MA232A - Euclidean and non-Euclidean Geometry Michaelmas Term 2015 Dr. David R. Wilkins The Axiom System of Book I of Euclid's Elements of Geometry

Postulates, Common Notions and Axioms: two Labelling Schemes

Editions of Euclid's Elements of Geometry that were published up to the latter part of the nineteenth century set out the axiom system as consisting of three postulates and twelve axioms. Later editions, such as those of T.L. Heath and R. Fitzpatrick are based on the edition of Heiberg, which is considered today to be a more authoritative rendering of Euclid, based on a wider selection of manuscripts than were known to earlier authors. In these more modern editions, the axiom system consists of five postulates and five `common notions'. The five extra axioms of the earlier editions are considered to be interpolations into the original text of Euclid.

The French scholar François Peyrard published, in two volumes in 1814 and 1816, a new edition of Euclid's Elements of Geometry based on Greek Vatican Manuscript 190, which had been taken from the Vatican Library on the orders of the French Emperor Napoleon I (see mathforum.org and historyofinformation.com). This manuscript was an important source for the later authoritative critical edition of Heiberg. Heiberg's critical edition in turn supplied the Greek text that was translated into English by T.L. Heath.

The Axiom System of Euclid, Book I, in Editions based on Heiberg

Postulates and Common Notions from Heath's Edition

Let the following be postulated:

Postulates and Common Notions from Fitzpatrick's Edition

The Axiom System of Euclid, Book I, in Editions based on Commandinus

Postulates and Axioms from Keill's Edition

Postulates and Axioms from Todhunter's Edition

Let it be granted,

Postulates and Axioms from Casey's Edition

Let it be granted that—

Playfair's Axiom

For information on Playfair's Axiom, see the discussion of the theory of parallels in Book I of Euclid's Elements of Geometry.

John Casey's annotations on Euclid's Axioms

Postulate 1: A right line may be drawn from any one point to any other point.

When we consider a straight line contained between two fixed points which are its ends, such a portion is called a finite straight line.

Postulate 2: A terminated straight line may be produced to any length in a right line.

Every right line may extend without limit in either direction or in both. It is in these cases called an indefinite line. By this postulate a finite right line may be supposed to be produced, whenever we please, into an indefinite right line.

Postulate 3: A circle may be described from any centre, and with any distance from that centre as radius.

If there be two points A and B, and if with any instruments, such as a ruler and pen, we draw a line from A to B, this will evidently have some irregularities, and also some breadth and thickness. Hence it will not be a geometrical line no matter how nearly it may approach to one. This is the reason that Euclid postulates the drawing of a right line from one point to another. For if it could be accurately done there would be no need for his asking us to let it be granted. Similar observations apply to the other postulates. It is also worthy of remark that Euclid never takes for granted the doing of anything for which a geometrical construction, founded on other problems or on the foregoing postulates, can be given.

Axiom 1 (Common Notion 1): Things which are equal to the same, or to equals, are equal to each other.

Thus, if there be three things, and if the first, and the second, be each equal to the third, we infer by this axiom that the first is equal to the second. This axiom relates to all kinds of magnitude. The same is true of Axioms 2, 3, 4, 5, 6, 7, 9; but 8, 10, 11, 12 are strictly geometrical.

Axiom 2 (Common Notion 2): If equals be added to equals the summs will be equal.

Axiom 3 (Common Notion 3): If equals be taken from equals the remainders will be equal.

Axiom 4: If equals be added to unequals the sums will unequal.

Axiom 5: If equals be taken from unequals the remainders will be unequal.

Axiom 6: The doubles of equal magnitudes are equal.

Axiom 7: The halves of equal magnitudes are equal.

Axiom 8 (Common Notion 4): Magnitudes that can be made to coincide are equal.

The placing of one geometrical magnitude on another, such as a line on a line, a triangle on a triangle, or a circle on a circle, &c., is called superposition. The superposition employed in Geometry is only mental, that is, we conceive one magnitude placed on the other; and then, if we can prove that they coincide, we infer, by the present axiom, that they are equal. Superposition involves the following principle, of which, without explicitly stating it, Euclid makes frequent use:—“Any figure may be transferred from one position to another without change of form or size.”

Axiom 9 (Common Notion 5): The whole is greater than its part.

This axiom is included in the following, which is a fuller statement:—
Axiom 9': The whole is equal to the sum of all of its parts.

(Axiom 9': The whole is equal to the sum of all its parts.)

Axiom 10: Two right lines cannot enclose a space.

This is equivalent to the statement, “If two right lines have two points common to both, they coincide in direction,” that is, they form but one line, and this holds true even when one of the points is at infinity.

Axiom 11 (Postulate 4): All right angles are equal to one another.

This can be proved as follows:—Let there be two right lines AB, AD, and two perpendiculars to them, namely EF, GH, then if AB, CD be made to coincide by superposition, so that the point E will coincide with G; then since a right angle is equal to its supplement, the line EF must coincide with GH. Hence the angle AEF is equal to CGH.

Axiom 12 (Postulate 5): If two right lines (AB, CD) meet a third line (AC), so as to make the sum of the two interior angles (BAC, ACD) on the same side less than two right angles, these lines being produced shall meet at some finite distance.

This axiom is the converse of Prop. XVII., Book I.

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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.