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ANSWERS TO MATHEMATICAL QUESTIONS.
QUESTION (III. Aug.) and 70. QUES
TION (IV. Aug.) not answered.
71. QUESTION (I. Sept.) anfwered by Mr. R. CARLISLE, the propofer.
PUT AC-a, CD=b, v the velocity gene.
rated by gravity in a given time, and let r A be that part of p which balances q, and s the remainder. Then, by Cor. 2, Prop. LVI. of Emerfon's Mechan. we have this proportion,
xv, the velo
city of p, generated by the force s in the fame
time. But r = and s=p-r, =1. =
Let Q be the center of gravity of the
two bodies, then CQ =
pa-qb, and the ve
dent that the accelerating force of the center of gravity must be deducted from the accelerating force of p+q, when unconnected with the machine, in order to find the preffure upon the axis. But the time being given, the velocities are as the
2 pa-qiz forces; hence this proportion, as v p+q:: pa2+9b2 p+q pa2+gb2 that part of p+q which is not fupported by the axis. Therefore, +9. pa-gbla a + b 2 pa2+962
× pq, the preffure upon the axis required.
In the example which Mr. Emerfon has given, the preffure upon the axis, by his method, comes out greater than the weight of both bodies, which is evidently impoffible.
only this with the title page re-printed). This edition, 1687, is made up of the following feparate publications. The first part from "Cleiveland's, Vindicia," containing only genuine pieces collected by J. L. and S. D. as is described in a former note: the fecond part from " J. Cleaveland revived: Poems, Orations, and Epiftles, and other of his genuine incomparable Pieces; with fome other exquifite Remains of the most eminent Wits of both Univerfities, that were his Contemporaries." This fecond edition, &c. Lond. 1660, 12mo. with a curious preface, figned E. Williamfon, Newark, Nov. 21, 1658, in which he speaks of "the intimacy he had with Mr. Cleaveland before and fince thefe civil wars," and of that poet's "ever-to-be-honoured friend of Gray's-Inn," who was probably the Mecenas mentioned by Wood. To thefe is added a third piece, being the Hiftory of Wat Tyler's Infurrection, under the quaint title of "The Ruftick Rampant," &c. In the fecond part of this edition, 1687, the notice is fuppreffed, that occurred in the original titlepage and preface, that this part contained "other Remains of eminent Wits, &c." which is indeed the cafe with most of the poems in it, only a few of them being of Cleiveland's own writing. For the verfes in page 186 were by Tho. Weaver. All that occur between page 200 and 265 are from R. Fletcher's " Ex otio Negotium, &c." being poems printed 1656, 8vo. and the following are from the "Poems of John Hall, 1646," 12mo. viz. thofe here printed in page 297, 298, 302, 309, 315, 334, 353, 358, 375, 377. The Elegy, in page 310, is by Jasper Mayne. The Song, in page 336 has been attributed to Sir J. Denham. And fome of the reft belong to other writers.
But, to show how popular Cleiveland was among his contemporaries, we fhall here enumerate feveral of his editions; which were printed with more or fewer of his pieces, in 1647, 1651, 1653, 1654 twice, 1658, 1659, 1660, 1661, 1662, 1665, 1667, 1668; and then in 1677 (which aft date, Wood fays, he has seen mifprinted 1617; but the writer of this note has now before him two copies of this edition, containing some variations, yet both rightly dated 1677.) Lastly, in 1687, 8vo.
72. QUESTION (II. Sept.) anfwered by Mr. G. SANDERSON.
Make DC equal to the given difference of the fegments of the bafe, which bifect in F; draw the indefinite perpendicular GFE, on which take FG equal to the given difference of the bafe and perpendicular. Alfo on CD take FH equal to half FG. Draw the indefinite right line GHÅ, then by Problem 15, p. 223, Simpson's Geometry, draw DA and CA to meet A GA in A, fo that their difference may be equal to the given difference of the fides, draw AEB parallel to DC, and make AE EB; laftly, draw CB, and ACB is the triangle required.
About the triangle defcribe the circle ADCB, and draw the perpendicular CI. Then, because AB is parallel to DC, and both are bifected by the perpendicular FE, it is manifeft that the circumference A,B,C, paffes through the point D, and therefore CB AD. It is alfo plain that DC is equal to the difference of AI and IB (the difference of the fegments made by the perpendicular) but GF=2FH (by conftruction.) Hence, by fimilar triangles, GE 2AE AB; confequently, GF is equal to the difference of AB and CI.
Note. If the triangle is to be conftructed fo that the perpendicular be greater than the bafe, Fg must be taken below DC, and the rest of the construction the fame as above.
This question was anfwered algebraically by the propofer, Tasso.
