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wife equal, will make the power weaker. Therefore a force which would he only tufficient for fuftaining the mafs Q (fig. 5.) by acting according to the direction PP, would be no more fo if it went out of that line, and much lefs at a greater diftance, by placing itfelf at the points c, d, e, f.

Q. How ought the fulcrum that fupports the lever to be considered ?

A. It ought to be confidered as a third power, equiponderating with the moving force or refiftance, or concurring with either for bearing the others's effort. In levers of the first kind, the fulcrum or point of fupport fuftains the effort of two oppofite for res: In thofe of the fecond and third kinds, it bears but a part of one of the two.

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As the point of fupport is not always fixed and immoveable, and as it often may be animated body, or other flexible bodies, fubject to be bruifed or broken, it is of fome moment to know how much the point of fupport, or that which ferves for one, is load. ed, when two other forces act against one another on the fame lever, in order that it may have a proportion with the effort it is to fuftain. On this head it will not be amifs to obferve,

That the effort which proceeds from the mals, and which may be called abfolute, is limited: A pound, or the action of a power equivalent to a pound, when it weighs upon the arm of a lever, in the most advantage ous direction, must be in æquilibrio with a like weight opposed to it under the fame circumftances. But the effort that proceeds from the distance to the point of fupport, may increase in infinitum; to that, if one of the two arms was 100 times as long as the other, a pound would become equivalent to too.

What will therefore be the load on the point of fupport, first, if there be an equilibrium with equality of mafs; zdly, if the maffes or forces are in æquilibrio by the inequality of their distances from the point of fupport?

The first queftion is anfwered by faying that, if the directions of the power and refiftance are parallel, the point of fupport is loaded with the fum of two abfolute forces, and its effort is in a parallel direction to thofe of the power and refiftance. But, if the directions of two oppofite forces are inclined to each other, the point of fupport bears but a part of their abfolute effort; it bears the lefs according as they are more inclined to the lever, and its refiitance tends to the point of concurrence of these two directions.

As to the fecond question, which is to know what the effort is that is made on the

point of support, when the power and the reliftance are in æquilibrio by unequal diftances between them and the point of fupport, I anfwer, that this effort is never greater than the fum of the opposite abfolute forces or maffes; that is, if the weight of one pound fuftains another of 12, because it acts by the arm of a lever 12 times longer than that of the other part, the point of fupport can never be loaded with more than 13 pounds, and not 24; and its effort is directed parallel to the directions of the forces it fuftains, if those directions are parallel among themselves, or directly to the point of their concurrence, if inclined to one another.

When it is known how great the effort is on a fulcrum, all accidents arifing from difproportions may be prevented, or an advantage made by a proper application of the forces regarded as infufficient.

Place, for example, a load of 200 lb. in the midit of a lever whereof the extremities reft on the shoulders of two men, these two fulcrums will be fufficient to the burden, if each of the porters is capable of fupporting 100 lb. But if one of them can carry only 50, tho' the other may be fufficient to fuftain an effort of 150lb. the weaker notwithstanding will be obliged to give out, as long as the burden is at an equal distance between him and the other; and both will become ufelefs, in regard to the business expected from them. But let the load be placed farther from the weaker porter, and let the arms of the lever now made unequal be in a reciprocal ratio of the efforts the two men are capable of; then the load will be fupported, as it fhould have been at first, by two other men who might be each equal to an effort of 100 lb.

Another example may ferve alfo to illus trate this matter: When a carpenter carries a joift, it is always for the most part by the middle of the length that he lays it on his fhoulder: By fo placing it, he carries only the weight of the piece of timber, because the two ends are reciprocally in æquilibrio, and the point of fupport is loaded with nothing more than the fum total of the two mafies. But, if he laid it at the two thirds or three fourths of its length, he would be obliged, to keep it from falling, to hold it with his arms by the fhorter end; and this effort would be equivalent to a weight that would make an equilibrium with the excefs of the length of the joist on the opposite side. The carpenter's fhoulder would therefore be to no purpose loaded with this greater quan. tity.

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Q. Why does a man, drawing a boat, or fome burden faftened to the end of a rope, bend forward? A. Becaufe

A. Because he joins to the action of the muscles a part of the weight of his body to conquer the refiftance against which he acts. Q. Why are ashes, or the like, thrown on frequented places which are covered with glazed froft?

