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A. L. Millan, giving an interesting ac. rather misapprehended the problem. The count of the Ille of France, where he defideratum stated is, to cut a given cone now is.

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 Oblervations 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 10. be formed shall pass tbrough the

Brunck, editor and tranflator of re- given point, has very much narrowed its veral of the Greek peers-Deparcieux, application : it being only poffible to pernaturalist-Docteville, 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 Ourcorrespondent's deduction of the equaMoitte, sculptor-Naigeon and Sedaine,

r2-b2

45372 men of letters-Parmentier, physician

tion #3+2kx

x27-12%

72 +62 624.12 Vincent and Vien, painters and Wailly, is perfe&tly 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=h, giving Le Voyage du jeune Anacbarsis, also re

the circular scation, the two others x= ceived a present of 3000 livres in the name of the Republic, a little before his b

x (62—3r24vbt-22r252—714) death.

2(12+12) The following have received TWO THOUSAND livres each :

being so only when b is not less than Schiveig-Haeufer; Berenger; Caftillon vu+82.', 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- fides of the cone instead of the perpendilan, author of the Antiquities of France cular.–For putting 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.

m=LV, we shall get z?- (20--%+ · FIFTEEN HUNDRED livres have been presented to each of the following :

a?r=b2; whose roots are z=å, giving Beffroi ; Defaulnais ; Imbert Lapla- the circular seĉtion, and z=; tiere; Lieble ; Soules, men of letters

X(azba Devoges; Ferlus, schoolmasters--Brion va_6u2b2+54), giving the elliptical and Robert Vaugondy, geographersą ones ;--the two latter roots being also Devoges; Renou; and Vanloo, painters- poffible 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

that no cone whose vertical angle is ralist-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 extiemeties 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 core is of the above value,

Miscellany, I observe, p. 394, an and becomes a right angle when the answer of Mr. Hickman, to question VI, latter is =0. from which I am inclined to think he has Mr. H's deduction, in his ift corollary,

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1796.]
Mathematical Correspondence.

475 feems a little miftaken. The roots of the To be Editor of the Montbly Magazine. g2 62 ba

SIR, equation 82+3bx tso, are 2+2 3

I

SEND answers to the Mathematical b

Questions in your last, and also to 3(647x37x(252—21-15–14672+ri

) Question VII, which I do not find has

yet been answered. I before fent an whose limits of poffibility are when bir answer with Queition XI, but the in

closed is somewhat more concise. the tangent or cotan

Your's, &c. 7+43

2016 June, 1796.

J.F--r. gent of 15°; that is, when the angle of che cone is 30° or 150°.-By a similar pro- Question VII (No. II). Answered by cess, we shall get for a maximum or mini.

Mr. 9. Fr

LET AB be xf2a-b27/«-422+), the pole, C the 3а

eye of the obferwhose limits are when a :b::I ver, DE che surWatu 3, the chord of 30° or 150°.- and FG the exMr. H. observes in his scholium, that the eremities of the Bl roots of the last-meniioned equations do image of the pole, not always indicate the greateit and least appearing by re- DA sections of which the cone is capable. flection.-By the The truth is, that when the angle of the laws of optics, the triangles AGD and CGE cone is under 30°, the plane TV in re

are similar, as are also the triangles BFD and solving about T, from H towards L, +CÉ (=39): DG+GE(330) :: CE (=13)

CFE; therefore, AD: DG :: CE: GE; or AD forms a series of ellipses whose areas con

: GE=10; and, in like manner, BD+CE ftantly 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 6 inches nearly. by the lesser value of these quantities, We shall also have (CGTAG: CG or) DE again'diminishing from thence to the ver- : GE ::4:11 inch, the breadth of the image at tex of the cone. --This maximum, when the end neare't 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 ellipse, but less if the QUESTION XI (No. III). - Anfwered by angle is between that value and 30°, in

7. Fwhich 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 =21-5 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

-Bt

C, &c. A, B, C, &c. fore fent you. The question giving some

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 p be put equal any number of boards lefs 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

foregoing series will give the answer, fore not deficient in candour.

