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Mathematical Correspondence.

797 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 alligarural æconomy, particularly in the vete tion, the mass becoming less by measure rinary arts.

after the mixture than the sum of the TEYSSIER read two other memoirs measures of the two liquorsThe 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 całe 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 æconomy 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 spondence 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 fpiIslands, 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 fol[ Some of the Memoirs will be given at length lowing formula for the folution of this in future Numbers of this Migazine. ]

problem :

Put the total volume or measure of the MATHEMATICAL CORRESPONDENCE. entire mass or compound

The mea

Jure of alcohol contained therein (supporQuestion XIV (Ņo. V).-Answered by ing alcohol of the fpec. gr. of 0.8 199 to

Mr. N. Bol vortb, of Peterborough. be pure or perfectly dephlegmated) =x. ON duly considering the nature and cir- The diminution of the volume or bulk of

cumitances 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) =(. 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 = f(x—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 luppose, it is unne augmentation of density does not at all ceffary here to explain, we have 7 miles change the absolute weight of the mais)


1-xxa+bx=1--461 +26x2 Xy ; 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, which is the mean velocity of the nail


0.5 Scy


Scy Toe same Quillion was anfeucreil by 0.1801 7. H.

9.2 3049

QUESTION XV (No. V).-Anfwered by VI +
Mr. J. I--r.

0.11529 0.2 If alcohol and water be mixed in any


I-01801X proportion, the specific gravity of the 1-46+40x2 1-0.1152x +0.1152x2,

5 I 2





per hour.


Monsieur Pouget gives several instanc- =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 35, lays, thai a tube whose length is three places of figures ; and concludes, to its diameter, as 100 to 97, takes in one that they may be relied on as fufficiently 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 al. cohol.

The sume anfvered by Mr. iv. Aitans, For the former of the two kinds of

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

and the field of view-one 20th of the he. 0.1801

misphere, thereforeis the segment's *-0.5+

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

191, first edition ; hence 20+30=217 is ✓ I-0.92 0.1801 +

the complete radius; and then ?V41,5 x in

-0.5 C.1152 X 0.92 0.2304 x 0.92

=13.1373636 is the internal diameter 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=0.5.


The same answered by Mr. John Haycock,

of Ware.

Let LNOM be the tube, A the aper+

-0.5 0.1152 X0.9 0.2 304X0.9

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

be one 20th of the hemisphere.- Since Tois Question was also answered by Mr. the surface of any segment is as the heigni, 7. H.

we have, as 19:20::20=AC:21;=rD; QUESTION XVII, misprinted XVI (No. hence NO=2NC=2V783

19 19
VI).- Anfwered by Mr. 0.6. Gr.Or. 2115600
In the

=13.14736 inches, the internal

19 figure an

dianister required. nexed, let BNOE be

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

The curved luperficies of the fegments representa

of a sphere being as the versed tines of section of

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


sent initance, b=the verfcd fine of 180 {phere, so ntuated that its axis ihall touch' ll' 39', half :he arc of the given portion the base LM of the tube LNIG, and its of the iplere ; and hence, as radius: tang. convex furface país through the circum- 189 ; 1' 39' : :20:6.573 inches the semiference of the other end No. Then (as diameter or the iube, or 13.147. the whole spherical segments are in furface propor- diameter required. tional to their altitudes) we muht have Toisaca was also anfredly Milfs, CD equal to one 20th of AD, or one W. Ciavil', 7. Rubiei, 7, Collins. L.J.D. 19th of AC, and the surface of ihe and 7. H. segment NDOC will be one zoth of the surface of the hemisphere : under which Now MATHEMATICAL QUESTION. conditions, an eye, locking through an QUESTION XXII--By Mr. O. C. Greaperture at A, would view one 20th of

gori: the celestial hemisphere. In the instance

Venus's greatest clongation from the before us, where AC is 20 inches, CD

Sun, as observed from the Earth, is found must be 17 inch ; and then, as may be to be 48° : from this the young astronoreadily found, the difference between CD and the axis BE is 41.1 inches ; be

mer is requested to point out an easy and tween which and CD, as geonetricians tance of that planet from the Sun,-the

expeditious ethod of finding the difhave proved, CN is a geometrical mean Earth’s distance from that luminary being proportional: hence 4175 X 11=6,57378 95 millions of miles,

20 X




( 799)

Enrolled in the Months of September and O&tober.

MR. LOWNDES's GYMNASTICON. This machine, the patentee conceives, On the 30th of September, letters pa- will be of the highest importance to me

tent were granted to Mr. FRANCIS dicine, by affording a degree of exercise LOWNDES, 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 muscular exercise, which he calls parts of the lystem. The diseases, wherein a Gymnofticon. Its cbject is to enable it may be applied with the greatest sucpersons to exercise themselves in any de- cets, he conceives to be all those which gree wished for, in their own chambers; arise from obstructions in the joints and generally in all parts of the body at once, in the abdominal viscera; and that it or 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, fuliciently large to admit the hu- dentary habirs 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, fimilar to those made ute of in On the 17th of Stptember, 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 square, 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 axle of the cranks, belonging to the keeper, confiits in the manner of applytreadles. The person, therefore, re- ing a renovating (pring; which mode of quiring exercise in both extremitics, has application is entirely original, and aponly to put the treadles in motin, by pears to be a considerable improvement. the weight of his feet; the treadles will By producing perfectly equal arcs of vithen turn their cranks, which, by a bration in the balance, the Patentee has band, communicate motion to the upper attained an object which all artiis have wheel and cranks, and by holding the latter had in view, but 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. HAKany segree, by varying the position of RISON, who applied a renovating spring the fece on the treadies, and the depth on the contrate wheel, wound up every of the upper cranks.

