chapter 1: euclid’s elements

Chapter 1: Euclid’s Elements

Or, It’s All His Fault

1MthEd/Math 362 Chapter 1Summer 2009

Ancient Geometry

• Early civilizations such as those in Egypt, Mesopotamia, and India were concerned mainly with the utility of mathematics in general and geometry in particular.

• Geometry means “earth measure” and reflects its use in measuring land, setting property values, figuring out taxes, splitting land among surviving children, etc.


Ancient Geometry

• Ancient civilizations were interested in building and buying and selling and so were interested in mathematics only to the extent that it helped in these endeavors. They did some quite clever things mathematically, but seldom knew and even more seldom cared why it all worked.


Ancient Geometry

• For example, Egyptian “rope stretchers” used knotted ropes to measure land boundaries and in particular used a closed rope with 12 knots to form a 3 – 4 – 5 triangle and hence a right angle when needed.

Summer 2009 MthEd/Math 362 Chapter 1 4

Ancient Mesopotamia


Mesopotamia

• 2400 BC first approximation of π at 3.125.• 2400 to 2200 BC - multiplication tables, tables

of square and cube roots.• 1950 to 1750 BC – linear and quadratic

equations, Pythagorean Theorem. • Babylonians knew how to determine areas

and volumes of various figures and solids, including circles. They could generate all Pythagorean triples.


Egypt

• 1900 BC – Moscow Papyrus: volume of a frustrum, other methods for finding areas, volumes.

• 1700 BC – Rhind Mathematical (Ahmes) Papyrus: multiplication and division by doubling and halving, use of unit fractions, fraction doubling tables, approximation of π, linear equations, use of cotangent, attempt to square the circle.

• 1300 BC – Berlin Papyrus: a quadratic equation and its solution.


Egypt• Example from the Moscow

Papyrus. If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4,result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take a third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right.


Egypt

• Example from the Rhind Papyrus:– A round field has diameter 9 khet. What is its

area? – Here is the solution as given by the scribe Ahmes:

Take away 1/9 of the diameter, namely 1; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore it contains 64 setat of land.

• Note that this implicitly uses the value


2563.16049

81

India

• 1000 to 500 BC - Yajnavalkya describes motion of sun and moon.

• 900 BC – Yajur Veda contains first idea of infinity • 800 to 600 BC – Sulbasutras – directions for

building altars – appear. The contain the Pythagorean theorem, procedures for squaring a rectangle, approximate procedures for squaring a circle and circling a square, an approximation of π correct to five decimal places, quadratic equations, dividing a segment into equal parts


Ancient Geometry

• In summary, geometry prior to about 600 BC was mainly a collection of methods, tricks, and seat-of-the-pants reckoning used to accomplish practical ends.

• The exception to this “practical” use of geometry was astronomy, which was tied to mystical and religious practice. Still, we have little or no record of mathematics for mathematics’ sake.


Ancient Greek Geometry• The Greek civilization that flourished between

about 600 BC and 400 AD seems to the first to care about the “why” of mathematics, and more particularly, to appreciate the need for proof. They put logical reasoning and proof at the center of mathematics.

• It is important to remember that in this context, “Greek” refers to a language of scholarship, a worldview, learned by people from Alexandria in Egypt (e.g. Euclid) to Syracuse in Italy (e.g. Archimedes).


Ancient Greek Geometry

• Among the important Greek mathematicians we have:– Thales of Miletus– Pythagoras– Eudoxus– Hippocrates of Chios– Euclid


Thales:• Thales of Miletus (about 600 BC) learned his

math in Egypt and Babylonia. He is sometimes referred to as the Father of demonstrative mathematics, because he not only stated mathematical facts and procedures, but proved them.

• Tradition holds that Thales first proved:– Vertical angles are equal– The angle sum of a triangle equals two right angles– The base angles of an isosceles triangle are equal– An angle inscribed in a semicircle is a right angle


Pythagoras• Pythagoras (585 – 501 BC) also studied in Egypt and Babylonia, and maybe

studied under Thales himself. Most importantly, he founded the Pythagorean Brotherhood (although women were equal members), a society of scholars with cult-like behaviors and beliefs.

• They:– never ate meat or beans (beans were sacred) – never hunted and used no wool, – dressed in white and drank no wine, – did not pick up anything that had fallen, – did not stir a fire with an iron, and – slept in white linen bedding.

