Book 2
1
Therefore etc. Q. E. D. the rectangle A, BC. From this point onward I shall translate thus in cases where Euclid leaves out the word contained (περιεχόμενον). Though the word rectangle is also omitted in the Greek (the neuter article being sufficient to show that the rectangle is meant), it cannot be dispensed with in English. De Morgan advises the use of the expression the rectangle under two lines. This does not seem to me a very good expression, and, if used in a translation from the Greek, it might suggest that ὑπό in τὸ ὑπό meant under, which it does not.
Proposition 2.
2
If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole.
2
For let the straight line AB be cut at random at the point C; I say that the rectangle contained by AB, BC together with the rectangle contained by BA, AC is equal to the square on AB.
2
For let the square ADEB be described on AB [I. 46], and let CF be drawn through C parallel to either AD or BE. [I. 31]
2
Then AE is equal to AF, CE.
2
Now AE is the square on AB;
2
AF is the rectangle contained by BA, AC, for it is contained by DA, AC, and AD is equal to AB; and CE is the rectangle AB, BC, for BE is equal to AB.
2
Therefore the rectangle BA, AC together with the rectangle AB, BC is equal to the square on AB.
2
Therefore etc. Q. E. D.
Proposition 3.
