Appendix A

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: 1872-1914*, 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 more in line with word usage today. In my modifications I used Heath's extensive notes on the translation in order to not 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.

*Definitions*

- A
is that which has no parts.*point* - A
is length without width.*line*

[We normally today use the term "" in place of "line".]*curve* - The ends of a line are points.

[It is not assume here that the line*has*ends (for example, see Definition 15 below); but, if the line does have ends then the ends are points.] - A
is a line that lies symmetrically with the points on itself.*straight line*

[The commonly quoted Heath's translation say "...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 on part or side of it from another."] - A
is that which has length and width only.*surface* - The boundaries of a surface are lines.
- A
is a surface which lies symmetrically with the straight lines on itself.*plane*

[The comments for Definition 4 apply here as well.] - A (plane)
is the inclination to one another of two lines in a plane which meet on another and do not lie in a straight line.*angle*

[What Euclid meant by the term "inclination" is not clear to me and apparently also to Heath.] - The angle is called
when the two lines are straight.*rectilinear*

[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
and the lines are called*right**angles*.*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
is an angle greater than an right angle.*obtuse angle* - An
is an angle less than a right angle.*acute angle* - A
of anything is that which contains it.*boundary* - A
is that which is contained by any boundary or boundaries.*figure* - A
is a plane figure contained by one line (called the*circle*) 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.*circumference* - The given point of a circle is called the
of the circle.*center* - A
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.*diameter* - A
is a figure contained by a diameter the part of the circumference cut off by it. The*semicircle*of the semicircle is the same as the center of the circle.*center* -
are those figures whose boundaries are made of straight lines:*Polygons*being those contained by three,*triangles*those contained by four, and*quadrilaterals*those contained by more than four straight lines.*multilateral* - An
is a triangle that has three equal sides, an*equilateral triangle*is a triangle which has only two of its sides equal, and a*isosceles triangle*is a triangle which has all three sides unequal.*scalene triangle* - A
is a triangle which has a right angle, an*right triangle*is a triangle which an obtuse angle, and an*obtuse triangle*is a triangle that has all of its angles acute.*acute triangle* - A
is a quadrilateral which is equilateral (has all equal sides) and right angled (has all right angles), a*square*is a quadrilateral is right angled but not equilateral, a*rectangle*is a quadrilateral which is equilateral but not right angled, a*rhombus*is a quadrilateral which has opposite sides and angles equal to one another but which is neither equilateral nor right angled. Let quadrilaterals other than these be called*rhomboid*.*trapezia* -
straight lines are straight lines lying in a plane which do not meet if continued indefinitely in both directions.*Parallel**Postulates* - A (unique) straight line which 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. 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 are the angles less than two right angles.

[See Chapter 10 for more discussion of this postulate.]*Common Notions* - Things which 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 which coincide with one another are equal to one another.
- The whole is greater any of its parts.