Greece. Chapter 5: Euclid of Alexandria

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1 Greece Chapter 5: Euclid of Alexandria

2 The Library at Alexandria What do we know about it? Well, a little history

3 Alexander the Great In about 352 BC, the Macedonian King Philip II began to unify the numerous Greek city states into one kingdom. After his death, his son, Alexander (the Great) continued the conquest of everything between Macedonia and India.

4 Alexander the Great s Empire

5 Alexandria Alexander planned and built the city of Alexandria in Egypt, on the west end of the Nile river delta. Although Alexander died before the city was complete, it remained the capital of Egypt for nearly a thousand years. Ptolemy, one of Alexander s generals, took over the Egyptian part of his empire. A son, Ptolemy II, built a library and museum, and Ptolemy III populated it with books (scrolls, really).

6 Euclid The Library of Alexandria, founded about 300 BC, became the center of learning. It had reading rooms, lecture rooms, meeting rooms, a dining hall, and gardens to walk in (sounds like a modern day college). It is to this library that Euclid came to study and teach.

7 Euclid s Elements What do we know about the source of our current version of the Elements? Original (lost) Commentary and copy by Theon of Alexandria ( CE) Copy at Oxford (888 CE) 900 CE, copy from version earlier than Theon s, in Vatican library 1880 s, comparison by J. L. Heiberg 1908, Thomas Heath, pretty much the standard today.

8 Euclid s Elements What can we remember about the content of the Elements? Geometry, plane and solid Ratios of numbers and of magnitudes Geometric algebra Number theory Polyhedra

9 Euclid s Elements Book I: Basic plane geometry Book II: Geometric algebra Book III: Circles Book IV: Inscribing and circumscribing figures Book V: Extending Eudoxus ideas of ratio Book VI: Similarity of figures Books VII IX: Number theory Book X: Incommensurable magnitudes Books XI XIII: Solid (3 dimensional) geometry

10 Euclid s Elements 13 books, 465 propositions in plane and solid geometry and number theory. Few if any results are original to Euclid; it is likely a compendium of already known results. In fact it has been suggested that the first books of Euclid s Elements may have been taken from the lost Elements of Hippocrates of Chios.

11 Euclid s Elements What is important is the logical structure of the books. He gave us an axiomatic development of geometry: 23 definitions 5 postulates 5 common notions

12 Euclid s Elements Most of each book consisted of propositions which were proved using only the definitions, common notions, and postulates, as well as any propositions previously proved. Thus Proposition I.3 may be proved using only the common notions, postulates, definitions, and Propositions I.1 and I.2.

13 Euclid s Elements Some of Euclid s definitions: A point is that which has no part (1). A line is breadthless length (2). A straight line is a line which lies evenly with the points on itself (4). 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 a perpendicular to that on which it stands (10). A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another (15).

14 Euclid s Elements Euclid s five postulates: To draw a straight line from any point to any point To produce a finite straight line continuously in a straight line. To describe a circle with any center and distance. That all right angles are equal to one another. That, if a straight line falling on two straight lines makes 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.

15 Euclid s Elements Euclid s Common Notions: Things which are equal the same thing are also equal to one another If equals be added to equals, the wholes are equal If equals be 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.

16 Euclid s Elements Proposition 1: On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line. It is required to construct an equilateral triangle on the straight line AB. Describe the circle BCD with center A and radius AB (Postulate 3). Again describe the circle ACE with center B and radius BA (Postulate 3). Join the straight lines CA and CB from the point C at which the circles cut one another to the points A and B (Postulate 1). C D A B E

17 Euclid s Elements C D A B E Now, since the point A is the center of the circle CDB, therefore AC equals AB (Definition 15). Again, since the point B is the center of the circle CAE, therefore BC equals BA (Definition 15). But AC was proved equal to AB, therefore each of the straight lines AC and BC equals AB. And things which equal the same thing also equal one another, therefore AC also equals BC (Common Notion 1). Therefore the three straight lines AC, AB, and BC equal one another. Therefore the triangle ABC is equilateral, and it has been constructed on the given finite straight line AB. Being what it was required to do.