Your's, &c. July stb, 1796.

J. F

The same answered by Mr. T. Hickman. Dr. Hutton, in his New Mathematical Dice sionary, Vol. I, page 303, has shewn that

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n. 1-T12.13 QUESTION XII (No III).-Answered by 1. 2. 3. 4.

Mr. Y. Fr.

It is evident that and..

the distances of 1. 2. 3. 4. 5

the point perpendim31.-4. n

will

cularly under the 1. 2. 3. 4. 5.

balloon from the severally express the number of combinations of stations, will be as r: things by ones, by twos, by threes, by fours, the respective coby fives, and by fixes. In the present case, tangents of the an. A we 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 í board in each.

Put, therefore, a, b, and the cotangents of 75 DÓ

by 2

DO

the angles of elevation observed at A, B, and C, DO

by 3

DO

respectively:--Find the line AD: AB::c:b, by 4

DO

and the line CD: BC :: 2:6.--About A and 6 Do

by 5

DO

C with the radii AD and CD, describe arcs Do

by i

Do

intersecting 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 same 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:6:: 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 known rule, but by experiment only.'

by one analogy, viz. Cotangent angle elevaany The most concise and plain method of treating tion : distance of perpendicular from station : : 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 :

892.708=AD; and b: a :: BC : 1818.9215 itinen -M for any number whatever. But

CD. as the positions of each single shutter is to be

Then, by plain trigonometry, we get the anadded, the rule will stand 29-1. For instance, if n=6, then 26-1563, the whole number of 6 gles BAD° 7' 3", and BCD=15° 33'15";

and hence BD=549-077. Also, things; if n=9, then 29-1=511, the whole

CD :: AB: 3333 ©2 3=AE. number of 9 things, &c. for any number what.

As BD: AB :: BC: 2732.127=BE. 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° : AE : ; radius: 887722, &c. yards, ways, with themselves and each other, both as

Cot. 180: BE the height of the balloon reto things themselves, and the order or position Cot. 20° : CE quired. of them, which some authors call the compofition of things; the general rule, then, is, nn.--I : X n. In this case, if n be 6, it will be

The fume an!wered by Mr. T. Hinckman. 65. -X6=559862 the number of compofi. In the annexed figure, let

o 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

2427

right line AC; OD a perthe 24 letters of the alphabet be

-X24; pendicular from the balloon

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

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1796.]
New Mathematical Questions.

477 put m=AB=1000, n=BC=1500, s=-t- QUESTION XIX.-By Philalethes. =AC=2500, a=tang. angle. A OD or 75',

Of seven numbers, in a continued geometrical b=tang. BOD or 720, catang. 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 :0::*: ax=AD; in like 381,250 : to find the seven numbers? manner is BD-cbx, and CD=ax. Again, as AC (s) : AD+CD (axtıcx) : : AD- CD The folutions of the above questions must be a2x2-(272

fent, at the latest, in the forft week of September ; (2x8) :

-=AF-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. a2x2 (2x2

722 two give AF--BF=AB=M=

ERRATA. P. 394, col.ii, 1. I,

for

read

r_2 b2x2 (2x2

_h from which equation we have

Ib. line 13 from the bottom, for 72 h2o

371 =887.53 yards, the

=Cd read 37=Ca. Pa. 396, col. i, 1. 7, for vaz

ABFX, read ABF+

Pa. 213, col. i, I. 24 from the bottom, for pendicular height of the balloon as required.

x+.x and amb read ata and x-b. cholium. The theorem above given, may be

Pa. 213, col. ii, l. 2, for propir read proposed.

Ib. col. ii, of considerable use to determine the height and

3, for 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-Nx Nx?,"&c. Pa. 305, col.i, 1.2, N. B. S-lutions 10 Question XIII. by Mr. J. F r, and Mr. 7. H. are deferred till our

31232

read "I X4=4*32674,the value,"&c. next for want of room.

