Math 161 - Notes

Neil Donaldson

Winter 2020

1 Geometry and the Axiomatic Method

1.1 The Early Origins of Geometry: Thales and Pythagoras

The word geometry comes from the ancient Greek terms geo meaning Earth, and metros meaningmeasure. The measurement of distance, area, height, angle, etc., had obvious practical benefits withregard to construction and the taxation of land. A second major driver of ancient geometry wasthe study of astronomy, with its applications to navigation, the calendar and as part of religiousunderstanding.

Condensed Overview of Geometric History

Ancient times (pre-500 BC) Egypt, Babylonia, China, India: basic rules for measuring lengths, ar-eas and volumes of simple practical shapes. Applications: surveying, tax collection, con-struction, religious symbology. Typically recorded as worked examples without general for-mulæ/equations.

Ancient Greece (from c.600 BC) Philosophers such as Thales and Pythagoras began the process ofabstraction. General statements (theorems) formulated and proofs attempted. Mathematics be-gins to move beyond the purely practical. Early scientific reasoning develops concurrently.

Euclid of Alexandria (c.300 BC) collected and developed much earlier work, especially from thePythagoreans. His masterwork, the Elements, became one of the most important books inWestern history, and remained a standard textbook until the early 1900’s. Began the axiomaticmethod on which modern mathematics is based.

Later Greek Geometry Archimedes’ (c.270–212 BC) work on areas and volumes included techniquessimilar to those of modern calculus. Ptolemy (c. 100–170 AD), publishes the Almagest, a treatiseon astronomy which began the study of trigonometry.

Post-Greek Geometry The Romans (who conquered much of the Greek empire), were much lessinterested in philosophy. With the fall of Rome, much of the knowledge of the Greeks was lostto Europe. However, during the Dark Ages, this learning was preserved and greatly enhancedby Indian and Islamic mathematicians who, in particular, developed trigonometry and algebra.The melding of algebra with geometry was completed by Descartes in mid-1600’s France, withthe advent of co-ordinate systems (analytic geometry).

Modern Development Non-Euclidean geometries help provide the mathematical foundation for Ein-stein’s relativity. Following Klein (1872), modern geometry largely follows the study of sym-metry using group theory.

Thales and Pythagoras Two of the earliest mathematicians are Thales of Miletus (c. 624–546 BC)and Pythagoras of Samos (570—495 BC)1 Both travelled/traded widely and absorbed knowledgefrom other cultures, in particular Egypt, Babylonia, etc. Their work formed some of the earliestexamples of deductive reasoning, if not quite from first principles.

Thales’ Theorems Thales was one of the first philosophers to state theorems. The ancient Greekword θεωρεω (theoreo) has several meanings, including, ‘to look at,’ ‘speculate,’ or ‘consider.’ In thissense, a theorem could be thought of as a proposition to be considered, or as something straightfor-ward to visualize. Early proofs were very pictorial. Here are five theorems, at least partly attributedto Thales, the last indeed is still known as Thales’ Theorem.

1. A circle is bisected by a diameter.

2. The base angles of an isosceles triangle are equal.

3. The pairs of angles formed by two intersecting lines are equal.

4. Two triangles are congruent if they have two angles and the included side equal.

5. An angle inscribed in a semicircle is a right angle.

Thales’ proofs were not what we’d understand by the term. His reasoning was part deductive andpart inductive, based on examples. Importantly, the statements are abstract and general: e.g. any circlewill be bisected by any of its diameters.

βα

α β

The pictures show Thales’ Theorems 1, 2 and 5. ‘Arguments’ for Theorems 1 and 2 could be as simpleas ‘fold.’ Theorem 5 follows from the observation that the radius of the circle (in green) splits the largetriangle into two isosceles triangles: Theorem 2 says that these have equal base angles (labelled), nowcheck that α + β is half the angles in a triangle, namely a right-angle.

The Pythagoreans In his youth, Pythagoras was almost certainly influenced by the ideas of Thales.After some years travelling the Mediterranean, Pythagoras settled in Croton, southern Italy, around530 BC where he founded a philosophical school.2 Amongst other things, Pythagoras and his fol-lowers studied number, music and geometry. Their influence was greater than their verifiable con-tributions to mathematics; their obsession with number and the ‘music of the universe’ inspiredmany later mathematicians/philosophers (in particular Euclid, Plato and Aristotle), who believedthey were refining and clarifying work begun by the Pythagoreans. It has been claimed, for instance,

1Miletus was a town/city-state on the west coast of modern-day Turkey. Samos is nearby island in the Aegean sea.2Perhaps cult is a better word. . .

