the latitude: in the cafe given. As tang. 51° 31'; cof. 54° 15′ :; rad. : tang. 24° 54'. This Queftion was also answered very ingeniously by Meff. Dale, Hellins, Sanderfon, and Webb. 8. QUESTION (VIII. July) answered by Mr. GEORGE SANDERSON. Let C, E, and D reprefent the places of the hip at the firft, fecond, and third obfervations, A and B the two islands. Because ACB is a right angle and AE=EB, if OE be drawn perpendicular to AB, then O is the center and AB the diameter of a circle paffing through the points A,C,B, and if CD be produced to meet it in P, and AP, BP be joined, the angles PAB and PBA are equal to the given angles PCB and PCA, whence the angles CAB and CBA may nearly be determined by the following method: On any right line AB defcribe a circle, and make the angle BAP=22° 30', draw AP to meet the circle in the point P, through which draw the indefinite right line LPE perpendicular to AB; and parallel thereto, through O, draw the indefinite line GHR; on AO, produced if neceffary, take OK to OF as 8 to 3, and on FL take PL to PF in the fame ratio of 8 to 3, through the points K and L draw the indefinite right lines MN, and LN, parallel to OR, and AB, to meet in the point N, with which, as afiymtotes, through the point O, defcribe the equilateral hyperbola OCQ, to cut the circle in C; join the points AC, CB, and the angles CBA and CAB are equal to the angles which the North and West rhumbs make with a line joining the two iflands. Draw CP cutting GO and OB in the points E and D, alfo draw CZ parallel to GO, cutting OK in V, and NL in Z, and draw CI perpendicular to MN. By a well-known property of the hyperbola the rest. CN rect. ON, wherefore IC (KV): KO::OR (EL): CZ (CV+FL), but KO: KF :: PL: FL (by const.) hence, KV: KF:: PL: CV+FL, and (by divifion, &c.) KV: PL :: FV : FP + CV::DF : FP (by fim. triangles); again (by alternation) KV : DF :: PL : PF ::KO: FO (by conft.) hence (by divifion and alter.) KO : FO :: VQ : DO :: CE : ED, by fim. triangles, but KO: FO :: 8:3 (by conft.). therefore CE: ED :: 8: 3. QE. D. The fame anfwered by APOLLONIUS, If the thing were done; and A and B the fituations of the two iflands, C that of the fhip when A bore due north and B due weft, D its fituation when equally diftant from them, and E its fituation when in a right line with them: moreover if DI be drawn at right angles to the line AB which joins them; it is manifeft that as DA DB, IA will be IB; and, confequently, by Euc. III. 31, a the line joining the two iflands is bifected by a femi-circle defcribed on DE; and, from this confideration, the following construction is evident: Draw the North rhumb CA, the N. N. W. rhumb CE, and the Weft rhumb CB. In CE, take CD equal 8, and CE equal to 11 miles; and on DE defcribe a femi-circle DIE. From D, as a center, defcribe feveral concentric circles, cutting CA in a, A, a, &c. and CB in b, B, b, &c. join the correfponding points a, b; A, B; a, b; &c. with the right lines ab, AB, ab, &c. and bifect thefe lines in the points i, I, i, &c. Then, if through thefe points the curve Dili be described, cutting the femi-circle DIE in I, and through the points I and E the ftraight line AB be drawn, cutting CA in A, and CB in B; it is manifeft that A and B will be the fituations of the two islands. This Queftion was alfo anfwered by Mr. Thomas Penny, pupil to the Rev. Mr. Smith of St. Budeaux. 9. Question (I. Auguft) anfwered by Mr. GEORGE SANDERSON. Draw the ordinate CG, and perpendicular to it ND; alfo produce PC to meet the axis in the point H. VE2+EN2 (DG2) =ND2 (EG2) +CD2 (Euc. 47. . I.) .. VE2—EG2-CD2— DG2, or VG × VE—EG = CG × CD—DG, (Simpson. Geom. B. 2, Theo. 7.) whence VG: CG :: CD~DG (CG−2DG): VE-EG :: VGx2VB: CGx2VB, but VG × 2VB-CG', by the property of the curve; CG: 2VB :: CG—2DG: VE-EG, whence (by alter. and divifion) CG: 2DG (IPA) :: 2VB : 2VB - VE +EG=VB+AG (because AE-EB by conft.) hence, CG: PA::VB: VB+AG :: GH; AH (AG+GH) by fimilar triangles; therefore HGVB= parameter by conft. Wherefore, by a well-known property of the curve, a tangent at the point C is perpendicular to PH; confequently PC is the fhorteft diftance between the point P and the curve, as required. N. B. This construction holds good when the perpen. PA falls above the vertex on the axis produced. This Question was alfo anfwered by Mr. Ifaac Dalby and J. Wallon. 10. QUESTION (II. Auguft) anfwered by Mr. JAMES WEBB. Let ZP be an arc of the meridian, where Z is the zenith, and P the elevated pole: alfo fuppofe S and s to be the places of the two ftars when the difference of their azimuths is a maximum or minimum; the angles PZS, PZs their azimuths, PS and Ps their polar diftances, and ZS and Zs their diftances from the zenith. Then, by what is done in the Answer to Question II. P: Z :: R3 : cof. PZ x R2 — S Z fin. PZxcof. Z × cot. SZ; and P: Ż: : R3 : cof. PZ x R2 fin, PZx cof. Zx cot. sz. Hence, because the fluxions of the two azimuths must be equal when their difference is a max. or min. fin. PZx cof. PZsxcot. sZ=fin. PZx cof. PZS x cot. SZ; or cof. PZs x cot. Zscof. PZS x cot. ZS. Therefore, when the difference of azimuths of two ftars is a max. or min. the cofines of their azimuths are directly as the tangents of their altitudes. 