A. Because a point of fupport is wanting to those that walk on flippery ground. Horfes for the fame reason are rough-fhod in frofty weather.

Q. What is it that fatigues horfes fo much when they draw a carriage up an alcent?

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A. Besides the weight of the load which is then lefs fupported by the ground, the inclination of the ground prefents them the point of fupport in a very oblique direction to that of their effort; for their legs, by being kept ftiff against the ground, incline the fame way, and the more they draw near to a parallelifm, the lefs they are fupported. Of the CENTER of GRAVITY, or WEIGHT; and the EQUILIBRIUM. Q. What is the center of gravity in a body?

A. It it the point about which all the parts of this body are in æquilibrio among themselves, in whatever fituation they are placed. When the center of gravity is fup.. ported, the body may remain at rest, because the oppofite parts are in æquilibrio. When nothing fupports the center of gravity, the body moves till the center is fupported; for it is only about this point that the oppofite parts are in æquilibrio. This point is not exactly in the middle, but in bodies perfectly homogeneous, and of a regular figure. In a round bowl of uniform denfity, all the femi-diameters are equal and of the fame weight; equal, by reafon of the perfectly spherical figure; of the fame weight, upon account of the homogeneity of parts: Every thing is therefore in æquilibrio about a point which is at the fame time the center of gravity and of figure. But the fame cannot be faid of an arrow, of which the end is tipped with iron, or of a writing-pen: If the length is divided into two equal parts, the one will be more heavy than the other, and the fection will not pafs through the center of the weight, though it does through that of the figure.

When feveral maffes weigh on the fame rope by cords that faften them to it, the common knot of the cords may be deemed the center of the particular weights. A, B, (fig. 6.) being therefore the centers of gravity of two fufpended bodies, their actions unite at C, or in any other chofen point of the line CD, provided the weight A be equal to the weight B; for if one of the two

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bowls was of wood, and the other of Atone, the center of the heavier would draw nearer to the line CD, and the line ab would be divided by the direction CD into two unequal parts, of which the longer would be to the fhorter, as the greater to the lefs weight. In whatever number thefe heavy bodies may be, if the center of gravity of each of them is known, the place where their forces unite is eafily determined, becaufe the diftances are known.

The point in any body whatsoever, or in a machine, that fupports the center of vity of a body, fupports the whole weight graof this body; and the whole force wherewith alfo this body tends towards the center of the earth, is, as it were, united in this

center.

The weight has a different intenseness, when the bodies are more or lefs diftant from the center of the earth where they tend to; but this difference is never fenfible in the extent of a machine. Thus a bucket full of water that weighs 100 lb. on the pulley of a well when it is drawn up, is deemed to weigh as much when it is abftracting from the weight of the rope. feet lower,

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The directions of two weights diftant from one another ought to be regarded as parallel, though in rigour they are fomewhat inclined, because all heavy bodies tend to one and the fame point, which is the center of the earth; but we are at too great a diftance from this center to be apprehenfive of any mifcalculation by making no account of this inclination.

Q. Why do certain edifices still remain ftanding, though they have loft their dicular? perpen

A. Because their center of gravity contiques fupported.

Q. Why does a rope-dancer gefticulate almolt always his arms?

A: The reafon is, that as he walks on a moveable plane, continually inclining, when he perceives that the center of his gravity is not fupported, he recovers it in the line of direction, by holding out the arm of the oppofite fide, as a lever whereof the weight is the more powerful, as its parts are farther from the center of their motion.

Q. Why do perfons of a large belly bend backwards?

A. Because, without that attitude, the center of gravity not fufficiently fupported would throw them on their face. Of PULLEYS and MOUFLES or moufled

PULLEYS.

Q. What is the pulley?

A. It is a kind of wheel or round body, ufually flat, of wood or metal, moveable on

its

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its center C, (fig. 7.) often channelled or made a little hollow gorge-wife throughout its circumference, the better to receive and hold, by means of this cavity, a rope or chain, to which on one fide is applied the power E, F, or G, and on the other the weight or refiftance R. The wheel or body of the pulley commonly moves in a fort of handle called the block, CD, which fuftains the axis; and this axis is a pin which traverses the wheel or rundle, and round which as many points as it afcends on one fide, fo many it defcends on the other. But as the rope must lead the pulley, or the pulley the rope, when there is room to apprehend that the rope may not flide upon the pulley, the gorge is hollowed in the form of an angle, or ftuck with points, (fig. 8.)

