Euclid’s ELEMENTS

Reference: Mathematical proof

Download: The First Six Books of the Elements of Euclid

Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek “axios” meaning “something worthy”), and used these to prove theorems using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

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EUCLID’S AXIOMS:

I.      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 ii., iii., iv., v., vi., vii., ix.; but viii., x., xi., xii., are strictly geometrical.

II.    If equals be added to equals the sums will be equal.

III.  If equals be taken from equals the remainders will be equal.

IV.   If equals be added to unequals the sums will be unequal.

V.     If equals be taken from unequals the remainders will be unequal.

VI.   The doubles of equal magnitudes are equal.

VII.   The halves of equal magnitudes are equal.

VIII.  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.”

IX.   The whole is greater than its part.

This axiom is included in the following, which is a fuller statement:—

IX’.  The whole is equal to the sum of all its parts.

X.    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.

XI.  All right angles are equal to one another.

This can be proven by superposition.

XII.   If two right lines meet a third line, so as to make the sum of the two interior angles on the same side less than two right angles, these lines being produced shall meet at some finite distance.

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NOTE 1: Some of the definitions that I like, though different from Euclid’s, are as follows:

  1. A point is a location in space. It does not extend in any direction, therefore, it is said to have no dimensions.
  2. A line is a path traced in space by a moving point. If a point moves without changing its direction it will describe a straight line. A straight line extends in one direction, therefore, it is said to have one dimension.  If the moving point continually changes its direction it will describe a curve; hence it follows that only one straight line can be drawn between two points.
  3. A surface is a path traced in space by a line that is moving in a direction that goes across the line.  If a straight line moves without changing its direction it will describe a plane. A surface extends in two directions, therefore, it is said to have two dimensions.
  4. An angle is the difference between the directions represented by two straight lines. For example the difference (angle) between the directions east and north is 90 degrees. The difference (angle) between two parallel lines (same direction) is zero.

NOTE 2: The perception of “straight” is always in the direction in which light  propagates, regardless of any curvature.

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