11. QUESTION (III. Auguft) anfwered by Mr. THOMAS TODD. Let x, y, and v, reprefent the number of apples bought by the first, fecond, and third boys refpectively; then will 9-x, 18-y, and 24- be the oranges. Let be the price of an apple, and m that of an orange. Then x+my+18m-my; from which y=9+ 152 m-n +x. Also, nx +9m —mx=nv+24m-m~; wherefore v≤15+- +x. Now it is evident that 9n mn and 157 the greatet value of x must be taken to anfwer the problem, and that will be when are the least whole numbers prime to each other, which will be the cafe when m is equal to 4 times n, or when m4, and n = 1, for then 152 = 3, and m-n 722-22 5, and thence y=12+x, and v= 20+x. Now the limit, or greatest value of x is 3; in which cafe y=15, and 23: for x cannot be 4, becauic would then be 24, and fo the third boy would have all apples, and no oranges, which is contrary to the question. Hence, if we put m=4, n=1, of farthings, half-pence, or any coin whatever, we fhall have the values of A, B, and C, apples and oranges, as required. The Same answered by Mr. ISAAC DALBY. Let x, y, and, reprefent the number of oranges bought by the firft, fecond, and third boys refpectively, then will 9-x, 18-y, and 24-2, be the apples; let a be the price of an orange, b that of an apple, then by queft. ax+9-x×b=ay+18—y ×6=az+24−2xb or ax-tx+9bay—by+18b=az bz + 24b, and putting d a-b we have dx = dy + 9b = dx + 15b, hence x = y + ==x+ 96 Now 16 d' and are the leaft poffible whole num 15h it is manifeft that when the terms bers, and d the greatest common measure of the coefficients, 9, and 15, d will be equal 3, and b= 1; therefore x = y + z = x + 5. Here it appears that x muft be greater than 5, for when x=5, 20: therefore, x 6; and, confequently, y=3, and 1, the numbers of oranges; and 3, 15, and 23, are the numbers of apples bought by the firft, fecond, and third boys, respectively. SCHOLIU M. It is evident the prices a and b may be any numbers in the ratio of 4 to 1 thus if the price of an orange be a penny, that of an apple will be a farthing, &c. MR. JOHN BLAKE Subftitutes x, y, and z, for the oranges bought by the firft, fecond, and third boys refpectively, and r to 1 for the ratio of the price of an orange to that of an apple. Then by a procefs fimilar to Mr. Dalby's, he finds r=4, x=6, y=3, and The Rev. Mr. Garnons, the Rev. Mr. Hellins, Mr. Sanderson, and Mr. Walfon alfo anfwered this Queftion. MATHEMATICAL QUESTION S. IN taking the distance of Arcturus and Lyra, and of Arcturus and Spica Virginis, the plane of the inftrument, paffing through the ftars, was inclined to the the horizon in angles of 56° 30', and 58° 40', refpectively; required the latitude of the place of obfervation, and the hour of the night on March the 10th, 1783. 28. QUESTION II. by Mr. GEORGE SANDERSON. Mr. Robertfon, in his Elements of Navigation, Art. 253. B. IV. has given the following rule to find the bafe of a fpherical triangle, the two fides, and the included angle being given. To twice the log. fine of half the given angle add the log. fines of the two containing fides; from half the fum of thefe logs. fubtract the log. fine of half the difference of the fides, and the remainder is the log. tangent of an arc; the log. fine of which are fubtracted from the faid half fum of the logs. leaves the log. fine of half the required fide. The demonftration of this rule is required. 29. QUESTION III. by GEOMETRICUS. Given the fam of the three fides, the line drawn from the vertical angle to bifect the bafe, and the angle which that line makes with the bafe, to determine the triangle. The answers to thefe queftions are to be fent, poft-paid, to Mr. Baldwin, in Paternofter-row, London, before the 1ft of February, 1784. LITERARY REVIEW. ARTICLE XXVII. LECTURES on Rhetoric and the Belles Lettres. By Hugh Blair, D. D. one of the Ministers of the High Church, and Profeffor of Rhetoric and Lelles Lettres in the University of Edinburgh. 4to. 2 Vols. Cadell, Creech, &c. IN the preface and introduction to thefe lectures, the very ingenious and worthy author tells us, that they are published nearly in the fame form, in which they were read by him, for twenty-four years, in the Univerfity of Edinburgh; that their prefent appear ance is in fome meafure occafioned by the circulation of imperfect copies, and the reafon he had to apprehend furreptitious publications of them: that they were originally intended, not merely to contain what had never before been fuggefted, but to exhibit fuch a comprehenfive fyftem of critical knowledge, as might convey moft ufcful inftruction to his pupils; and that their object is to apply the principles of reafon and good fenfe to compofition and difcourfe, in the place of artificial and fcholaftic rhetoric; to diftinguifh accurately between the fpecious and the folid, between affected and natural ornament; to bring into view the chief beauties that ought to be ftudied, and the principal faults that ought to be avoided; and thereby to contribute to enlighten tafte, and to lead genius, from unnatural