Q. How is the pulley ufed?

A. Two different ways, it is fixed or moveable. It is faid to be fixed, though the wheel run round the pin, whenever the block is fixed and immoveable, and moveable, -when the block is not fixed to one point, but follows the direction of the weight that is hung to it. The fixed pulley is a true balance, because one may conceive each point of the wheel as the extremity of a line or radius terminated at the pin, and corresponding with a like line on the other fide. The two lines, or radii, make together two arms, or the equivalent of the beam of a balance. But the beam must be taken horizontally, if we would form a true judgment of the equality of weights, The moveable pulley is a true lever: The reafon is, because, in all the points which compofe the wheel, we need only regard the two extremities of the line that croffes the wheel and axis: Thefe being properly the points which receive the preffion of the powers, and which may be confidered as prolonged by means of ropes, and immediately applied to the two ends of the line that cut the point of fufpenfion.

The pulley may be used as a lever of the firft kind, whereof the arms are equal, and on which two powers, whofe abfolute forces are equal, remain always in æquilibrio, whatever their directions may be. The applied powers act with greater force, as their diftance from the axis is greater. The axis is loaded with the fum total of the power, and resistance, and its effort is in a direction parallel to theirs, and tending to their point of concurrence.. Some experiments will clear up this matter.

Let there be a machine, (fig. 9.) compofed of two posts a, b, raised and fixed upon a fhelf cc of greater length than breadth. One of thefe pofts, a, bears up a metal pulley; and the other, b, an angular lever, whose arms are equal, and which turns very

freely on its rivet, and in the fame plane as the pulley. A cord is firft put about the pulley, at the ends of which two equal weights P, R, are fastened, and let to act in directions parallel and vertical, as A P and B R. Afterwards the weight R is tranfported to the cord that keeps hold of the arm D of the angular lever, and the cord of the pulley is placed, as PA, FE. Lastly, the weight R being put in its first place, and the angular lever being turned, so as that D may be at d, and E at e, the weight P is faftened to the end of the cord dp, and the cord of the pulley that bore it, to the arme of the turning lever.

In this experiment the two weights P, R, are always in æquilibrio, not only when they are both in parallel and vertical directions, but also when one of them acts horizontally on the pulley. The pulley AF B may be regarded as an affemblage of the firft kind, of which the arms are equal, and which have a point of fupport, common to the center where the axis is. When the cord is vertical on both fides, if it cannot flide upon the pulley, it ought to have the fame effect as if it were of two pieces, whereof one was fastened at A, and the other at B. There is therefore an equilibrium between the two weights P, R, because they act at equal distances from the point of fupport, and each of them exerts its effort in a direction perpendicular to the arms A or B. All the rest is eafily explained and needs no demonftration.

Let us fuppofe a pulley (fig. to.) compofed of feveral circular planes, which leave breadths between them, and of which the circumferences are hollowed gorge-wife. The diameters, and confequently the radii of thefe circles are among themfelves as the number 1, 2, 3. On the leaft of the three circumferences a cord is placed, from which are fufpended two weights of fix ounces each; and two other cords are fixed at a and b which embrace the two other circumferences, and hang perpendicular to the points 2 and 3. When the two weights are at H and I, there is an equilibrium between fix ounces on one fide, and as many on the other. If that at H is taken away, another weight of three ounces does the fame at K; and, when this alfo is taken away, two ounces placed at L fupport the weight of fix ounces at I, which is thus proved;

The radius c 1 being equal to cd, there is an equilibrium between two equal weights, because their efforts are at equal distances from the point of fupport. But, c2 being double cd, the equilibrium ought to be be tween two maffes that are in a reciprocal ratio of these two lengths: Thus three oun

ces

ces fupport fix; and, by the fame reason, two ounces are fufficient for a distance that equals three times cd.

