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breaking in their poffeffion." Mr. Selwyn moved to omit the claufe, which requires that perfons taken with inftruments of the kind defcribed fhall be fuch perfons only as come within the defcription of the act of the 17th Geo. II. commonly called the Vagrant Act; and he added, that he found that the bill, as it at prefent flood, had given alarms, which he thought unneceffary. Mr. Fox faid, the bill was crudely and infufficiently worded, and objected to the claufe which offers rewards, comparing the whole of that fyftem of thief-takers by profeffion, to the fituation of fportfmen, who always increafe and preferve the game within their range and circle.

Mr. Sawbridge alfo oppofed the bill, and faid that although it meant to defcribe fufpicious perfons, it was very plain that no man with half a crown in his pocket to give to the conftable would be deemed a fufpicious perfon. Mr. Pitt faid that fome fuch bill was neceffary. Many other members fpoke, but all against it. The motion being made that the chairman leave the chair, the numbers were Ayes 22-Noes 29. The Houfe reported a progrefs.

On Thurfday, May 1, in the Houfe of Lords, the order of the day being read for the fecond reading of the EaftIndia Company's indemnity bill, Lord Walfingham objected to the bill. Lord Fitzwilliam fpoke in favour of it, and the question being put, it was committed for to-morrow.

Friday, May 2. The order of the day being read for the fecond reading of the American manifeft and intercourfe bill, Lord Thurlow rofe to make fomne objections; it appeared to him to require fome alterations. What the neceffity for thefe alterations was, he would in fairnefs ftate to them, in order that the Houfe might be apprized of them in fufficient time, fo that the nobie Lords would be enabled to think on them before the bill came into the committee, which would be the proper moment for difcution, adopting, or rejetting them. And first he defired their lordfhips attention to the laft claufe of the bill, upon which he had a queftion to propofe that appeared to him exceed

ingly material. The laft claufe ftated, that the act, as to the exercife of the powers and authorities thereby given to his Majefty, was to continue to be in force for the space of fix weeks, and no longer. Was he to understand from this, that the orders iffued by his Majety in council were to continue in force no longer than fix weeks, or not? or, in other words, was he to underftand that the powers vefted in the crown by the bill were to exift for fix weeks only, and that the acts done under the authority, and by virtue of those powers were to continue in force, and to be binding upon the fubject for ever? He fhould hope not; but he wished to have that afcertained. His lordship repeated his question, and looking at Lord Bathurit, faid, does the noble lord tell me that they are not?

Lord Bathurst declared he knew nothing of the matter, but ftated his reafons for thinking there must be fome amendment made in the committee. The Duke of Portland declared, that he was obliged to the noble and learned lords for their candour in having taken the opportunity of fuggefting the obfervations upon the bill that the House had heard. He owned he did not fee much force in many of their objections to the bill; but as none of them went to its principle, it would be time enough for him to go into a difcuffion of them when the bill was in a committee; for the prefent, he prefumed their lordships would not object to his moving that the bill be committed. This was agreed to.

Friday, May 2. In the Houfe of Commons, Lord Newhaven moved, That the order for the copy of the minute of the examination of Meff. Powel and Pembridge, cathier and accountant of the Pay-Office, before the Lords of the Treafury, on Thurfday the 24th ult. be difcharged. His lordship faid, he understood fome profecutions at law were going on againit thofe gentlemen, on account of their conduct in office; before, therefore, he moved to have the order he had moved on a former day difcharged, he wished to hear from fome perfon in authority, if the fact was not fo.

Mr. Sheridan, in anfwer, faid fomething which indicated that proceedings at law were carrying on against them.

Mr. W. Pitt objected to an irregular converfation upon a fubject not properly before the Houfe, and more efpecially in the abfence of certain perfons, who had it in their power to give the Houfe authentic information with regard to the proceedings refpecting Mr. Powell and Mr. Bembridge. He defended the conduct of the last adminiftration, in ordering the discharge of the cashier and accountant from their offices, and owned he thought their reftitution to their places fuddenly, and without any explanation refpecting their conduct, an extraordinary proceeding.