382337 Ib. col. ii, !. 25, for 31(m+»)-(02-1)^2

X!, New MATHEMATICAL QUESTIONS.

3n (9---)-(721)m

3n(n+m)—(MP-11.72 To be answered in No. VIII, the Mag. for

Xr.

31(2-1)-(2-1)m2 Sefirember.

P. 393, col. ii, I. 36, for fee do you," read QUESTION XVI.-By Mr. 0. G. Gregory, of seem to." Ib. col. ii, 1. 44, for fummoned' Yuxley.

read “ summed." Ib. col. ii, 1.47, for "parts

read “ points.” THE dimensions of a cylindrical tube are such, that, having a plate of tin at one end, with an

P. 394, col. i, 1. 28, for aperture in its centre, the other end being open and turned towards the heavens, the field of

b

read V view it takes in, is one-twentieth of the hemi- P. 395, col. i, l. 40, for 3V

3ac

Зас sphere :-The young perusers of the mathema

Ib. col. ii, 1. 38, for there read then. 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 fhall certainly termine the height of the mountain ?

appear in our near. ANECDOTES AND REMAINS

1. 2

1

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EMINENT PERSONS. [This article is devoted to the reception of Biographical Anecdotes, Papers, Letters, &c. and

we request the Communication of such of our Readers as can alist us in these obječts.] ANECDOTES OF PERSONS CONNECTED Bossuet, and minister of the marine, on WITH THE FRENCH REVOLUTION. the recommendation of Condorcet*. He [Continued from our lajt.]

was an honest and virtuous, but dull and MONGE,

plodding, ORIGINALLY a stone-cutter at

Mezieres, in Champagne, became * « On ne favoit 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

a

2

plodding, man; tetally incapacita:ed, by light emitted from the Satellites of Junature and education, to act the impor- piter, published in 1971, that he told the tant part assigned to him by friend ihip, 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 aniwers 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 posterity.” 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, affiduity, and thip-building, this might have been a talents, united in one and the same perfufficient qualification ; but when, in- fon ; it was a great misfortune, both for stead of contending with the pallive signs himself and his country, that he should of triangles and pirailelograms, 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 fleets, he was quite During his mayoralty, he was induced, bewildered. The result was, accordingly, by Lafayette, to host the red flag, the what might have been expected the symbol of insurrection, on the top of the French marine became almost annihi- Hotel de Ville, and thus countenance the lated, during the administration of a mal Icre, 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 twa Buzor

years afterwards, before a tribunal, stain: Was 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 trofe of Greece, America, and Swit- (uth November) 1793, in the 57th year zceland, instead of a republic, one and ind wjzinc, cost him his life. How much

CHAMPAGNEUX mut 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 fa. his caracter, by the opprobrious nick- ther of a numerous family; a man of unn!'re of le roi Buzo!! But this was at a impeached morals, and was attached to pasli, and the cufom is not yet abo. liberty from principle, at a time, and in a luned, when naughty children were country, when it was not unusual to be wnip ed by their parents for being les fo from mere speculation ! He was sepetits orifioirats!

lected by Roland on account of his inBAILLY,

dustry and talents; and was put by him The firî mayor of Paris, was a man of at the head of the principal division of the science rather than a politician. He home department. In short, during his distinguished himself by his History of adminiftration, he became, what is termed Astronomy, in 5 vols. 4to; by his The- in England, under-fecretary of fate. ory of the Satellites of Jupiter, which

Camus. had engaged his attention ever since 1763; and by severai learned Memoirs, does great credit to his discernment. Soon

This is another of Ruland's élèves, and inle ted in the proceedings of the Aca- after the resignation of his friend, he quitdemy of Sciences. Jerome Lalande, one of the first astronomers of the present member of the Convention, and is now

ted the home department, was elected a day, and who, at this moment, prefides over the National Observatory, was fo Arcbevis to the present legislature. He much plealed 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 foles de la Ville of his legislative functions (for he is one de Berne, & c.-Azpel de Mad. Roland. of the two-thirds) and, on the first vacancy,

is

of his age.

C.

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