The fimple pulley may be alfo confidered as a lever of the fecond kind; and, indeed, it has its properties when, the weight or the refiftance R (fig. 11.) being fastened to the block, one of the ends of the cord holds to a fixed point a, org, whilft the other is drawn or upheld by the power P, or d. And then either the directions of the power and refiftance are parallel, as c I, dE; or inclined, as Pk, ck. In the firft cafe, the power bears but the half of the weight of the refiftance; in the fecond, the effort of the power diminishes, and the point of fupport has a direction to the point of concurrence of the directions of the power and resistance, that is, at k.

Let there be two fmall pegs of the length of three inches, A, B, (fig. 12.) that may flide into two grooves made in the two arms of the machine G. The firft ferves as a fixed point to a cord that goes about a pulley C, loaded with a weight D, and of which the other end is faftened to the arm of a balance from which one fcale has been taken, and which has been put in æquilibrio with itfelf by the means of a small weight faftened at H. This balance is fufpended from the other peg B. The two pegs may be firft placed at fuch a distance from one another, that the two ends of the cord coming from the palley may be parallel. After wards, by removing the pegs, the two ends of the cord may be made to take directions inclined contrary ways; and in both cafes the bason of the balance may be loaded as much as is neceffary to keep the beam in an horizontal fituation.

It is manifeft that, the pulley and its weight D weighing together eight ounces, four only will be required in the bafon of the balance to make the equilibrium, when the two ends of the cord are parallel and in a vertical direction; but when they are inclined, as Pl, gm, of the foregoing figure, the bafon of the balance must have a greater load to keep it in æquilibrio. Confidering the arm H of the balance, as the power that fupports the pulley and its load, after that the other end of the cord is fixed at A, the weight put into the bafon expreffes the effort made on the power, when the whole is in æquilibrio. It appears from the premiffes:

1. That, the directions of oppofite forces being parallel, the power fupports but the half of the effort of the refiftance; for in the firft cafe of the experiment, wherein the two ends of the cord are parallel, e I, (fig. 11.) the direction of the refiftance is alfo parallel de, which is that of the power, and four

ounces in the bason of the balance support eight at D, (fig. 12.)

2. That, the directions of oppofite forces being no longer parallel, the power is alto no longer equal to the half of the effort of the refiftance, and that the direction of the point of fupport paffes to the point of concurrence of the two other directions; for in the fecond cafe, wherein the power acts obliquely as Pk (fig. 11.) four ounces in the bafon of the balance are no longer fufficient to make an equilibrium, and the angle gke is equal to that on the other fide Pc.

When the two ends of the cord are parallel as a b, de, they may be confidered as faftened to the two extremities of the diameter be: When they are oblique as Pl, gm, they may be conceived as holding to the points of tangency 1, m: But the two lines eb, ml, are two levers of the fecond kind, both divided into two equal arms by the direction c I of the refiftance: The cord, fufpended at a or at g, tranfporting the fixed point to bor m, it may be immediately feen that the power applied at e orl, acts always at a distance eb, or 1 m, from the point of fupport, double of that of the refiftance placed at c or I. Now four ounces, at a double diftance from the point of fupport, are capable of fupporting eight, according to what has been faid concerning the lever. But, when the power has an oblique direction, it is no longer fufficient for the fame effects as before; becaufe the direction, perpendicular to the arm of the lever, is the most advantageous of all, and confequently all others are lefs fo.

Let us now proceed to the application, which is easily done. When the cord of the pulley has been fixed at A, (fig. 12.) a force of four ounces only is required at H, to fupport another of eight at D; and a force of four ounces is always the fame, whether it acts from above downwards, or vice versâ by the means of a balance: One may therefore fubftitute to the beam HK another beam L or I (as in fig. 13.) which will alfo execute the function of a lever of the first kind, and there will never be more to be fupported at M or at m, than an effort of four ounces.

If, in order to refift this effort of four ounces, the cord is lengthened from M to N, (fig. 14.) and made to pafs untier a third pulley (3) NO, this, like the first (1), will become a lever of the fecond kind, wherein the power O, once farther from the point of fupport N, than the refiftance that loads the axis, will need but an abfolute force that is the half of its own: Therefore nothing more than an effort of two ounces will be required from below upwards, and, if it be, M m

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more commodious to draw from above downwards, a fourth pulley (2) will give, as the fecond (4), that direction. The fecond (2) and fourth (4) pulleys, that ferve as a drawback for changing the direction, may be placed in the fame block; and, if this block is fixed upwards, its lower part may ferve as a fixed point for the first end of the cord fastened at F.

