Should the Pythagorean Theorem Actually be Called the ‘Pythagorean’ Theorem?

by

Amreen Moledina

A thesis submitted in conformity with the requirements for the degree of Master of Science - Mathematics

Department of Mathematics University of Toronto

© Copyright by Amreen Moledina 2013

ii

Should the Pythagorean Theorem Actually be Called the ‘Pythagorean’ Theorem?

Amreen Moledina

Master of Science - Mathematics

Department of Mathematics University of Toronto

2013

Abstract

This paper investigates whether it is reasonable to bestow credit to one person or group

for the famed theorem that relates to the side lengths of any right-angled triangle, a

theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the

first-documented occurrences of the theorem, along with its first proofs. In addition, proofs

that stem from different branches of mathematics and science are analyzed in an effort to

display that credit for the development of the theorem should be shared amongst its many

contributors rather than crediting the whole of the theorem to one man and his supporters.

iii

Acknowledgments

There are a number of people that I want to acknowledge for your support during this

journey to complete my thesis and degree. Each and every one of you are in my heart. I

begin by thanking God, without whom, none of this would have been possible.

I would like to thank my thesis advisor, Dr. Joe Repka, for always giving me the time to

discuss my ideas and progress, and for providing your insight as to how to improve my

paper. Your prompt, professional advice and feedback was invaluable and truly inspiring,

and certainly was a major reason as to why my paper was completed successfully. I am

truly blessed to have had the opportunity to work with you and receive guidance from such

a helpful, intelligent man; I cannot thank you enough for everything.

I also would like to extend my sincere thanks to my friend and colleague, Eric Corlett, for

being my sounding board with respect to my mathematical analysis and ideas, and for

having the faith and confidence in me and my work. Thank you for all of your time and

efforts.

I would like to send a warm thank you to all of my professors at the University of Toronto.

It has been an honour and pleasure to learn from you all. Thank you for imparting your

knowledge on me to equip me with a solid foundation that played a significant role in the

completion of this thesis.

I cannot overlook the support of my wonderful colleagues at Emily Carr S.S., past and

present. You have become a second family to me, and I thank you for your encouragement

iv

and sharing in the joy of seeing this thesis to its completion. I am very grateful to work

alongside all of you.

I especially need to thank my family. To my parents: you have been there with me

throughout everything, and have provided more love, support, encouragement and prayers

than anyone could ever ask for or expect. The belief you displayed in me was the

motivation and courage I needed and relied upon. Words cannot express my gratitude for

everything you have given to me. To my grandmothers: I am grateful for all of your love

and countless prayers. To my sisters, brothers and niece; thank you for sharing every

moment of this journey with me and being my biggest fans. I also want to extend my

appreciation to my new family for rooting me on towards the finish line and being so proud

of me, offering your endless support, prayers and love. To my love, Nadim: thank you for

joining me on this road. From listening to me, offering me your opinions, spending

countless hours reviewing my numerous drafts, and for all of your support and work

towards finalizing and submitting my paper; I appreciate it and love you dearly. The smiles

and laughs you brought to me when I needed to pull my mind away from my thesis was

more therapeutic than I ever realized; thank you for everything.

To all of my family and friends who supported me throughout this journey: know that your

support has been vital and always sincerely appreciated. You have often been my rock,

biggest supporters and advisors, whether you did so vocally or silently, and I truly am

grateful. Whether I needed a push to go to the library or a person to vent to when I hit a

road block, or help with editing, you were by my side and behind me with encouragement,

time, smiles, solutions and most of all, love. I am truly blessed to have such amazing people

v

around me. I love you all dearly, and while words cannot do justice to your efforts, rest

assured that I appreciate each and every one of you for every single thing that you did for

and with me.

