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1795.]

Mathematical Correspondence. In a second memoir, he proposed a me- compound will be greater than would be thod of organizing public inftruction in inferred from the limple rule of alliga. rural æconomy, particularly in the vete- tion, the mass becoming less by nieasure rinary arts.

after the mixture than the sum of the TEYSSIER read two other memoirs ineasures of the two liquors.-The greaton the same subjects. In the first, he est diminution of the whole mass takes traces the abuses arising from the ex- place when equal parts by measure of alceflive encouragement granted to the cohol and water are mixed; in which case persons who clear the forests, and the in- (the alcohol used being extremely decalculable mischiefs which arise from the phlegmated and of the specific gravity of extension of the practice, affecting the 0.8199) the diminution of the mass after foil of the most useful forests, mea- the mixture was found to be 0.0283 dows, &c. In the second, TEYSSIER of the whole.-This diminution of the explained his project of extending the mass, and the consequent increase of its science of rural economy into other specific gravity, takes place in a ratio, the parts of the world, by a general corre- relation of which to the proportions of fpondence he has established, and thus the mixture, it has been hitherto found contributing to its perfection in France. difficult to ascertain ; and it has therefore He then entered into a long detail on the long been a defideratum to ascertain the real practice of agriculture in the Canary proportion of alcohol in any weaker spiTands, and noticed the high degree of rit from its specific gravity.-M. Pouget, prosperity into which those islands are of Montpellier, in a letter to Mr. Kircapable of being raised. — TEYSSIER wan, dated 12th May 1783, and inserted is thus employed in tracing a sort of in the 3d volume of the Transactions of grand Agrarian chart, a kind of rural the Royal Irish Academy for 1789, page map of the world, the execution of which 157, after a great deal of most ingenious has been so long wanted, and which will reasoning, and a detail of several expericontribute so much to the progress of the ments made with alcohol of the abovefirst and most useful of the arts.

mentioned specific gravity, gives the folSome of the Memoirs will be given at length lowing formula for the solution of this in future Numbers of this Migazine.]

problem:

Put the total volume or measure of the MATHEMATICAL CORRESPONDENCE. entire mass or compound =1. The mea.

Jure of alcohol contained therein (supporQuestion XIV (No. V).- Answered by ing alcohol of the fpec. gr. of 0.8 199 to Mr. N. Bosworth, of Peterborough. be pure or perfectly dephlegmated) =x.

The diminution of the volume or bulk of ON duly considering the nature and circumitances of the Question, it will

a mixture of equal parts of alcohol and very plainly appear, that the nail in the water found by experiment (equal as betire of the wheel will

, in every revolution fore stated to 0.0288) =c. Then, he says, of it, describe a cycloid ; and that the sum

the diminution of the volume of a mixof the length of all the cycloids thus form- ture containing x measures of alcohol will ed will be in exact proportion to the num- be =4cx—40x2. And putting the specific ber of revolutions made by the wheel, and gravity of water (=1.0000) =a, that of to the distance travelled. Therefore, ac

alcohol (=0.8199). ==b, that of the uncording to the well-known properties of

known mixture = y, it will be (as the the cycloid, which, I fuppose, it is unne- augmentation of density does not at all ceffary here to explain, we have 7 milcs change the absolute weight of the mats)

,
3.1416 1-*Xa+bx=1--4c1+36x2xy ;

and X458.912656 miles, the sum of the from hence he deduces the following lengths of all the curves described by the equations : nail, while the coach is going 7 miles,

a-b which is the mean velocity of the nail

1=0.5

+

0.5

8cy 46 Scy Toe fame Question was answered by

0.1801 7. H.

0.5
p.2 3049

0.1801 QUESTION XV (No. V).--- Anfeuered by VI

+

- y= Mr. 7. F--r.

0.1152 0.2304 If alcohol and water be mixed in any

amaxfbx

1-O.1801x proportion, the specific gravity of the 1–467+40x2 1-0.1152x+0.1152.x2.

5 I 2

Monsieur

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per hour.

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20

Monsieur Pouget gives several instance =CN, and the diameter LM or NO, of es of the coincidence of the deductions the tube, is 13.14736 inches. from the foregoing formulæ, with the re

N.B. Mr. Emerson, in his Astronomy, sults from actual experiment, to about page 36, lays, that a tube whole length is three places of figures ; and concludes, to its diameter, as 100 to 97, takes in onc that they may be relied on as sufficiently tenth of the hemisphere : but his propornear for practice, where the mixture does tion does not seem to be very accurate. not contain less than half its bulk of alcohol.