The name of moufles, or moufled pulleys, is given to that affemblage of feveral pulleys fo placed in the fame block, either parallel to or above one another. These machines are much used for raifing great burdens, and they are commodious by taking up little roon, and alfo, as one without embaraffment may augment at pleasure the action of the fame power; but this is not done, as in all other machines, but at the expence of a greater velocity in the power; for if the pulley, which is loaded by the refiftance, rifes to the line da (fig. 11.) the power that produces this effect, makes twice as much

way at the fame time, because the two parts ab, de, of the cord whereby it acts, ought to be above the line da, when the center of the pulley reaches it. But these two lengths ab, de, are equivalent to twice the height cb.

The moufles ought to be so disposed, that the directions of the cords might be parallel as much as poffible. If they depart from the parallelifm by concurring, the help the power would receive from thefe machines, would be fomewhat lefs, because in this cafe the effort of the power would be divided, by drawing the weight partly upwards, and partly on the fide towards which it inclines.

To make the multiplication of pulleys more profitable, it will be neceffary to join the fixed to the moveable. The affemblage of moveable pulleys is called a moveable moufle, and that of fixed pulleys a fixed moufle. The fixed pulleys are all inclosed in the fame block, and the moveable in like manner in another block.

MEMOIRS of the Siege of QUEBEC From the Journal of a French Officer on Board the CHEZINE Frigate, taken by his Majesty's Ship RIPPON.

N the roth of May, 1759, Monfieur de Bougainville arrived at Quebec from Old France, in the Chezine, Captain Duclos, foon after which, we had an account of the arrival in the river of 15 merchantmen, under the convoy of Monfieur Kanou; and, on the zoth, counted 23 fail in the bafon of Quebec. Thafe veffels came in very good time, for the English fleet was foon after them, and, on the night of the 23d, the fires on Point Levi gave us notice of its approach to the Bec. Thefe fignals were confirmed by a Courier, who brought intelligence, that 14 fhips were already come to an anchor at St. Bernebie.

Monfieur de Montcalm, who was then at Quebec, immediately difpatched an exprefs to Monfieur de Vaudreuil with this account; who inftantly repaired to the garrifon, and both Generals made the neceffary difpofitions for a vigorous defence. The Beauport fide of the coaft was fortified all along from the river St. Charles to the falls of Montmorency; a bridge of boats was built over this river, and the head of the bridge defended by a horn-work; an intrenchment was thrown up in the meadow, which was carried on from St. Rock to the bridge; the entrance of the river St. Charles was fecured by a boom, and this boom defended by two hulks with cannon, which were run afhore a little within the chain; feveral bateaux were put upon the stocks, fome of which were to carry a twelve, and others a fourteen pounder: A kind of floating battery

was likewise begun upon, of twelve embra. fures, to carry cannon of twelve, eighteen, and twenty-four pounders, and 90 men; and the command given to Captain Duclos, of the Chezine, who was the inventor of it. Batteries en barbette were erected on the Quay du Palais; and those on the ramparts, and in the lower town, were repaired, completed, and confiderably enlarged. Eight veffels were likewife fitted out as firefhips, which did no execution, owing to the ill management of the Officers who had the direction of them: Fire-ftages were likewise built, but met with as little fuccefs as the fhips. A ftreet was opened in the garden of the Bishop's palace, for the easier communication between the town and ramparts; the paffage that led to the Lower Town was blocked up, and the walls of the houfes pulled down that were adjoining to it. The breaches in the city walls were all filled up; and fuch of them as could not be finished with masonry, for want of time, were secu、 red by a palifade from any fudden attack. The fhips which were not likely to be wanted during the fiege were ordered up the river, as far as Batifcan, and all the feamen taken out, but fuch as were abfolutely neceffary for working them; the reft were employed at the batteries; and all persons who could be of no fervice in the fiege, fuch as Ladies and others, were defired to withdraw from the city: This request, being confi→ dered by most people as an order, was fubmitted to, but not without reluctance.

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