2

that they first classified the regular (Platonic) solids, and gave the first proof of what is now known asPythagoras’ Theorem. This last claim is almost certainly incorrect.3 Nonetheless, the work done bythe Pythagoreans, in particular their dedication to number and geometry as a pure, unsullied formof knowledge, exerted great influence on later Greek mathematicians.

A simple proof of Pythagoras’ Theorem4

1.2 The Axiomatic Method

It is believed that a primary motivation for Euclid’s writing of the first book of the Elements was torigorously prove Pythagoras’ Theorem. In so doing, he essentially invented the axiomatic method.Before considering Euclid’s approach, we describe the revolutionary structure Euclid followed, thatof a deductive or axiomatic system.

Definition 1.1. An axiomatic system comprises four pieces:

1. Undefined terms: concepts accepted without definition/explanation. In the setting of basic ge-ometry these might include line or point.

2. Axioms/Postulates: logical statements regarding the undefined terms, which are accepted as truewithout proof. For example: There exists a line joining any two given points.5

3. Defined terms: concepts defined in terms of 1 and 2. For instance a triangle could be defined interms of three non-collinear points.

4. Theorems: logical statements deduced from 1–3.

We consider two simple examples of a deductive system: the first is a game—many such can beconsidered deductive systems—while the second is more traditionally mathematical.

3Other cultures’ frequent use of the result suggests that a general statement was understood earlier. Indeed the earliestknown references (without proof) to the Theorem come from China several hundred years before Pythagoras where thestatement is known as the gou gu. By modern standards, it it not believed that the Pythagoreans had a rigorous proof.

4The second picture will animate if you click on it while open in a full-feature pdf reader like acrobat.5In Euclid, an axiom is considered to be somewhat more self-evident than a postulate, but the distinction is not impor-

tant to us. For instance, one of Euclid’s axioms is, ‘Things which are equal to the same thing are also equal to one another,’essentially the transitivity of equality. This feels more general than the postulate regarding a line joining two points.

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Chess

1. Undefined terms: Pieces (as black/white objects) and the board.

2. Axioms/Postulates: Rules for how each piece moves.

3. Defined terms: Concepts such as check, stalemate or en-passant.

4. Theorems: For example, Given a particular position, Black can win in 5 moves.

Monoid

1. Undefined terms: A set G and a binary operation ∗.

2. Axioms/Postulates: (A1) Closure: ∀a, b ∈ G, a ∗ b ∈ G.(A2) Associativity: ∀a, b, c ∈ G, a ∗ (b ∗ c) = (a ∗ b) ∗ c.(A3) Identity: ∃e ∈ G such that ∀a ∈ G we have a ∗ e = e ∗ a = a.

3. Defined terms: Concepts such as square a2 = a ∗ a, or commutativity a ∗ b = b ∗ a.

4. Theorems: For example, The identity is unique.

Given that the concept of a set and all the required notation is also required, one could argue thatmany more axioms are really required!

Properties of Axiomatic Systems

Definition 1.2. A model of an axiomatic system is an interpretation of the undefined terms such thatall the axioms/postulates are true.

Example (G, ∗) = (Z,+) is a model of a monoid, where e = 0.6

The big idea of models and axiomatic systems is this:

Any theorem proved within in an axiomatic system is true in any model of that system.

Several great mathematical discoveries have hinged on the realization that two seemingly separatediscussions can be described in terms of models of a common axiomatic system.

Definition 1.3. Certain properties are desirable in an axiomatic system:

Consistency Contradictions are impossible within the system. No two statements (theorems/axioms)contradict one another. To show its consistency, it is sufficient to give a (concrete) model of anaxiomatic system. An inconsistent system is essentially useless from a mathematical perspec-tive, though demonstrating that a system is inconsistent might be very challenging.

6Strictly this is an abstract model in that the integers themselves are defined in terms of another axiomatic system! Aconcrete model is one that can be embedded in reality, where contradictions are assumed impossible(!!). Thus a single dot• on the page is a concrete model of a monoid when equipped with the operation • ∗ • = •. The subtleties of thesedistinctions are left for a course on formal logic. . .

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Independence An axiom is independent if it is not a theorem of the others. An axiomatic system isindependent if all its axioms are. We can show that an axiom is independent by exhibiting twomodels: one in which all axioms are true, and one in which only the considered axiom is false.

