turn the jewels to his unfortunate brother; but this application, even from fo great an interceffor, would have failed of fuccefs, with a man equally confpicuous for his immenfe wealth, and a parfimony that would degrade the meaneft character, had not the King of Sweden called to his affiftance the powerful influence of the Pope. The jewels were returned, and part of them feld by the Chevalier. Thus the Cardinal, who carries the love of money fo far as to exact of his fifter-in-law, the Princefs de Stolberg, 500 crowns for the hire of part of a palace which he never inhabits himfelf, was forced, by the apprehenfion of incurring the difgrace of his holiness, to do his brother that justice which neither nature, humanity, nor the interference of an amiable monarch could have extorted from him. Since that time the Chevalier is faid to have totally forfaken that debafing habit of drinking which has degraded him in the opinion of his beft friends. With fobriety his peace of mind, natural good fenfe, and understanding, are returned, and his royal friend is highly pleased both with his conduct and converfation. It is worthy of remark, that the Chevalier, notwithstanding his natural forbearance, and the humiliations he has experienced, affumes the imperious ftyle of a fovereign, in the letters written by him to Monf. de Vergennes, concerning Lady Charlotte. He does not request the King to legitimate her, &c. but does it of his own authority, and only expreffes a with that the King may not withhold his confent, which has been readily granted." The following is the progreffive increase of the revenue of the PoftOffice:-In 1644, Mr. Edmund Prideaux, who was inland poft-mafter, was fuppofed to collect about 3000l. per annum. In 1654, the parliament farmed it to Mr. Manley, at 10,000l. per annum.—In 1664, D. O'Neil, Efq. farmed it at 21,500l.--In 1674, it was let at 43,000l.-In 1685, it was eftimated at 65,000l.-In 1698, the amount was 76,3181.-In 1637, it was 90,5051. In 1710, it was allowed to be111,4611.-In 1715, the grofs amount was 145,2271.In 1744, the inland office amounted to 198,2261. but the total amount of both inland and foreign offices, which can alone demonftrate the extent of our correfpondence, was in that year 235,492l.-In 1764, the grofs amount was 432,0481. and fince that period it has frequently amounted to upwards of 600,000l. His Majefty has been pleased to incorporate the furgeons of Dublin into a college, to be called henceforward The Royal College of Surgeons of Ireland, with authority to examine and grant letters teftimonial to all fuch perfons as fhall be deemed qualified to practife furgery in that kingdom, Mr. Samuel Croker King is elected prefident, and is to be affifted by five cenfors, a fecretary, &c. Anecdote of an extraordinary emigration. A very fingular event took place about two years ago:-A Scotch gentleman, in the ifle of Herries, one of the Weftern Ifles, having been very much croffed in love, fold his eftate, which produced him upwards of 7000l. with which he fitted out two good fhips, embarking at Glafgow himself, and fixty families of his old vaffals, with every article neceffary for the establishment of a fort and colony, and fet fail, defigning for New Zealand. His intention was to enter the river Thames of Captain Cook, and to navigate his fhips into fome very fecure creek, where they might be fixed to remain, in the vicinity of a rock, to ferve as a fort. He took every fort of cattle and feed of England, birds, &c. &c. Being a man of great temper and prudence, there is little doubt but he will entirely conciliate the affections of the natives, by doing them good offices; and fhould that be the cafe, he will, in a few years, be fovereign of that noble iland: fhould the fcheme fail, he is provided for building, if neceffary, other hips. The great mifery of the natives arifes from a want of cultiva tion. He will be able, when he has made fome progrefs in their language, to explain fully the importance of a ANSWERS TO MATHEMATICAL QUESTIONS. 69. QUESTION (III. Aug.) anfwered by Mr. ISAAC DALBY. F two right lines MQ, KF cut each other, and if a given triangle (AOB, ANB) be made to move between them, fo that two of the angular points (A,B) are always in these lines, then the other angular point (0,N) will defcribe an ellipfis, a circle, or a right line. For, if the angle ANB twice the angle ACB, the point N will defcribe a circle, and if the angle AOB the comp. of the angle ACB to two right ones, the point O will defcribe a right line; in all other cafes it will defcribe an ellipfis : this needs no demonstration here, as it is the very property on which the elliptical compaffes are founded. Fig. 1. h N F b a A D M P M A P K E To apply this to the queftion. Let MK, MQ, KF, be the three lines given in pofition; take any right line AB between the lines MQ, KF, that form the leaft angle (for it is eafily proved that the given angle will