1795.]

In a fecond memoir, he propofed a method of organizing public inftruction in rural economy, particularly in the veterinary arts.

TEYSSIER read two other memoirs on the fame fubjects. In the first, he traces the abuses arifing from the exceffive encouragement granted to the perfons who clear the forefts, and the incalculable mifchiefs which arife from the extenfion of the practice, affecting the foil of the moft ufeful forefts, meadows, &c. In the fecond, TEYSSIER explained his project of extending the fcience of rural economy into other parts of the world, by a general correfpondence he has eftablished, and thus contributing to its perfection in France. He then entered into a long detail on the practice of agriculture in the Canary Islands, and noticed the high degree of profperity into which thofe iflands are capable of being raised. -TEYSSIER is thus employed in tracing a fort of grand Agrarian chart, a kind of rural map of the world, the execution of which has been fo long wanted, and which will contribute fo much to the progress of the firft and moft ufeful of the arts.

[Some of the Memoirs will be given at length in future Numbers of this Magazine.]

MATHEMATICAL CORRESPONDENCE.

QUESTION XIV (No. V).-Anfwered by

Mr. N. Boforth, of Peterborough. ON duly confidering the nature and circumftances of the Queftion, it will very plainly appear, that the nail in the tire of the wheel will, in every revolution of it, defcribe a cycloid; and that the fum of the length of all the cycloids thus formed will be in exact proportion to the number of revolutions made by the wheel, and to the distance travelled. Therefore, according to the well-known properties of the cycloid, which, I fuppofe, it is unne7 miles ceffary here to explain, we have 3.1416 X4 8.912656 miles, the fum of the lengths of all the curves defcribed by the nail, while the coach is going 7 miles, which is the mean velocity of the nail per hour.

The fame Quefiion was anfwered by

J. H.

797

compound will be greater than would be inferred from the fimple rule of alligation, the mafs becoming lefs by mcafure after the mixture than the fum of the measures of the two liquors.-The greateft diminution of the whole mafs takes place when equal parts by measure of alcohol and water are mixed; in which cafe (the alcohol ufed being extremely dephlegmated and of the fpecific gravity of 0.8199) the diminution of the mafs after the mixture was found to be 0.0288 of the whole.-This diminution of the mafs, and the confequent increafe of its fpecific gravity, takes place in a ratio, the relation of which to the proportions of the mixture, it has been hitherto found difficult to afcertain ; and it has therefore long been a defideratum to afcertain the real proportion of alcohol in any weaker fpirit from its specific gravity.-M. Pouget, of Montpellier, in a letter to Mr. Kirwan, dated 12th May 1783, and inferted in the 3d volume of the Tranfactions of the Royal Irish Academy for 1789, page 157, after a great deal of moft ingenious reafoning, and a detail of feveral experiments made with alcohol of the abovementioned fpecific gravity, gives the following formula for the folution of this problem:

Put the total volume or measure of the entire mafs or compound =1. The mea"Jure of alcohol contained therein (fuppofing alcohol of the fpec. gr. of 0.8199 to be pure or perfectly dephlegmated) =x. The diminution of the volume or bulk of a mixture of equal parts of alcohol and water found by experiment (equal as before ftated to 0.0288) c. Then, he fays, the diminution of the volume of a mixture containing a measures of alcohol will be =4x-4cx2. And putting the specific gravity of water (=1.0000) a, that of alcohol (0.8199), that of the unknown mixture=y, it will be (as the augmentation of denfity does not at all change the abfolute weight of the mais) 1-Xa+bx=1—4ca+÷cx2 Xy; from hence he deduces the following equations:

0.1801

Mr. J. Fr.

If alcohol and water be mixed in any proportion, the fpecific gravity of the

and

=

y=

1-0.1801X

0.11521 0.2304 a-ax+bx 14+4x2 10.1152x+0.1152x2. 5 I 2

Monfieur

fphere, fo ftuated that its axis shall touch the bafe LM of the tube LMON, and its convex furface pafs through the circum. ference of the other end NO. Then (as fpherical fegments are in furface proportional to their altitudes) we must have CD equal to one 20th of AD, or one 19th of AC, and the furface of the fegment NDOC will be one 20th of the furface of the hemifphere: under which conditions, an eye, looking through an aperture at A, would view one 20th of the celeftial hemifphere. In the infiancé before us, where AC is 20 inches, CD must be 1 inch; and then, as may be readily found, the difference between CD and the axis BE is 41 inches; be tween which and CD, as geometricians have proved, CN is a geometrical mean proportional; hence 416.57378

CN, and the diameter LM or NO, of the tube, is 13.14736 inches.