vi

Table of Contents

Abstract………………………………………………………………..………………………………………………………..ii

Acknowledgements…………………………………………….………………………………………………………….iii

List of Figures…………………………………………………………………………………………………………………v

Introduction………………………………………………………...…………………………………………………………1

Chapter ONE: History…………………………………………...…………………………………………………………1

1.1 Greece ~ 530 BCE (The Pythagoreans)…….……………………………………………..………1

1.2 Mesopotamia ~ 1800-1600 BCE (The Babylonians)…………………………………………6

1.3 Egypt ~ 1300 BCE………………………………………………………………………………...………10

1.4 India ~ 600 BCE……………………………………………………………………………………………11

1.5 China ~ 6th Century BCE……………..…………………………………………………………………13

1.6 Euclid’s Elements ~ 300 BCE………………………………………...………………………………16

Chapter TWO: Types of Proofs………………………………………………………………………………………24

2.1 The Other Proofs…………………………………………………………………………………..………24

2.1.1 Quarternionic Proofs……………………………………………………………………….25

2.1.1.1 Cosine of an Angle……………………………………………………...………25

2.1.1.2 Linearity of the Dot Product……………………………………….………27

2.1.1.3 Back to the ‘Pythagorean Theorem’…………………………….………28

2.1.2 Dynamic Proof…………………………………………………………………………..…….29

2.1.3 Calculus Proof…………………………………………………………………………………32

Chapter THREE: Conclusion…………………………………………………………………………………………..35

References……………………………………………………………………………………………………………………39

vii

List of Figures

Figure 1: Side lengths of a right-angled triangle……………..…………………………………………………2

Figure 2: Square ABCD……………………………………………………………………………………………………2

Figure 3: Square ABCD with interior Square KLMN…………………………………………………..………3

Figure 4: Figure 3 with copied and translated triangles…………………………………………………….3

Figure 5: Figure 3 with non-45-45-90 triangles………………………………………………………………..4

Figure 6: Translated triangles NC and L within square ABC ………………………..………4

Figure 7: Map of Mesopotamia…………………………………………………………………………………...……6

Figure 8: Clay Tablet YBC 7289……………………………………………………………………………………….6

Figure 9: Translated Clay Tablet YBC 7289………………………………………………………………………7

Figure 10: Clay Tablet Plimpton 322………………………………………………………………………………..8

Figure 11: Translation of Plimpton 322……………………………………………………………………………9

Figure 12: Arbitrary Right-angled triangle…………………………………………………………………..…11

Figure 13: Square with side length a + b………………………………………………………………………..12

Figure 14: Hsuan-thu iagram………………………………………………………………………………………13

Figure 15: Redrawn hsuan-thu………………………………………………………………………………………14

Figure 16: One triangle of hsuan-thu……………………………………………………………………………...14

Figure 17: imensions of interior square formed in Figure 15………………………………………..15

Figure 18: iagram to accompany proof of the lemma……………………………………………………17

Figure 19: Diagram highlighting BA and FAC…………………………………………………………...18

Figure 20: Diagram highlighting BA and CA …………………………………………………………...19

Figure 21: Diagram highlighting FAC and FA ……………………………………………………………20

Figure 22: Diagram outlining the construction of areas BCN and BKE ………………………..21

Figure 23: ABC with altitude C …………………………………………………………………………………..22

Figure 24: Two similar triangles, ABC and EF………………………………………………………..…25

Figure 25: ABC on the unit circle………………………………………………………………………………….26

Figure 26: A , B , C and A B …………………………………………………………………………………………...27

Figure 27: pro A …………………………………………………………………….……………………………………27

Figure 28: Outlining the various projections………………………………………………………..…….….28

Figure 29: ABC with right-angle at C……………………………………………………………………………28

viii

Figure 30: Configuration of a particular system……………………………………………………………...29

Figure 31: The same configuration as Figure 30 but with OP T and O OT ………………31

Figure 32: Region R………………………………………………………………………………………………………32

1

Introduction

The Pythagorean Theorem is arguably one of the most fundamental theorems in

mathematics. It has far-reaching applications in nearly all branches of science1 and is the

foundation of trigonometry2. The theorem, which commonly states, “the sum of the squares

of the legs of a right triangle is equal to the square of the hypotenuse” is named after

Pythagoras of Samos, the famed Greek mathematician. This paper endeavours to illustrate

that this theorem, which bears Pythagoras’ name, should be credited instead to several

authors, as evidenced by various proofs that have been discovered in different branches of

mathematics. Furthermore, it will be argued that he may not have been the first to

demonstrate the universal validity of it. This paper will begin by analysing the

developments that Pythagoras made with respect to this theorem and then continue to

examine where else this theorem has surfaced through history and across the globe. This

will be followed by an examination of what a valid proof is and the different types of proofs

possible in an effort to discover who should be credited with proving the theory we have

come to know as the ‘Pythagorean Theorem’.

Chapter ONE: History

1.1 Greece ~ 530 BCE (The Pythagoreans)

The Pythagoreans were an elite group of scholars founded by Pythagoras of Samos, after

whom both the group and the theorem at the centre of this paper were named. Although

said to have been born in Samos, Pythagoras did not found his group there; rather he

founded it in the Greek outpost of Croton. This group is famous for its theorem that relates

the length of the sides a and b of a right-angled triangle to its hypotenuse c, by equating

.3

1 Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007) p. xi. 2 Posamentier, Alfred S., The Pythagorean Theorem, (Amherst: Prometheus Books, 2010) p. 37. 3 Henceforth, a, b and c will retain these meanings unless otherwise stated.

2

Figure 1: Side lengths of a right-

angled triangle.

However, the question remains: did they prove it for all cases – and enough so for the

revolutionary theorem to be named after their group and their leader? Let us analyse this

by discussing what evidence has been reported by later generations of writers in regards to

the Pythagoreans’ knowledge of the theorem.

The following is a geometric proof of the specific case of the 45-45-90° right isosceles

triangle, which some evidence suggests can be attributed to the Pythagoreans.

Consider the square ABCD:

Figure 2: Square ABCD.

A B

C D

a

b

c

3

Determine the midpoints of each edge of the square, and label them K, L, M and N. Join

these midpoints to their adjacent and opposite sides.

Figure 3: Square ABCD with

interior Square KLMN.

With this, the square KLMN is formed and the square is split into 4 congruent triangles.

Consider two of these triangles: copy, move and combine them such that they form two

more squares along and .

Figure 4: Figure 3 with copied and

translated triangles.

A B

C D

K

L

M

N

A B

C D

K

L

M

N

4

Examining , in Figure 4, specifically the areas of the squares formed at each of its

sides it can be seen that . This proves the ‘Pythagorean Theorem’ for

right-angled isosceles triangles.

What about the general case? Did the Pythagoreans have a proof for it? Indeed there is a

general proof credited to Pythagoras which is as follows:

Consider Figure 3 again except with square KLMN tilted, such that it does not form 45-45-

90° triangles.

Figure 5: Figure 3 with non-45-45-

90 triangles.

By translating and to form rectangles with and , the following is

obtained.

Figure 6: Translated triangles

and within square

ABCD.

A B

C D

K

L

M

N

R

S

A B

C D

K

L

N

P

5

Note that by extending the line segment originating at K until it intersects with , two

squares are formed, where and . Consider

: the square of the length of is equivalent to the area of the square LDPS and the

square of the length of is equivalent to area of the square CNRP. The sum of these two

areas is equivalent to the area of the original square KLMN, seen in Figure 5. Upon closer

examination, this is square formed at the hypotenuse of , which leads to

.

