Appendix A

Postulates, and

Common Notions

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 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.

**Definitions**

*A***point**is that which has no parts.*A*[Heath translates this as "line," but today we normally use the term "**curve**is length without width.

" in place of "line".]*curve**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*[The comments for Definition 4 apply here as well.]**plane**is a surface that lies symmetrically with the straight lines on itself.

*An*[What Euclid meant by the term "inclination" is not clear to me and apparently also to Heath.]**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.

*The angle is called*[Of course, we (and Euclid in most of the**rectilinear**when the two curves are straight lines.

*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.**Postulates***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.]

**Common Notions***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.*