N.B. Mr. Emerfon, in his Aftronomy, page 36, fays, that a tube whose length is to its diameter, as 100 to 97, takes in one tenth of the hemifphere: but his proportion does not feem to be very accurate.

The fame anfwered by Mr. W. Adams, jun. of Wooturn School.

The length of the tube being 20 inches, and the field of view one 20th of the hemifphere, therefore 9 is the fegment's height, by Dr. Hutton's Menfuration, p. 191, firft edition; hence 20+0=214 is the complete radius; and then 2/41× 1 13.1473636 is the internal diameter of the tube, by a well-known property of the circle.

The fame anfwered by Mr. John Haycock, of Ware.

Let LNOM be the tube, A the aperture, AC the axis, CD the verfed fine or height of the vifible fegment, which is to be one 20th of the hemifphere.- Since the furface of any fegment is as the height, we have, as 19:20::20 AC: 211=AD,

The fame anfwered by J. Fr. The curved fuperficies ct the fegments of a fphere being as the verfed fines of half their arcs, we fhall have, in the prefent inftance, the verfed fine of 189 11' 39', half the arc of the given portion of the fphere; and hence, as radius: tang. 1891 39:20:6.573 inches the femidiameter of the tube, or 13.147. the whole diameter required.

This Queflion was alfo anfwered by Meffis, W. Claviy, J. Rub.er, J, Collins. L.W.D. and J. H.

NEW MATHEMATICAL QUESTION. QUESTION XXII-By Mr. O. C. Gregory.

VENUS's greatcft clongation from the Sun, as obferved from the Earth, is found to be 48° from this the young aftronomer is requested to point out an eafy and expeditious method of finding the diftance of that planet from the Sun,--the Earth's diftance from that luminary being 95 millions of miles,

NEW

1796.]

NEW

MR. LOWNDES'S GYMNASTICON.

ON the 30th of September, letters patent were granted to Mr. FRANCIS LOWNDES, of St. Paul's Church-yard, London, for his invention of a machine for mufcular exercife, which he calls a Gymnafticon. Its object is to enable perfons to exercife themfelves in any degree wifhed for, in their own chambers; generally in all parts of the body at once, or partially as the cafe may require.

The machine confifts of an upright frame, fufficiently large to admit the human body. The action is produced on the lower extremities by means of two treadles, fimilar to thofe made ufe of in turning lathes. The action of the upper extremities is produced by means of two cranks of any required depth; which cranks are put in motion by a wheel, connected by a band, with a wheel on the axle of the cranks, belonging to the treadles. The perfon, therefore, requiring exercise in both extremities, has only to put the treadles in motion, by the weight of his feet; the treadles will then turn their cranks, which, by a band, communicate motion to the upper wheel and cranks, and by holding the latter with the hands, every joint and mufcle in the body is put in action. It is obvious, that this action may be increafed to any degree, by varying the pofition of the feet on the treadles, and the depth of the upper cranks.

The foregoing defcription applies to motion, fimilar in its effect to walking or running, and may be eafily varied, fo as to produce a degree of exercife equal to two, or even to ten miles per hour.

By very fimple contrivances, Mr. LOWNDES has conftructed the apparatus in fuch a manner, as to enable the perfon while taking the exercife, to read or write; alfo, in cafes wherein the patient from debility is unable to ftand, he is enabled to fit down, and receive at the fame time any degree of mufcular action that may be neceflary. In cafes of extreme weaknefs, in which a patient is unable to communicate motion to the Treadles, or in any other cafe in which it may

be neceffary, the machine may be put in motion, and all the exercise procured, by means of a hand-turn or winch, which requires fo fmall a degree of power, that a child may perform the operation,

This machine, the patentee conceives, will be of the highest importance to medicine, by affording a degree of exercife which cannot otherwite be obtained, and by directing the exercife into particular parts of the fyftem. The difeafes, wherein it may be applied with the greatest fuccefs, he conceives to be all thofe which arife from obftructions in the joints and in the abdominal vifcera; and that it might therefore be highly ufeful in female boarding fchools; to perfons of fedenary habits and employments; to the paralytic, the gouty, and the rheumatic.