deviations, into its proper channel. Such is the view which our author gives the plan of his Lectures, and it would be the height of injustice not to allow that he has executed his plan in a mafterly manner. Though there are many excellent detached pieces of criticifin in the English language, yet we know of no work upon fo comprehenfive and liberal a plan as that now before us, or fo proper to be put into the hands of youth, and those who are defirous of forming their tafte in compofition. Dr. Blair never amufes his readers with fanciful theories, nor bewilders them in the mazes of metaphyfics, but leaving fubtleties and refinements to the merely fpeculative theorift, proceeds in the plain path of reafon and common fenfe; lays down rules that are cafily underflood, and illuftrates them by a great variety of pertinent examples, taken from the works of fome of the most eminent writers, both antient and modern. His style is admirably fuited to a didactic work, being plain, perfpicuous, and and unaffected. In a word, good fenfe, found judgement, correct and elegant tafte, a truly philofophical fpirit, and a zeal for the caufe of virtue and literature are the characteristical excellencies of Dr. Blair's Lectures. They are divided into five parts. The firft contains fome introductory differtations on the nature of tafte, and the fources of its pleafures. In the the fecond, the author treats of language; in the third, of ftyle; in the fourth, of eloquence properly fo called, or public fpeaking in its different kinds; and in the laft, we have a critical examination of the most diftinguished fpecies of compofition, both in profe and verfe. The first lecture contains fome very juft obfervations on the advantages to be derived from the ftudy of criticifm. Among nations in a civilized ftate, we are told, no art has been cultivated with more care than that of language, ftyle, and compofition, and the attention paid to it may be affumed as one mark of the progrefs of fociety towards its moft improved period. Accordingly, we find, that in all the polifhed nations of Europe, the ftudy of language and eloquence has been treated as highly important, and has poffeffed a confiderable place in every plan of liberal education." "Of those who perufe the following Lectures (fays our author) fome, by the profeffion to which they addict themselves, or in confequence of their prevailing inclination, may have the view of being employed in compofition, or in public fpeaking. Others, without any profpect of this kind, may with only to improve their tafte with refpect to writing and difcourfe, and to acquire principles which will enable them to judge for themfelves in that part of literature called the Belles Lettres. "With refpect to the former, fuch as may have occafion to communicate their fentiments to the public, it is abundantly clear that fome preparation of ftudy is requifite for the end which they have in view. To fpeak or to write perfpicuoufly and agreeably, with purity, with grace and ftrength, are attainments of the utmoft confequence to all who purpofe, either by speech or writing, to addrefs the public. For without being master of those attainments, no man can do justice to his own conceptions; but how rich foever he may be in knowledge and in good fenfe, he will be able to avail himself less of thofe treafures, than fuch as poffefs not half his ftore, but who can display what they poffefs with more propriety. Neither are thefe attainments of that kind for which we are indebted to nature merely. Nature has, indeed, conferred upon fome a very favourable diftinction in this refpect, beyond others. But in thefe, as in most other talents fhe beftows, fhe has left much to be wrought out by every man's own induftry. So confpicuous have been the effects of study and improvement in every part of eloquence; fuch remarkable examples have appeared of perfons furmounting, by their diligence, the difadvantages of the moft untoward nature, that among the learned it has long been a contested, and remains ftill an undecided point, whether nature or art confer moft towards excelling in writing and difcourfe. "With refpect to the manner in which art can inoft effectually furnish affiftance, for fuch a purpose, there may be diverfity of opinions. I by no means pretend to fay that mere rhetorical rules, how juft foever, are fufficient to form an orator. Suppofing natural genius to be favourable, more by a great deal will depend upon private application and study, than upon any fyftem of inftruction that is capable of being publicly communicated. But at the fame time, though rules and inftructions cannot do all that is requifite, they may, however, do much that is of real ufe. They cannot, it is true, infpire genius; but they can direct and affift it. They cannot remedy barrennefs; but they may correct redundancy. They point out proper models for imitation. They bring into view the chief beauties that ought to be ftudied, and the principal faults that ought to be avoided; and thereby tend to enlighten tafte, and to lead genius, from unnatural deviations, into its proper channel. What would not |