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HISTORY OF
MATHEMATICS
GREECE
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 1 / 140
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The School of Athens
Raphael School of Athens (1509–1511) Vatican
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 2 / 140
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Ancient Greece
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 3 / 140
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Ancient Greece
Next we must speak of the development of this science during
the present era… we say, as have most writers of history, that
geometry was first discovered among the Egyptians and
originated in the remeasuring of their lands. This was necessary
for them because the Nile overflows and obliterates the boundary
lines between their properties. It is not surprising that the
discovery of this and the other sciences had its origin in necessity,
since everything in the world of generation proceeds from
imperfection to perfection. Thus they would naturally pass from
sense-perception to calculation and from calculation to reason.
Just as among the Phoenicians the necessities of trade and
exchange gave the impetus to the accurate study of number, so
also among the Egyptians the invention of geometry came about
from the cause mentioned.
Proclus (5th century CE)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 4 / 140
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Pre-Socratic Thinkers: Thales (b. ca. mid 620s BCE)
Only through later writers including historian Herodotus (5th century
BCE), mathematical commentator Proclus (5th century CE), and
historian Diogenes Laertius (3rd century CE)
“Thales, who had travelled to Egypt, was the first to
introduce [geometry] into Greece. He made many
discoveries himself and taught the principles for many others
to his successors, attacking some problems in a general way
and others more empirically.” (Proclus)
The first to search for a logical foundation of geometrical theorems
Search for explanation of nature based on single ultimate substance
▶ Water
Search for essence and foundations also pervaded to mathematics
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 5 / 140
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Pre-Socratic Thinkers: Thales (b. ca. mid 620s BCE)
Proclus ascribes to him five propositions in geometry:
▶ A circle is bisected by its diameter
▶ The base angles of an isosceles triangle are equal
▶ If two straight lines intersect, the vertical angles are equal
▶ Two triangles are congruent if they have one side and two adjacent
angles equal
▶ Every angle inscribed in a semicircle is a right angle
Idea of an order to mathematics and a rational organisation
Emergence of a deductive sequence
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 6 / 140
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Pythagoras (ca. 560–475 BCE)
Following upon these men,
Pythagoras transformed
mathematical philosophy into a
scheme of liberal education,
surveying its principles from the
highest downwards and
investigating its theorems in an
immaterial and intellectual
manner. He it was who discovered
the doctrine of proportionals and
the structure of the cosmic
figures. (Proclus)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 7 / 140
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Pythagoras (ca. 560–475 BCE)
Founder of a religious and a
philosophical school which
flourished from about sixth
to fourth century CE
Inner circle called
mathematikoi from verb
manthanō ‘to learn’
“All things are number”
Mathematics provides an
explanation for the universe
Majority of scholars question
whether Pythagoras even
discovered the famous
theorem bearing his name
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 8 / 140
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Pythagoras and Incommensurability
Incommensurable magnitudes very important challenge to
Pythagorean dictum ‘all is number’.
Issue: Given any two lines segments, is it possible to find a third that
can mark off the whole of each segment exactly?
Not always! Such a ratio was called alogos: “irrational”
Hippasus of Metapontum is said to have discovered these (and as
legend has it was subsequently thrown overboard from the ship!)
Formal proof in Euclid’s Elements of the incommensurability of the
side and diagonal of the square
▶ Book X
▶ reductio ad absurdum argument
▶ if diagonal and side were commensurable, the same number would be
both odd and even
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 9 / 140
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Plato (ca. 427–348 BCE)
Founder of a very important
center of learning in Athens:
The Academy
Platonic dialogues
Special status of mathematical
knowledge: a paradigm for
philosophical thought
Dialogue Timaeus: ultimate
creator is a mathematician
Platonic forms
Mathematics and its role in
education: The Quadrivium
▶ Arithmetic
▶ Plane and solid geometry
▶ Astronomy
▶ Harmonics
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 10 / 140
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Plato: On mathematical language
The science of geometry is in direct contradiction with the
language employed by its adepts…Their language is most
ludricrous…for they speak as if they were doing something and as
if all their words were directed towards action … They talk of
squaring and applying and adding and the like…whereas in fact
the real object of the entire subject is…knowledge…of what
eternally exists, not of anything that comes to be this or that at
some time and ceases to be.
Republic Book VII (527a)
Geometry: eternal and unchanging items in the world of Being
Dynamic language is not appropriate to describe its operations
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 11 / 140
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Plato: On mathematics and knowledge
Platonic dialogue about virtue: The Meno
Geometrical demonstration is a paradigm for all knowledge acquisition
How is inquiry possible?
Paradox:
▶ If you know what you are looking for, inquiry is unnecessary
▶ If you don’t know what you are looking for, inquiry is impossible
How do we come to know about mathematics?
Theory of recollection: learning is actually recollecting something that
is already innate
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 12 / 140
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Plato: The Meno
The characters:
Socrates Meno Slave boy
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 13 / 140
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Plato: The Meno
Meno asks Socrates about his theory of recollection.