ten seconds. This application took off The foregoing description applics to such a proportion of the imperfections of motion, fimilar in its effe&t to walking the main tpring, and of the iregularities or running, and may be eanly varied, to of power,' arising from the train of as to produce a degree of exercile equal wheels, as to obtain a degrec of accuracy to two, or even to ten milcs per hour. which intitled him to very considerable

By very limple contrivances, Mr. reward from the board of longitude. Lowndes has constructed the apparatus Mr. Harrison's time-keeper did not, how, in such a manner, as to enable the per- ever, produce rhe desired and expeċied fon while taking the exercise, to read or

effcét, namely, the obtaining of equal write; also, in cases wherein the patient arcs of vibration. He, therefore, afterfron 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 fame time any degrée of muscular ac- and fmall arcs of vivration, 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 motion tv the whole inachine, as constructed by Mr. freadles, or in any other case in 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 artempted the making of cured, by means of a hand-turn or winch, one; and he only constructed two during which requires sofmall a degree of power, the whole of the last fifteen years of his that a child may perform the operation, life,


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 ipring 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 inftant 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 vi. rect to ascertain the longitude at fca. bration, by means of the resisting force

In the new time-keeper of Mr. Ha of the pendulum spring, when again the LEY, the scapement confits of a ba small pallet on the verge unlocks the relance wheel, with straight teeth, resem novating spring, and receives the next bling those used to detached (capements, impulse. between which and the verge, is placed It, consequently, 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 (pring, cannot alter from

any external cause whatever, it mult there. The verge, the helical or renovating fore always produce equal arcs of vibraspring, and the wheel, are each distinctly tion; and no additional force of the inain detached from each other. On winding spring, were it poflible 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 train 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, rest retical reason can be established, by which ing on a detant, which holds ir up. there can be ałcertained 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 likewife is stopped from proceeding any time-keeper may be relied upon, as farther by a detant which receives the keeping an uniform rate. As long, he tooth of the wheel, after it has left the

obferves, as it does go, it must necessarily pallet of the renovating spring.

go correćtly, and it cannot poil.bly fail, It may be here oberved, 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 pofiibly lose any degree of the renovating spring : that it does not its claftic force, never being ftrained to require the alliltance 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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kingdom in Europe, and of the latter, in ONE of the most difficult points in the United States of America.

the formation of a conftitution, is to A learned and elegant writer*, in his ascertain the best and safeít depositary of cclebrated essay on the “ Idea of a perfect the exicutive power. In a monarchy, commonwealth,” confers the executive this invariably resides in the prince, and on a “senate,” consisting of “one huneven in a deinecracy, 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.


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 shall be forty years of age ac 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 ministers. 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 insitutions, and the experience of year; and an interval of five years must all former ages before them, posseffed take place, before the individual retiring considerable advantages, when they laid can be re-admitted. the foundation of their modern colossus. 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 commonwealth, and America after ihak- 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 tivo extremes, and equally functionaries, from among their own farejecting the supreme direction of “ one, mily, relations, or immediate connecand of v 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 selves from council more than five days we ourselves are familiar in the manage at a time, or retire beyond four myriament of our different boards, particu- metres ,eight French leagues) from their larly the Adiniralty, the most, perhaps residence, the only; fourishing 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 edifice; and the departments, nor even the notorious pro- annual appointments of each are fixed at feffional ignorance of its chiefs, have as the current value of 50,000 myriagramyet 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 fuperintends all the unless dressed in costume *. others. It forins treaties of all kinds, The following is a list of the present lubject, 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 profession of declaration of peace and war, in which arins; it is also to be observed, as not a it poffeffes the initiative, or decision in little remarkable, that two of them are the firit instance. It directs the opera- of a proscribed clals, no longer recognized tion of the, armics of the republic, auminates to a variety of places and offices * Le diveEisire exécutif aura deux costumes ; pointed out by the conftitution ; receives l’un pour ses fonctions ordinaires, l'autre pour les the ambassadors of foreign states, and cor- représentations dans les fêtes nationales, &c. refpond's, by means of cnvoys or diploma

COSTUME ORDINAIRE. tic agents, with all the governments of

Habit-manteau à revers & à manches, couleur the world.

macarat double-de-blanc, richment brolé en or fur Title VI, of the new Conftitution,

leiterieur et les revers. points out the manner of the election, and

Vifte longue et croisée, blanche et brodée d'or. the extent of the authority of the exe• pantalon blanc (le tout en soie).

L'écharpe en ceinture bleue à franges d'or, le

Le chapeau i.oir rond, retroussée d'un colé, et 1. The five members are chosen by orné d'une panache tri-color.

L'épée portée en baudrier sur la vese; la con

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

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

cutive power :

teau macerat.

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