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### Table of contents

- Definitions (2)
- Propositions (14)

- Guide to Book II
- Logical structure of Book II

### 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 acutc angle.

Proposition 14.
  To construct a square equal to a given rectilinear figure.

### Guide to Book II

The subject matter of Book II is usually called "geometric algebra." The first ten propositions of Book II can be easily interpreted in modern algebraic notation. Of course, in doing so the geometric flavor of the propositions is lost. Nonetheless, restating them algebraically can aid in understanding them. The equations are all quadratic equations since the geometry is plane geometry.

II.1. If y = y1 + y2 + ... + yn, then xy = x y1 + x y2 + ... + x yn. This can be stated in a single identity as

x (y1 + y2 + ... + yn) = x y1 + x y2 + ... + x yn.

II.2. If x = y + z, then x2 = xy + xz. This can be stated in various ways in an identity of two variables. For instance,

(y + z)2 = (y + z) y + (y + z) z,
or
x2 = xy + x (x - y).

II.3. If x = y + z, then xy = yz + y2. Equivalent identities are

(y + z)y = yz + y2,
and
xy = y(x - y) + y2.

II.4. If x = y + z, then x2 = y2 + z2 + 2yz. As an identity,

(y + z)2 = y2 + z2 + 2yz.

II.5 and II.6. (y + z) (y - z) + z2 = y2.

II.7. if x = y + z, then x2 + z2 = 2xz + y2. As an identity,

x2 + z2 = 2xz + (x - z)2.

II.8. If x = y + z, then 4xy + z2 = (x + y)2. As an identity,

4xy + (x - y)2 = (x + y)2.

II.9 and II.10. (y + z)2 + (y - z)2 = 2 (y2 + z2).

The remaining four propositions are of a slightly different nature. Proposition II.11 cuts a line into two parts which solves the equation a (a - x) = x2 geometrically. Propositions II.12 and II.13 are recognizable as geometric forms of the law of cosines which is a generalization of I.47. The last propostion II.14 constructs a square equal to a given rectilinear figure thereby completeing the theory of areas begun in Book I.

### Logical structure of Book II

The proofs of the propositions in Book II heavily rely on the propositions in Book I involving right angles and parallel lines, but few others. For instance, the important congruence theorems for triangles, namely I.4, I.8, and I.26, are not invoked even once. This is understandable considering Book II is mostly algebra interpreted in the theory of geometry.

The first ten propositions in Book II were written to be logically independent, but they could have easily been written in logical chains which, perhaps, would have shortened the exposition a little. The remaining four propositions each depend on one of the first ten.

Dependencies within Book II
6 | 11
4 | 12
7 | 13
5 | 14

Elements Introduction - Book I - Book III.

© 1996
D.E.Joyce
Clark University