they appear. intrusive. Mr. Hickman will not, I am If n be put equal any number of boards less fure, be displeased at the liberty which I than the whole, and it be required to find how have taken with his answer; he appears many fignals may be made in each of which a man of science, and is probably there- boards Thall be displayed; the pth terms of the fore not deficient in candour. foregoing series will give the answer, Your's, &c. July stb, 1796. J. F The fame answered by Mr. T. Hickman. Dr. Hutton, in his New Mathematical Dice sionary, Vol. I, page 303, bas shewn that 2 4 -r. 1 11. I 1. I. 1-2 I. 2 15 DO 20 DO Is Do by 4 6 DO I QUESTION XII (No. III). Answered by 1. 2. 3 1. 2. 3. 4. Mr. J. Fr. 11. 1-I. 2. 1.-3. 94 It is evident that and... the distances of 1. 2. 3. 4. 5 the point perpendin. 1 I. 1-2, 1-3. n - 4. - 5 will cularly under the 3. 4. 5. balloon from the feverally express the number of combinations of stations, will be as r: things by ones, by twos, by threes, by fours, the respective co. by fives, and by fixes. In the present case, tangents of the anuwe have n=6, and the numbers resulting from gles of elevation tak en at each of thein. the above theorem, are D 6 Signals by exhibiting 1 board in each. Put, therefore, a, b, and the cotangents of by 2 DO the angles of elevation observed at A, B, and C, by 3 Do respectively. Find the line AD: AB ::6:6, DO and the line CD: BC::a:b.-About A and by Do C with the radii AD and CD, describe arcs Do by I interf-cting in D, and draw BD.--Find AE : AB :: CD:BD, and CE : CB :: AD: BD, 63 the total number, agreeable to the re and with these as radii, describe arcs about A marks of J. C. at page 296. and C, whose intersection £ 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 The fame answered by Mr. 7. H. and CBD are similar; as is also in like manner In answer to Question XI, in No. III, of proyed of the triangles CBE and ABD.--But your Monthly Magazine, a variety of authors a:b:: CD: CB :: AE : BE, and c:b:: AD have given general rules for the doctrine of : AB ::CE : BE. Therefore, AE, BE, CE are combinations, permutations, &c. notwithstand- as a, b, and c, respectively; and, consequently, ing your ingenious correspondent, J. C. has said, the point E is rightly found -From which “ these combinations are not to be ascertained by the height is obtained either by construction, or any known rule, but by experiment only." by one analogy, viz. Cotangent angle elevaThe most concise and plain method of treating tion : distance of perpendicular from Itation : : that of combinations, is that of Dr Hutton's, radius : height required. in his valuable Mathematical Dictionary, which Cor. If a circle be described about ADC, and he comprises under two heads : First, DB be produced to cut its circumference on the Having given any number of things, with fide E of AC, the point of intersection will also the number in each combination, to find the be the point E required; which gives another number of combinations. This comes under method of construction, as easy as the former. the changes in the Telegraph; and the gene In the case given, we shall have for calcural rule is, (if n be the number of thutters) lation the following analogies; 6:0:: AB : itinn -Mi for any number wha:ever. But 892.708=AD; and b:a :: BC: 1818.9213 CD. as the positions of each single shutter is to be Then, by plain trigonometry, we get the anadded, the rule will itand 2.1-1. For instance, if n=6, then 26-»=63, the whole number of 6 gles BAD=33° 7' 3", and BCD=15° 33'15"; and hence BD=549-077. Also, things; if n=9, then 2'-1=511, the whole number of 9 things, &c. for any number what. SCD:: AB: 3313*023=AE. As BD: AB :: BC : 2732.127=BE. ever. Second, to find the number of changes or alterations which any number of quantities and from any one of these, AD:: BC,: 2438.994=CE.. can undergo, when combined in all possible Cot. 15° : ÅE : : radius : 887.722, &c. yards, ways, with themselves and each other, both as Cot. 180: BE the height of the balloon re quired. of them, which some authors call the composition of things; the general rule, then, is, n. I X 11. In this case, if n be ==-6, it will be The fume an!wered by Mr. T. Hinckman, 65. -X6=55986 the number of composi. In the annexed figure, let represent the balloon ; tions that can be made out of 6 different things. A, B, and C, the places of And fo may the whole different ways of placing observation, being in the right line AC; OD a perthe 24 letters of the alphabet be X24; pendicular from the balloon B В 24-I to the horizontal plane. Draw AD, BD, CD, amounting to 34 figures. in the same plane ; and DF perp. to AC, then put 11-I 6-I 2426 11 . n a2. 