We give another method of demonstrating this truth, merely for the beauty of the demonstration.

Let AK be the diameter of a semicircle, and also the side of a parallelogram whose width is the radius of the semicircle.

Join the center of the semicircle to either extremity of the parallelogram, as CB, CL. Now conceive the parallelogram to revolve on AK, and it will describe a cylinder; the semicircle will describe a sphere, and the triangle ABC will describe a cone.

R

In AC, take any point, D, and draw DH parallel to AB, and join CO. Then, as CA-AB, CD-DE. In the right angled triangle CDO, we have,

But,

CD2+DO2=CO2 (1)

CRD2=DE2, and CO2=DH2

Substituting these values in equation (1), and we have,

DE2+DO2=DH2 (2)

Multiply every term of this equation by я,

Now, the first term of this equation, is the measure of the surface of a plane circle, whose radius is DE; the second term is the measure of a plane circle, whose radius is DO; and the second member is the measure of the surface of a plane circle, whose radius is DH. Let each of these surfaces be conceived to be of the same extremely minute thickness; then the first term is a section of a cone, the second term is a corresponding section of a sphere, and these two sections are, together, equal to the corresponding section of the cylinder; and this is true for all sections parallel to CR, which compose the cone, the sphere, and the cylinder; therefore, the cone and sphere, together, are equal to the cylinder; but the core described by the triangle ABC, is of the cylinder described by AR (th. 10, b. 7); therefore, the corresponding section of the sphere, is the remaining two-thirds, and the whole sphere is twothirds of the whole cylinder described by the parallelogram AL.

Q. E. D.

ELEMENTARY PRINCIPLES OF PLANE

TRIGONOMETRY.

TRIGONOMETRY in its literal and restricted sense, has for its object, the measure of triangles. When the triangles are on planes, it is plane trigonometry, and when the triangles are on, or conceived to be portions of a sphere, it is spherical trigonometry. In a more enlarged sense, however, this science is the application of the principles of geometry, and numerically connects one part of a magnitude with another, or numerically compares different magnitudes.

As the sides and angles of triangles are quantities of different kinds, they cannot be compared with each other; but the relation may be discovered by means of other complete triangles, to which the triangle under investigation can be compared.

Such other triangles are numerically expressed in Table II, and all of them are conceived to have one common point, the center of a circle, and as all possible angles can be formed by two straight lines drawn from the center of a circle, no angle of a triangle can exist whose measure cannot be found in the table of trigonometrical lines.

The measure of an angle is the arc of a circle, intercepted between the two lines which form the angle-the center of the arc always being at the point where the two lines meet.

The arc is measured by degrees, minutes, and seconds, there being 360 degrees to the whole circle, 60 minutes in one degree, and 60 seconds in one minute. Degrees, minutes, and seconds, are desigThus 27° 14' 21", is read 27 degrees, 14 minutes, and 21 seconds.

nated by

"

All circles contain the same number of degrees, but the greater the radii the greater is the absolute length of a degree; the circumference of a carriage wheel, the circumference of the earth, or the still greater and indefinite circumference of the heavens, have the same number of degrees; yet the same number of degrees in each and every circle is precisely the same angle in amount or

measure.

As triangles do not contain circles, we can not measure triangles by circular arcs; we must measure them by other triangles, that is, by straight lines, drawn in and about a circle. from the center.

Such straight lines are called trigonometrical lines, and take particular names, as described by the following

DEFINITIONS.

1. The sine of an angle, or an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end. Thus, BF is the sine of the arc AB, and also of the arc BDE. BK is the sine of the arc BD, it is also the cosine of the arc AB, and BF, is the cosine of the arc BD.

N. B. The complement of an 'arc is what it wants of 90°; the supplement of an arc is what it what it wants of 180°.

2. The cosine of an arc is the perpendicular distance from the center of the circle to the sine of the arc, or it is the same in magnitude as the sine of the complement of the

arc. Thus, CF, is the cosine of the arc AB; but CF-KB, the sine of BD.

3. The tangent of an arc is a line touching the circle in one extremity of the arc, continued from thence, to meet a line drawn through the center and the other extremity.

Thus, AH is the tangent to the arc AB, and DL is the tangent of the arc DB, or the cotangent of the arc AB.

N. B. The co, is but a contraction of the word complement.

4. The secant of an arc, is a line drawn from the center of the circle to the extremity of its tangent. Thus, CH is the secant of the arc AB, or of its supplement BDE.

5. The cosecant of an arc, is the secant of the complement. Thus, CL, the secant of BD, is the cosecant of AB.

6. The versed sine of an arc is the difference between the cosine and the radius; that is, AF is the versed sine of the arc AB, and DK is the versed sine of the arc BD.

For the sake of brevity these technical terms are contracted thus: for sine AB, we write sin.AB, for cosine AB, we write cos.AB, for tangent AB, we write tan. AB, &c.

From the preceding definitions we deduce the following obvious consequences:

1st, That when the arc AB, becomes so small as to call it nothing, its sine tangent and versed sine are also nothing, and its secant and cosine are each equal to radius.

2d, The sine and versed sine of a quadrant are each equal to the radius; its cosine is zero, and its secant and tangent are infinite. 3d, The chord of an arc is twice the sine of half the arc. Thus the chord BG, is double of the sine BF.

4th, The sine and cosine of any arc form the two sides of a right angled triangle, which has a radius for its hypotenuse. Thus, CF, and FB, are the two sides of the right angled triangle CFB.

Also, the radius and the tangent always form the wo sides of a right angled triangle which has the secant of the arc for its hypotenuse. This we observe from the right angled triangle CAH. To express these relations analytically, we write

From the two equiangular triangles CFB, CAH, we have

The ratios between the various trigonometrical lines are always the same for the same arc, whatever be the length of the radius; and therefore, we may assume radius of any length to suit our convenience; and the preceding equations will be more concise, and more

readily applied, by making radius equal unity. This supposition being made, the preceding becomes

The center of the circle is considered the absolute zero point, and the different directions from this point are designated by the different signs and On the right of C, toward A, is commonly marked plus (+), then the other direction, toward E, is necessarily minus (-). Above AE is called (+), below that line (-).

If we conceive an arc to commence at A, and increase continuously around the whole circle in the direction of ABD, then the following table will show the mutations of the signs.

The chord of 60° and the tangent 45° are each equal to radius; the sine of 30° the versed sine of 60° and the cosine of 60° are each equal to half the radius.

(The first truth is proved in problem 15, book 1).

On C, as radius, describe a quadrant; take AD=45°, AB =60°, and AE-90°, then BE-30°.

Join AB, CB, and draw Bn, perpendicular to CA. Draw Bm, parallel to AC. Make the angle CAH-90°, and draw CDH. In the AABC, the angle ACB=60° by hypothesis; therefore, the sum of the other two angles is (180-60)=120°. But B CB CA, hence the angle CBA the angle CAB, (th. 15 b. 1), and as the sum of the two is 120°, each one must be 60°; therefore, each of the angles of triangle ABC, is 60°

=