Completeness Every proposition stated in terms of the undefined and defined terms is decidable(can be proved or disproved). Showing completeness is typically very difficult. To show in-completeness, one requires an undecidable statement; it is possible to do this by exhibiting twomodels, one in which the statement is true, the other in which it is false. Alternatively, a newaxiom postulating that the undecidable is true should be independent of the others.7

Example The axiomatic system for a monoid is:

Consistent We have a (concrete) model: see above.

Independent Consider three models:

• (N,+) satisfies axioms A1 and A2 but not A3.

• ({e, a, b}, ∗) defined by the following table satisfies A1 and A3 but not A28

∗ e a be e a ba a e ab b b a

• (Z \ {1},+) satisfies axioms A2 and A3 but not A1.

Incomplete The proposition ‘A monoid contains at least two elements’ is undecidable just from the ax-ioms. This should be clear from the existence of two models; ({0},+) with only one elementand (Z,+) with infinitely many.

Alternatively, we could ask if all elements have an inverse. This is the same as saying that thefollowing axiom is independent of the first three.

(A4) Inverse: ∀g ∈ G, ∃g−1 ∈ G such that g ∗ g−1 = g−1 ∗ g = e.

The new system defined by the four axioms is also consistent and independent: this is the struc-ture of a group. Even the axiomatic system for a group is incomplete; for instance, we could adda new axiom of commutativity. . .

The details of this discussion are mostly beyond the level of this course. Our primary goal is to coverjust enough about axiomatic systems to understand what Euclid did and how he motivated latermathematicians. See the handout in class for another example of an axiomatic system, and considerreading the examples on the following site:

http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxSysWorksheet.htm

7There is enormous subtlety here, and we cannot do justice to it!. A statement can be valid in a theory, but might not beprovable. It might be the case that there is essentially only one model for a theory, but that there are still statements whichare true within the model but which cannot be proved/disproved.

8E.g., a ∗ (b ∗ b) = a ∗ a = e 6= a = a ∗ b = (a ∗ b) ∗ b.

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Godel’s incompleteness theorems In 1931, the German mathematician/logician Kurt Godel provedtwo revolutionary results about axiomatic systems.

Theorem 1.4. 1. Any consistent system containing the natural numbers is incomplete.

2. The consistency of such a system cannot be proved within the system itself.

Godel’s theorems have caused philosophers a great deal of anguish, since mathematicians can’t domuch without the natural numbers! Godel essentially says that we will never be able to find anultimate axiomatic system within which any mathematical question we ask can be answered; therewill always be undecidables.9

Godel’s second theorem fleshes out the difficulty in proving the consistency of an axiomatic system.If a system is sufficiently complex to describe the natural numbers, we can only prove its consistencyrelative to some other axiomatic system. if you want more information.

1.3 Euclid’s Axioms

Euclid’s axiomatic system for geometry, as laid out in the Elements (c.300 BC) is the first seriousattempt at axiomatization. Here is a summary:

Undefined Terms E.g., point, line, etc.10 In Euclid, a line was finite in extent (a line segment).

Axioms/Postulates A1 If a = b and b = c then a = c (= is transitive)A2 If a = b and c = d then a + c = b + dA3 If a = b and c = d then a− c = b− dA4 Things that coincide are equal (in magnitude)A5 The whole is greater than the partP1 We can create a line by joining two pointsP2 We can extend a finite line continuouslyP3 We can create a circle given a point and a radiusP4 All right angles are equalP5 If a straight line crosses two others and the angles on one side sum to less than two right

angles then the two lines (when extended) meet on that side

For more detail, you can find an English translation of Euclid’s proofs at

http://math.furman.edu/~jpoole/euclidselements/euclid.htm

You can also read a version of the Elements from 1847 at

http://math.uci.edu/~ndonalds/Elements-I-VI.pdf

This is a very large file! It is worth noting that this version contains a long list of definitions, insertsmany more axioms, and relabels propositions 4 and 5 as axioms (pages xviii–xxiii).

9Perhaps this is reassuring; we could interpret this as saying that mathematics will never be finished! A famous exampleof an undecidable statement from standard set theory is the Continuum Hypothesis which states that there is no uncountableset with cardinality strictly smaller than that of the real numbers. Both the hypothesis and its negation have been proven tobe independent of the standard axioms of set theory. Abstractly, the undecidable statements cooked up by Godel to provehis theorems are analogues of the Liar Paradox ‘This sentence is false.’ It is therefore a matter of debate as to whether theconsideration of such undecidable statements is truly useful!