Although this is an intuitive proof, it is a valid one and from this it seems logical that

Pythagoras should get the credit; however, the original proofs and the circumstances

around which they were discovered by Pythagoras have not survived. This is due to the fact

that the Pythagoreans did not believe in writing their discoveries down; rather they only

believed in the oral transmission of knowledge. What evidence we have of their knowledge

is written by writers from generations later and thus is susceptible to embellishments and

misunderstandings as well. To add, in honour of their leader, history suggests that the

Pythagoreans had a tendency to credit many of their discoveries to Pythagoras himself,

potentially leaving the true discoverer a mystery.4 It is also interesting to note that before

settling in Greece, Pythagoras travelled the major centres of the civilized ancient world.

Thus, one could suggest that it is possible for some of the knowledge of the theorem that

bears his name to have been acquired, rather than discovered, during these travels. It is for

these reasons that it is a worthy task to delve back into history to see if this theorem was

proven before the Pythagoreans. We start with the Babylonians.

4 Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007) p. 18.

6

1.2 Mesopotamia ~ 1800-1600 BCE (The Babylonians)

Figure 7: Map of Mesopotamia,

the land that extended between

the Euphrates and Tigris Rivers;

Corresponds to Modern-day Iraq.

Home of Old Babylon5

There is evidence that the people of the Old Babylonian period had some knowledge of

what was later called the ‘Pythagorean Theorem’. This evidence dates as far back as the

Hammurabi dynasty (from 1800-1600 BCE). Evidence is found in the form of a clay tablet

referred to as “YBC 7289” (i.e. tablet #7289 of the Babylonian Collection at Yale

University).

Figure 8: Clay Tablet YBC 72896

5Roosevelt, Kermit. War in the Garden of Eden. New York: 1919. HTML file.

http://www.gutenberg.org/files/13665/13665-h/13665-h.htm, p. 8. 6 Image: Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007) p. 5.

7

The tablet depicts cuneiform character etchings over a square with its two diagonals. These

characters are in fact part of the Babylonian sexagesimal number system. The following

figure dissects the tablet in more detail by translating the sexagesimal numbers.

Figure 9: Translated Clay Tablet

YBC 72897

The square depicted above has a side length of 30 (1) and diagonal length of 42.426389

(3). The number 1.414213 (2) suggests the Babylonians had knowledge of the

‘Pythagorean Theorem’. The reason that this is evidence that the Babylonians knew of the

theorem is that . 2 2. 2 9. Working from the ‘Pythagorean Theorem’

, call the length of the diagonal of the square d; this is hypotenuse of the right

angled triangle with the legs, a and b, being the side lengths of the square. Keeping in mind

that √2 . 2 , the Pythagorean Theorem simplifies to:

2

√2

√2

7 Original Image: Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007) p. 5.

With translations insert by the author of this paper.

2

. 2 (2)

2 2

2. 2 89 (

30 (1)

8

This is essentially what the Babylonians demonstrated except with actual numbers,

generalized arithmetic and through false proposition8. Rudman states that generalized

arithmetic is after all elementary algebra in that it utilises an algorithm that follows a

sequence of arithmetic operations, as opposed to random guessing.

This is not the only evidence that has been found that supports the position that the

Babylonians were aware of the relationship between the side lengths of a right angled

triangle. Other the evidence was found in the form of Plimpton 322 (tablet #322 in the G. A.

Plimpton Collection at Columbia University).

Figure 10: Clay Tablet Plimpton

3229

This 12.7 by 8.8 cm tablet clearly outlines a four-column table of numbers. As one can see, a

portion of the upper leftmost column of the tablet is broken. The following is a table that

translates the etching found on the tablet into our modern base-10 numbers. 10

8 Here the concept of false proposition is used when the problem is not solved directly. This is to avoid solutions involving variables and avoid large numbers. For example, if we have a right-angled triangle that has a hypotenuse of 10 and one side length is ¾ of the other, then we can use false propositions to find the unknown side lengths. Assume one side is 1 (false proposition) and the other false side is ¾. The false

diagonal is √( (

)

. But the true diagonal is √ . Thus the

8. Therefore one side is 8 8 and the other is 8

.

9 Image: Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007) p. 8. 10 Translation composed from Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press,

2007) p. 7-11. as well as Rudman, Peter S., The Babylonian Theorem, (Amherst: Prometheus Books, 2010) p.

93-96. ‘Typos’ and interpretations that account for context have been made in the translation.

9

Figure 11: Translation of Plimpton

322

So the question is, how does this show the Babylonians’ knowledge of the ‘Pythagorean

Theorem’? By analysing the table, one will notice that the fourth column merely labels each

row of the table from 1 to 15. Columns two and three represent two sides of a right-angled

triangle, the shortest side and the hypotenuse, respectively. Further analysis will uncover

that these represent two integers that form what is now known as a ‘Pythagorean Triple’.

In other words, a ‘Pythagorean Triple’ is a set of three integers that form right-angled

triangles. The third integer of each ‘Triple’ is not explicitly outlined in the table, rather is

implied in the first column. The first column is the squared ratio of the number found in

column three divided by the last integer of that ‘Triple’ which is not found in the table. This

strongly suggests that the scribe must have had the last integer of each set in order to

formulate the first column. This demonstrates once again that the Babylonians did have a

working knowledge of the theorem at the centre of this paper.

Although these discoveries date back to well before Pythagoras, are they sufficient to call

the famed theorem the ‘Babylonian Theorem’? This is hardly the case, as neither etching

proves the theorem for the general case; rather they show the relationship of specific cases.