MR. HALEY'S TIME-KEEPER.

On the 17th of September, letters patent were granted to Mr. CHARLES HALEY, of Wigmore-ftreet, Cavendishfquare, for his invention of an improved MARINE TIME KEEPER.

The principal invention in this timekeeper, confifts in the manner of applying a renovating spring; which mode of application is entirely original, and appears to be a confiderable improvement. By producing perfectly equal arcs of vibration in the balance, the Patentee has attained an object which all artills have had in view, but hitherto have only obtained in a partial manner.

The firit attempt of the kind, attended with any fuccefs, was that of Mr. HARRISON, who applied a renovating spring on the contrate wheel, wound up every ten feconds. This application took off fuch a proportion of the imperfections of the main fpring, and of the iregularities of power, arifing from the train of wheels, as to obtain a degree of accuracy which intitled him to very confiderable reward from the board of longitude. Mr. Harrifon's time-keeper did not, however, produce the defired and expected effect, namely, the obtaining of equal arcs of vibration. He, therefore, afterwards, added another contrivance, the cycloid pin, by which to produce large and fmall arcs of vibration, to be performed in equal times. The latter contrivance fucceeded but partially, and the whole machine, as conftructed by Mr. Harrifon, was fo complicated and expenfive, that no artift except Mr. KENDAL, ever attempted the making of one; and he only conftructed two during the whole of the last fifteen years of his life.

Mr.

The late Mr. MUDGE, fenfible of the advantages arifing from Harrison's invention, purfued it ftill farther, and brought it to a much greater degree of perfection. He gave in his time-keeper, the impulfe to the balance from the renovating fpring, which is wound up every beat. The accuracy of his performance encouraged him to become a candidate for the remaining reward due to thofe perfons who might be able to conftruct a time-keeper, fufficiently correct to afcertain the longitude at fea.

In the new time-keeper of Mr. HaLEY, the scapement confifts of a balance wheel, with ftraight teeth, refembling thofe ufed to detached fcapements, between which and the verge, is placed a cylindrical arbor, or axis, on which is a pallet, and a helical fpring, wound up every beat.

The verge, the helical or renovating fpring, and the wheel, are each diftinctly detached from each other. On winding up the time-keeper, the balance wheel acts on the pallet, and alfo immediately winds up the renovating fpring, to the required degree of tenfion. In which fituation the balance wheel leaves it, refting on a detant, which holds it up. When the wheel has carried the renovating fpring into this pofition, the wheel likewife is ftopped from proceeding any farther by a detant which receives the tooth of the wheel, after it has left the pallet of the renovating fpring.

It may be here obferved, that the fpring is never wound higher than one fixth part of the revolutions of which It is capable of being wound; it therefore cannot pofiibly lofe any degree of its claftic force, never being trained to its higheft ftate of tenfion.

By a vibratory motion being given

to the balance, it difcharges, by means of a fall pallet on the verge, the renovating ipring from its detant, and the verge is impelled by it immediately. This force drives the balance with rapidity, and the renovating fpring continues in a circular motion, until it unlocks the detant belonging to the wheels. The inftant the wheel is at liberty, it winds up the renovating fpring again. This latter operation is performed long before the balance returns from its vibration, by means of the refifting force of the pendulum fpring, when again the fmall palict on the verge unlocks the renovating fpring, and receives the next impulfe.

It, confequently, from the principle of this invention, as the impelling power of the renovating fpring, cannot alter from any external caufe whatever,it muft therefore always produce equal arcs of vibra tion; and no additional force of the main fpring, were it poffible to increase it to double its power, nor any irregularities in the train of wheels, can make the leaft alteration in its rate of going.

