MA232A  Euclidean and nonEuclidean Geometry

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.
Let the following be postulated:
Heiberg  Commandinus  

Postulate 1  To draw a straight line from any point to any point.  Post. 1 
Postulate 2  To produce a finite straight line continuously in a straight line.  Post. 2 
Postulate 3  To describe a circle with any centre and distance.  Post. 3 
Postulate 4  That all right angles are equal to one another.  Ax. 11 
Postulate 5  That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.  Ax. 12 
Common Notion 1  Things which are equal to the same thing are also equal to one another.  CN. 1 
Common Notion 2  If equals be added to equals, the wholes are equal.  CN. 2 
Common Notion 3  If equals be subtracted from equals, the remainders are equal.  CN. 3 
Common Notion 4  Things which coincide with one another are equal to one another.  Ax. 8 
Common Notion 5  The whole is greater than the part.  Ax. 9 
Heiberg  Commandinus  

Postulate 1  Let it have been postulated to draw a straightline from any point to any point.  Post. 1 
Postulate 2  And to produce a finite straightline continuously in a straightline.  Post. 2 
Postulate 3  And to draw a circle with any center and radius.  Post. 3 
Postulate 4  And that all rightangles are equal to one another.  Ax. 11 
Postulate 5  And that if a straightline falling across two (other) straight lines makes internal angles on the same side (of itself whose sum is) less than two rightangles, then the two (other) straightlines, being produced to infinity, meet on that side (of the original straightline) that the (sum of the internal angles) is less than two rightangles (and do not meet on the other side).  Ax. 12 
Common Notion 1  Things equal to the same thing are also equal to one another.  CN. 1 
Common Notion 2  And if equal things are added to equal things then the wholes are equal.  CN. 2 
Common Notion 3  And if equal things are subtracted from equal things then the remainders are equal.  CN. 3 
Common Notion 4  And things coinciding with one another are equal to one another.  Ax. 8 
Common Notion 5  The whole [is] greater than the part.  Ax. 9 
Commandinus  Heiberg  

Postulate 1  Grant that a Right Line may be drawn from any one Point to another.  Post. 1 
Postulate 2  That a finite Right Line may be continued directly forwards.  Post. 2 
Postulate 3  And that a circle may be described about any Centre with any Distance.  Post. 3 
Axiom 1  Things equal to one and the same Thing, are equal to one another.  CN. 1 
Axiom 2  If to equal Things are added equal Things, the Wholes will be equal.  CN. 2 
Axiom 3  If from equal Things equal Things be taken away, the Remainders will be equal.  CN. 3 
Axiom 4  If equal Things be added to unequal Things, the Wholes will be unequal.  — 
Axiom 5  If equal Things be taken from unequal Things, the Remainders will be unequal.  — 
Axiom 6  Things which are double to one and the same Thing, are equal between themselves.  — 
Axiom 7  Things which are half one and the same Thing, are equal between themselves.  — 
Axiom 8  Things which mutually agree together, are equal to one another.  CN. 4 
Axiom 9  The Whole is greater than its Part.  CN. 5 
Axiom 10  Two Right Lines do not contain a Space.  — 
Axiom 11  All Right Angles are equals between themselves.  Post. 4 
Axiom 12  If a Right Line, falling upon two other Right Lines, makes the inward Angles on the same Side thereof, both together, less than two Right Angles, those two Right Angles, infinitely produc'd, will meet each other on that Side, where the Angles are less than Right ones.  Post. 5 
Let it be granted,
Commandinus  Heiberg  

Postulate 1  That a straight line may be drawn from any one point to any other point:  Post. 1 
Postulate 2  That a terminated straight line may be produced to any length in a straight line:  Post. 2 
Postulate 3  And that a circle may be described from any centre, at any distance from that centre.  Post. 3 
Axiom 1  Things which are equal to the same thing are equal to one another.  CN. 1 
Axiom 2  If equals be added to equals the wholes are equal.  CN. 2 
Axiom 3  If equals be taken from equals the remainders are equal.  CN. 3 
Axiom 4  If equals be added to unequals the wholes are unequal.  — 
Axiom 5  If equals be taken from unequals the remainders are unequal.  — 
Axiom 6  Things which are double of the same thing are equal to one another.  — 
Axiom 7  Things which are halves of the same thing are equal to one another.  — 
Axiom 8  Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.  CN. 4 
Axiom 9  The whole is greater than its part.  CN. 5 
Axiom 10  Two straight lines cannot enclose a space.  — 
Axiom 11  All right angles are equal to one another.  Post. 4 
Axiom 12  If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.  Post. 5 
Let it be granted that—
Commandinus  Heiberg  

Postulate 1  A right line may be drawn from any one point to any other point.  Post. 1 
Postulate 2  A terminated straight line may be produced to any length in a right line.  Post. 2 
Postulate 3  A circle may be described from any centre, and with any distance from that centre as radius.  Post. 3 
Axiom 1  Things which are equal to the same, or to equals, are equal to each other.  CN. 1 
Axiom 2  If equals be added to equals the summs will be equal.  CN. 2 
Axiom 3  If equals be taken from equals the remainders will be equal.  CN. 3 
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  Magnitudes that can be made to coincide are equal.  CN. 4 
Axiom 9  The whole is greater than its part.  CN. 5 
Axiom 10  Two right lines cannot enclose a space.  — 
Axiom 11  All right angles are equal to one another.  Post. 4 
Axiom 12  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.  Post. 5 
For information on Playfair's Axiom, see the discussion of the theory of parallels in Book I of Euclid's Elements of Geometry.
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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.