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Euclid of Alexandria: Elementary Geometry

Euclid of Alexandria:Elementary Geometry

Waseda University, SILS,History of Mathematics


Outline

Introduction

Systematic deductive mathematicsStructure in the Elements

The Elements, Book ITheory of congruencyTheory of parallels and theory of areaElements I.47


Introduction

Ancient cultures around the Mediterranean


Introduction

The Hellenistic Period (around 330–30 BCE)

I The Hellenistic Period began with the conquests ofAlexander (”the Great”) III of Macedon (356–323 BCE).

I At the age of 19, Alexander inherited a strong country thatalready controlled much of mainland Greece. Until hisdeath at 32, near Babylon, he devoted himself to conquest.

I When he died, his generals fought over his vast empireand carved it up into a number of Greek-ruled monarchiesthat persisted until the arrival of the Roman armies.

I This led to a flourishing of Hellenic culture in the easternMediterranean and Middle East.

I With regards to sciences and mathematics, the Hellenisticperiod was a high point for Greek-speaking culture.


Introduction

The Hellenistic kingdoms


Introduction

The city of Alexandria

I Founded by Alexander in the delta of the Nile in 331 BCE.I Alexandria became a principal center of Greco-Roman

culture.I It had an important institute of higher learning (the

Museum) and the largest library in Antiquity (the Libraryof Alexandria).

I Many famous philosophers, physicians, mathematicians,poets, etc., worked, or were educated, in Alexandria.

I One of the great legends of antiquity is the burning of thelibrary. Everyone likes to blame it on their enemies(Pagans, Christians, Muslims, etc.), but what happened toall the other libraries?


Introduction

Euclid (c. early 3rd century BCE)

I We know almost nothing about Euclid.I We are not even certain that he lived and worked in

Alexandria, but we assume this is the case, since he iscalled Euclid of Alexandria by later authors.

I Nevertheless, we have reason to believe he lived beforeArchimedes (c. 287–212 BCE), and we are told (by Pappusin the 4th century CE) that Apollonius studied inAlexandria with Euclid’s students.

I He wrote works in a number of the exact sciences:mathematics (geometry, ratio theory, number theory,spherical geometry), astronomy (spherical astronomy),optics (geometric optics), and (perhaps) music theory.


Systematic deductive mathematics

Structure in the Elements

Stucture in the Elements

Euclid may not have been a brilliant mathematical discovererbut his works demonstrate great attention to the details ofmathematical and logical structure. One of Euclid’s principleconcerns seems to have been to set mathematics on a surelogical foundation.

I Each book is arranged around a particular topic or theme.I Each book begins with a series of (1) definitions, (2)

permitted geometric or arithmetic procedures (oroperations), and (3) common assumptions.

I It then proceeds by developing (1) theorems, which showwhat facts are true, and (2) “problems,” which show whatprocedures are possible (and how to do them). We use theword proposition for both theorems and “problems.”




The structure of a Euclidean proposition

A Euclidean proposition has a very specific structure.1. Enunciation: A general statement of what is to be shown.2. Exposition: A statement setting out a number of specific

objects with letter names.3. Specification: A restatement of what is to be shown in

terms of the specific objects.4. Construction: Instructions for drawing new objects that

will be required in the proof, but which were notmentioned in the enunciation. (Relies on postulates andprevious construction propositions (“problems”).)

5. Proof: A logical argument that the proposition holds.(Relies on definitions, common notions and previoustheorems.)

6. Conclusion: A restatement, in general terms, of what hasbeen shown.




The structure of a theory

I A theory develops propositions that all relate to aparticular property of some set of objects.

I The goal is to develop a full picture of the properties froma limited set of assumptions.

I For example, we might want to be able to show how wecan check for the existence (or absence) of a property (suchas tangency, or parallelism, etc.), so that it can be used infurther mathematical work.

I We will look at the examples of the theory of thecongruency of triangles, and the theory of parallel lines andequal area in Elements I.




The overall structure of the Elements

I The goal of the Elements is to develop certain individualmathematical theories by relying on a limited set ofassumptions (starting points), and deriving everythingfrom these.

I Books I–IV: elementary plane geometry (I: congruence oftriangles, theory of parallels, II: application of areas, III:circles, IV: regular figures)

I Book V: ratio theoryI Book VI: proportionality in geometric figuresI Book II–IX: number theoryI Book X: incommensurable (irrational) lengthsI Book XI–XIII: three dimensional geometry


The Elements, Book I

The overall structure of Elements Book I

I The book begins with (1) definitions (ex. line, plane, circle,right angle, parallel line, etc.), (2) postulates (ex. to draw aline, circle, all right angles are equal, 5th post.1), and (3)common notions (equals to equals are equal, the whole isgreater than the individual parts, etc.)

I The propositions start with a series of construction“problems” and develop a theory of the congruency oftriangles (SAS (I.4), SSS (I.8), AAS (I.26)).

I Next, a theory of parallel lines, involving the 5th postulate,leads to a theory of the equality of certain areas.

I Finally, these two theories are combined to demonstratethe so-called Pythagorean Theorem (I.47).

1“If a straight line falling on two lines makes the interior angles on thesame side less than two right angles, then the two lines, if produced, willmeet on the side on which the angles are less than two right angles.”



Theory of congruency

Theory of congruency

I We begin with some construction “problems” (equilateraltriangle (I.1), moving a line (I.2 & I.3), etc.).

I We show SAS (side-angle-side) congruency (I.4) and SSS(side-side-side) congruency (I.8).

I Using the fact that a straight line is 2R (= 180◦) (I.13 & 14),we show that vertical angles are equal (I.15), and that theexterior angle of a triangle is greater than the two interiorangles (I.16).2

I We use all this to show AAS (angle-angle-side) congruency(I.26).

I All these theorems hold with or without the 5th postulate.32This theorem is not a theorem of “absolute geometry.” Why? What

happens on a sphere?3With the exception of I.16, we say these theorems belong to “absolute

geometry,” because they hold in both Eclidean and non-Euclidean spaces.



Theory of parallels and theory of area

Theory of parallels and theory of area

I We develop some theorems about angles related to parallellines (alternate angles (I.27), exterior and interior angles(I.28), and conversely (I.29)).

I We show that the exterior angle of a triangle is equal to thesum of the the two opposite interior angles (I.32).4

I We show that parallelograms on the same base betweentwo parallels are equal (I.35), which leads us to show thatany parallelogram between two parallels is twice the areaof a triangle on the same base between the same parallels(I.41).

4How is this related to I.16? Since I.32 is a stronger result, why did webother to show I.16?



Elements I.47

Elements I.47 (The so-called Pythagorean Theorem)

I We begin with 4ABC anddraw squares BFGA, AHICand CEDB so that lines BAHand CAG are straight.

I We argue that4ABD = 4FBC.

I Therefore, rectangleBKLD = ABGF.

I The same argument showsthat KCEL = ACIH.



Elements I.47

Final considerations

I Why does Euclid divide his approach into differenttheories?

I Why does he avoid using the 5th postulate until as late aspossible.

I How can the Elements I be taken as a ideal for allmathematical practice? Until the 19th century, the Elementswas still held out as a sort of model for mathematicians.

I and so on...

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Euclid of Alexandria: Elementary Geometry - 早稲田大学 · PDF filedeath at 32, near Babylon, he ... Euclid may not have been a brilliant mathematical discoverer ... Euclid of