The fume ansevered by Mr. iv. dans, For the foriner of the two kinds of

jun. of looturn School. brandy proposed in the Question, we shall The length of the tube being 20 inches, have, by the above formula,

and the field of view-one 2oth of the he0.1801

misphere, therefore 39 is the feginent's x=0.5to

height. by Dr. Hutton s Mensuration, p. 0.2 304 X 0.92

191, first edition; hence 20+2=21; is I~0.92

0.1801 +

--0.5

the complete radius; and then 2V/41,5* in C.1152 X 0.92 0.2304 x 0.92

=13.1.473636 is the internal diametir of 0.5836, the proportion of alcohol or pure the tube, by a well-known property of spirit in the liquor. And for the latter the circle.

0.1801 kind, x=0.54

t0.2394 X 0.9

The Same answered by Mr. Yobn Haycock,

of Ware. ♡ 1-0.9 0.1301

Let LNOM be the tube, A the aper+

-0.51 0.1152 X 0.9 0.2304X0.9

ture, AC the axis, CD the versed fine or 0.6836, its proportion of alcohol or pure height of the vifible fegment, which is to spirit.

be one 20th of the hemisphere.- Since This Question was also answered by Mr. the furface of any segment is as the heiglii, 7. H.

we have, as 19:20::20=AC:21,5=D; QUESTION XVII, misprinted XVI (No. hence NO=2NC=20780 =...

19 19 VI).- Answered by Mr. 0. 6. Gregory. 2W 15600 In the

=13.14736 inches, the internal N

19 figure an

diameter required. nexed, let BNOE be

The same answered by 7. F--r. supposed to

The curved luperficies ct the fegments representa

of a spbere being as the versed tines of fe&tion of

half their arcs, we shall have, in the prehemi-B

M

sent initance, b=the vericd fime of 189 1phere, so sinated that its axis ihall touch' 11' 32', half :he arc of the given portiun the base LM of the tube LMOS, and its of ihe iphere; and hence, as radius: tang. convex surface pass through the circum. 18° 11' 39' ::20:6.573 inches the semiference of the other end NO. Then (as diameter or the iube, or 13.147. the whole spherical fegments are in surface prepor- diameter required. tional to their altitudes) we muft have Tois uti was also an uredly Milfs. CD equal to one 20th of AD, or one WV. Ciawil', 7. Richici, 7, Collins. L.I.D. 19th of AC, and the surface of ihe und 7. H. fegment NDOC will be one 2oth of the surface of the hemisphere : under which Now MATHENATICAL QUESTION. conditions, an eye, locking through an QUESTION XXII--By Mr. 0. C. Greaperture ar A, would view one 20th of

girl'. the celestial hemisphere. In the infancé before us, where AC js 20 inches, CD

VENUS's greatest clongation from the

Sun, as observed from the Earth, is found must be li's inch ; and then, as may be to be 48° : from this the young astronoreadily found, the difference between CD and the axis BE is 41 inches ; be

mer is requested to point out an easy and

expeditious method of finding the diftween which and CD, as geometricians have proved, CN is a geometrical mean

tance of that planet from the Sun,--the

Earth’s distance from that luminary being proportional: hence 4175X11=6.57378 95 millions of miles.

an

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NEW

1796.]

( 799)
NEW PATENTS
Enrolled in the Months of September and Oétober.

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MR. LOWNDES's GYMNASTICON. This machine, the patentee conccives, ON the 30th of September, letters pa- will be of the higheit importance to me

tent were granted to Mr. FRANCIS dicine, by affording a degree of exercise LOWN DES, of St. Paul's Church-yard, which cannot otherwite be obtained, and London, for his invention of a machine by directing the exercise into particular for mu?cular exercise, which he calls parts of the lystem. The diseases, wherein a Gimnafticon. Its cbject is to enable it may be applied with the greatest fucpersons to exercise themselves in any de- cels, he conceives to be all those which gree withed for, in their own chambers; ariti from obstructions in the joints and generally in all parts of the body at once, in the abdominal visccra; and that it er partially as the case

may require. might therefore be highly useful in feThe machine consists of an upright male boarding schools ; to persons of feframe, sufficienily large to admit the hu- dentary habits and employments; to the man body. The action is produced on paralytic, thu gouty, and the rheumatic. the lower extremities by means of two MR. HALEY's TIME-KEEPER. treadles, similar to those made use of in On the 17th of September, letters turning lathes. The action of the up- patent were granted to Mr. CHARLES per extremities is produced by means of HALEY, of Wigmore-street, Cavendishtwo cranks of any required depth; which fquare, for his invention of an improved cranks are put in motion by a wheel, MARINE TIME KEEPER. connected by a band, with a wheel on The principal invention in this time. the axie of the cranks, beionging to the keeper, confiits in the manner of apply. treadles. The person, therefore, re- ing a renovating {pring; which mode of quiring exercise in both extremities, has application is entirely original, and aponly to put the treadles in motiin, by pears to be a confiderable improvement. the weight of his feet; the treadles will By producing perfe&tly equal arcs of via then turn their cranks, which, by a bration in the balance, the Patentee has band, communicate motion to the upper attained an object which all artiiis have wheel and cranks, and by holding the latter had in view, bù: hitherto have only obwith the hands, every joint and muscle tained in a partial manner. in the body is put in action. It is obvi- The firit attempt of the kind, attended ous, that this action may be increased to with any succes, was that of Mr. HANany kegree, hy varying the position of RISON, who applied a renovating spring the feet on the treadics, and the depth on the contrate wheel, wound up every of the upper cranks.

