The fame anfwered by the Rev. Mr. HELLINS, Teacher of the Mathematics and Natural Philofophy. ANALYSIS. Let abcd reprefent the fih-pond, and ABCD the outside bounds of the walk. Then, fince the walk is of the fame breadth on every fide, it is -evident that if any two correfponding angular points be joined, the line which joins them will bife&t thofe angles. Thus the line Aa bifects the angle DAB, and the line Bb bisects the angle ABC, &c. If now Aa and Bb be produced, till they meet in there will be given the triangle aeb. In like manner the three other triangles bfc, cgd, and tha become known. The probiem, then, is reduced to this: To four given triangles to add as many spaces, by producing the fides of thofe triangles until A a D B they meet four right lines drawn parallel to their bafes, which four spaces, taken together, fhall be equal to a given fpace: to facilitate the conftruction of which I thall premise the following LEMMA. If through two triangles of equal bafes, and between the fame parallels, a line be drawn parallel to their bafes, it will cut off equal spaces from thofe triangles. This is fufficiently evident from Euclid I. 38. & Ant. COROLLARY. If the fides of two triangles, having equal bafes and altitudes, be produced to two fines, drawn parallel to, and at equal distances below their bafes, the spaces added will be equal to each other. CONSTRUCTION Let the two indefinite lines Is, in form a right angle at 1; in is take Imab, the bafe of the triangle arb in fig. 1, and draw un fo that the altitude of the triangle Inm may be equal to the altitude of the triangle aeb. In like manner, in the fame right line, take mo, oq, qs, respectively equal to the bafes of the other three triangles, making the altitudes of the triangles mpo, org, qts, formed on them, equal to the altitudes of their correspond x Produce s, and ing ones in the given trapezium. Produce st until it cut la in v. make the triangle lxu, by Prob. IV. p. 218 of Simp. Geom. fimilar to the triangle lus, and equal to the given area of the walk. Join xv, and in vl, produced, take vy=vx, and ly will be the breadth of the walk. DEMONSTRATION. 2 Produce vs till it cuts a line drawn through y, parallel to Is in z. Then (Euc.? VI. 19) the areas of the fimilar triangles yuz, lus, and I are as the fquares of the fides yv (=xv) lu, and lx; and fince yv2 (=xv2) =/v2+x2; it is evident the area of the triangle yuz is equal to the fum of the areas of the triangles lus and Ixu; and, confequently, that the quadrilateral ls is equal to the triangle lxu. Produce now nm, po, and rq to meet the line y in the points zu, i, k, respectively; then, by the foregoing Lemma, the quadrilateral law is equal to its correfponding fpace AabB; and fo are the others, mulo, oikq, qkzs to their correfponding ones in the first figure, and their fum, or the quadrilateral ysz, is therefore equal to the area of the walk, Q. E. 1) SCHOLIUM I. It is evident that this conftruction may be used for any multilateral. SCHOLIUM II. If a circle be defcribed, the diameter of which bears the fame proportion to the perimeter of the given trapezium that radius bears to the fum of the four cotangents to the four half angles of it; and if a tangent be drawn to this circle, equal to the fide of a fquare which has the fame proportion to the area of the walk that radius has to the fum of the said four cotangents; the difference between the radius of this cicle and the fecant to that tangent will be the breadth of the walk. Another ANSWER to the fame, by Mr. GEO. SANDERSON. ANALYSIS. Suppofe thething done, and ABCD the pond, EFGH the outward boundary of the walk, AJEK, BLFM, &c. trapezia, made by perpendiculars from the angles of the pond on the fides of the walk. Then becaufe the perpendiculars are equal, the trapezia, as well as the angles E, F, G, and H, are bifected by the lines AE, BF, &c. but the angles are given, therefore the Patios of the perpendiculars to the correfponding fides of the trapezia are given. Moreover it is manifeft that a rectangle on one of the equal perpendiculars, K and the fum of the fides IE, LF, &c. of the trapezia, together with a rectangle under the fame perpendicu Jar, and the fum of the fides of the pond is equal to the area of the walk; whence the following ΕΙ CONSTRUCTION, Q H R Make ak equal to the fum of the fides of a the pond, and let R be the fide of a fquare that is equal to the area of the walk. To ak draw the indefinite perpendicular kr, on which take kg to ak in the given ratio of IE to AI, and gb ak: LF: LB (IA); hf: ak :: NG: ° CN and fr: ak:: PH: DP, join ar, and on ak take ae a third proportional to ak and R; then by Problem 3. Book I, of Mr. Wales's Deter. Section, cut ea in o, fo that the fquare on ao may be to the rect. contained by eo and ak in the ratio of ak to kr; and having erected the perpendiculars AI, BL, CN, and DP (fig. 1)each equal to ao, through the points I,L,N,P, draw EH, EL, FG, and GH parallel to the fides of the pond, meeting in the points E, F, G, and H, and the thing is done, e k g b DEMONSTRATION. Join ag, ab, af, and draw og parallel to kr, cutting them in the points 1, n, p, q. fimilar triangles, and the conftr. ao: ol:: ak: kg :: AI: IE, but AI ao by conftruction; therefore IE. And by the fame cafoning, LFla, NG=np, and PH 19. And becaufe o and AIE are right angles, a rect. on ao and of is. equal to a rectangle under AI and IE trapezium AJEK, and a rect, under ao and In (twice triangle lan) trapezium BLFM, . a ect. under ao and og is equal to the fum of the trapezia AIEK, BLFM, &c. Again ak : kr :: ao2 : oe × ak (by const.) = ae-ao x akae × ak-ak × ao; but ae × ak R2 by conft. therefore ak : kr:: ca2 : R2 —ak x ao ¦¦as: og:: ac2 : oq xao, wherefore R2=ak × ao+oq×ao, but ak is equal to the fum of the fides of the pond by conft. and ao AIBL, &c. Therefore, the rectangles AL, BN, &c. together with the fum of the four trapezia are equal to R2 the given area of the walk, as required. MATHEMATICAL MATHEMATICAL QUESTIONS. 