fall in the line oppolite the leaft angle formed by the given lines) on which fuppofe a feg. inent of a circle to be defcribed that will contain the given angle (AOB) let N be the center of that circle, then AN= BN, and when AC=BC, the point N will fall in n; make Ca Cb, and abAB; alfo make the A aNb▲ ANB, then to the femi-conjugate CN, and femi-tranfverfe Cn, defcribe an ellipfis for the locus of the point N, or center of the circle. Through the centre C draw the diameter hk parallel to MK, and to the point draw the tangent kV; alfo draw CN parallel to KV, and NV parallel to Ck; then VN will be a tangent to the ellipfis in the point N. About N with the rad. NB (NA) defcribe the arc BOA, to which draw the tangent mg parallel to NV, draw AO, BO, and through O draw CP to meet MK in P; draw PD parallel to OA, and PS parallel to OB, join DS; then will the angle DPS be = the given angle (AOB), and the line DS a minimum. For the tangents VN, gm, being parallel, it is evident that the point N is the nearest in the ellipfis to the tangent mg, and confequently the pofition of the given line AB is fuch, that the lines AO, BO, drawn to meet mg, will form the greatest angle (AOB) poffible; but if AB is given, and the angle AOB a Fig. 2. W D F maximum maximum, the converfe is evident; that is, if the angle AOB is given, then AB will be a minimum. Therefore, that the angle DPS is AOB the given angle, and DS a min. follows from the fimilarity of the trapeziums CAOB, CDPS. When the locus of the point N, or center of the circle, is a right line or circle, the construction will be very fimple, as is evident from the foregoing analyfis. If MK, one of the lines, inftead of cutting the other two, be parallel to one of them, the construction will be fimilar to the foregoing. When the lines are parallel to each other (Fig. 2) the conftruction may be performed by the circle and right lines only; or thus: draw DA perpendicular to the lines; take any point D in WF, the outermoft of the two nearest of the three lines; from this point draw two lines DE, DG, to ineet QS, on thefe make ifoceles tri angles DBG, DRE, fo that the angles DBG, DRE, are each double of the given angle, if it is acute, or twice the complement of the given angle to two right ones if obtufe; through B,R draw WK, take DF-DA, and bifect DA in O; then, to the focus D, vertex O, and ordinate DF, defcribe the parabola OF, and the point C, where it cuts WK, will be the cent. of a circle which will pass through D, and touch MK. From the point of contact P, draw PD, PS; join DS, and DPS will be the given angle, and DS a minimum. When the three lines meet in a point, the prob. evidently admits of no answer. 74. QUESTION (I.) and 75. QUESTION (II.) for October, not answered. 76. QUESTION (III. O&.) anfwered by Mr. G. SANDERSON. Put n equal to the number of terms, S equal to the fum of all the terms; then 2NI. 2x+1.2x+3·2n +5.2n +7 is equal to the nth term, by the progression of the feries; and the next term or 3 is equal to 2n+1. 2n+3.21+5.21+7. 22+9. Put x2+1; then x2222, and Sxxxxx, whofe integral S This question was alfo answered by Taffo, the propofer. MATHEMATICAL QUESTIONS. The prize queftion in the Ladies Diary, for 1784, being not completely folved in the Diary for 1785, a more perfect folution of it is required. 66 **The question is this: If two bodies, A and B, connected by a ftring or otherwife, at the fame invariable diftance from each other, move, the one A along a given right line with a given uniform celerity, the other B fo, that its velocity in the direction of the connecting line AB, may always be equal to that in a direction perpendicular to it. I demand the afymptote, equation, quadrature, and rectification of the path of E, its center of curvature, and the quadrature of the path of that center.” TO THE EDITOR. SIR, The two following theorems appear to me to be of ufe in the projection of the fphere: if they appear fo to you, you will undoubtedly infert them in your Magazine. I am, &c. THOMAS MOss. THEO. I. being 88. QUESTION II. If the diameters AB and CD of a circle cut each other at right-angles, and any chord EF be drawn parallel to DC, cutting AB in K, and the right lines or go. QUESTION IV. by Mr. S. HAMILTON. In a plane triangle there is given one angle, B, the fum of AC (its oppofite fide) and AB (one of its adjacent fides) equal to M, and the fum of the other adjacent fide, BC, and a line, DE, drawn parallel to it, and intercepted between the other two fides AB, AÇ (or thofe fides produced) equal to N, to construct the triangle. The anfwers to thefe queftions may be directed (poft-paid) to Mr. Baldwin, in Paternofter-row, London. ASTRONOM Y. ON A METHOD OF DESCRIBING THE RELATIVE POSITIONS ROM fome alterations which have FROM of late years been difcovered, in the relative pofitions and apparent magnitudes of a few of the ftars we called fixed, it feems not unreasonable to conclude, that there may