found. Thus, if C be the content, 7 the length, b the breadth, and h the height, we have

The following Examples will sufficiently illustrate the preceding Theory, for our present purpose:

Ex. 1.

If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. (EUCLID, Book 11. Prop. 7.)

Let a and b represent the two parts into which the given line is divided; then a + b is the whole line,

and (a + b)2 + a2 = the squares of the whole line, and of one of the parts; = a2+2ab+b2 + a2

the other part.

=2a2+2ab+b2
=2(a + b) a + b2

= twice the rectangle &c. together with the square of Q.E.D.

Ex. 2. To find the radius of a circle inscribed in a given triangle.

See Euclid's diagram, Book III. Prop. 4; let r be the radius of the inscribed circle, and a, b, c the sides of the triangle respectively opposite to the angles A, B, C. Then (Art. 475)

1

r.a+ r.b+ r.c= whole area of the triangle,

1 2

1 2

r(a+b+c) = ax the perpend' (p) upon a from the opposite angle;

To find p, let the segments into which a is divided by it be a and a-x; then (EUCLID, Book 1. Prop. 47)

(a+b+c) (a + c − b) (a + b −c) (b + c − a);

Ex. 3. To find the area of the square inscribed in a given circle; and also of the square circumscribed about a given circle.

(1). Let r be the radius of the circle; then (see EUCLID, Book IV. Prop. 6.)

g2 + p2 = AD2,

(2). Again, r2 + go2 + p2 + p2 or 4r2 = circumscribed square,

Ex. 4. To find the area of the equilateral and equiangular hexagon inscribed in a given circle.

Let r be the radius of the given circle, then (EUCLID, Book IV. Prop. 15.) also r = the side of the inscribed hexagon; and the area of the hexagon = the sum of the areas of the six equal equilateral triangles of which it is composed,

Ex. 5. The depth of water in a cistern (whose form is a rectangular parallelopiped) is h feet, and the base contains a square feet. Find (1) the number of cubic feet of water; and (2) the depth of the same quantity of water in another cistern whose base contains b square inches.

It is not necessary here to multiply Examples, because the subject is now of sufficient importance to form a separate treatise, to which, in the regular course of reading, the student's attention will be next directed. Dr. Hymers' Treatise on Conic Sections and the Application of Algebra to Geometry is the book in most general use for teaching Algebraic Geometry.

APPENDIX.

EQUATIONS.

In the infinite variety of equations which ingenious persons may put together, it is not to be supposed that any general Rules can be laid down for every operation necessary to their solution. There are, however, peculiar artifices of more frequent occurrence than others, which shall be exhibited in the following Examples.