ten seconds. This application took off The foregoing description appliis to fuch a proportion of the imperfections of motion, similar in its effect to walking the main fpring, and of the iregularities or running, and may be eanly varied, to of power,' arifing from the train of as to produce a degree of exercile equal wheels, as to obtain a degree of accuracy to two, or even to ten miics per hour. wnich intitled him to very considerable

By very simple contrivance, Mr. reward from the board of longitude. Low Dis has constructed the apparatus Mr. Harrison's time-keeper did not, how, in such a manner, as to enable the per- ever, produce the defired and expected fon while taking the exercise, to read or effect, namely, the obtaining of equal write; allo, in cases wherein the patient arcs of vibration. He, therefore, afterfrorn debility is unable to stand, he is wards, added another contrivance, the enabled to sit down, and receive at the cycloid pin, by which to produce large same time any degree of muscular ac- and fmall ares of viuration, to be pertion that may be necessary. In cales of formed in equal times. The latter conextreme weakness, in which a patient trivance succeeded but partially; and the is unable to communicate metion tu the whole machine, as constructed by Mr. freadles, or in any other case which Harrison, was so complicated and exit may be necessary, the machine may be pensive, that no artist except Mr. KENput in motion, and all the exercile pro- DAL, ever aftempted the making of cured, by means of a hand-turn or winch, one; and he only conftructed two during which requires fofmall a degree of power, the whole of the last fifteen years of his that a child may perform the operation, life,

Mr.

a

The late Mr. MUDGE, sensible of the to the balance, it discharges, by means advantages arising from Harrison's in- of a imall pallet on the verge, the revention, pursued it still farther, and novating spring from its detant, and the brought it to a much greater degree of verge is impelled by it immediately. perfection. He gave in his time-keeper, This force drives the balance with rathe impulse to the balance from the re- pidity, and the renovating spring connovating spring, which is wound up tinues in a circular motion, until it unevery beat." The accuracy of his per- locks the detant belonging to the whee:s. formance encouraged him to become a The instant the wheel is at liberty, it candidate for the remaining reward due winds up the renovating {pring again. to those persons who might be able to This latter operation is performed long construct a time-keeper, fufficiently cor- before the balance returns from its virect to ascertain the longitude at sca. bration, by means of the resisting force

In the new time-keeper of Mr. Ha- of the pendulum fpring, when again the LEY, the scapement consists of a ba- mall pallet on the verge unlocks the relancé wheel, with straight teeth, refem- novating spring, and receives the next bling those used to detached fcapements, impulse. between which and the verge, is placed It, confequently, from the principle of a cylindrical arbor, or axis, on which is this invention, as the impelling power of a pallet, and a helical spring, wound up the renovating spring, cannot alter from

any external cause whatever,it mult there. The verge, the helical or renovating fore alw produce equal arcs of vibra. spring, and the wheel, are each distinctlý tion; and no additional force of the inain detached from each other. On winding spring, were it possible to increale it to up the time-keeper, the balance wheel duuble its power, nor any irregularities acts on the pallet, and also immediately in the tra:n of wheels, can make the least winds up the renovating spring, to the alteration in its rate of going. required degree of tension. In which The patentee conceives, that no theo. situation the balance wheel leaves it, reft- retical reason can be established, by which ing on a detant, which holds it up. there can be a certained any alteration in When the wheel has carried the renovat- the rate of going in a given period; and ing spring into this position, the wheel therefore, in a very long voyage, his likewise is stopped from proceeding any time-keeper may be relied upon, farther by a detant which receives the keeping an uniform rate. As long, he tooth of the wheel, after it has left the observes, as it does go, it must necessarily pallet of the renovating spring.

go correctly, and it cannot poil.bly fail, It may be here občerved, that the unless it wholly stops, from ill treatspring is never wound higher than one sixth part of the revolutions of which The patentee adduces another arguit is capable of being wound; it there- ment in favour of his mode of applying 'fore cannot posibly lose any degree of the renoyating spring : that it does not its clastic force, never being strained to require the alliance of oil to the baits highest state of tenlion.

lance wheel teeth, to take off the friction By a vibratory motion being given which attends the winding of it up.

every beat.