10In fact Euclid attempted to define these: ‘A point is that which has no part,’ and ‘A line has length but no breadth.’

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Consistency, Independence and Completeness? Euclid’s system is not the best place to answerthese questions. It is certainly incomplete, for Euclid makes a great many hidden assumptions. Forinstance he does not assume the existence of points and makes frequent use of the notion of continu-ity of lines/circles without justification. Euclid’s postulates have been shown to be independent. Todo so, one must exhibit five models, each of which obeys all but one of the postulates. For example,remove the concept of circle from Euclidean geometry and postulate 3 is irrelevant. Deleting either ofpostulates 2 and 4 is quite difficult; you can read about this elsewhere. For centuries, mathematiciansattempted to prove that postulate 5 (the parallel postulate) is dependent on the others. Eventually,with the discovery of non-Euclidean geometries (such as hyperbolic geometry), it was seen to be in-dependent. Even Euclid seemed to find the parallel postulate awkward, using it sparingly: he waitsuntil the 29th theorem of Book I before using it!

Eventually mathematicians such as David Hilbert (1899) created axiomatic systems with the goalof properly axiomatizing Euclidean Geometry. We’ll consider this later. In modern times, Euclid isbetter thought of as an inspiration rather than a fully realized ideal of an axiomatic system.

Basic Theorems a la Euclid Most theorems/propositions were phrased in a constructive manner; asa problem. For example, the first proposition of Book I states:

Theorem (I. 1). Problem: to construct an equilateral triangle on a given line segment.

Euclid then provides a series of steps whereby the required triangle is constructed, before proving thathis triangle really is equilateral. Compare our proof with that in either of the links on the previouspage. We use the notation AB for the line (segment) joining the points A and B.

Proof. Given a line segment AB:

• By postulate 3, construct the circles centered at A andB with radius AB.

• Call one of the intersection points C; by postulate 1,construct AC and BC.

• Then4ABC is equilateral.

A B

C

To prove this, note that AB and AC are radii of the circle centered at A, while AB and BC are radii ofthe circle centered at B. Thus (axiom 1) all three sides of4ABC are equal.11

These constructive proofs are the hallmark of ancient Greek geometry: in modern times, pictures cre-ated using Euclid’s first three postulates are known as ruler and compass constructions, about which agreat many theorems have been proved. To the Greeks, and to many philosophers, an existence proofwas barely considered a proof at all: one needed to provide a method of constructing an object! TheGreeks used geometric constructions in lieu of basic algebraic operations, such as addition, squaringand taking square-roots. Two triumphs of this study are Gauss’ construction of a regular 17-gon, andthe eventual proof that, given a circle it is impossible to construct a square with the same area. Youcan read about these problems elsewhere.

11Euclid uses the work equal where we’d say congruent in modern mathematics. We’ll discuss this more in the nextchapter, but for now we’ll follow Euclid’s approach.

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Here are a select few other theorems from the early books of Euclid.

• To bisect a line segment (Thm I. 10)

• If two lines cut one another, opposite angles are equal (Thm I. 15)

• Thales’ Theorem: An angle in a semicircle is right-angle (Thm III. 31)

For instance, Euclid proves Thales’ Theorem12 in essentially the same manner as we did on page 2

Proof. Given a triangle in a semi-circle, join the apex to the center. We now have two isosceles trian-gles which necessarily have equal base angles (I. 5). Since the sum of the angles in a triangle equals astraight-edge (I. 32), we conclude that the angle at the apex is half this: a right-angle.

Parallel lines, their construction and uniqueness

It is Euclid’s discussion of parallels that has the most use for us, and underpins much of this course.

Definition. Two lines are parallel if no extensions of the lines intersect.

Theorem (I. 16 Exterior Angle Theorem). If one side ofa triangle is extended, then the exterior angle is larger thaneither of the opposite interior angles.In the language of the picture (although Euclid did not quan-tify angles), we have δ > α and δ > β.

α

β δ

Proof. Draw the bisector BM of AC (Thm I. 10).

Extend BM the same distance beyond M to E (Thm I. 2).

Connect CE (P1).

The opposite angles at M are equal (Thm I. 15).

The Side-Angle-Side Theorem (I. 4) says that the trian-gles4AMB and4CME are equal.

α

α

A

B C

M

E

It follows that the angle indicated at C is also α. Clearly this is less than δ.

Bisecting BC and repeating the argument shows that β < δ.