be many changes among others of them we little fufpect. This thought has led me into a wifh, that fome method were adopted whereby to detect fuch motions. The firft idea which occurred to me was, to make a proposal to aftronomers in general; that each fhould undertake a strict examination of a certain district in the heavens; and, LOND. MAG. Feb. 1785. not only by a re-examination of the catalogues hitherto published, but by taking the right ascension and declination of every star in their several allotments, to frame an exact map of it, with a correfponding catalogue; and to communicate their obfervations to one common centre. This is what I could be glad to fee begun. Every aftronomer muft with it, and therefore every one fhould be ready to take his fhare in it. Such a plan, undertaken with fpirit, and carried on gradually with care, would, by the joint labours and emulation of fo many aftronomers N as as are now in Europe, produce a celeftial Atlas far beyond any thing that has ever yet appeared. one. But this would be a work of time, and not within the compafs of every What I mean now to propofe is more immediate; and not out of the reach of any who amuse themfelves with viewing the heavenly bodies. Meridian altitudes and tranfits can be taken but once in twenty-four hours; and, though accurate, are therefore tedious. Neither can any re-examination of them be made, but with the fame labour as at the firft, Equatorial fectors are in the hands of few; and require great fkill. Some more general method feemed wanting; to difcover variations, which, when detected or only furmised, should be configned immediately to a more ftrict inveftigation. Turning this in my thoughts, I confidered, that the noting down at the time the exact appearance of what one fees would be far more fimple, and fhew any alterations in that appearance more readily, than any other method. A drawing once made would remain, and could be confulted at any future period; and if it were drawn at firft with care, a tranfient review would difcover to one whether any fenfible change had taken place fince it was laft examined. Catalogues, or verbal defcriptions of any kind, could 'not anfwer that end fo well. To do this with eafe and expedition was then the requifite: and a telescope with a large field, and fome proper fub-divifions in it, to direct the eye and affiit the judgement, feemed to bid moft fair for fuccefs. The following is the method which, after various trials, I have adopted, and think I may now venture to recommend. To a night-glafs, but of Dollond's improved conftruction, which magnifies about fix times, and takes in a field of just about as many degrees of a great circle, I have added crofs wires, interfesting each other at an angle of 45°. More wires may be croffed in other directions; but I apprehend thefe will be found fufficient. This tele fcope I mount on a polar axis. One coarfely made, and without any divifions on its circle of declination, will anfwer this purpose, fince there is no great occafion for accuracy in that refpect: but as the heavenly bodies are more readily followed by an equatorial motion of the telescope, fo their relative pofitions are much more eafily difcerned when they are looked at conftantly as in the fame direction. An horizontal motion, except in the meridian, would be apt to mislead the judgement. It is fcarcely neceffary to add, that the wires must stand fo as for one to defcribe a parallel of the equator nearly. Another will then be a horary circle; and the whole area will be divided into eight equal fectors, Thus prepared, the telescope is to be pointed to a known ftar, which is to be brought into the centre or common interfection of all the wires, The relative pofitions of fuch other stars as appear within the field are to be judged-of by the eye: whether at 1, or, or from the centre towards the circumference, or vice versa; and fo with regard to the neareft wire refpectively. Thefe, as one fees them, are to be noted down with a black lead pencil upon a large meffage card held in the hand, upon which a circle, fimilarly divided, is ready drawn. (One of three inches diameter feems moft convenient.) The motion of the heavenly bodies in fuch a telescope is fo flow, and the noting down of the ftars fo quickly done, that there is moft commonly full time for it without moving the telescope. When that is wanted, the principal ftar is eafily brought back again into the centre of the field at pleasure, and the work refumed. After a little practice, it is aftonishing how near one can come to the truth in this way: and, though neither the right afcenfions nor the declinations are laid down by it, nor the distances between the ftars mea fured; yet their apparent fituations being preferved in black and white, with the day and year, and hour if thought neceffary, written underneath, each card becomes a register of the then appearance of that fmall portion of |