22. QUESTION I. by MATHEMATICUS, of Greenwich. 293 It is required to determine that parallel of latitude, in which if two places be fituated that differ 180° in longitude, the distance between them, reckoned on the parallel, may exceed their distance on the meridian by the greatest quantity poffible. 23. QUESTION II. by ASTRONOMICUS. To find the declination of that ftar whofe change in azimuth is the greatest or leaft poffible in paffing from one given almicanter to another given one, in a given latitude, 24. QUESTION III. by NUMERICUS. A father on his death-bed divided his cash, confifting of a number of gui neas, among his children in the following manner: I guinea and 4th part of what remained; the fecond to take 2 guineas and th He ordered the firft to take of what remained; the third to take 3 guineas and 4th of what remained; and fo on, fucceffively, for the others. Now, this diftribution being made, it was found that each child had an equal portion, What number of guineas did the old man diftribute, and how many children had he? 25. QUESTION IV. by Mr. REUBEN ROBBINS. In a plane triangle, there is given the rectangle of the fides about the vertical angle, the perpendicular on the bafe, and the difference of the fegments of the bafe, made by it, to conftruct the triangle. 26. QUESTION V. by Mr. GEORGE SANDERSON. Suppofe AEB a given femi-circle, the center of which is C; and let D be a given point in the diameter: now, if the point E be fuppofed to move, in the circumference, with an equable celerity; it is required the angular velocities of the two lines ÉD, EC are equal. to find its place when The anfwers to thefe queftions must be fent, poft-paid, to Mr. Baldwin, in Paternofter-row, London, before the 1st of January, 1784. MISCELLANEOUS PAPERS. FOR THE LONDON MAGAZINE. ON THE CHARACTER OF CORDELIA, IN SHAKSPEARE'S KING LEAR. DISINTERESTED principles are of different kinds. Of confequence, the actions that flow from them are more or lefs beneficial, and more or lefs entitled to praife. We are moved by inconfiderate impulfe to the performance of beneficent actions, as we are moved by inconfiderate impulfe to the perpetration of guilt. You fee an unhappy perfon: you difcern the vifitations of grief in his features: you hear them in the plaintive tones of his voice: you are warmed with fudden and refitlefs emotion: you never enquire concerning the propriety of your deportment, or the merits of the fufferer; and you haften to relieve him. Your conduct proceeds from mere feeling. lity, but not of reflection. It entitles you to the praise of fenfibiagain in the fame fituation. But the You are fymptoms of diftrefs do not produce on you the fame ardent effects. You are moved with no violent agitation: perceive diftrefs, you are convinced that and you feel little fympathy. But you the fufferer fuffers unjustly. You know that you are bound to relieve him; and afford him relief. Your conduct proin confequence of thefe convictions you ceeds from fenfe of duty, and though it entitles you to the credit of rational 3 humanity, humanity, it does not entitle you in this inftance to the praise of fine fenfibility. Thofe who perform beneficent actions from immediate feeling or impetuous impulfe have a great deal of pleafure. Their conduct, too, by the influence of fympathetic affection, imparts pleasure to the beholder. The joy felt both by the agent and the beholder is ardent, and approaches to rapture. There is alfo an energy in the principle, which produces great and uncommon exertions. Yet both the principle of action, and the pleasure it produces are tranfitory. Beauteous as the morning cloud or the early dew," like them, too, they pafs away. The pleafure arifing from fenfe of duty is lefs impetuous; it has no approaches to rapture, it feldom makes the heart throb, or the tear defcend; and as it produces no tranfporting enjoyment, it feldom leads to uncommon exertion. But the joy it affords is uniform, fteady, and lafting. 66 As the conduct is most perfect, fo our happiness is most complete when both principles are united: when our fenfe of duty is animated with fenfibility, and fenfibility guided is by fenfe and duty. No happinefs can be more defirable than that which is both ardent and lafting. It is indeed to be regretted that feeling and a fenfe of duty are not always united. It is deeply to be regretted, that unlefs fenfibility be regulated by that fenfe of duty which arifes from reflection on our own condition, and knowledge of human nature, it may produce unhap piness both to ourfelves and others; but chiefly to ourselves. Shakspeare, in his character of Cordelia, has given us a fine example of exquifite fenfibility, governed by reafon, and guided by the golden rule of propriety. This amiable character indeed, is conceived and executed with no lefs kill and