Col. I - … diagonal …

Col. II – solving number of the width

Col. III – solving number of the diagonal

Col. IV – its name

[1].983 119 169 1 [1].949 3367 4825 2 [1].919 4601 6649 3 [1].886 12709 18514 4 [1].815 65 97 5 [1].785 319 481 6 [1].720 2291 3541 7 [1].692 799 1249 8 [1].643 481 769 9 [1].586 4961 8161 10 [1].563 0.75 1.25 11 [1].489 1679 2929 12 [1].450 161 289 13 [1].430 1771 3229 14 [1].387 56 106 15

10

Moreover, tablet YBC 7289 is an example of only an isosceles triangle. Thus, there is not

enough evidence to definitively call this theorem the ‘Babylonian Theorem’, and so we must

look elsewhere. With that, we now move 500 miles to the southwest of Mesopotamia, along

the Nile Valley to the ancient civilization of Egypt.

1.3 Egypt ~ 1300 BCE

The two civilizations of Mesopotamia and Egypt co-existed in harmony from 3500BCE to

the time of the Greeks. It is not surprising that evidence of the ‘Pythagorean Theorem’ is

also found in the ancient civilization of Egypt as well. The first piece of evidence is found in

so called ‘Harpedonapts’ or ‘rope stretchers’. These rope stretchers had ropes with 12

equidistant knots that they used to form right-angled triangles with side lengths of 3, 4 and

5. They used these ropes to rebuild rectangular farming fields on the banks of the Nile after

the annual floods.11 The second piece of evidence is found on a fragment of the Berlin

Papyrus 6610 #1. On this fragment of papyrus, which dates back to 1300BCE, the following

problem is written:

Two quantities are given: one is ¾ of the other. The sum of the squares of the

quantities is 100. What are the quantities? (Rudman, 2010)

The Egyptians also used the concept of false proposition, as the Babylonians did, to arrive

at a solution with the quantities being 6, 8, and 10 – another Pythagorean Triple. Although

the result is a Pythagorean Triple, nowhere on the papyrus fragment is there evidence to

suggest that the quantities in question relate to the side lengths of a right-angled triangle,

nor is there a diagram to suggest as such. Rudman states, however, that there was such a

triangle, due to the similarities between the Egyptian and Babylonian calculations.

However, we are forced to draw the same conclusion as with the Babylonians – these two

pieces of evidence are not enough to suggest that the Egyptians should be given credit for

the famed theorem.

11

Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007).

11

1.4 India ~ 600 BCE

Evidence of knowledge of the ‘Pythagorean Theorem’ can be found in a section of literature

called the Śulbasūtras, which comes from the Vedic era. The Śulbasūtras was a collection of

material that described the intricate sacrificial system of the priests – the bearers of the

religious traditions – which eventually led to Hinduism. It is in the Baudhāyana

Śulbasūtras, from around 600 BCE, where we find the following:

The areas of the squares produced separately by the length and the breadth

of a rectangle together equals the area of the square produced by the

diagonal. This is observed in rectangles having sides 3 and 4, 12 and 5, 15

and 8, 7 and 24, 12 and 35, 15 and 36.12

A formal geometric proof of this, however, did not come until the mid-sixteenth century.

Jyesthadeva wrote a proof for this in the Yuktibhāsā. It is similar to the following

explanation.

Consider the following right-angled triangle,

Figure 12: Arbitrary Right-angled

triangle

12 Katz, V. A History of Mathematics an Introduction (3rd ed.). (Boston: Addison-Wesley, 2009).

a

b

c

12

Create four copies of this right-angled triangle and arrange them to form a square of side

length a + b,

Figure 13: Square with side length

a + b

The total area of the outer square formed is ( (1).

The outer square’s area is also equal to the area of the interior square13, , plus the four

triangles, (

) 2 (2).

Equating (1) and (2), we get the following:

( 2

Expanding and rearranging:

2 2

13

Note that the interior quadrilateral formed is indeed a square because along the edge of

the outer square where the sides of two of the triangles meet three angles combine to form

a straight angle. Two of these angles are angles found in the original triangle in Figure 3.1.

Specifically, neither is the 90° angle in the right-angled triangle and thus by the sum of

interior angles of a triangle theorem these two angles must sum to 90° as well. Returning

back to the three angles that meet to form a straight angle, we find that the third angle, the

interior angle of the quadrilateral, must then equal 90°. This, combined with all the sides of

this quadrilateral being equal to c, proves that it is indeed a square.

a

b

13

The Baudhāyana Śulbasūtras does date back to before Pythagoras, but once again it shows

knowledge of the relationship yet no proof. The geometric proof did not come until the

sixteenth century. In addition, this proof does not directly relate the areas of the

and ; rather it uses the area of the square with side length and the relation,

is just a happy by-product. Nevertheless, one could argue that Pythagoras was

inspired by what was written in Baudhāyana Śulbasūtras, during his travels before settling

in Croton. As result, perhaps it was not his original thought. In and around the same time as

Baudhāyana Śulbasūtras, if we move further east to China, there we also find evidence of

knowledge of the famed theorem.

1.5 China ~ 6th Century BCE

The theorem in question also appears in the oldest Chinese mathematical text, Chou Pei

Suan Ching, (The Arithmetical Classic of the Gnomon and the Circular Paths of

Heaven). Some estimate that the Chou Pei Suan Ching ranges as far back as 1100 B.C.,

however it is more widely accepted that the text is from the time of Confucius, the sixth

century BCE.14 The text reflects the mathematical knowledge accumulated in China until

that time. A proof for this theorem is described in a diagram called the hsuan-thu, see

Figure 14.

Figure 14: Hsuan-thu Diagram15

14

Swetz, Frank J. and Kao, T. I., Was Pythagoras Chinese (The Pennsylvania State University Press, 1977)

15 Abraham, Ralph. "The hsuan-thu diagram." Ralph Abraham. 25 April 1996. Ralph Abraham . 30 August 2013. <http://www.ralph-abraham.org/courses/math181/math181.S96/lectures/lecture.3w/china/diagram.html>.

14

To better illustrate the proof outlined in the above figure, consider Figure 15.