73. QUESTION (III. Sept.) anfwered by SENEX, the propofer.
The force of a particle at Q, urging it from AUE in a direction parallel to a line joining the centers O and C of the two bodies,
rection of the
tangent QF, is nearly = 2fm × RD, as computed by Mr. Simpson; QR, RD
being perpendicular to OC, OQ; and OC being = d, OP=1, OR=x. But,
which that befides that force, there is another, in the direction QR,= gentleman has not confidered: and from this laft-mentioned force arifes an addifm
tional one= × RD, in the direction QF. Therefore, instead of : d3
XRD, we have : × RD ::ƒ: 3fm xRD; and consequent
Hence, our author having fhewn that OP2 will be to OA2 as 1 to
1+B, we find OP-OA nearly = × OP and thus the tides at the body O, by the action of the body C, appear to be greater in the proportion of 3 to 2 than his computation makes them. The body O is taken as a perfect sphere, except D 2
by fo much as it differs therefrom through the cause under confideration (which will caufe no fenfible error in the folution); and the quantity of matter in that body O, to the quantity of matter in the body C, is fuppofed as 1 to m. The accele. rative force of the body O on a particle at Q, in the direction QO, is denoted by f.. The force which Mr. Simpfon has omitted is derived (by refolution) from that of the body C in the direction QC.
Mr. Emerfon, at p. 421 of the second edition of his Fluxions, has computed the height of the tides. Is his computation right or wrong? If wrong, please to fhew how it may be rectified.
84. QUESTION II. by A. M.
In the peal of Grandfire, or Plain Bob, upon fix bells, to prove that, according to the law laid down for regulating the changes, no two changes can be alike in the whole peal of 720 changes.
85. QUESTION III. by SANCHO PANCHO.
Given the difference of the fegments of the bafe of a plane triangle, made by the perpendicular, and the ratio of the fides, to conftruct the triangle when the area is a maximum,
86. QUESTION IV. by SENEX.
Peter and John play with a box and two dice. Peter plays firft: and if he brings 6, 7, or 8, he wins; if 5 or 9 he lofes; if 2 or 12 he throws again if 3, 4, 10, or 11, he paffes the box to John. If then John brings 6, 7, or 8, he wins; if 2 or 12 he throws again; if 3, 4, 5, 9, 10, or 11, he paffes the box again to Peter, who then throws again: and fo they continue to play till one of them wins. Quere, their refpective probabilities of winning?
The anfwers to thefe queftions may be directed (poft-paid) to Mr. Baldwin, in Paternofter-row, London.
TO THE EDITOR OF THE LONDON MAGAZINE,
IF you think the following letters will afford any pleasure to your readers,
MISS CATHERINE T
TO THE HON, MISS C
but unfortunately it does not come
is not my fault that you do not: for when at any time fome of his oddities have peculiarly ftruck my fancy, I have writ you whole volumes about him. However, that you may not be forced to recollect how I have formerly tired you, I will repeat, that, with one of the honefteft hearts in the world, he has one of the oddest heads that ever dropped out of the moon. Extremely well verfed in coins, he knows hardly any thing of mankind; and you may judge what kind of education fuch an one is likely to give to four wild girls, who have had no female directress to polish their behaviour, or any other habitation than a great rambling manfion-house in a country village. As, by his little knowledge of the world, he has ruined a fine eftate that was, when he firft had it, 2000l. per ann. his prefent circumftances oblige him to an odd kind of frugality, that fhews itself in the flovenlinefs of his drefs, and makes him think London much too extravagant an abode for his daughters, at the fame time that his zeal for antiquities makes him think an old copper farthing very cheaply bought with a guinea, and any journey properly undertaken, that will bring him to fome old cathedral on the faint's day to which it was dedicated. As, if you confine the natural growth of a tree, it will fhoot out in a wrong place, in fpite of his expenfivenefs, he appears faving in almoft every article of life that people fhould expect him otherwife in; and, in fpite of his frugality, his fortune, I believe, grows worfe and worfe every day. I have told you before he is the dirtieft creature in the world; fo much fo, that it is quite difagreeable to fit by him at table. He makes one fuit of clothes ferve him at leaft two years; and then his great coat has been tranfmitted down, I believe, from generation to generation, ever fince Noah. On Sunday he was quite a beau. The Bishop of Gloucefter is his idol; and if Mr. W. was Pope, St. Martin, as he calls him, would not wait a minute for canonization.
To honour laft Sunday as it deferved, after having run about all the morning
to all the St. George's churches, whose difference of hours permitted him, he came to dine with us in a tie wig that exceeds indeed all defcription. It is a tie wig (the very colour of it is inexpreffible) that he has had, he fays, thefe nine years; and of late it has lain by at his barber's, never to be put on but once a year, in honour of the Bishop of Gloucefter's birth-day. But you will fay, what is all this to my engagement this morning? Why, you muft know, Be diftinguifhes his four daughters into the lions and the lambs. The lambs are very good, and very infipid: they were in town about ten days, that ended the beginning of laft week, and now the lions have fucceeded them, who have a little fpirit of rebellion, that makes them infinitely more agreeable than their fofter fitters. The lambs went to every church that Be pleafed every day; the lions came to St. James's church on St. George's day. The lambs thought on no higher entertainment than to fee fome collections of fhells; the lions would fee every thing, and go every where. The lambs dined here one day, were thought good aukward girls, and then were laid cut of our thoughts for ever. The lions dined with us on Sunday, and were fo extremely diverting, that we spent all yefterday morning, and are engaged to fpend all this in entertaining them, and going ourselves to a comedy that I think has no ill-nature in it, for the fimplicity of thefe girls has nothing blameable in it, and the contemplation of fuch unaffifted nature is infinitely amufing.