18 Euclid s Elements Some problems: Definitions: The first few are vague and intuitive (that which has no part? breadthless length?). Later definitions sometimes leave out parts (what does it mean for an angle to be greater than another?) Many he never uses or even refers to later in the book. The fix: some terms are defined, while some remain undefined. Gaps are filled in definitions.

19 Euclid s Elements More Problems Proofs: Use unstated assumptions. For example, in the proof of Proposition 1, how do we know that the point C exists at the intersection of two circles? What does it mean for one point to be between two other points on a line? Diagrams can make these ideas clear and convincing, but it does not meet modern standards of rigor. The fix: make all unstated assumptions explicit and prove them first if possible. Add postulates if necessary.

20 Aside: Fixing Euclid s Elements In the late 1800's David Hilbert developed a set of 20 axioms that made explicit all the assumptions needed to complete Euclid s program. Like Euclid s axioms, Hilbert s were synthetic, that is, they did not depend on properties of the real numbers made explicit though coordinates (as in Descartes analytic geometry). In 1932, Birkhoff developed a system of axioms that made explicit use of real numbers (e.g. made lines essentially equivalent to realnumber lines). In the 1960's the School Mathematics Study Group (SMSG) developed set of axioms somewhat like Birkhoff s, but especially suited to high school study. They differed in that they were not a minimal list of axioms but included some axioms that could be proved from the others in order to avoid more tedious technical proofs.

21 Other Axiom Systems for Geometry Developer Axiom Set Comments Hilbert 20 Axioms Fixed Euclid Does not explicitly depend on Real Numbers, Set Theory, etc. Many Axioms Detailed Technical Proofs of the Obvious Implicitly depends of the real numbers and Set Theory Birkhoff MacLane Only 4 Axioms Incorporates Real Numbers and Measurement More accessible 14 Axioms including continuity axiom Only Euclidean (fourth axiom is about similar triangles) SMSG 22 Axioms Designed for High School Some unnecessary axioms Geometry UCSMP Lots of Axioms Redundancy Uses Reflection Venema (a current text) 12 Axioms Explicitly uses real numbers and set theory. Probably closest to Birkhoff s.

22 1862 (Königsberg) 1943 (Göttngen) Grundlagen der Geometrie (1899) 23 Unsolved Problems (1900) Modern Axiomatics and Proof Theory Wir müssen wissen. Wir werden wissen. David Hilbert

23 Is it just me, or do these two look alike?

24 OK, Back to the Elements As an overview of Book I, let s look at how theorems build to the proof of the Pythagorean Theorem in Proposition 47.

25 Moving Toward I.47 I.4 Prop. I.4 If two triangles have two sides of one triangle equal to two sides of the other triangle plus the angle between the sides that are equal in each triangle is the same, then the two triangles are congruent

26 Moving Toward I.47 I.14 Prop. I.14 Two adjacent right angles make a straight line. Definition 10 asserted the converse, that a perpendicular erected on a straight line makes two right angles.

27 Moving Toward I.47 I.41 Prop. I.41 The area of a triangle is one half the area of a parallelogram with the same base and height.

28 Moving Toward I.47 I.31 Prop. I.31 Given a line and a point not on the line, a line through the point can be constructed parallel to the first line.

29 Moving Toward I.47 I.46 Prop. I.46 Given a straight line, a square can be constructed with the line as one side.

30 And Now: Proposition I.47 In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

31 Proof of Proposition I.47 Draw a line parallel to the sides of the largest square, from the right angle vertex, A, to the far side of the square subtending it, L. Connect points F & C, and A & D, making FBC and ΔABD

32 Proof of Proposition I.47 The two shaded triangles are congruent (by Prop. I.4) because the shorter sides are respectively sides of the constructed squares and the angle between them is an angle of the original right triangle, plus a right angle from a square.

33 Proof of Proposition I.47 The shaded triangle has the same base (BD) as the shaded rectangle, and the same height (DL), so it has exactly half the area of the rectangle, by Proposition I.41.

34 Proof of Proposition I.47 Similarly, the other shaded triangle has half the area of the small square since it has the same base (FB) and height (GF). (Here is where he needed that G, A and C were all on one line.

35 Proof of Proposition I.47 Since the triangles had equal areas, twice their areas must also be equal to each other (Common Notion 2), hence the shaded square and rectangle must also be equal to each other.