Theorem I. 16 essentially constructs a parallel line to AB through a given point C not on the line. Itremains to prove that this line really is parallel.

12Somewhat amazingly, Euclid doesn’t prove this until Book III of the Elements!

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Theorem (I. 27). If a line falls on two other lines in such a waythat the alternate angles (labelled α) are equal, then the two orig-inal lines are parallel.

α

α

Proof. If the lines were not parallel, then they must meeton one side. WLOG suppose they meet on the right side ata point C.The angle α at B, being external to4ABC must be greaterthan the angle α at A (Theorem I. 16): contradiction.

α

α

A

B CIt follows that the line CE constructed in the proof of Theorem I. 16 really is parallel to AB. Thefollowing theorem is an immediate corollary.

Theorem (I. 28). If a line falling on two other lines makes thesame angles, then the two original lines are parallel.

α

α

Thusfar Euclid only uses the first four of his postulates. In any model for which the first four pos-tulates are true, it is therefore possible to create parallel lines using the construction of the ExteriorAngle Theorem. Euclid now proves the converse to Theorem I. 27, invoking the parallel postulate toshow that this is the only way to create parallel lines.

Theorem (I. 29). If a line falls on two parallel lines, then the alternate angles are equal.

Proof. Given the picture, we must prove that α = β.Suppose not and WLOG that α > β.But then β + γ < α + γ, which is a straight edge.By the parallel postulate, the lines `1, `2 meet on the leftside of the picture, whence `1 and `2 are not parallel.

`1

`2

αγ

β

Angles in a triangle Euclid is finally in a position to prove the most famous result about triangles:that the interior angles sum to a straight edge. Euclid words this slightly differently.

Theorem (I. 32). If one side of a triangle is protruded, the exte-rior angle is equal to the sum of the two interior opposite angles.

In the language of the picture, ∠ACD = ∠BAC +∠ABC.For clarity we’ll label the angles with Greek letters.

Proof. Construct CE parallel to BA (Theorem I. 16).Then ∠ACE = ∠CAB (Theorem I. 29) and∠ABC = ∠ECD (Corollary of I. 29).

A

B C D

E

β βα

α

But then ∠ACD = ∠ACE +∠ECD = ∠CAB +∠ABC.

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Non-Euclidean Geometries

Euclid’s proof of Theorem I. 32 (that the angles in a triangle sum to 180°) relies on I. 29 and thereforeon the parallel postulate. The fact that he waited until theorem 29 before invoking it shows that hefound the parallel postulate uncomfortable and messy; plainly he was trying to establish as much ashe could about triangles and basic geometry in its absence. This attitude persisted for centuries, withmany mathematicians assuming that the fundamental fact about angles in a triangle must be truewithout having to assume the parallel postulate. Much effort was expended in attempting to provethat the parallel postulate was not an independent axiom of Euclid’s first four axioms.

It was not until the 17–1800’s, with the development of non-Euclidean geometries, that the indepen-dence of the parallel postulate was finally proved. Examples were found of models (in particular ofhyperbolic geometry) in which Euclid’s first four postulates hold but for which the parallel postulateis false. Later in the term we shall see that every triangle in hyperbolic geometry has angle sum lessthan 180°!

To get this far in hyperbolic geometry will require a lot of work! For amore-easily visualized non-Euclidean geometry consider elliptic geometry,whose simplest example is the round sphere. Imagine stretching a rubberband between three points on a sphere: the result is a spherical triangle. Inthe picture, one corner of an equilateral triangle is placed at the top of thesphere and the other two corners lie on the equator: the sum of the anglesis clearly 270°!

Elliptic geometry is not enough to show that the parallel postulate is inde-pendent in Euclidean geometry since it does not satisfy all of the first fourof Euclid’s postulates (one cannot extend a line indefinitely for instance).

Playfair’s Postulate

Euclid’s parallel postulate is stated in the negative (angles don’t sum to a straight edge, thereforelines are not parallel). Euclid probably chose this formulation in order to facilitate proofs by contra-diction, but the result is to make the parallel postulate difficult to understand. Here is a more moderninterpretation:

Axiom 1.5 (Playfair’s Postulate). Given a line ` and any pointP not on `, there exists exactly one parallel line m through P.

m

`

P

It should immediately be clear that the parallel postulate im-plies Playfair’s postulate.

• Let A, B ∈ ` and construct the triangle4ABP.

• The Exterior Angle Theorem (I. 16) constructs Q andthus a parallel m to ` by Theorem I. 27.