2 n 192 1796.] 477 put m=AB=1000, na =BC=1500, samt Question XIX.-By Philalethes. =AC=2500, a=tang. angle. AOD or 75°, Of seven numbers, in a continued geometrical b=tang. BOD or 720, sztang. COD or 70°; progression, having given the sum of the two and x==the common perp. OD. Then, by least =90, and the sum of the two greatest = trigonometry, as 1:a::*: ax=AD; in like 381,250 : to find the seven numbers? manner is BDbx, and CD=ax. Again, as AC (s) : AD+CD (axfcx) : : AD- CD The folutions of the aborre questions must be а?х?—с2х2 fent, at the latest, in the first week of September ; (ax-x): FAF-CF; and in like but the fooner the better. And all Communications manner as BC : BD+CD : : BD-CD : must be post paid, and direEted, For the Monthly 62x2(2x2 Magazine, at Mr. Johnson's, Buokseller, St. -=BF-CF; the difference of these Paul's Church Yard, London. -(2x2 72_h2 two give AF-BF=AB=m= ERRATA. P. 394, col.ii, 1. for read r22 62x2 (272 5 from which equation we have Ib. line 13 from the bottom, for 37} y? <h2! -887.53 yards, the =Cd read 37}=Ca. Pa. 396, col. i, 1. 7, for per ABFX , read ABF+. Pa. 213, col. i, I. 24 from the bottom, for pendicular height of the balloon as required. xtx and amb read xta and x-b. cholium. The theorem above given, may be Pa. 213, col. ii, l. 2, for proper read proposed. of considerable use to determine the height and Ib. col. ii, 1. 3, quadratures read quadraties. distance of a cloud, a meteor, or other objects Pa. 304, col i, l. 11 and 12, read, “ that is in motion. N-Nxtar Nx?,"&c. Pa. 305, col. i, l.2, N. B. S-lutions 10 Question XIII. by Mr. J. 1. 2 Fr, and Mr. 7. H. are deferred till our 31232 read "=1 “next for want of room. -X4=4:32674,the value," &c. 382337 X!, 3n ( 7:)-(2-1) m2 34(n+m)-(x2 -1)..? To be answered in No. VIII, the Mag. for Xr. 31(2:---m)--(2-1)m2 September. P. 393, col. ii, I. 36, for “ see do you," read QUESTION XVI.-B; Mr. 0. G. Gregory, of seem to." Ib. col. ii, 1. 44, “summoned' Yuxley. read “ summed.” Ib. col. ii, 1.47, for "parts" THE dimensions of a cylindrical tube are such, read “ points.” that, having a plate of tin at one end, with an -h2 P. 394, col. 1, 1.28, for ;*?, read, aperture in its centre, the other end being open 12+12 and turned towards the heavens, the field of b b view it takes in, is one-twentieth of the hemi- P. 395, col. i, l. 40, for 3V 3ас fphere :-The young peruters of the mathema- Ib. col. ii, 1. 38, for there read then. Зас tical 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, N.B. For want of room, we have this month Given the difference of the times of sun-rising been compelled to defer the communication relative at the top and bottom of a mountain, situated to Mr. Frend's Principles of Algebra. We are in a given latitude, and on a given day ; to de- obliged to the writer, and his favour fall certainly termine the height of the mountain ? ophear in our nexr. ANECDOTES AND REMAINS OF EMINENT PERSONS, . [This article is devotrd to the reception of Biographical Anecdotes, Papers, Letters, &c. and we request the Communication of such of our Readers as can alift us in these obje&ts.] ANECDOTES or PERSONS CONNECTED Bossuet, and minister of the marine, on WITH THE FRENCH REVOLUTION. the recommendation of Condorcet*. He [Continued from our laji.] was an honest and virtuous, but dull and MONGE, plodding, RIGINALLY a stone-cutter at Mezieres, in Champagne, became * « On ne savoit qui mettre à la marine; Cona mathematician of some celebrity, in dorcet, parla de Monge, parce qu'il l'avoit ou consequence of the liberality of the Abbé résoudre des problèmes de géométrie à l'Académie des read read V plodding, man; tetally incapacita:ed, by light emitted from the Satellites of Junature and education, to act the impor- piter, published in 1771, that he told the tant part asligned to himn by friend thip, author, then in the height of his glory, on one hand, and the want of able and that he would