36 Proof of Proposition I.47 By the same reasoning, triangles constructed around the other non right vertex of the original triangle can also be shown to be congruent.

37 Proof of Proposition I.47 And similarly, the other square and rectangle are also equal in area.

38 Proof of Proposition I.47 And finally, since the square across from the right angle consists of the two rectangles which have been shown equal to the squares on the sides of the right triangle, those squares together are equal in area to the square across from the right angle.

39 Proof of Proposition I.47

40 Just for fun, James Garfield s proof: Start with a right triangle with legs of length b and a, and hypotenuse of length c. Extend the side of length b to length a + b, and construct a perpendicular of length b. Form segments as shown. a b b c c a

41 Garfield s Proof Now the area of the trapezoid formed can be calculated in two different ways: as a trapezoid, the area is ½ the sum of the bases, times the height: a b c b c a

42 Garfield s Proof On the other hand, we can calculate the area of the half square of length c and add it to the sum of the two triangles: a b c c b a

43 Garfield s Proof b So, a c So, ya think Bush or Obama or Mitt or Newt or could do that? b c a

44 Book II: Geometric Algebra An example of geometrical algebra is finding the square root of a magnitude by geometric construction: D x Let AB = a, and extend AB to C with BC = 1., so. A a M B 1 C

45 Proposition II 5 If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half. Huh?

46 If a straight line Proposition II 5

47 Proposition II 5 If a straight line is cut into equal and unequal segments, A M C B y a x

48 Proposition II 5 If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole A M C B y a x x x y

49 Proposition II 5 If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section A M C B y a x x x y a a

50 Proposition II 5 If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the A M C B half. y a x x x x y a a x

51 Proposition II 5 Why? Well, since the length of segment AM is the same as x+a, this is an x by x+a rectangle. A M C B y a x x x x y a a x

52 Proposition II 5 Why? This is also an x by x+a rectangle. A M C B y a x x x x y a a x

53 Proposition II 5 Why? So the rectangle on the two unequal segments, the left hand purple together with the brown rectangle, added to the small grey square, will add up to the square on the half (right hand purple together with brown and A M C B grey). y a x x x x y a a x

54 OK, Where s the Algebra? Suppose we have the two equations which are typical of Babylonian problems that gave rise to what we would call quadratic equations.

55 OK, Where s the Algebra? In the diagram, D is the area of the brown rectangle, and b is the length of the line segment. A M C y a x B a

56 OK, Where s the Algebra? Also, and says that, or. Then II 5. A M C B y a x a

57 OK, Where s the Algebra? So we have and written in terms of (given) and. So, to find our and we just need to construct the value a somehow. This can be done with ruler and compass.

58 Finding : x M D 1

59 Finding a: 2 2

60 Finally, Since and it is now easy to construct segments of these two lengths.

61 Ratios and Proportion in Euclid One notion, discussed extensively in Book V, was Eudoxus notion. This was applied to magnitudes in Book VI where Euclid deals with similar figures. For both magnitudes and numbers, there was a different notion, as discussed in Books VII and X. This used Euclidean algorithm, and is our old friend anthyphairesis.

62 Number Theory, Books VII IX Prime, composite. Every composite number is measured by some prime number. Primes are the basic measures of other numbers. Many results of elementary number theory. The infinitude of primes (IX 20).

63 The Infinitude of Primes You should really know this proof for the good of your soul.

64 Areas and Volumes Circles are to one another as the squares on their diameters. Spheres are to one another in the triplicate ratio of their respective diameters. Any cone is a third part of a cylinder which as the same base with it and equal height. Notice: Not formulas.

65 Regular Polyhedra Tetrahedron Cube Octahedron Dodecahedron Icosahedron

66 The Elements Why are the Elements so important? What legacy did they give us? 1. Axiomatic development of mathematics. 2. Next to the Bible, probably the most published book ever. 3. Numerous mathematicians first met mathematics by reading Euclid.

Greece: The Beginnings to Euclid. Chapters 2 and 3

Greece: The Beginnings to Euclid Chapters 2 and 3 Timeline and Overview Archaic Old Kingdom Int Middle Kingdom Int New Kingdom EGYPT 3000 BC 2500 BC 2000 BC 1500 BC 1000 BC Sumaria Akkadia Int Old Babylon