The patentee conceives, that no theoretical reafon can be established, by which there can be ascertained any alteration in the rate of going in a given period; and therefore, in a very long voyage, his time-keeper may be relied upon, keeping an uniform rate. As long, he obferves, as it does go, it must neceffarily go correctly, and it cannot poflibly fail, unlefs it wholly ftops, from ill treat

ment.

as

The patentee adduces another argument in favour of his mode of applying the renovating fpring: that it does not require the affittance of oil to the balance wheel teeth, to take off the friction which attends the winding of it up.

1796.]

borrowed from the policy of Venice, a government the most uniform in its operations, the most invariable in its views, the most permanent in its nature, of any in Europe, and against which there can be urged only two trifling objections: the existence of a ftate inquifition, and the confideration, that the nobles are every thing, and the people nothing.

The French, with the defects of all other inftitutions, and the experience of all former ages before them, poffeffed confiderable advantages, when they laid the foundation of their modern coloffus. Whatever the architects of republican Greece and Rome, of England while a commonwealth, and America after thaking off her yoke, were able to achieve, have been feen and examined by them. Avoiding the two extremes, and equally rejecting the fupreme direction of "one," and of "one hundred," they have formed an executive of FIVE, an idea well known to the Dutch *, and with which we ourselves are familiar in the management of our different boards, particularly the Admiralty, the moft, perhaps the only, flourishing establishment among us, and which-owing to its republican form-neither the degeneracy of the times, nor the corruption of the other departments, nor even the notorious profeffional ignorance of its chiefs, have as yet bereft of its native vigour.

To the Executive Directory of France is entrusted the power of the nation; it is the first of the conflituted authorities, and that which fuperintends all the others. It forms treaties of all kinds, fubject, however, to the ratification of the legiflature, to whom it also presents the declaration of peace and war, in which it poffeffes the initiative, or decifion in the first inftance. It directs the operation of the armies of the republic, nominates to a variety of places and offices pointed out by the conftitution; receives the ambaffadors of foreign ftates, and correfponds, by means of envoys or diplomatic agents, with all the governments of the world.

Title VI, of the new Conftitution, points out the manner of the election, and the extent of the authority of the exccutive power:

1. The five members are chofen by

*The Dutch council confifted of feven, one for each province; and this number is fill preferved in their new constitution.

801

the legislature, then executing the functions of an electoral body in the name of the nation.

2. After the 9th year of the republic, the directors fhall be forty years of age at leaft, and felected from among the citizens who have either been legislators or minifters.

3. The directory is partially renewed by the election of a new member every year; and an interval of five years muft take place, before the individual retiring can be re-admitted.

4. It cannot deliberate unless three members at leaft are prefent.

5. The minifters chofen by it, for the home, foreign, war, &c. departments, muft not be under thirty years of age, and they cannot nominate thefe, or any other functionaries, from among their own family, relations, or immediate connections.

6. The members cannot abfent themfelves from council more than five days at a time, or retire beyond four myriametres eight French leagues) from their refidence,

7. They fhall receive military honours; be entitled to a guard, confifting of 120 men; shall be lodged, at the expence of the public, in a national edifice; and the annual appointments of each are fixed at the current value of 50,000 myriagrammes of wheat (10,222 French quintals).

9. Each fhall be attended in public by two guards of honour, and they cannot appear in the exercife of their functions unlefs dreffed in coftume *.

The following is a lift of the prefent directory, two of whom have been bred to the bar, and three to the profeffion of arms; it is alfo to be obferved, as not a little remarkable, that two of them are of a profcribed clais, no longer recognized

* Le directoire exécutif aura deux coftumes; l'un pour ses fonctions ordinaires, l'autre pour les reprefentations dans les fétes nationales, &c.

COSTUME ORDINAIRE.

Habit-manteau à revers & à manches, couleur macarat double-de-blanc, richment brodé en or fur l'extérieur et les revers.

Vifte longue et croifée, blanche et brodée d'or. L'écharpe en ceinture bleue à franges d'or, pantalon blanc (le tout en foie).

le

Le chapeau noir rond, retrouffée d'un côté, et orné d'une panache tri-color.

L'épée portée en baudrier fur la vefte; la conleur du baudrier macarat.

GRAND COSTUME. L'habit-manteau bleu, et par deffus un man

teau macarat.