The First Six Books of the Elements of Euclid

by Euclid; John Casey

Now the sum of the three last rectangles, viz. 30, 24, 18, is 72. Hence the rectangle A.BC = A.BD + A.DE + A.EC.

The Second Book is occupied with the relations between the segments of a line divided in various ways. All these can be proved in the most simple manner by Algebraic Multiplication. We recommend the student to make himself acquainted with the proofs by this method as well as with those of Euclid. He will thus better understand the meaning of each Proposition.

Cor. 1.—The rectangle contained by a line and the difference of two others is equal to the difference of the rectangles contained by the line and each of the others.

Cor. 2.—The area of a triangle is equal to half the rectangle contained by its base and perpendicular.

Dem.—From the vertex C let fall the perpendicular CD. Draw EF parallel to AB, and AE, BF each parallel to CD. Then AF is the rectangle contained by AB and BF; but BF is equal to CD. Hence AF = AB.CD; but [I. xli.] the triangle ABC is = half the parallelogram AF. Therefore the triangle ABC is =

AB.CD.

PROP. II.—Theorem.

If a line (AB) be divided into any two parts (at C), the square on the whole line is equal to the sum of the rectangles contained by the whole and each of the segments (AC, CB).