Figure 15: Redrawn hsuan-thu

Figure 16 shows one of the right-angled triangles from Figure 15 labeled with a, b and c

representing the side lengths of the right-angled triangle:

Figure 16: One triangle of hsuan-

thu

The area of two of these triangles is equal to 2 (

) , thus the area of all four

triangles is 2 . Consider the interior square16 formed in Figure 15, its dimensions are

( by ( , thus its area is ( .

16

Using the straight angle theorem it is clear that the interior quadrilateral is a square.

a

b

c

15

Figure 17: Dimensions of interior

square formed in Figure 15

Thus the total area of all of the triangles and the interior square is 2ab + (a – b)2. Notice,

however, that the square formed from all of the triangles, and the interior square has a side

length of c, therefore it has an area of c2. Hence,

2 (

Expanding,

2 2

The mathematical statement now commonly known as the ‘Pythagorean Theorem’ is thus

before us.

From this, we are led to conclude that this theorem should be credited to the Chinese and

should be called the Gougu theorem, as it was, according to this evidence, founded in

ancient China. It should be called the Gougu theorem because it is in ancient Chinese

shadow reckoning where one finds the use of this theorem. “Gu” means vertical gauge and

a – b

a

b

16

“Gou” refers to the shadow cast by the sun.17 Thus we have shown that knowledge of this

theorem predates the Pythagoreans, as does a proof at least as valid as the one attributed

to them (in that they are both intuitive). However, neither strictly follows the current

model for presenting proofs, which did not come about until Euclid and his famed work,

Elements. Thus, it is now to Euclid we turn to analyze what constitutes a proof.

1.6 Euclid’s Elements ~ 300 BCE

Elements is Euclid’s famous work that boasts 465 theorems in subjects ranging from

geometry, number theory, and (non-symbolic) algebra. It encompasses the knowledge of

mathematics at the time of its writing. This famous work is the model for mathematical

writing to this day with its style of definitions, axioms, theorems, and proofs.

It is within Euclid’s famed Elements that we find two proofs for the theorem being

investigated. Before these two proofs are presented in this paper, it is important to

understand the format of Elements. Elements is a work that was written in thirteen books,

which we now call parts. It has no introduction or preface; rather it gets to the heart of the

matter from the first sentence. It opens with twenty-three definitions of fundamental

concepts; these are followed by what we now refer to as axioms. These axioms are split

into two groups of five. The first group deals with geometric concepts, “postulates” and the

second, “common notions” deal with arithmetic concepts.

The first proof of presented in Elements is found in I 47 (Book I, Proposition

47)

The square described on the hypotenuse of a right-angled triangle is equal to the

squares described on the other two sides. (Mackay, 1885)

Before delving into the proof of this, Euclid presents the proof to the following lemma:

17

Posamentier, Alfred S., The Pythagorean Theorem, (Amherst: Prometheus Books, 2010) p. 31.

17

In a right triangle, the square on a side has the same area as the rectangle formed by

the hypotenuse and the perpendicular projection of the same side on the

hypotenuse. (Maor, 2007)

To better understand what this lemma says, consider Figure 18.

Figure 18: Diagram to accompany

proof of the lemma.

18

What the lemma is saying is that area of square ACHG is equal to the area of rectangle

ADEF. The proof of this is as follows; is a right triangle with a right angle at C. Square

ACHG is built alongside . As a result 9 , thus 8

making an extension of . is the perpendicular projection of onto , the

hypotenuse. Rectangle ADEF is built such that and are equal to and

perpendicular to it as well. Connect to form and connect to form .

Figure 19: Diagram highlighting

and

19

by SAS (side-angle-side) as

∵ they are both sides of square ACHG

∵ 9

and ∵ 9

by the construction of rectangle ADEF

But by Proposition 38 of Book I of Elements, which states

Triangles on equal bases and between the same parallels are equal in area. (Mackay,

1885)

Here the line segment that connects vertices B and C is parallel to their common base of AG.

Figure 20: Diagram highlighting

and

20

Using the same Proposition, it can be found that ,

Figure 21: Diagram highlighting

and

Thus, (1) since is half the area of ACHG and is half

the area of ADEF.

21

Figure 22: Diagram outlining the

construction of areas BCNM and

BKED

This same process can be replicated for the other side of to show that

(2)

Adding (1) and (2),

– QED.

The second instance where the theorem makes an appearance is VI 31 (Book 6 Proposition

31).

In right-angled triangles the figure on the side subtending the right angle is equal to

the similar and similarly described figures on the sides containing the right angle.

(Maor, 2007)

22

Notice that the only differences between this Proposition and I 47 is that in this

Proposition, “figure” replaces “square” and “similar and similarly described figures”

replaces “squares”. Essentially VI further generalises I 7. The proof that accompanies

this Proposition utilises similarity and starts off with a right-angled with a right

angle at vertex C. From vertex C line segment is formed such that is perpendicular to

and D is on .

Figure 23: with altitude

Clearly, ACB = ADC = 90°

As well DAC + DCA + ADC = 180°

DAC + DCA + 90° = 180°

DAC + DCA = 90°

But DCB + DCA = 90°.

Thus DCB = DAC

Therefore

From this similarity we have,

and

(1)

Cross multiplying, we get

and (2)

Adding equations (1) and (2), we have

(3)

A B

C

D

23

Factoring equation (3),

( ) (4)

But so equation (4) becomes,

So here we have two examples of proofs for the ‘Pythagorean Theorem’ given in the

accepted model.

Thus, until this point in this paper, we have seen an (intuitive) proof of the theorem that

predates Pythagoras and we have discussed the method by which a proof should be

presented, but what about the different types of proofs? It is the opinion of the author of

this paper that delving into the other types of proofs is a valid endeavour in trying to

discover what this renowned theorem should be known as, since different types of proofs

represent different lines of mathematical thought.

