water AB. So that, if AB be about 34 feet, which is equal to the force of the atmosphere, then CD will be equal to CE; but is AB be double of that, or 68 feet, then CD will be CE; and so on. And hence, by knowing the depth AF, to which the vessel is sunk, we can easily find the point D, to which the water will rise within it at any time. For, let the weight of the atmosphere at that time be equal to that of 34 feet of water; also, let the depth AF be 20 feet, and the length of the tube CE 4 feet; then, putting the height of the internal water DE = x. A G B F root is x = 1414 of a foot, or 17 inches nearly; being the height DE to which the water will rise within the tube. 345. But if the vessel be not equally wide throughout, but of any other shape, as of a bell-like form, such as is used in diving; then the altitudes will not observe the proportion above, but the spaces or bulks only, will respect that proportion, namely, 34+ AB: 34: capacity CKL capacity CHI, if it be common or fresh water; and 33 AB: 33: capacity CKL capacity CHI, if it be sea-water. From which proportion, the height DE may be found, when the nature or shape of the vessel or bell CKL is known. A G B F H D OF THE BAROMETER. 346. THE BAROMETER is an instrument for measuring the pressure of the atmosphere, and elasticy of the air, at any time. It is commonly made of a glass tube, of near 3 feet long, close at one end, and filled with mercury. When the tube is full, by stopping the open end with the finger, then inverting the tube, and immersing that end with the finger into a basin of quicksilver, on removing the finger from the orifice, the quicksilver in the tube will descend into the basin, till what remains in the tube be of the same weight with a column of the atmosphere; which is commonly between 28 and 31 inches of quicksilver; and leaving an entire vacuum in the upper end of the tube above the mercury. For, as the upper end of the tube is quite void of air, there is no pressure downwards but from the column of quicksilver, and therefore that will be an exact balance to the counter pressure of the whole column of atmosphere, acting on the orifice of the tube by the quicksilver in the basin. The upper three inches of the tube, namely, from 28 to 31 inches, have a scale attached to them, divided into inches, tenths, and hundredths, for measuring the length of the column at all times, by observing which division of the scale the top of the quicksilver is opposite to; as it ascends and descends within these limits, according to the state of the atmosphere. So the weight of the quicksilver in the tube, above that in the basin, is at all times equal to the weight or pressure of the column of atmosphere above it, and of the same base with the tube; and hence the weight of it may at all times be computed; being nearly at the rate of half a pound avoirdupois for every inch of quicksilver in the tube, on every square inch of base; or more exactly, it is of a pound on the square inch, for every inch in the altitude of the quicksilver: for the cubic inch of quicksilver weighs just lb., or nearly a pound, in the mean temperature of 55° of heat. And consequently, when the barometer stands at 30 inches, or 2 feet high, which is the medium or standard height, the whole pressure of the atmosphere is equal to 143 pounds, on every square inch of the base. And so in proportion for other heights. OF THE THERMOMETER. 347. THE THERMOMETER is an instrument for measuring the temperature of the air, as to heat and cold. It is found by experience, that all bodies expand by heat, and contract by cold and hence the degrees of expansion become the measure of the degrees of heat. Fluids are more convenient for this purpose, than solids: and quicksilver is now most commonly used for it. A very fine glass tube, having a pretty large hollow ball at the bottom, is filled about half way up with quicksilver: the whole being then heated very hot till the quicksilver rise quite to the top, the top is then hermetically sealed, so as perfectly to exclude all communication with the outward air. Then, in cooling, the quicksilver contracts, and consequently its surface descends in the tube, till it come to a certain point, correspondent to the temperature or heat of the air. And when the weather becomes warmer, the quicksilver expands, and its surface rises in the tube; again contracts and descends when the weather becomes cooler. So that, by placing a scale of any divisions against the side of the tube, it will show the degrees of heat, by the expansion and contraction of the quicksilver in the tube; observing at what division of the scale the top of the quicksilver stands. And the method of preparing the scale, as used in England, is thus:-Bring the thermometer into a temperature of just freezing, by immersing the ball in water just freezing, and or in ice just thawing, and mark the scale where the mer- This division of the scale, is commonly called Fahrenheit's. According to this division, 55 is at the mean temperature of the air in this country; and it is in this temperature, and in an atmosphere which sustains a column of 30 inches of quicksilver in the barometer, that all measures and specific gravities are taken, unless when otherwise mentioned; and in this temperature and pressure, the relative weights, or specific gravities, of air, water, and quicksilver, are as 12 for air, 1000 for water, and 13600 for mercury; and these also are the weights of a cubic foot of each, in avoirdupois ounces, in that state of the barometer 30 10. 100 25 and thermometer. For other states of the thermometer, each of these bodies expands or contracts, according to the following rate, with each degree of heat; viz. OF THE MEASUREMENT OF ALTITUDES BY THE BAROMETER AND THERMOMETER. 