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ID, by Euc. III. 35; therefore the rectangle contained by OD and DG is equal to that contained by FD and ID; that is, to the rectangle contained by OD and ND, by construction : confequently GD is equal to ND; GA is, therefore, equal to KA, and the rectangle AHFD is the greatest that can be iloibed in the triangle FGK, by Prop. VIII. p. 199, of Simp. Geom. but tbe rectangle thus inscribed in the triangle FGK, is always equal to the triangle ABC inscribed in the circle ABEC, by Euc. I. 42 ; the triangle ABC is, therefore, the greatest that can be inscribed in the given circle BACE, when the difference of the segments of the base is equal to the given line CE,

An elegant construction was also given to this question by Mr. W. Richards.

48. QUESTION (III. March) answered by ELTONIENSIS. Take Maclaurin's example, page 185: viz. *3-—px? +9x_roo, and let the greater root (or value of x) be a, the mean b, and the least c. Then (hy articles 13 and 19) a+b+c=p; ab +ac + bc=9; and abc=r; : q2-2pr=a2 b2 + azca +62 c2; but a is greater than c or b; : 324 is greater than az b2+a(2 +62 cm, (q2_zpr) therefore, by putting e+=q2-2pr, =ab2 +a (2+62c3, * is less than

3 is less than a, as was to be investigated. V 3 In the second theorem, x-px-+qxy--rx—3+389-4, &c. by sect. 13, and the doctrine of combinations, the terms in g (or number of products that can be made by multiplying any two of the roots) are equal to n x ?; and in r (of

e

at or

2

nI

three) are equal to n x

X

i

also in s, the number of products formed

3 by multiplying any four of the roots, is equal to n x whence it is manifest that the number of terms a: b2 + a2ca, &c. in g“, is equal to

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n3

72 Х

Х 3

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2

4

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; and the number of terms in 2pr (azbe +a-bc+baac+b?ac, &c.) is

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nI

n-3

72 Х

3

X

&c. is equal to 2n X

; from whence it appears, that the rule fails when the roots are nearly eçual, and n greater than 3.

For example, in the biquadratic xtpx3+9x?-r*+s=0; grab tartad, &c.

2

4

n1 ton X

2

(6) terms, and q2 =ahz a2c2 + a? d?, &c. to n2 x

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terms; and those of 2pr--a2bc+a’dd, &c. to 2n?

(32); and 25

n-3 =abcd tabcd, to 2n x

Х (2) terms: : the. Dunber of 3

4 terms in q2-2pr +25, when the roois are all equal to a; is manifeftly equal to 6,

92-2pr+25 and

6a4 and the 4 växa = is greater than a,which is 4 contrary to the rule. In the surfolid equation x5-2x4+qx-rx? +$x-10:

92-2pr+25

5 4 10 as, or

= 4dio

And the same might be thewn for all the 5 higher powers.

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49. QUESTION

104.

45. QUESTION (IV. March) answered by A COBBLER. Let L represent the place where the thips met,

с D

B B and C the ports they failed from, LD the differs ence of latitude, LB, LC the distances run by the two ships, and which, consequently, are to each other as 5 to 3, or as i to }, which put = m: make p=260 (not 250, as was printed in the question) the sum of the three fides, LB, LC, and BC; a=LD,=64, and x=LB. Then, because 1 : m :: x:mx, =LC; and, by the question, x+mx+ V x2-a? + m2x2-a2 = p. Hence, V x2--22

+v m2x2 - a2 = p-a-mx; and, by squaring both fides of the equation, and making proper re

L duction, vx--a-xmox:- a2 = 1 p+a?--px-pmx+mx?. Put b={pata?, and n=p+1m, and again, Squaring both sides, we obtain 2mnx}~*Xa?-ama-maarta’m2 +12 + 2bm

2b71x 02-24 2bm+abnx=b2-a+; or x

xx? +

which in numbers is xl-448,92179487x2 +63160x=2843208,33, &c. Now, in order to resolve this equation, it may be considered that as LD is 64, LC cannot be leis than 64; and, as LC is to LB as 3 to 5, LB (x) cannot less than 1063.. Again, as LB cannot be lets than 1063, and LD is 64, DB cannot be less than V1063-642, = 85,4: consequently, LB (x) cannot be so great as 110,6. Having thus got LB within such narrrow limits, we readily find x= 106,83, by the common methods of approxiination : LC, the distance run by the other thip, is, therefore, = 64,1; and the distance of the ports 89,07 miles. The course of one of them is S. 53° 12' W. and of the other S. 3° 12' E.

Q. E. I.

= 2mn

2 MN

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MATHEMATICAL QUESTIONS.

64. QUESTION I. by NUMERICUS. What three numbers are those, the sum of which is a cube number; and if this cube be increased by half the product of the two least, the sum will then be a square number: moreover, the sum of the squares of the two least is equal to the square of the greatest.