• Theorem I. 29 invokes the parallel postulate to provethat this is the only such parallel.

`

m

B A

P Q

α

α

10

In fact the postulates are exactly equivalent: to be precise. . .

Theorem 1.6. In the presence of Euclid’s first four postulates, Playfair’s postulate and the parallel postulateare equivalent.

Proof. We proved that the parallel postulate implies Playfair above. The converse (Playfair =⇒ P5)requires more work.

We prove the contrapositive: we start by assuming Euclid’s postulates P1–P4 are true and that P5is false. We therefore need to consider the negation of the parallel postulate. With reference to thepicture,13 the parallel postulate being false means:

∃ a pair of lines `, m and a crossing line n, suchthat α + γ < straight edge and `, m are parallel.

The assumption is without loss of generality: if α + γ weregreater than a straight edge, simply use the angles on theother side of n.

`

m

γ

α

n

Now use the standard construction of the Exterior AngleTheorem (I. 16) to build a parallel line ˆ to ` through theintersection point P of m and n.

ˆ

`

mPα

γ

α

Our construction requires only up to Theorem I. 28 (before Euclid’s first use of the parallel postulate),and so is perfectly legitimate in this context.

Now observe that the lines ˆ and m are distinct since α + γ is less than a straight edge. It followsthat there exists a line ` and a point P not on that line, though which pass (at least) two parallels to `:Playfair’s postulate is false.

We conclude the contrapositive, that Playfair’s postulate implies the parallel postulate.

The negation of Playfair’s postulate in fact says that there might be none or many parallels though agiven point not on a given line. Later we shall see that hyperbolic geometry satisfies Euclid’s firstfour postulates but not the fifth: there will turn out to be infinitely many parallels though a point noton a given line.

At the other extreme, in spherical geometry (see the previous page) lines are great circles on the sphere:no parallel lines exist since any two great circles meet at two antipodal points. There is no contradic-tion here: spherical geometry only satisfies Euclid’s first and fourth postulates, the Exterior AngleTheorem is false on the sphere14 whence we have no method for constructing parallels!

13Using quantifiers, the parallel postulate may be stated:

∀ pairs of lines `, m and ∀ crossing lines n, α + γ < straight edge =⇒ `, m not parallel.

14For a challenge, consider the constructions in the proof of the Exterior Angle Theorem applied to the spherical triangleon the previous page: What happens? Where does the argument go wrong?

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Pythagoras’ Theorem

The crowning achievement, and penultimate result, of Book I is Pythagoras’ (Thm I. 47). This stronglysuggests that Book I was structured in order to efficiently prove this result without relying on theerroneous assumptions of the Pythagoreans. Here is how Euclid approaches the problem.

• Theorems I. 33–41 develop the idea that parallelogramswith the same base and height are equal (in area), andmoreover double any triangle with the same base andheight. Note that Euclid did not define area numerically,he only compared ratios of areas.The picture illustrates Theorem I. 41: parallelogramABCD is double (the area of)4ADE.

• Theorem I. 42–46 involve constructions of parallelo-grams and squares. A

B C

D

E

Proof of Pythagoras’ Theorem (I. 47). Consider the picture below, where the given triangle4ABC is as-sumed to have a right angle at A.

1. Construct squares on each side of4ABC (Thm. I. 46)and a parallel AL to BD (e.g. Thm. I. 16).

2. We have AB = FB and BD = BC (sides of squaresare equal). We also have ∠ABD = ∠FBC since bothcontain ∠ABC and a right angle.

3. By Side-Angle-Side (Thm. I. 4), 4ABD and 4FBCare equal (identical up to rotation by 90°).

4. By Thm I. 41,

Area(�ABFG) = 2 Area(4FBC)= 2 Area(4ABD)

= Area(

BOLD)

5. Similarly Area(�ACKH) = Area(

OCEL)

.

6. Sum the rectangles to obtain the square�BCED andcomplete the proof.

A

B CO

LD E

F

G

H

K

Euclid finishes Book I of the Elements with the converse.

Theorem (I. 48). If the squares on two sides of a triangle equal the square on the third side, the triangle has aright angle opposite the third side.

Look up the proof: it should be easy to follow. . .

The remaining books of the Elements contain further geometric constructions, including three dimen-sional constructions, as well as discussions of basic number theory such as what is now known as theEuclidean algorithm. This is enough pure-Euclid for us: we now turn to a more modern descriptionof Euclidean geometry, courtesy of David Hilbert.

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