24

Chapter TWO: Types of Proofs

Loomis states in his book, The Pythagorean Proposition, that there are four types of proofs

for the ‘Pythagorean Theorem’, or as he called it, “The Pythagorean Proposition” (Loomis,

1927):

1. Algebraic Proofs – based on linear relations

2. Geometric Proofs – based on comparison of areas

3. Quarternionic Proofs – based on vector operations

4. Dynamic Proofs – based on mass and velocity18

He further states that there are no trigonometric proofs. Posamentier in The Pythagorean

Theorem further elaborates that this is because “Trigonometry … is essentially based on

this important theorem” and thus “proofs using trigonometry … would result in circular

reasoning” (Posamentier, 2 .

Using the definition that geometric proofs compare areas, all the proofs presented here,

other than that of Elements VI 31, are therefore geometric. Elements VI 31 however results

from similar triangles, which Loomis categorises as algebraic in nature. What about the

other types of proofs?

2.1 The Other Proofs

For this proportion of the paper, the author will explore each type of proof defined by

Loomis as well as a proof derived from calculus. Since geometric and algebraic proofs have

already been presented in this paper, this section will focus on the remaining two types

listed by Loomis, Quarternionic and Dynamic, and then the Calculus proof.

18

Loomis, Elisha S., The Pythagorean Proposition, (Ohio: Masters and Wardens Association of the 22nd Masonic District of the Most Worshipful Grand Lodge of Free and Accepted Masons, 1927) p. 7.

25

2.1.1 Quarternionic Proofs

Recall that Loomis defines quarternionic proofs as those that involve vector analysis.

Vector analysis is the branch of mathematics that assigns both magnitude and direction to

line segments. Loomis presents four such proofs for the ‘Pythagorean Theorem’ but also

states that these are not the only proofs possible. The proof given here is the first that

Loomis presents, and it is credited to Arthur Hardy. However, before presenting this proof

it is necessary to cover to concepts: the definition of the cosine of an angle and a proof of

the linearity of the dot product. These concepts are well known in mathematics but they

are often based on the ‘Pythagorean Theorem’, thus the author wishes to define them

without it so that the proof that follows is not circular in nature.

2.1.1.1 Cosine of an Angle

Consider two similar right-angled triangles, see Figure 24.

Figure 24: Two similar triangles,

and with right-angles

at C and F, respectively.

A C

B

θ

D F

E

θ

26

We define the cosine (cos) of an angle, θ as

Thus from we have

and from we have

but

since and therefore

. However we know this already from Elements

VI 4. As a result

is well defined.

What of the extreme cases where or 9 ? For this we reconsider such

that the length of and it is placed within the unit circle centred at the origin (see

Figure 25).

Figure 25: on the unit circle

with vertex A at the centre, origin

and vertex B falling on the circle.

For the case of

where , therefore .

For the case of

where 9 , therefore 9 .

A C

B 1

27

2.1.1.2 Linearity of the Dot Product

The proof that Loomis presents uses the idea that the dot product is linear; in other words

( . This will be demonstrated using projections. Given vectors ,

and , such that they are arranged as in Figure 26.

Figure 26: , , and

Define cos ( , where is the magnitude of and is the angle between

and . Note that results in a scalar and is the size of the projection of along –

to see why, note the definition of cosine given previously. Extend a line from the tip of

down to such that it meets at a right angle and thus forms a right-angled triangle with

, the length of the adjacent to will be , see Figure 27.19

Figure 27:

So ( .

Thus it is sufficient to show [ ( )]| | | | (1) which then

implies ( . Simplifying (1) by factoring we obtain,

( ) , which is what we will endeavour to show. Consider

Figure 28.

19

Recall that cosine has already been defined without the ‘Pythagorean Theorem’

𝐴

�� 𝐴 ��

𝐶

𝐴

�� 𝐴 ��

𝐶

𝑝𝑟𝑜𝑗𝐶𝐴

28

Figure 28: Outlining the various

projections

As a result from Figure 28 it can be seen that ( ) ,

and

. Since , , and is a rectangle,

. Also

,

which implies ( ) and so the dot product is linear, at least

for addition, and we have ( .

We are now ready for the proof presented by Loomis. Consider a right-angled, triangle

formed by three vectors, , and , see Figure 29.

2.1.1.3 Back to the ‘Pythagorean Theorem’

Figure 29: with right-angle

at C

Consider :

| || | 9

Similarly,

�� 𝐴

𝐶 𝐵 𝐴

𝐶

𝐴

�� 𝐴

��

𝐶

𝑝𝑟𝑜𝑗𝐶𝐴 𝑝𝑟𝑜𝑗𝐶 ��

𝑝𝑟𝑜𝑗𝐶(𝐴 ��

𝑃 𝑃

𝑃 𝑃 𝑂

29

By vector addition we have . Dotting with itself, we get:

( (

By definition, when dotting a vector with itself we obtain the vectors’ magnitude squared.

In other words:

| |

| | | |

– QED

2.1.2 Dynamic Proofs

The final type of proof for the ‘Pythagorean Theorem’ that Loomis discusses is the Dynamic

Proof. He defines these proofs as those that stem from problems involving masses and their

velocities, thus they are proofs that utilise the laws of physics. Once again, one such proof

will be presented here. This proof requires a basic understanding of kinetic energy and

angular velocity.

Consider Figure 30,

Figure 30: Configuration of a

particular system

O

P

S

T

30

Let the distances of , and . Let there be two massless rods, one

which spans along and the other along . The rod along can rotate about O and the

midpoint of is P. At S and T lie two equal masses, m and m’, respectively, again each

equidistant from P since , call this distance r.