348. FROM the principles laid down in the Scholium to prop. 66, concerning the measuring of altitudes by the barometer, and the foregoing descriptions of the barometer and thermometer, we may now collect together the precepts for the practice of such measurements, which are as follow: First, Observe the height of the barometer at the bottom of any height, or depth, intended to be measured; with the temperature of the quicksilver by means of a thermometer attached to the barometer, and also the temperature of the air in the shade by a detached thermometer. Second, Let the same thing be done also at the top of the said height or depth, and at the same time, or as near the same time as may be. And let those altitudes of barometer be reduced to the same temperature, if it be thought necessary, by correcting either the one or the other, that is, augment the height of the mercury in the colder temperature, or diminish that in the warmer, by its part for every degree of difference of the two. Third, Take the difference of the common logarithms of the two heights of the barometer, corrected as above if necessary, cutting off three figures next the right hand for decimals, the rest being fathoms in whole numbers. Fourth, Correct the number last found for the difference of temperature of the air, as follows:- -Take half the sum of the two temperatures, for the mean one; and for every degree which this differs from the temperature 31o, take so many times the part of the fathoms above found, and add them if the mean temperature be above 31o, but subtract them if the mean temperature be below 31°; and the sum or difference will be the true altitude in fathoms; or, being multiplied by 6, it will be the altitude in feet. 349. EXAMPLE 1.-Let the state of the barometers and thermometers be as follows; to find the altitude, viz. 350. EXAMPLE 2.-To find the altitude, when the state of the barometers and thermometers are as follows, viz. OF THE RESISTANCE OF FLUIDS, WITH THEIR FORCES AND ACTION ON BODIES. PROP. LXVIII. 351. If any body move through a fluid at rest, or the fluid move against the body at rest; the force or resistance of the fluid against the body, will be as the square of the velocity and the density of the fluid. That is, R∞ dv2. FOR, the force or resistance is as the quantity of matter or particles struck, and the velocity with which they are struck. But the quantity or number of particles struck, in any time, are as the velocity and the density of the fluid. Therefore the resistance or force of the fluid, is as the density and square of the velocity. 352. Corol. 1. The resistance to any plane, is also more or less, as the plane is greater or less; and therefore the resistance on any plane, is as the area of the plane a, the density of the medium, and the square of the velocity. That is, R a adv. 353. Cordl. 2. If the motion be not perpendicular, but oblique to the plane, or to the face of the body; then the resistance, in the direction of motion, will be diminished in the triplicate ratio of radius to the sine of the angle of inclination of the plane to the direction of motion, or as the cube of radius to the cube of the sine of that angle. So that Radv3s3, putting 1= radius, and s = sine of the angle of inclination CAB. C B For, if AB be the plane, AC the direction of motion, and BC perpendicular to AC; then no more particles meet the plane than what meet the perpendicular BC, and therefore their number is diminished as AB to BC, or as 1 to s. But the force of each particle, striking the plane obliquely in the direction CA, is also diminished as AB to BC, or as 1 to s; therefore the resistance, which is perpendicu lar to the face of the plane, by art. 52, is as 12 to s. But again, this resistance in the direction perpendicular to the face of the planes, is to that in the direction AC, by art. 51, as AB to BC, or as 1 to s. Consequently, on all these accounts, the resistance to the plane when moving perpendicular to its face, is to that when moving obliquely, as l3 to s3, or 1 to s3. That is the resistance in the direction of the motion, is diminished, as 1 to s3, or in the triplicate ratio of radius to the sine of inclination. PROP. LXIX. 354. The real resistance to a plane, by a fluid acting in a direction perpendicular to its face, is equal to the weight of a column of the fluid, whose base is the plane, and altitude equal to that which is due to the velocity of the motion, or through which a heavy body must fall to acquire that velocity. 1 THE resistance to the plane moving through a fluid, is the same as the force of the fluid in motion with the same velocity, on the plane at rest. But the force of the fluid in motion, is equal to the weight or pressure which generates that motion; and this is equal to the weight or pressure of a column of the fluid, whose base is the area of the plane, and its altitude that which is due to the velocity. 355. Corol. 1. If a denote the area of the plane, v the velocity, n the density or specific gravity of the fluid, and g = 16 feet, or 193 inches. Then, the altitude due to the velocity v being ,the whole resistance, or motive force R, will be a Xn x 356. Corol. 2. v2 4g If the direction of motion be not perpendicular to the face of the plane, but oblique to it, in an angle whose sine is s. Then the resistance 357. Corol. 3. Also, if w denote the weight of the body, whose plane face R a is resisted by the absolute force R; then the retarding force f, or will be > W anv3ss 4gw 358. Corol. 4. And if the body be a cylinder, whose face or end is a, and radius r moving in the direction of its axis; because then s=1, and a = pr2, where p = 3·1416; the resisting force R will be pn, and the retarding force 4g |
