Euclid
Math 390 Lecture 6
Background
Euclid of Alexandria
- Active around 300 BCE and following
- One of the first scholars active at the Museum and Library of Alexandria
- Known for The Elements; oldest complete copy dated to 888
The Elements
- Books I-VI: geometry (magnitude)
- Books VII-IX: Number theory
- Book X: commensurability and incommensurability
- Book XI: 3D geometry
- Book XII: method of exhaustion
- Book XIII: five regular polyhedra
Why study The Elements?
- Lawful regularity of Creation ensures their continued (though contingent) truth
- Heavily influenced mathematical development
- Model of rigorous thinking
In the course of my law-reading I constantly came upon the word demonstrate. I thought, at first, that I understood its meaning, but soon became satisfied that I did not…. I consulted Webster’s Dictionary. That told of “certain proof,” “proof beyond the possibility of doubt;” but I could form no idea what sort of proof that was. I thought a great many things were proved beyond a possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood “demonstration” to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means;” and I left my situation in Springfield, went home to my father’s house, and staid there till I could give any propositions in the six books of Euclid at sight. I then found out what “demonstrate” means, and went back to my law studies.
Book I
Structure of Book I
- Postulates and Common Notions
- Definitions, theorems, proofs—no motivation or exposition
- Used propositional logic rather than Aristotle’s syllogisms
- Goal: prove the Pythagorean Theorem
Postulates and Common Notions (Axioms)
Postulates
- To draw a straight line from any point to any point.
- To produce a finite straight line continuously in any direction.
- To describe a circle with any center and distance.
- That all right angles are equal to one another.
- That, if a straight line intersecting 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 the angles are less than two right angles.
Common Notions
- Things which are equal to the same thing are also equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal to one another.
- The whole is greater than the part.
The Pythagorean Theorem
Theorem 1 (Pythagorean Theorem (I-47)) In right-angled triangles the square on the hypotenuse is equal to the sum of the squares on the legs.
Remark. What does it mean for two plane figures to be “equal”? Euclid doesn’t say, but he apparently meant area.
Book II: Geometric Algebra
Context
- Subject of debate
- Propositions used infrequently elsewhere in the Elements
- Representation of algebraic concepts through geometric figures, e.g., squares of side length \(a\) represent \(a^2\), etc.
- Medieval Islamic mathematicians applied these results to solve quadratics
- A definition to start: any rectangle is said to be contained by the two straight lines forming the right angle.
Example: Prop II-1
Proposition 1 If there are two straight lines, and one of them is cut into any number of segments whatsoever, the rectangle contained by the two straight lines is equal to the sum of the rectangles contained by the uncut straight line and each of the segments.
Example: Prop II-4
Proposition 2 If a straight line is cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
Example: Prop II-5
Proposition 3 If a straight line is cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.
Circles
- Circle: The most perfect of plane figures
- Sphere: The most perfect of the solid figures
- Important to the Greek understanding of astronomy
- Book IV: construction of polygons inscribed in/circumscribed about the circle
- Concludes with the construction of the regular pentagon, hexagon, and 15-gon in a circle.
Number Theory
Magnitude and Number
Recall Aristotle:
- Magnitude: infinitely divisible (e.g., lines, surfaces, time)
- Number: discrete, composed of indivisible elements
- Book VII: first book of the Elements on number theory
Euclidean Algorithm
If \(a > b\) are two numbers, we can find their greatest common “measure” (divisor):
- Subtract \(b\) from \(a\) as many times as possible
- If there is a remainder \(c\), subtract \(c\) from \(b\) as many times as possible.
- Continue in this manner until you come to a number \(m\) which “measures” (divides) the one before (Prop VII-2) or the unit (Prop VII-1)
- Euclid proves this is the greatest common measure of \(a\) and \(b\).
Proportionality
Definition 1 Four numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
Attempted to extend to magnitude:
- Ratios of magnitude must be of the same kind (line to line, surface to surface, etc)
- Modern symbolism: if \(a: b = c: d\), then given any positive integers \(m,n\), whenever \(ma > nb\), then \(mc > nd\), and whenever \(ma = nb\), also \(mc = nd\), and whenever \(ma < nb\), also \(mc < nd\).
Definitions from Book VII
- A unit is that by virtue of which each of the things that exist is called one.
- A number is a multitude of units
- A prime number is that which is measured by the unit alone.
Proposition 4 (VII-31) Any composite number is measured by some prime number.
Proposition 5 (VII-31) Any number either is prime or is measured by some prime number.
Proposition IX-20
Prime numbers are more than any assigned multitude of prime numbers.
Proof. Let \(A, B, C\) be primes and \(N = ABC+1\). If \(N\) is prime, then a prime other than those given has been found. If \(N\) is composite, then it is divisible by a prime \(p\). Then \(p\) is distinct from \(A,B,C\) as none of these divides \(N\). Thus a new prime \(p\) has been found.
Irrational Magnitudes: Book X
- Longest, most important, best organized
- One motivation: characterize the edge lengths of regular polyhedra
- Goal: compare the edges of the icosahedron and the dodecahedron to the diameter of sphere in which they are inscribed (Book XIII)
- Led to a more elaborate classification presented in Book X
- Proposition X-9: Generalizes the incommensurability of the diagonal of a square.
Solid Geometry
Solid Geometry
- Book XI-XIII: Solid Geometry
- 3D analogues of 2D results from Books I and VI.
- Definition 14: When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure to be comprehended is a
- sphere.
- Book XII: use of a limiting process, known as the method of exhaustion
- Developed by Eudoxus
Proposition XII-2
Circles are to one another as the squares on the diameters.
Idea: “Exhaust” the area of the circle by inscribing it in polygons of increasingly many sides.
Book XIII: Five Regular Polyhedra
The Six Platonic Solids: xkcd 2781
Plato made the solids, and five were gifted to the mathematicians. But in secret Plato forged a sixth solid to rule over all the others.