When the system revolves about O as a centre, at the point P there will be a linear velocity

of

where ds is the portion of the arc described in time dt, da is the

differential angle through which the rod along turns and W is angular velocity.

Point P is the centre of mass between m and m’ as they are equidistant from point P and

have equal masses. Assuming that the rod that these masses are at the ends of and that

spans is free to rotate about point P, then ’

(2 , where E’ is the

combined kinetic energy between m and m’. However, this is not the total kinetic energy

since the rod along also revolves around O with angular momentum W. Rotational

kinetic energy is in addition to any translational kinetic energy the object has if its center of

mass is moving.20 Thus if the rod along is said to be attached at P but free to move about

it, then this rod also has angular velocity about P equal to W. The additional kinetic energy

posed by this angular velocity is then equal to ”

(2 . Therefore the total kinetic

energy, E is:

2(2

2(2

( (1)

Let be the individual kinetic energy of mass m situated at S , then:

20 Knight, Randall D., Physics for Scientists and Engineers with Modern Physics: A Strategic Approach. (San Francisco: Pearson Addison Wesley, 2004)

31

2

Similarly the individual kinetic energy of mass m’ situated at T is:

2

Therefore from this perspective the total kinetic energy must be:

2

2

( (2)

Equating equations (1) and (2):

(

2 (

( (3)

Consider if however then see Figure 31

Figure 31: The same configuration

as Figure 30 but with and

Equation (3) then becomes

(4)

But X is the length of the hypotenuse of right-angled triangle OSP and R and r are the

lengths of the other two sides. Therefore equation (4) is the ‘Pythagorean Theorem’.

O

P

S

T

32

2.1.3 Calculus Proof

The proof stemming from the concepts of calculus presented here is a result of the

divergence theorem which states the integral over a region, of the gradient of a scalar

function, is equivalent to the integral over the boundary of of that same function

multiplied by the unit normal vector directed outside of :

∬

∫

21,22

Define the region such that it is a right-angled triangle with a hypotenuse of length and

the other two sides of length and respectively. Place this region on the Cartesian plane

with the right-angle at the origin, see Figure 32.

Figure 32: Region R

21 Note that a line integral of a vector function over a curve ( ( , ( , ∫

∫ |

|

Expanding [

,

]

|

,

| but [

,

]is just [

,

]under a 90° rotation. Thus |

,

| |

,

| . Setting

we have

∫

∫

∫

[

,

]

|

,

| ∫

[

,

]

|

,

| ∫

[

,

]

|

|

∫ [

,

]

|

|

|

|

∫ [

,

] . Therefore the vector lengths do not need to be known in the proof of the divergence

theorem, and so we can apply it without using the ‘Pythagorean Theorem’. 22 Consider 2-dimensions, , a scalar function and a vector function . Let , and , .

Looking at first by the divergence theoerem, ∬ ∫ (1). Also

. Then left

side of (1) becomes ∬ ∬

and the right side becomes ∫ ∫ , [ , ]

∫ . Thus ∬

∫ . Similarly, ∬

∫ . Consider ∫ ∫ [ , ]

∫[ , ] ∬ [

,

] ∬ . Therefore ∬ ∫ as desired.

33

Let the function be defined as , then so the divergence theorem states:

∬

∫

∫

∫

∫

∫

For :

( where ,

where | |

,

∫

∫ ,

, ∫

,

Similarly for

∫

∫

∫

where is the normal to side that points away from

Lastly for

∫

∫ ,

, ∫

,

Thus

, ,

, ,

,

Since the vectors are equal to another then their magnitudes must also be equal.

34

,

√

– QED23

These three proofs, as well as the proofs presented earlier in this paper, demonstrate that

there are proofs for the ‘Pythagorean Theorem’ that originate from different branches of

mathematics. Not only do proofs exist from many branches of mathematics, but the proofs

are plentiful. It is not the aim of this paper to present all of them; rather to present a

handful to illustrate that the Pythagoreans were, and are, not the only ones to prove the

theorem.

23 Brodsky, Alexander. "Part 08 Triangular Balloon and Pythagorean Theorem." Calculus +. 17 April 2012. .

28 August 2013. <https://sites.google.com/site/calcuplus/video-lectures/calculus-iii-2012-spring/lecture-

2012-04-17/part-08>.

35

Chapter THREE: Conclusion

The purpose of this paper was to call into question the legitimacy of naming the theorem at

the centre of this paper after one man, Pythagoras of Samos. It is rare that such

monumental discoveries are made by one person. They usually surface throughout history

and around the globe. As noted earlier in this paper, it is believed that Pythagoras travelled

a fair bit of the ancient world, including Egypt and Persia, where he studied their literature,

religion, philosophy and mathematics. Considering that there is some evidence that the

Egyptians knew the concept of the theorem, Pythagoras could have been influenced by

them to study it further. In addition, once Pythagoras settled and founded his group, his

members, out of respect for him, often credited Pythagoras for their discoveries – leaving

the true authors potentially unknown.24 One could argue that by calling the theorem the

‘Pythagorean Theorem’ we are giving the Pythagoreans credit not Pythagoras himself.