rather have composed that patriotic competitors, on the other--for Memoir, than been president of the all those appertaining to the ancient States-General ; “ for,” added he," there marine-royal, from the minister of the are, assuredly, more citizens, worthy of department down to the enseigne, which being mayor of Paris, or of filling the anlwers to our midshipman, was, at this chair of the National Assembly, but there period, notoriously counter-revolutionary. are not ten men in all Europe, capable Monge had fulved several difficult of writing such a dissertation as that ; it problems while a boy, before the Aca. will, therefore, of course, become a more demy of Sciences, a circumstance which certain passport to the notice of pofterity.” had captivated the regard of the secre- Jean Silvain-Bailly exhibited a rare tary. As the inspector of a feminary for instance of modesty, zeal, alsiduiry, and thip-building, this might have been a talents, united in one and the same per• fufficient qualification, but when, in- son ; it was a great misfortune, both for stead of contending with the passive signs himself and his country, that he should of triangles and pirallelograms, the ma- have quitted the retreats of science, and thematician was to enter upon active life, embarked on the ftormy sea of politics. and regulate men and feets, he was quite During his mayoralty, he was induced, bewildered. The result was, accordingly, by Lafayette, to hoist the red flag, the what might have been expected--the symbol of insurrection, on the top of the French marine became almost annihin Hotel de Ville, and thus countenance the lated, during the administration of a maj.cre, as it was called, of the Champ minister, an adept indeed in geometry, de Mars, which ensued. but an ignoramus in respect to mankind. He was tried for this upwards of two BuzoT years afterwards, before a tribunal, stainWas one of the Girondists, and his at. ed with blood; and executed, by the untachment to a federative republic, such sparing guillotine, on the 21st Brumaire, as trole of Greece, America, and Swit- (uth November) 1793, in the 57th year zerland, instead of a republic, one and indivjanie, cost him his life. How much CHAMPAGNEUX muit the idea of royalty have been Was the editor of one of the three-score dreaded in France, when his enemies newspapers, that imparted the revolucould undermine his reputation, and ruin tionary stimulous to France. He is the fahis character, by the opprobrious nick, ther of a numerous family; a man of unnaire of le roi Buzo!! But this was at a impeached morals, and was attached to periori, and the custom is not yet abo. liberty from principle, at a time, and in a Tihed, when naughty children were country, when it was not unusual to be whiped by their parents for being les fo from mere speculation ! He was sepenus arisiorats! lected by Roland on account of his inBAILLY, dustry and talents; and was put by him The first mayor of Paris, was a man of at the head of the principal division of the science rather thin a politician. He home department. In short, during his distinguished h'mself by his History of adminiftration, he became, what is termed Astronomy, in 5 vols. 4to; by his The- in England, under-secretary of state. ory of the Satellites of Jupiter, which had engaged his attention ever since Camus. This is another of Ruland's élèves, and 1763 ; and by severai learned Memoirs, infeited in the proceedings of the Aca” does great credit to his discernment. Soon demy of Sciences. Jerome Lalande, one after the resignation of his friend, he quitof the first astronoiners of the present ted the home department, was elected a day, and who, at this moment, presides member of the Convention, and is now over the National Observatory, was so Arcbevist to the present legislature. He much pleased with the paper on the was one of the deputies delivered over by Dumouriez to, and confined by, the des Sciences, & Morge fut élu. C'est un espèce Prince de Cobourg. From an Austrian d'original qui feroit lien des fingeries à la manière prison he has been restored to the exercise des ours que j'ai vus jouer dans les folles de la Ville of his legislative functions (for he is one de Berne, &C. Apel de Mad. Roland, of the two-thirds) and, on the first vacancy, of his age. IS |