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ORIGINAL ANECDOTES AND REMAINS

OF

EMINENT PERSONS.

THE DIXECTORY.

kingdom in Europe, and of the latter, in ONE of the most difficult points in the United States of America. the of a

A learned and elegant writer*, in his ascertain the best and safest depositary of celebrated essay on the “ Idea of a perfect the executive power. In a monarchy, commonwealth,” confers the executive this invariably resides in the prince, and on a “ fenatc,” consisting of " one hun even in a deinccracy, it is sometimes con- dred members :" this idea is evidently fided to the fidelity of a single person : we have instances of the former in every

* Hume, Vol. I. Essay xvi.

borrowed

1796.]
Original Anecdotes.-The Directory.

801 borrowed from the policy of Venice, a the legislature, then executing the funcgovernment the most uniform in its ope- tions of an electoral body in the name of rations, the most invariable in its views, the nation. the most permanent in its nature, of any 2. After the gth year of the republic, in Europe, and against which there can the directors fhall be forty years of age at be urged only two trifling objections: the least, and selected from among the citiexistence of a state inquisition, and the zens who have either been legislators or consideration, that the nobles are every

minjiters. thing, and the people nothing

3. The directory is partially renewed The French, with the defects of all by the election of a new member every other institutions, and the experience of year; and an interval of five years must all former ages before them, possessed take place, before the individual retiring considerable advantages, when they laid can be re-admitted. the foundation of their modern coloflus. 4. It cannot deliberate unless three Whatever the architects of republican members at least are present. Greece and Rome, of England while a 5. The ministers chosen by it, for the common ivealth, and America after shak- home, foreign, war, &c. departments, ing off her yoke, were able to achieve, must not be under thirty years of age, and have been seen and examined by them. they cannot nominate these, or any other Avoiding the two extremes, and equally functionaries, from among their own farejecting the supreme direction of “ one,' mily, relations, or immediate connecand of one hundred,” they have form- tions. ed an executive of FIVE, an idea well 6. The members cannot absent themknown to the Dutch *, and with which felves from council more than five days we ourselves are familiar in the manage

at a time, or retire beyond four muria. ment of our different boards, particu- metres ,eight French leagues) from their larly the Admiralty, the most, perhaps refidence, the only; flourishing establishment among 7. They shall receive military honours; us, and which—owing to its republican be entitled to a guard, consisting of 120 form-neither the degeneracy of the men ; shall be lodged, at the expence of times, nor the corruption of the other the public, in a national edisice ; and the departments, nor even the notorious pro- annual appointments of cach are fixed at feffional ignorance of its chiefs, have as the current value of 50,000 myriagruiryet bereft of its native vigour.

mes of wheat (10,222 French quintals). To the Executive Directory of France 9. Each shall be attended in public by is entruited the power of the nation; it two guards of honour, and they cannot is the firit of the confiituted authorities, appear in the exercise of their functions and that which superintends all the unless dressed in costume *. others. It forins treaties of all kinds, The following is a list of the present Tubject, however, to the ratification of the directory, two of whom have been bred legislature, to whom it also presents the to the bar, and three to the profeffion of declaration of peace and war, in which arins; it is also to be observed, as not a it poffefies the initiative, or decision in little remarkable, that two of them are the first instance. It directs the opera- of a proscribed clals, no longer recognized tion of the, armies of the republic, auminates to a variety of places and oñices * Le diveElcire exécutif aura deux costumes ; pointed out by the conftitution ; receives l'un pour les forétions ordinaires, l'autre pour les the ambassadors of foreign states, and cor- représentations dans les fites nationales, Ec. responds, by means of envoys or diploma

COSTUME ORDINAIRE. tic agents, with all the governments of

Habit-manteau à rouers & à manches, correur the world.

macarat double-de-blanc, richment brulé' en or jur

l'extérieur et les revers. Title VI, of the new Conftitution, points out the manner of the election, and

Vifte longue et croisée, blanche et brodée d'or.

L'écharpe en ceinture bleue à franges d’or, le the extent of the authority of the exc

pantalon blanc (le sout en foie): cutive power :

Le chapeau zoir rond, retrouffée d'un côté, et 1. The five members are chosen by orné d'une panache tri-color.

L'érée portée en baudrier sur la veste; la cort

leur du baudrier macaral. * The Dutch council confifted of seven, one

GRAND COSTUME. for each province ; ard this number is still L'habit-mantcut bleu, et par desus un manpreserved in their new constitution.

by

2

trau macarat.

J

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