Mr. Sheridan rofe again, and faid, the delay of the legal proceedings against Meff. Powell and Bembridge was by no means chargeable as matter of cenfure against the prefent miniftry. The Attorney-General had commenced the bufinefs, and there had been time enough elapfed for him to have gone on with it.

Mr. Kenyon upon this rofe, and explained his conduct refpecting Mefirs. Powell and Bembridge. He faid he could only judge from the cafe that had been laid before him; as far as that went, it appeared to him that they were enormous offenders

Mr. Burke took fire at the words enormous offenders, and argued upon the injuftice of applying a term to men unconvicted of any offence whatever. Mr. Burke faid, he held himfelf refponfible for thofe he employed in of fice, and appealed to the tenour of his life, in proof of the improbability of his countenancing men guilty of enormous offences.

Mr. Martin faid he knew nothing of the gentlemen who had been difcharged, and restored to their places in the PayOffice; but as they had been reftored without one fyllable being faid of the matter, he could not but confider the fact as a daring infult to the public

Mr. Burke rofe in great heat, and cried out "I fay it is not a daring infult to the public"-when the noife becoming general, and the cry of bear, bear, coming from fome parts of the

Houfe, Mr. Fox, who fat next to Mr. Burke, pulled him down by his fleeve. Sir Edward Aftley spoke to the fame purpofe as Mr. Martin, when

Mr. Fox rofe, and faid, that if the honourable baronet and honourable gentleman who fpoke before him confidered a moment, they furely would think, that calling what his honourable friend had done a daring infult to the public, was an expreffion not more harsh than unjuftifiable. With regard to the honourable gentleman (Mr. Martin) he had on fome occafion touched on humanity; furely, if that honourable gentleman would reflect ever fo little, he would fee that it was the fixed principle of human juftice, to prefume every perfon innocent till fome criminality was proved against him. Mr. Powell, he faid, had ever the character of a man of the ftrictest honour and integrity, and he faw no reafon, therefore, for accepting the accufation, even (if accufation there was) against fuch a character for proof, nor of condemning him unheard, any more than of condemning any other perfon, accufed of any other offence, before he had been tried. With regard to the degree of refponfibility belonging to the offices of cashier and accountant, he declared he was wholly ignorant; his honourable friend was refponfible to the public, not only for his own conduct, but for that of every clerk under him, and, therefore, it was not to be prefained, that his honourable friend would have reftored two perfons to their offices under him, of whofe unimpeachable condut he was not in his own mind perfectly convinced. But that his having done fo was a daring infult to the public, was furely not only a very harsh affertion, but an affertion by no means true. It was poffible for the late paymafter to have feen the fame conduct in a reprehenfible point of view, and for his honourable friend to have feen it in a different point of view. His honourable friend could never have been fo weak as to fuppofe, that the act of reftoring the cafhier and accomptant would país unnoticed, or that it would not call forth obfervations, and provoke enquiry. Undoubtedly |

obvious, that notice would be taken of it in that House, and he had no manner of doubt but that his honourable friend would be able to fhew that he had not, by any imprudent and hafty meafure, done a thing fo culpable as fome gentlemen chofe to fuppofe it. Mr. Fox declared, that, for his own part, he knew nothing of the two gentlemen being restored, till his honourable friend told him of it, as he was entering the clofet at St. James's. With regard to an enquiry, it was a matter which concerned him more nearly than any other person whatever. Mr. Powell was the acting executor of his father, and faid, that if contrary

to his general character, contrary to what he believed, Mr. Powell should not turn out a man of honour and honefty, the House must fee, that in fo large, fo complicated a tranfaction as the executorfhip of his father's affairs, there muft have been great opportunity for wronging his family, and though 40,000l. or 60,000l. was a trifle with regard to the public, yet when it came to be the cafe of an individual, the confideration was a very large one.

After this, a defultory converfation took place, which the Speaker ended, by reminding the House that it was diforderly, and that no motion was before them on that business.