invention than that of her father. Treated with rigour and injuftice by Lear, the utters no violent refentment; but expreifes becoming anxiety for reputation. yet befeech your Majesty That you make known It is no vicious blot, murder, or foulness, She difplays the fame gentleness, accompanied with much delicacy of reproof, in her reply to a mercenary lover. Peace be with Burgundy! Since that refpects of fortune are his love I thall not be his wife. Even to her fifters, though fhe has perfect difcernment of their characters, and though her misfortune was owing to their diffimulation, fhe expreffes nothing virulent nor unbecoming. She expreffes, however, in a fuitable manner, and with no improper irony, a fenfe of their deceit, and apprehenfions of their difaf fection to Lear. Ye jewels of our father with wash'd eyes And like a fifter am most loth to call Towards the clofe of the tragedy, when the receives complete information concerning the violent outrages committed against her father, the fufferings he has undergone, the ruin of his understanding, and has the fullest evidence of the guilt and atrocity of her fifters, the preferves the fame confitency of character: notwithstanding her wrongs, the feels and is affected with the deepest forrow for the misfortunes of Lear: fhe has the most entire abhorrence of the temper difplayed by Goneril and Regan: yet her forrows, her refentment, and indignation are guided by that fenfe of propriety, which does not in the smallest degree, impair her tenderness and fenfibility, but directs to that conduct and demeanour, which are feitable, amiable, and interefting. Tendernefs, affection, and fenfibility, meking into grief, and mingled with fentiments of reluctant difapprobation, were never delineated with more delicacy than in the defcription of Cordelia, when he receives intelligence of her father's misfortunes. Kent. Did your letters pierce the Queen to any demonftration of grief? Gent. I fay the took 'em, read 'em in my And now and then an ample tear trill'd down Kent. Which thould exprefs her goodlieft; you have feen Kent. Made the no verbal question? She heav'd the name of father, Pantingly forth, as if it prett her heart. 295 and watch with anxious concern over We regard it with refpectful attention, our whole behaviour, left by any impropriety we fhould difturb that concerted tranquillity, which it requires fo great an effort to fupport."-Cordelia, full of affection, feels for the diftrefs of her father: her fenfe of propriety impofes reftraint on her expreflions of forrow: the conflict is painful: full of fenfibility, and of a delicate ftructure; Cry'd, Sifters! Sifters! what? i'th torm of the conflict is more than fhe can endure; night? Let pity ne'er believe it! then the shook Minds highly enlightened, contemplating the fame object, both reason and are affected in a fimilar manner. The tone of thought in the following paffage in "the theory of moral fentiments," accords perfectly with Shakfpear's account of Cordelia. "What noble propriety and grace do we feel in the conduct of thofe who, in their own cafe, exert that recollection and felf-command which conftitute the digaity of every paffion, and which bring it down to what others can enter into? We are difgufted with that clamorous grief, which, without any delicacy, calls upon our compaffion with fighs and tears, and importunate lamentations. But we reverence that referved, that filent, and majestic forrow, which difcovers itfeif only in the fwelling of the eyes, in the quivering of the lips and cheeks, and in the diftant, but affecting, coldnefs of the whole behaviour. It impofes the like filence upon us. fhe muft indulge her emotions; her fenfe of propriety again interpofès; the muft vent them in fecret, and not with loud lamentation. She hakes The holy water from her heavenly eyes, There are few inftances in any poet where the influences of contending emotions are so nicely balanced and diftinguifhed: for while in this amiable picture, we difcern the corrected feverity of that behaviour which a sense of propriety dictates, mitigated and brought down by fine fenfibility, and the foftnefs of female character, we alfo fce this foftnefs upheld, and this fenfibility rendered fill more engaging, by the influences of a fenfe of propriety. We may, therefore, deduce from the whole, that the conduct is most perfeet, and the demeanour moft engaging where" fenfibility is guided by a fenfe of duty and propriety; and the fenfe of propriety animated or foftened by fine fenfibility." ACCOUNT OF THE EARTHQUAKES IN CALABRIA, AND VARIOUS PARTS OF SICILY. COMMUNICATED TO THE ROYAL SOCIETY BY SIR WILLIAM HAMILTON, (Continued from page 228.) FROM Terra Nuova I went to Oppido. This city is fituated on a mountain of a ferrugineous fort of gritty ftone, unlike the clay foil of its neighbourhood, and is furrounded by two rivers in a ravine deeper and broader than that of Terra Nuova. Instead of the mountain on which Oppido was fituated having fplit in two, and by its fall on the rivers ftopped their courfe, and formed great lakes, as we are told; it was (as at Terra Nuova) huge pieces of the plain on the edge of the ravine, that had been detached into it, nearly filled it up, and flopped the courfe of the rivers, the waters of which are now forming two great lakes. It is true, that part of the rock on which Oppido ftood was detached with feveral houses into the |