However, even what we know of the proofs presented by the Pythagoreans is questionable,

as they believed in the oral transmission of knowledge. Therefore, the work that is credited

to them is privy to embellishments and editing as they were written years later by those

who may have wished to glorify or embellish them.25 26 27 Neugebauer goes as far as to say

that “traditional stories of discoveries made by Thales or Pythagoras must be discarded as

totally unhistorical”28. This is due to the fact that little is known about Pythagoras or his

work, which is a result of the shroud of secrecy that followed the Pythagoreans and

Pythagoras himself.29

So who should the theorem be named after? The notion of the theorem dates back as early

as the Babylonians, so should it be called the ‘Babylonian Theorem’? It should not be titled

as such, since, even though there is evidence that the Babylonians knew of the relationship

24 Boyer, Carl B. A History of Mathematics. (New York: Wiley, 1968) 25 Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007) 26 Burkert, Walter. Lore and Science in Ancient Pythagoreanism. (Cambridge: Harvard University Press, 1972). 27

Swetz, Frank J. and Kao, T. I., Was Pythagoras Chinese (The Pennsylvania State University Press, 1977) 28 Neugebauer, Otto. The Exact Sciences of in Antiquity. (New York: Dover, 1969) 29

Maor, Eli, The Pythagorean Theorem, (Princeton: Princeton University Press, 2007)

36

between the lengths of the sides of a right-angled triangle, they did not prove it for all

cases. Rather they merely knew of some examples that worked. No proof or demonstration

was given until around the sixth century BCE by the Chinese in the form of an intuitive

proof seen in the hsuan-thu. Heath and Neugebauer both claim that this type of intuitive

proof was completely foreign to the Greeks with Neugebauer going further to say that

Greek mathematics was too primitive.30 31

Swetz and Kao, in their work Was Pythagoras Chinese? explore the authorship of the

‘Pythagorean Theorem’ and explore the possibility that it should be credited to the Chinese

instead. He presents strong evidence that it should be accredited to the Chinese as:

…the hsuan-thu diagram of the Chou pei represents the oldest recorded proof

of the “Pythagorean” Theorem. With simple elegance, the configuration and

implications of this diagram appeal to both intuition and an aesthetic sense

in proving that the sum of the squares of the sides of a right triangle equal

the square of the hypotenuse. (Swetz, 1977)

Although he leans to the notion that credit should be given to the Chinese, he refrains from

staking that claim and instead opts for the reader to choose. It is the opinion of this paper

that although the Chinese did indeed have the hsuan-thu as a proof, it is merely an intuitive

proof as even Swetz and Kao put it. Even though the Chinese proof is intuitive, it still has as

much validity as the Pythagorean proof. The currently accepted concept of how a proof

should be outlined did not come until Euclid and his Elements. As Maor states, “Its terse,

rigorous style-definitions, axioms, theorems, and proofs-is a model for mathematical

writing to this day.” (Maor, 2007) Within Elements, Euclid does present two proofs for the

‘Pythagorean Theorem’ but Elements is a compilation of the mathematical knowledge up

until the time it was written and is not a compilation of Euclid’s own discoveries. As such, it

30 Heath, Sir Thomas L. The Thirteen Books of Euclid’s Elements.3 vols. (Cambridge: The University Press, 1926). 31

Neugebauer, Otto. The Exact Sciences of in Antiquity. (New York: Dover, 1969)

37

would be difficult to claim that the ‘Pythagorean Theorem’ should be named after Euclid

instead.

With Euclid however, not only do we establish the model for the proper format of a proof,

but we also gain the concept that a non-geometric proof for the ‘Pythagorean Theorem’ is

possible. This calls into question ‘what other types of proofs are possible?’ and ‘who

discovered them?’ As Kaplan and Kaplan state in Hidden Harmonies: The lives and times of

the Pythagorean Theorem, “Some people collect Ketchikan beer coasters, some Sturmey

Archer three-speed hubs, others wives or ailments. Jury Whipper collected proofs of the

Pythagorean Theorem. He wasn’t the first…” (Kaplan and Kaplan, 2 They go on to list

others that “collected” proofs for the theorem with the largest “collection” lying with Elisha

Scott Loomis. Within Loomis’ work, The Pythagorean Proposition, he compiled 367 proofs

for the famed theorem with just as many authors, including James A. Garfield, a former

president of the United States of America. Not only did he compile such a number of proofs,

but it is noteworthy that they stem from different branches of mathematics as well.

However, while Loomis may have the largest collection, he by far does not have them all, as

Posamentier even states, “…to present day-there arise what are believed to be new proofs

or demonstrations of the Pythagorean Theorem published in professional journals”

(Posamentier, 2010) Loomis did not ever claim that a calculus proof was even possible

which means that this type, like the one presented in this paper, must have come after he

published his work.

So the question still remains, who should get credit for this famed and fundamental

theorem that relates the side lengths of a right-angled triangle? The Babylonians, for being

the first to show some knowledge of the theorem? The Chinese, for having the first intuitive

proof? Euclid, for presenting the first proof in the rigorous model used today? Loomis, for

collecting such a large number of proofs? Garfield, because he was a US president? It is the

opinion of the author of this paper that it should either be all or none of them, and I submit

that all of the aforementioned authors receiving credit would be more equitable, since

arbitrarily selecting a single person or group and plastering their name on the theorem will

unfairly take credit from some of the parties to whom it is, at least partially, due. It is for

38

this reason that I respectfully state that the theorem ought to be renamed to something

more general, such as ‘the right-angled theorem’. While this would remove an individual’s

name and thus credit from a glance at the theorem’s title, it would avoid the problem of not

giving credit where it is due. And while some may argue that it is unwise to remove the

source for future generations and their understanding of the source of education, I submit

that those interested in the theorem will investigate it, so there is limited to no loss or

unfairness in my proposition. After all, how is it just to give one man credit when so many

have made contributions to the development of this theorem until even today. They have

all added such a great deal to the importance and prominence of the theorem that it is

difficult to bestow such an honour on one man alone. As Sir Isaac Newton even said, “To

explain all nature is too difficult a task for any one man or even for any one age. 'Tis much

better to do a little with certainty, and leave the rest for others that come after you, than to

explain all things by conjecture without making sure of anything.”32

32 Statement from unpublished notes for the Preface to Opticks (1704) quoted in Never at Rest: A Biography

of Isaac Newton (1983) by Richard S. Westfall, p. 643.

39

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