65. QUESTION II. by R. M*. Given the base of a plane triangle, and the sum of the sides and perpendicular, to determine the triangle when the vertical angle is a maximum.

66. QUESTION III. by Mr. THOMAS Moss. If upon any indefinite right line, DO, two circles be described whose diameters DB, DC, are in

F any given ratio to each other, and two other circles be defcribed upon another indefinite D right line, dq, whose diameters,

B

Q db and dc are likewise in the same given ratio of DB to DC; and if from the points B and b, as centers, two other circles be

fo This gentleman is requested to send answers to such questions as yet remain with the editor, as the motives for proposing those questions do not appear to him without them.

R

f

77 fo defcribed as to cut the peripheries of the two larger circies in G and g, equidiftant from the points D and d: then if any two lines be drawn froin the two points D, d, cutting the peripheries of the circles in R, S, E, and F, and in r, s, e, and f; and so as to make DS = ds: I say that the corresponding chords DR, dr, and SE, sf; as also the dištances RS, ys, and RF, rf, intercepted by the two peripheries will be respectively equal to each other.

The answers to these questions may be directed (post-paid) to Mr. Baldwin, in Paternoster-row, London, before the itt of October.

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BIOGRAPHY. THE LIFE OF CHARLES CHURCHILL. CHARLES CHURCHILL", an the next day, he brought his exercise

eminent fatiric poet, was born fininei in such a mammer, that he in Vine-Street, in the parish of St. received the public thaoks of the John's, Weitminster, in the year 1731. maiters of the school. This initance His father, who was a very respectable of his sensibility, and of the applause, clergy man, was curate and lecturer of that resulted from it, was not followed the parish, and was poflefled, besides, by a complete reformation of conlust. of a living in the country. Young The vivacity of his imagination, and Charles, as might be expected, from the diflipation of his temper, still prethe vicinity of his situation, received vented his walking regularly forward his granınatical education at Wert in the trammeis of a scholaíic educa-, miniter fchcol; in which he soon di- tion. When, ther:fure, he was sent by, stinguished himself fo fir, as to make his father to the University of Oxford, his tutors fentible that he was a lad of he was refused an admittance into considerable abilities. His application, that illustrious seat of literature, upon however, as is too frequently the case account of his want of a proper skill in with youths of lively parts, by no the learned languages. This, no doubt, means kept pace with his natural tä- was a great mortification to himself, lents; fo that the chief character he, as weil as a severe disappointment to a obtained was, that he was a boy who worthy parent. Churchill, in the subcould do well if he would. One day, sequent parts of his life, often menhaving been enjoined to make an exer- tioned his repulie at Oxford; and the cise, he failed in bringing it at the time following turn was given to it by appointed; for which reafon his masier himself and his friends. He and they not only chastised him with some fe- frequently asserted, that he could have verity, but even charged him with answered the college examination had stupidity. The last reproach made a he thought proper; but that he fo. strong imprellion

lijon Charles much deipised the trifling, questions Churchill's mind, and the fear of which were proposed to him, that, shame wrought an effect which the initead of returning fuitable replies, he fear of stripes could not produce. On only launched out into fatical re. LOND, MAG. July, 1784.

D

flections * This life is abftracted from tha: infcrted in the new volume of the Biograpbia Britannica,

flections on the abilities of the gentle having taken a little house, he applied man whose office it was to make the himself to the duties of his station with trial of his literary improvements. If assiduity and chearfulness. His bethis was really the truth of the case, haviour gained him the love and esteem Mr. Churchill's conduct, to say the of his parishioners; and his sermons, Jeast of it, was highly imprudent. though somewhat raised above the level Whoever wishes to receive the benefit of his audience, were commended and of an university education must comply followed. What chiefly disturbed him with the customary forms of admission; was the smallness of his income, and it would be perfectly ridiculous which would, indeed, have been too for a young man to have it in his own narrow for the support of a family, power to prescribe in what mode he even where a much greater degree of îhould be examined, previously to his economy was exercised than was fuitmatriculation. Churchill's rejection able to Mr. Churchill's natural dispofrom Oxford will supply one very pro- fition. To supply, therefore, the bable reason for the severity with deficiency of his scanty falary, he which, in the course of his writings, entered into a branch of trade, which he hath sometimes treated that famous he hoped might raise him to compeseminary.