MATHEMATICS.

ANSWERS TO MATHEMATICAL QUESTIONS. 1. QUESTION (I. July) answered by the Propofer.

A 12+ 512

SSUME x2+n × x3+mx+r, = x5 + m+n.x3 + rx2 + mnx +'nr = x5 + x3+x+x+, the given expreffion. Then, by comparing the ho

rt

mologous terms, we shall have tnr, or n=, and mns, orm=2,

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==; and,

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xx3+= x+r=0. Confequently two of the values of x are ✔

37" t

- and

and the other three may be had from the refolution of the cubic equation x3 + =x+r=0.

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t

This question was alfo answered by Mr. James Webb.

2. QUESTION (II. July) answered by NAUTICUS.

P

Let ZPH reprefent an arch of the meridian, where Z is the zenith, P the pole, and H the point of the horizon which is of the fame name with the latitude. Moreover, let HA be the horizon, S the place of the given ftar, and

that of the required one; and Z✶ SA the vertical circle they are on when they país each other in azimuth. Then, by the queftion, the difference of the fluxions of the angles SZP and * ZP muft be a maximum. Now, by Theo. 21, of Cotes's traet De Eflimatio Errorum, &c. in the fpherical triangle SZP, P: Z Rx in. ZS: fin. PS x cos. S; and by fubftituting in this proportion for fin. FS, fin. 25, and cei. S their equals, derived from the principles of fpherical

S

H

trigonometry,

trigonometry, we shall have P : Ż :: R3 : coг. PZ x R2-fin. PZx cof. Z x cotan. ᎦᏃ . In like manner we may derive P: Z:: R3 : cof. PZx R2—fin. PZ x cos. Z× cotang. Z. Hence, if radius be taken equal to unity, the fluxion of the angle PZS will be equal to P x cos. ZP—P × fin. ZP × cof. Z × cotan. SZ; and the fluxion of PZ will be equal to Pxcof. ZP-Pxfin. ZP× cof. Zxcotan. *Z; and, confequently, their difference, or P x fin. ZP x cof. Zx cot. ZS-Px fin. ZPxcof. Z x cotan. Z must be a maximum; or, because 2 x fin. PZ is conftant, cof. Zx tot. ZS-cot. Z will be a maximum. Now, as the cofine of Z can never exceed unity, and as the difference of the cotangents of ZS and Z *: that is, the difference of the tangents of A and AS will be infinite when the required ftar is in the zenith; it is manifeft that the cof. Zxcot. ZS-cot. Z will be a maximum when the declination of the required ftar is equal to the latitude of the place.

3. Queftion (III. July) answered by Mr. JAMES WEBB.

Let CE be the given hyperbola, IQ the oppofite hyperbola, and I the given point in it.

From I, as a center, conceive

a circle CEF to be described, which touch-
es the hyperbola CE in the point C; this
point, it is evident, is that which is fought,
and may be determined as follows: Let
OP be the femi-conjugate axe of the twohy-
perbolas,c, the femi-tranfverfe EO=0Q
being: draw IC, and produce it to
meet the axe, produced, in G, and draw
IR, and CD perpendicular to EQ produced.
Let OD be put = x, IR a, and OR=b;
then by the properties of the hyperbola, 12
: c2 :: x : c^x = DG, 12 : c2 : 1+xxx-1!

DT

F

P

R

2xx-1=DC2; and by fimilar triangles c2x : c2x+x+b::e √ X2 = X 1 x. fequently, multiplying means and extremes, and reducing the equation

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of which is unity,

ac

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; and if x be now confidered as the fecant of an arc, the radius

x

will be the cofecant; and the excefs of the cofecant, above the fecant drawn into a given quantity, is known. Hence x is readily found by the method pointed out at p. 470. Philofoph. Tranja. for 1781.