• Their symbol was pentagram, and they used various rituals to strengthen solidarity.

• They engaged in frequent exercise, silent contemplation, and study of mathematics.

• They believed in transmigration of souls. • They also became politically strong, in their local area, leading to the burning of

their meeting house about 501 BC.


Pythagoreans• Attributed to the Pythagoreans (and ultimately to Pythagoras) are:

– The Pythagorean Theorem (duh!) – Work with ratios, particularly incommensurable ratios. They tended

to work with ratios rather than direct measurements. Thus they would say that the areas of two circles had the same ratio as the squares produced from their two radii.

– Two segments are commensurable if there is a smaller segment that divides evenly into both. The ratio of commensurable lengths is a rational number. The Pythagorean Hippasus discovered that the diagonal of a square is not commensurable with its side, contrary to the Pythagorean belief that any two segments were commensurable. Legend has it that they tossed him from a boat.

– What we would today interpret as irrational numbers caused some distress, and in fact the Greek preference for line segments, instead of their lengths, was prevalent throughout the 1000 years of Greek mathematics. This meant that Greek mathematics tended to be very geometrical rather than arithmetic or algebraic.


Eudoxus

• Eudoxus of Cnidos (about 370 BC) put forth a theory based on ratios of magnitudes, but magnitudes themselves were left undefined. This pushed number back into the realm of geometry, where it tended to stay for the remainder of the Greek period.

• Eudoxus was also responsible for the method of exhaustion, reminiscent of ideas from our modern calculus, which was later used by Archimedes to determine the area of a circle.


Hippocrates of Chios• Hippocrates of Chios (460 – 380 BC) wrote a book of

“Elements” that we have no copy of. He also performed the quadrature of a particular lune. Quadrature in general is the art of finding a square with the same area as a given (less regular) figure. Notice that the problem is not to find numbers, but to construct, with compass and straightedge, an actual square. Hippocrates was able to “square,” or perform the quadrature of, 3 different lunes.

• The next person to have any success in this was Euler, about 2000 years later, who performed the quadrature of 2 more lunes (although some claim those two were done by Martin Johan Wallenius in 1766 as well).

• Finally, it was proved in 1994 that the rest cannot be squared.


The Five Squarable Lunes


Euclid

• 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.


Alexander the Great’s Empire


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).


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.


Alexandria After Euclid

• Since we’re going to focus on Euclid’s Elements for the rest of the time, I’ll briefly outline the Library at Alexandria’s life after Euclid:– 287 - 212 BC Archimedes: Area of circle, π, buoyancy,

areas and volumes by exhaustion, levers.– 240 BC Eratosthenes: Sieve, circumference of the

Earth– 250 BC Apollonius: Conics– 75 AD Heron: Metrica, area of triangle


Alexandria after Euclid

– 150 AD Ptolemy: Amlagest, astronomy, geography– 250 AD Diophantus: Arithmetica, equations with

rational roots.– 350 AD Pappus: Mathematical Collection,

commentary– 375 AD Theon: Commentaries on Elements,

Amlagest– 400 AD Hypatia: (daughter of Theon):

Commentaries.– 641 AD Library at Alexandria burned


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 be taken from the Elements of Hippocrates of Chios.


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


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.


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


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).


Euclid’s ElementsEuclid’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.


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.


Euclid’s ElementsProposition 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

A BD

E

Euclid’s Elements

• 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.


C

A BD

E

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.


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.


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 real-number 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.


Other Axiom Systems for Geometry

Developer Axiom Set Comments

Hilbert 20 Axioms “Fixed” EuclidDoes not explicitly depend on Real Numbers, Set Theory, etc.

Many AxiomsDetailed Technical Proofs of the ObviousImplicitly depends of the real numbers and Set Theory

Birkhoff Only 4 AxiomsIncorporates Real Numbers and MeasurementMore accessible

Only Euclidean (fourth axiom is about similar triangles)

MacLane 14 Axioms including “continuity axiom”

SMSG 22 Axioms Designed for High School Geometry

Some unnecessary axioms

UCSMP Lots of Axioms RedundancyUses Reflection

This Book 12 Axioms Explicitly uses real numbers and set theory


chapter 1: euclid’s elements

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