tence, and, perhaps, to riches; but After this event, Mr. Churchill which, in fact, involved him in debts continued to prosecute his studies at that long involved him in perplexity Westminster school; and there can be no and trouble. The business in which cause to doubt, but that he would soon he engaged was that of keeping a have been esteemed properly qualified cyder-warehouse, with a view of vendfor an entrance into one of our learned ing that commodity in the different universities, if his views of this kind parts of the neighbouring country. A had not been prevented by an act of man of genius and a poet was but ill imprudence, which had a considerable qualified for such an undertaking. Mr. effect upon the colour of his future life. Churchill could not descend to the paWhen he was little more than seven- tience and frugality which are necessary teen years of age, he contracted an in the common course of merchandise, intimacy with a young lady in the where small gains are to be quietly neighbourhood, which sprang up into expected, and carefully accumulated. a warm affection, and was followed by A'kind of rural bankruptcy was, therea hafty marriage. This, like many fore, the consequence of the attempt. others, was a match which began in The ill success of Mr. Churchill's passion and ended in difguft. Their trading scheme brought him back to regard, however, for each other, which London, and his father foon after in its origin was mutual and sincere, dying, he succeeded him as curate and was preserved in its purity and ardour lecturer in the parish of St. John's. for å nuinber of years. In the se. The emoluments of his situation not questered life which Mr. Churchill was amounting to a full hundred pounds a row obliged to lead, he made such a year, in order to improve his finances, progress in literature, and sustained so he undertook to teach young ladies good a character, that, notwithstanding to read and write · Englith with bis want of an university education, propriety and correctness, and was he was thought worthy of being ad. engaged for this purpose in the boardmitted into holy orders, at the usual ing-school of Mrs. Dennis, a governess, age of obtaining them, and accordingly who had the honour of being one of was ordained by Dr. Sherlock, at that the first introducers of a Jaudable time Bimop of London. The first custom, which hath fince been adopted preferment he received in the church in many of the reputable seminaries of svas a very trifling one, being only a female education. Mr. Churchill conImall curacy of thirty pounds a-year, ducted himself in his new employment in Wales. "To this remote part of the with all the decorum becoming his kingdom he carried his wife, and, clerical profession. Still, however, his

method

method of living bore no proportion the orchestra. From this place he to his income; so that he contracted a thought that he could beft difcern the sariety of debts, which he was totally real workings of the passions in the incapable of paying; and a jail, the players, or the artifices which they subterror of indigent genius, seemed ready itituted in the room of genuine nature to clofe upon his miseries. From and feeling. As Mr. Churchill was this wretched situation he was relieved thus qualiñed, by judgement and expeby the benevolent interposition of Dr. rience, for delineating the excellencies Lloyd, the second master of Weft- and defects of the actors, so the vigour minster-school, and father of Robert of his fancy, and the strength of his Lloyd, the poet. The Doctor under- conceptions, enabled him to do it in took to treat with Churchill's creditors, the most lively colours. In the month and succeeded in engaging them to of March 1761, the “Rosciad" appearconsent to a composition of five shil. ed. The first edition stole as it were lings in the pound. In an inftance into the world, being very little adverwhich fell under the knowledge of the tifed, and published without a name. writer of the present article, as an A second impression was soon called executot and a guardian, Mr. Churchill; for, in the title page of which the when he had obtained money by his author asserted his claim to his own perpublications, voluntarily came, and formance. Scarcely ever was there an paid the full amount of the original instance of a poet's rising so suddenly debt. It is highly probable, from this from the most perfect obscurity to the unsolicited and unexpected act of equi- greatest celebrity. To this the players table retribution, that his conduct was themselves contributed more than any the same in some other cases.

other set of men. They ran about the The time now approached for Mr. town like so many stricken deer; and Churchill's appearing in the world as while they strove to extract the arrow an author. Hitherto nothing had come from the wound, by communicating the from him in this character, though he knowledge of it to their friends, spread was known among his acquaintance to abroad more and more the fame of the be a man of a very vigorous imagina. piece. It was pleasant enough to obtion, and a stroog understanding; and ferve how artfully some of them, who though he was in the habits of intimacy were, in fact, the most hurt, pretended with Thornton, Colman, and Lloyd, to be unaffected by the injury done to who had already begun to make a con- themselves, but to feel extremely for the siderable figure in the republic of obloquy thrown upon others. letters. With the last of these gentie- (exclaimed one of these disinterested men he was connected in the ties of persons) should this man attack Mr. the closest friendihip. Mr. Lloyd had Havard? I am not concerned at all printed a poem, entitled the A&tor, for myself; but what has poor Billy whidi met with a very favourable re- Havard done that he must be treated fo ception from the public, and justly cruelly?” -“ And pray (replied a genprocured him a considerable degree of tleman who was present at this artiseputation. By the success of his ficial declaration of benevolence) what friend, Mr. Churchill is supposed to has Mr. Havard done too, that he have been stimulated (how truly we cannot bear his misfortunes as well as know not) to exert his poetical talents another?" Whilst the actors, in dif. upon a subject of a similar kind, though ferent ways, expressed their resentment, more appropriated and personal. The the public enjoyed their distress. The theme he pitched upon was admirabiy Rosciad was regarded, in general, as a suited to his genius and his taste. He pleasant and reasonable retaliation for had long been a frequenter of the the mirth which the stage had conti. theatre, and had bełtowed inceffant nually excited, by the representation attention on stage representation. of the follies and frailties of mankind, The scene of his observations was The poem was not wholly employed usually the first row of the pit, next to in satire. Mr. Garrick was commended

" Why

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