## Theorems One and Two, with important Definitions and Postulates

Translated by Alex Pearson

Euclid's 23 Definitions for plane geometry: The definitions begin the Elements.

 1 A point is that of which there is no part. 2 A line is a widthless length. 3 A line's ends are points. 4 A straight line is one which lies evenly with the points on itself. 5 A surface is that which has only length and width. 6 The ends of a surface are lines. 7 A plane surface is one which lies evenly with the lines on it. 10 When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called perpendicular to that on which it stands. 15 A circle is a plane figure contained by one line [which is called the circumference], and all the straight lines coming from one point of those lying within the figure and falling [upon the circumference of the circle] are equal to one another. Note: An interpolation is suspected in the brackets. Manuscripts have the words, but the other ancient sources, namely the commentators, do not. Nor does papyrus Herculanensis No. 1061., the oldest source on The Elements. Heath notes (The Thirteen Books 184), "The words were doubtless added in view of the occurrence of the word "circumference" in Deff. 17, 18 immediately following, without any explanation. But no explanation was needed ... Euclid was perfectly justified in employing the word if Deff. 17, 18 and elsewhere, but leaving it undefined as being a word universally understood and not involving in itself any mathematical conception." 23 Parallel lines are whatever straight lines which, being in the same plane and being shot out to infinity in both directions, fall together with one another in neither direction.

Euclid's 5 Postulates: These come right after his 23 definitions. The word for "postulate" means "demand." In other words, these statements are true by insistence, not by proof.

 1 Let it be demanded that from every point to every point a straight line be drawn, 2 and that a limited straight line extend continuously upon a straight line, 3 and that to every point and every distance a circle be drawn, 4 and that all right angles are equal to one another, 5 and 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 the two right angles. Note on Postulate 5 from Dunham, 53-60: This postulate has caused controversy since ancient days. Many mathematicians have tried to argue that this assertion can be proved in a theorem instead of "demanded" as true in a postulate. In the 19th c., famous mathematicians developed an alternate geometry, called non-Euclidean geometry, which rejected this postulate and then demonstrated the logical results. Some mathematicians postulated that more than one line can be drawn parallel to a line, some that no parallels can be drawn. The results were counterintuitive but no less logically correct than Euclid's; however, they appear not to apply to our world. The Hungarian mathematician Johann Bolyai (1802-1860) published a piece on non-Euclidean geometry in 1832. He wrote of his discovery, "Out of nothing, I have created a strange new universe." (Dunham 56) The same discovery was published simultaneously but independently in Russia by Nikolai Lobachevski in 1829. When Bolyai realized that he was not the first to publish non-Euclidean ideas, he wrote, "it seems to be true that many things have, as it were, an epoch in which they are discovered in several places simlutaneously, just as the violets appear on all sides in springtime." (Dunham 57) In 1868, an Italian mathematician, Eugenio Beltrami, proved that non-Euclidean geometry was as valid as Euclidean.

Euclid's 5 Common Notions

 1 Things equal to the same thing are also equal to one another. 2 And if equal things should be added to equal things, the wholes are equal. 3 And if from equal things equal things should be taken away, the remaining things are equal. 4 and things fitted to one another are equal to one another. 5 And the whole is greater than the part. Note on common notion 4: What Euclid apparently means by "fitting" one thing to another is imaginarily picking up, for instance, a triangle and placing it down upon a corresponding triangle to see if all the points correspond to one another. Many mathematicians feel that this "fitting" belongs to the physical sciences and as such is incompatible with the metaphysical, hypothetical science of geometry. Euclid uses it in his fourth theorem.

The First Theorem (my translation)

upon a given, straight, bounded line to construct an equilateral triangle

Let there be the given, straight, bounded line, AB. It is necessary, then, upon the straight line, AB, to construct an equilateral triangle. With center A, then, and with radius AB, let a circle be drawn, the circle BCD. And again, with center B, then, and radius BA, let a circle be drawn, the circle ACE. And from point C, through which the circles cut one another, to the points A, B let there be joined straight lines, the lines CA, CB. And since the point A is the center of the circle CDB, the line AC is equal to the line AB. Again, since the point B is the center of Circle CAE, the line BC is equal to BA. It has been shown that CA is also equal to AB. For indeed, both of the lines CA and CB are equal to the line AB. Things, then, that are equal to the same thing are also equal to one another. And the line CA is surely equal to the line CB. The three lines, then, CA, AB, BC are equal to one another. Equilateral, then, is the triangle ABC. And it has been constructed upon the given, straight, bounded line, AB. [Upon the given, straight, bounded line, then, an equilateral triangle has been constructed.] The very thing it was necessary to make.

Notes: The description of the circles depends upon postulate 3. The joining of the points into lines depends on postulate 1. Definition 15 is the key to the theorem: that the radii of the circle are all equal. Common Notion 1, transitive property of equality, is the coup de grace. One objection to this theorem has been that it takes for granted that the circles do meet.

The Second Theorem (my translation)

upon a given point to place a straight line equal to a given straight line

Let there be the given point A, and then the given straight line BC. It is necessary, then, upon the point A to place a straight line equal to the straight line BC.

Then let there be joined from the point A to the point B the straight line AB, and let there be constructed upon it the equilateral triangle DAB, and at the straight lines DA and DB, let the straight lines AE and BF be extended; and with the center B, and then with the distance BC, let there be drawn the circle CGH, and again with the center D and the distance DG, let a circle be drawn, GKL.

Since, therefore, the point B is the center of [the circle] CGH, BC is equal to BG. Again, since the point D is the center of the circle GKL, the line DL is equal to the line DG, of which the line DA is equal to the line DB. The remainder, then, the line AL is equal to the line BG. But it has also been shown that the line BC is equal to the line BG. Each of the lines AL and BC, therefore, is equal to the line BG. Things which are equal to the same thing are also equal to one another. The line AL, therefore, is equal to the line BC.

To the given point A, therefore, lies the line AL equal to the line BC: the very thing which was necessary to make.

A note from Sir Thomas Heath: "We should insist here upon the restrictions imposed by the first three postulates, which do not allow a circle to be drawn with a compass-carried distance; suppose the compasses to close of themselves the moment they cease to touch the paper. These two propositions (1.2, 1.3) extend the power of construction to what it would have been if all the usual power of the compasses had been assumed; they are mysterious to all who do not see that postulate iii does not ask for every use of the compasses." (The Thirteen Books of the Elements, 246, quoting De Morgan, 1849)

Think about this process as mentally discovering lines instead of mechanically drawing them. The lines and figures are already there, they just need to be described. Without theorems 2 and 3, we could not say, "draw line A equal to line B," nor "cut off segment B from A."

This theorem uses post. 1, of course, to draw lines between points; theorem 1 to make the equilateral triangle; post. 2 to produce the lines from the segments; post. 3 to make the circles; c.n. 3 to establish the equality of remainders; c.n. 1 transitive equality.