They follow Mifs Jenny's rule, of never being strange in a ftrange place, yet in them this is not boldnefs. I could fend you a thousand traits of them, if I was fure they would not lofe by being wrote down, but there is no imitating that inimitable naiveté that is the grace of their character.
They were placed in our feat on Sunday. (Alas! I was ufed to feeing it filled with people that were quite indifferent to me, till feeing you in it once has thrown a fresh melancholy upon it.) I wondered to have heard
orders within fix years after his being elected fellow: but he was admitted on the law line (as the phrafe there is) and afterwards on that of phyfic, which excufed him from complying with this obligation; though it does not appear that he made either law or phyfic his profeffion; for remaining at college, he became the rhetoric reader there, and was ufually employed by the fociety in compofing their fpeeches and epiftles to eminent perfons* (of which fpecimens may be feen in his works) being in high repute, at that time, for the purity and terfenefs of his Latin ftyle. He alfo became celebrated for his occafional poems in English, and, at the breaking out of the civil wars, is faid to have been the firft champion that appeared in verfe for the royal caufe; which he alfo fupported by all his perfonal influence: particularly by exerting his intereft in the town of Cambridge, to prevent Oliver Cromwell (then an obfcure candidate, but ftrongly fupported by the Puritan party) from being elected one of its members. Cromwell's ftronger genius in this, as in every other pur
fuit, prevailing, Cleiveland is faid to have fhown great difcernment, by predicting, at fo early a period, the fatal confequences that long after enfued to the caufe of royalty t. The parliament party carrying all before them in the eaftern counties, Cleiveland retired to the royal army, and with it to the King's-head quarters at Oxford, where he was much admired and careffed for his fatyrical poems on the oppofite faction, efpecially for his fatire on the Scottish covenanters, entitled, The Rebel Scott. In his abfence he was deprived of his fellowship, the 13th of February, 1644, by the Earl of Manchefter, who, under the authority of an ordinance of parliament for regulating and reforming the Univerfity of Cambridge, ejected fuch fellows of colleges, &c. as refused to take the folemn league and covenant. From Oxford Cleiveland was appointed to be judge-advocate in the garrifon at Newark, under Sir Richard Willis the governour, and has been commended for his fkilful and upright conduct in this difficult office, where he alfo diftinguished his pen occafionally, by returning
their names, J. L. and S. D. to Francis Turner, D. D. then Mafter of St. John's-College, but afterwards fucceffively Bishop of Rochester and Ely, who is believed to have been a pupil of Cleiveland's alfo.
* One of thefe was fpoken before the King (Charles I.) and his fon, the Prince of Wales, at St. John's College, in Cambridge: with which the King was fo well pleafed, that after it was over, his Majefty called for him, and (with great expreflions of kindnefs) gave him his hand to kifs, and commanded a copy to be fent after him to Huntingdon, whither he was haftening that night.” This, according to Winftanley, was in 1642. But a MS. dates it in 1641.
For this fact we are indebted to the authors of his life, prefixed to his works, in 1677, who having obferved, that "no man had more fagacious prognofticks," tell us, that after the election was over, Cleiveland faid, "with much paffionate zeal, That fingle vote had ruined both church and kingdom." Whence it fhould feem, that Cromwell gained his feat in parliament by the majority of one vote only.
The fame writers mention another inftance of his being "Vates in the whole import of the word, both poet and prophet." When the King withdrew from Oxford, and furrendered himself to the Scots army, upon fome private intelligence three days before the King reached them, Cleiveland forefaw the pieces of filver paying upon the banks of Tweed, and that they were the price of his fovereign's blood, and predicted the tragical events."
Cleiveland had been before at Oxford, in the year 1637, and was then incorporated Master of Arts, with feveral other Cambridge men. But now his farcaftic attacks on the oppofite party would make him exceedingly popular there, efpecially the fatire above-mentioned. Of which we have the following proof: while he was now at Oxford he had his portrait painted by Fuller (a threequarter's length, now in poffettion of his great-nephew, the Bishop of Dromore) wherein he is drawn holding a paper, infcribed The Rebel Scot. An engraving from it is prefixed to the feventh volume of Nichols's "Select Collection of Mifcellany Poems, 1781," 12mo. where feveral of Cleiveland's poems are reprinted.
§"His next ftage was the garrifon of Newark, where he was judge-advocate, until the furrender and by an excellent temperature of both, was a juft and prudent judge for the King, and a faithful advocate for the country."
The Bishop of Dromore has in his poffeffion an authentic copy of the commiffion (figned by King Charles I. with his own band) dated at "our Court at Newarke," 12th October, 1645, by which Sir Richard Willis the governor, and other commiffioners therein mentioned, are empowered