Cor. Had the femi-tranfverfe EO, instead of being equal unity, been taken equal to any given quantity ; the final equation would then have been- 1x

12-2

ac

126

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== ; and the only difference that would have arifen in finding the value

ac

of x, by the method pointed out above, would have been in taking as the fecant of an arc, the radius of which is the given quantity 7.

Answered alfo by Mr. James Ealtwood, and the proposer Mr. W. Kay.

4. QUESTION (IV. July) anfwered by NAUTICUS,

CONSTRUCTION.

In the given angle H conflitute the parallogram HGAI, equal to twice the given area; and in HI, produced, take IK équal to the given difference of the including fides, alfo KE fo, that KEXIE may be equal IAXAG. Through E and A draw EF meeting HG, produced, in F: then completing the parallograms, as in the figure, and joining BC; BAC will be the triangle required.

LOND. MAG. OA. 1783.

H

G

B

D

I

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PP

DEMON

DEMONSTRATIO N.,

The angle BAC the oppofite vertical angle GAI (Euc. I. 15) = the given angle GHI (Euc. I. 34); and the, triangle BAC half the parallelogram ABCD (Euc. I. 34) half the parallelogram HIAG (Euc. I. 43) the given area, by conftruction. Laftly, fince IF: AG:: AI: KE, by contruction, and GF: GA:: AI: IE by fim. triangles, ex æquo perturbato, IE: GF:: IE: KE. Now, the antecedents being here the fame, the confequents must be equal: that is GF (= AB)= KE; and AC (=IE) exceeds AB (KE) by IK, the given difference. Q.E. D. Mr. Geo. Sanderton and the propofer alfo gave elegant conftructions to this question.

5. QUESTION (V. July) answered by Mr. ISAAC DALBY.

ANALYSIS.

Suppofe the thing done. and let ABCD be the fish. pond, and EFGHDCBAD the walk of equal breadth furrounding it; through A, D, C, E, draw Em, Hm, Gn, Fn, which, it is evident, will bifect the angles of all the trapeziums EG, AC, &c. whofe fides are parallel, and whofe angles fall in thefe lines, and confequently equally diftant all round. Make the trapezium ÍKLM fo that its fides may be to the fides of the given trapez. and produce EH, AD, and IM; take Hg=HG, De= DC, and MI=ML, draw gel, which must be a right linebecaufe HDM and GCL are right lines, and (letting fall the perp. ch, Dt. Drv) D-Dt-ch; hence it follows from Euc. I. 36, 37, 38, that the quadrilaterals DHgc DHGC, MD MLCD, therefore HMg= HMLG; and, consequently, if gl was produced to meet Hm, the formed thereby on the bale Hg would be equal in area to the AHZG.

In like manner, if there be taken gf GF, cb=CB, Ik=LK, and fe—FE, ba➡BA, ki➡KI, the quadrilateral gtk LGFK, and kƒei- KFEI, and therefore the locus of the points E, A, I, &c. when transferred to the points e, a, i, &c. is a right line, and the quadrilateral. EAae the area of the walk (by hypoth.) hence this easy

CONSTRUCTION.

Take IKLMI and equidiftant from ABCDA, and draw AI, DM, produced both ways at pleafure; produce AD, IM, and take Da=DC+CB+BA, alfo Mi =ML + LK + KI, that is Aa, lithe perimeters; B through a, i draw aR to meet AI produced, then is the ▲ aKA given; make the rectang. AO≈ ▲aRA, and the rectang. aN=▲aRA+area of the walk, take aP a mean proportional between aA and as, erect the perp. PE to meet IA produced, and PE is the breadth of the walk required; and drawing Ee | Aa, the quadrang, Ae is its area: For rectang. AO (▲ aRA): rectang. aN aA aS :: ▲ ARa: A ERe (Euc. VI. coral. 19.) therefore rectang. aNA ERe▲aRA+ area of the walk (by conftruc.) therefore EAae area of the walk.

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COROL. I. This method of folution holds good in any polygon, regular or irregular,. or confifting of any number of fides.

COROL. II. When the given figure is a regular polygon, the points m, Z, n, R, fæll

in its center,

The

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