of the mercury in the colder temperature, or diminish that in the warmer, by its o 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 31°, but subtract them if the mean temperature be below 31o; 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, Rx 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. Corol. 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 Rx adv3s3, putting 1 = radius, and s = sine of the angle of inclination CAB.

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 dimin

B

ished 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 s2. 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 13 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 fluïd 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.

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,

v2 4g

356. Corol. 2. 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 anv3s3 4gw

w

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 pnr, and the retarding force

4g

359. Corol. 5. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction of motion. But were its face an elliptic section, or a conical surface, or any other figure every where equally inclined to the axis, or direction of motion, the sine of inclination being s: then, the number of particles of the fluid striking the face being still the same, but the force of each, opposed to the direction of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resistpnriv22 ing force R will be 4g

PROP. LXX.

360. The resistance to a sphere moving through a fluid, is but half the resist ance to its great circle, or to the end of a cylinder of the same diameter, moving with the same velocity.

A

H

F

LET AFEB be half the sphere, moving in the direction CEG. Describe the paraboloid AIEKB on the same base. Let any particle of the medium meet the semi-circle in F, to which draw the tangent FG, the radius FC, and the ordinate FIH. Then the force of any particle on the surface at F, is to its force on the base at H, as the square of the sine of the angle G, or its equal the angle FCH, to the square of radius, that is, as HF to CF. Therefore the force of all the particles, or the whole fluid, on the whole surface, is to its force on the circle of the base, as all the HF to as many times CF". But CF is = CA AC. CB, and HF' = AH. HB by the nature of the circle; also, AH. HB: AC. CB :: HI: CE by the nature of the parabola; consequently the force on the spherical surface, is to the force on its circular base, as all the HI's to as many CE's, that is, as the content of the paraboloid to the content of its circumscribed cylinder, as 1 to 2.

361. Corol. Hence, the resistance to the sphere is R =

B

being the

half of that of a cylinder of the same diameter. For example, a 9lb iron ball, whose diameter is 4 inches, when moving through the air with a velocity of 1600 feet per second, would meet a resistance which is equal to a weight of 132 lbs., independent of the pressure of the atmosphere, for want of the counterpoise behind the ball.

ANALYTICAL

PLANE TRIGONOMETRY.

CHAPTER I.

INTRODUCTION AND DEFINITIONS.

PLANE TRIGONOMETRY, as the name imports, was originally employed solely in determining, from certain data, the sides and angles of plane triangles. In modern analysis, however, its objects have been much extended, and the formulæ of this branch of Mathematics are extensively employed as instruments of calculation in almost every department of scientific investigation. From this circumstance, some writers wishing to change its designation to one which might more fully express its nature and applications, have proposed to term it the Arithmetic of Sines, others, the Calculus of Angular Functions, but the original appellation is still retained by the great majority of authors upon these subjects.

In treating of angular magnitude, we have hitherto confined ourselves to the consideration of angles less than two right angles; but in trigonometry it is frequently necessary to introduce angles which are greater than two, than three, or even than four right angles. We may take the following method of illustrating the generation of angular magnitude.

Let Aa be a fixed straight line, and let a line CP be supposed to revolve round the point C in Aa, and to assume in succession the different positions CP1, CP 2,

When CP coincides with CA, there is no angle contained between CP and CA, or the angle CAP is 0.

When CP begins to revolve round C, and comes into the position CP1, it forms with CA an angle PCA less than a right angle.

1P2

P3

Pi

a

P6

When CP has performed one-fourth part of an entire revolution, and has thus reached the position CP2, where CP2 is perpendicular to CA, it forms with CA the angle PCA, which is a right angle.

As CP continues its revolutions, it will assume the position CP3, forming with CA the angle P3 CA, greater than one, and less than two right angles.

When CP coincides with Ca, it has performed one

half of an entire revolution, and forms with CA the

angle aCA, equal to two right angles.

CP having passed Ca, will assume the position CP4, forming with CA the angle PCA, greater than two, and less than three, right angles.

The dotted space indicates the angle which we are considering.

A

When CP has performed three-fourths of an entire revolution, it assumes the position CP5, where CP, is perpendicular to Aa, forming with CA the angle P¿CA, equal to three right angles.

Passing beyond CP5, the revolving line assumes the position CP6, forming with CA the angle P. CA, greater than three, and less than four, right angles.

Finally, when the line CP has completed an entire

revolution, it will return to its original position CA, having formed with CA an angle equal to four right angles.

If we suppose the revolution to recommence, it is manifest that CP may be conceived to form with CA angles greater than four, than five, or than any given number of right angles.

It is convenient in trigonometrical investigations, to draw two straight lines at right angles to each other,

and from their point of intersection to describe a circle, with any radius cutting these lines in any points A, B, a, b.

The circumference of the circle will thus be divided into four equal arcs AB, Ba, ab, bA, each of which, be

ing a fourth part of the whole circumference, is called

a quadrant, and subtends a right angle at the centre of the circle.

B

AB is called the first quadrant, Ba the second quadrant, ab the third quadrant, and bA the fourth quadrant.

If each of these right angles be divided by straight lines CP1, CP2, 2, ........... into 90 equal angles, the whole circumference will be divided into a correspond ing number of equal parts, each of which is called a degree. The whole circumference will thus contain 360 degrees, and each quadrant will contain 90 degrees.

The angles themselves, and the arcs which subtend them, are called degrees indifferently.

Ps

Pa

Pi

A

Angles are usually designated by the number of degrees which they contain; thus, a right angle is called an angle of 90 degrees; two right angles, an angle of 180 degrees, &c.

If each degree be divided into 60 equal parts, each of these smaller angles is called a minute.

If each minute be divided into 60 equal parts, each of these smaller angles is called a second.

Thus, four right angles, or the entire circumference of a circle, contains 360 degrees; 360 X 60, or 21,600 minutes; 360 × 60 × 60, or 1,296,000 seconds.

Degrees are expressed in writing by placing a small cypher immediately above the number to the right; thus 90°, 45°, 63°, signify 90 degrees, 45 degrees, 63 degrees, &c.

Minutes are expressed by placing one accent in the same manner above the number, and seconds by placing two accents: thus, 35′, 40′, &c. signify 35 minutes, 40 minutes, &c.; and 35", 40′′, signify 35 seconds, 40 seconds, &c.

Any lower subdivision of a degree is usually expressed in decimal parts of a second.