Euclid's Elements : Book II :

Geometric algebra
### Definitions

- Definition 1
- Any rectangular parallelogram is said to be contained by the
two straight lines containing the right angle.
- Definition 2
- And in any parallelogrammic area let any one whatever of the
parallelograms about its diameter with the two complements be called a gnomon.

### Propositions

- Proposition 1
- If there are two straight lines, and one of them is
cut into any number of segments whatever, then the
rectangle contained by the two straight lines equals the
sum of the rectangles contained by the uncut straight line and each of the segments.
- Proposition 2
- If a straight line is cut at random, then the sum of the
rectangles contained by the whole and each of the segments equals the square on the whole.
- Proposition 3
- If a straight line is cut at random, then the rectangle contained by the whole and one
of the segments equals the sum of the rectangle contained by the segments and the square
on the aforesaid segment.
- Proposition 4
- If a straight line is cut at random, the square on the whole equals the squares on the
segments plus twice the rectangle contained by the segments.
- Proposition 5
- If a straight line is cut into equal and unequal segments,
then the rectangle contained by the unequal segments of the whole
together with the square on the straight line between the points of
section equals the square on the half.
- Proposition 6
- If a straight line is bisected and a straight line is
added to it in a straight line, then the rectangle contained by the
whole with the added straight line and the added straight line
together with the square on the half equals the square on the straight
line made up of the half and the added straight line.
- Proposition 7
- If a straight line is cut at random, then the sum of the square
on the whole and that on one of the segments equals
twice the rectangle contained by the whole and the said segment plus the
square on the remaining segment.
- Proposition 8
- If a straight line is cut at random, then four times the rectangle contained
by the whole and one of the segments plus the square on the remaining segment
equals the square described on the whole and the aforesaid segment as on one straight line.
- Proposition 9
- If a straight line is cut into equal and unequal segments, then the sum of the
squares on the unequal segments of the whole is double the sum of the square on the
half and the square on the straight line between the points of section.
- Proposition 10
- If a straight line is bisected, and a straight line is added to it in a straight line,
then the square on the whole with the added straight line and the square on the
added straight line both together are double the sum of the square on the half
and the square described on the straight line made up of the half and the added
straight line as on one straight line.
- Proposition 11
- To cut a given straight line so that the rectangle contained by the whole
and one of the segments equals the square on the remaining segment.
- Proposition 12
- In obtuse-angled triangles the square on the side opposite the
obtuse angle is greater than the sum of the squares on the sides containing
the obtuse angle by twice the rectangle contained by one of the sides about the
obtuse angle, namely that on which the perpendicular falls, and the
straight line cut off outside by the perpendicular towards the obtuse angle.
- Proposition 13
- In acute-angled triangles the square on the side opposite the acute
angle is less than the sum of the squares on the sides containing the acute angle
by twice the rectangle contained by one of the sides about the acute angle,
namely that on which the perpendicular falls, and the straight line cut off within by the
perpendicular towards the acute angle.
- Proposition 14
- To construct a square equal to a given rectilinear figure.

Euclid's Elements: Book I <--- Book II ---> Euclid's Elements: Book III