At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world.
Bertrand Russell (1883),
Autobiography: 18721914, Allen & Unwin, 1967, p. 36
The following are the definitions, postulates, common notions listed by Euclid in the beginning of his Elements, Book 1. These are Heath's translations from [AT: Euclid, Elements] except that I modified them to make the wording and usage more in line with word usage today. In my modifications I used Heath's extensive notes on the translation in order not to change the meanings involved. But, remember that we can not be sure of the exact meaning intended by Euclid any translation should be considered only as an approximation.
- A point is that which has no parts.
- A curve is length without width.
[Heath translates this as "line," but today we normally use the term "curve" in place of "line".]
- The ends of a curve are points.
[It is not assumed here that the curve has ends (for example, see Definition 15 below); but, if the line does have ends then the ends are points.]
- A (straight) line is a curve that lies symmetrically with the points on itself.
[The commonly quoted Heath's translation says "...lies evenly with the points...", but in his notes he says "we can safely say that the sort of idea which Euclid wished to express was that of a line ... without any irregular or unsymmetrical feature distinguishing one part or side of it from another."]
- A surface is that which has length and width only.
- The boundaries of a surface are curves.
- A plane is a surface that lies symmetrically with the straight lines on itself.
[The comments for Definition 4 apply here as well.]
- An angle is the inclination to one another of two curves in a plane that meet on another and do not lie in a straight line.
[What Euclid meant by the term "inclination" is not clear to me and apparently also to Heath.]
- The angle is called rectilinear when the two curves are straight lines.
[Of course, we (and Euclid in most of the Elements) call these simply "angles".]
- When a straight line intersects another straight line such that the adjacent angles are equal to one another, then the equal angles are called right angles and the lines are called perpendicular straight lines.
[As discussed in the last section of Chapter 4, cones give us examples of spaces where right angles (as defined here) are not always equal to 90 degrees.]
- An obtuse angle is an angle greater than a right angle.
- An acute angle is an angle less than a right angle.
- A boundary of anything is that which contains it.
- A figure is that which is contained by any boundary or boundaries.
- A circle is a plane figure contained by one curve (called the circumference) with a given point lying within the figure such that all the straight lines joining the given point to the circumference are equal to one another.
- The given point of a circle is called the center of the circle.
- A diameter of a circle is any straight line drawn through the center and with its end points on the circumference, and the straight line bisects the circle.
- A semicircle is a figure contained by a diameter and the part of the circumference cut off by it. The center of the semicircle is the same as the center of the circle.
- Polygons are those figures whose boundaries are made of straight lines: triangles being those contained by three, quadrilaterals those contained by four, and multilaterals those contained by more than four straight lines.
- An equilateral triangle is a triangle that has three equal sides, an isosceles triangle is a triangle that has only two of its sides equal, and a scalene triangle is a triangle that has all three sides unequal.
- A right triangle is a triangle that has a right angle, an obtuse triangle is a triangle that has an obtuse angle, and an acute triangle is a triangle that has all of its angles acute.
- A square is a quadrilateral that is equilateral (has all equal sides) and right angled (has all right angles), a rectangle is a quadrilateral that is right angled but not equilateral, a rhombus is a quadrilateral that is equilateral but not right angled, a rhomboid is a quadrilateral that has opposite sides and angles equal to one another but that is neither equilateral nor right angled. Let quadrilaterals other than these be called trapezia.
- Parallel straight lines are straight lines lying in a plane, which do not meet if continued indefinitely in both directions.
- A (unique) straight line may be drawn from any point to any other point.
- Every limited straight line can be extended indefinitely to a (unique) straight line.
- A circle may be drawn with any center and any distance.
- All right angles are equal.
[Note that cones give us examples of spaces in which all right angles are not equal, see Chapter 4. Thus this postulate could be rephrased: "There are no cone points."]
- If a straight line intersecting two straight lines makes the interior angles on the same side less than two right angles, then the two lines (if extended indefinitely) will meet on that side on which the angles are less than two right angles.
[See Chapter 10 for more discussion of this postulate.]
- Things that are equal to the same thing are also equal to one another.
- If equals are added to equals, then the results are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things that coincide with one another are equal to one another.
- The whole is greater than any of its parts.