MENO: What do you mean when you say that we don’t learn anything, but
that what we call learning is recollection? Can you teach me that it is so?
SOCRATES: I have just said that you’re a rascal, and now you ask me if I
can teach you, when I say there is no such thing as teaching, only
recollection. Evidently you want to catch me contradicting myself straight
away.
MENO: No honestly Socrates, I wasn’t thinking of that. It was just habit.
If you can in any way make clear to me that what you say is true, please
do.
SOCRATES: It isn’t an easy thing, but still I should like to do what I can
since you ask me. I see you have a large number of retainers here. Call
one of them, anyone you like, and I will demonstrate it to you.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 14 / 140
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Plato: The Meno
Socrates poses the following question to the boy:
Given the side of some square, can you find me the side of a
square that has double the area of the original square?
The boy’s challenge is to find the side of the square whose area is double this
square.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 15 / 140
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Plato: The Meno
SOCRATES: Now boy, you know that a square is a figure like this?
BOY: Yes.
…
SOCRATES: Such a figure could be either larger or smaller, could it not?
BOY: Yes.
SOCRATES: Now if this side is two feet long, and this side is the same, how
many feet will the whole be?
…
BOY: Four
SOCRATES: Now could one draw another figure double the size of this, but
similar, that is, with all its sides equal like this one?
BOY: Yes
SOCRATES: How many feet will its area be?
BOY: Eight.
SOCRATES: Now then, try to tell me how long each of its sides will be. The
present figure has a side of two feet. What will be the side of the double sized
one?
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 16 / 140
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Plato: The Meno
When asked what the side of the square double the area will be, the
boy comes up with a possible solution:
BOY: It will be double, Socrates, obviously.
SOCRATES: You see, Meno, that I am not teaching him anything, only asking.
Now he thinks he knows the length of the side of the eight-feet square.
MENO: Yes.
SOCRATES: But does he?
MENO Certainly not.
…
SOCRATES: Now watch how he recollects things in order—the proper way to
recollect. You say that the side of double length produces the double-sized figure?
Like this I mean, not only long this way and short that. It must be equal on all
sides like the first to get it from doubling the side, that is eight feet. Think a
moment whether you still expect to get it from doubling the side.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 17 / 140
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Plato: The Meno
SOCRATES: Now watch how he recollects things in order—the proper way to
recollect. You say that the side of double length produces the double-sized figure?
Like this I mean, not only long this way and short that. It must be equal on all
sides like the first to get it from doubling the side, that is eight feet. Think a
moment whether you still expect to get it from doubling the side.
The boy’s first attempt—double the side of the original side
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 18 / 140
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Plato: The Meno
Socrates now guides the boy to recollect why this is not the case.
He compels the boy to draw a diagram.
SOCRATES: Let us draw four equal lines using the first as a base. Does this not
give us what you call the eight feet figure?
BOY: Certainly.
SOCRATES: But does it contain these four squares, each equal to the original
four-feet one?
BOY: Yes.
SOCRATES: How big is it then? Won’t it be four times as big?
BOY: Of course.
…
SOCRATES: Then how big is the side of the eight-feet figure? This one has given
us four times the original area, hasn’t it?
BOY: Yes.
SOCRATES: Good. And isn’t a square of eight feet double this one but less than
that?
BOY: Yes.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 19 / 140
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Plato: The Meno
The boy tries again at the insistence of Socrates. He tries a side
length half way between the original square and the square that was
four times too big, in this case, he tries three feet.
SOCRATES: Then the side of the eight-feet figure must be longer than two feet but
shorter than four.
BOY: It must.
SOCRATES: Try to say how long you think it is.
BOY:Three feet
…
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 20 / 140
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Plato: The Meno
SOCRATES: If it is three feet this way and three that, will the whole area be three times
three feet?
BOY: It looks like it.
SOCRATES: And that is how many?
BOY: Nine.
SOCRATES: Whereas the square double out first square had to be how many?
BOY: Eight.
SOCRATES: But we haven’t yet got the square of eight feet even from a three-feet side?
BOY: No.
SOCRATES: Then what length will give it? Try to tell us exactly. If you don’t want to
count it up, just show us on a diagram.
BOY: It’s no use, Socrates, I just don’t know.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 21 / 140
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Plato: The Meno
Having brought the boy to this state of perplexity, Socrates reflects
on the process thus far, and relates it to his ideas about recollection
with Meno.
SOCRATES: Observe Meno, the stage he has reached on the path of recollection.
At the beginning he did not know the side of the square of eight feet. Nor indeed
does he know it now, but then he thought he knew it and answered boldly, as was
appropriate—he felt no perplexity. Now however he does feel perplexed. Not only
does he not know the answer; he doesn’t even think he knows.
MENO: Quite true.
SOCRATES: Isn’t he in a better position now in relation to what he didn’t know?
MENO: I admit that too.
…
SOCRATES: Do you suppose then that he would have attempted to look for, or
learn, what he thought he knew (though he did not), before he was thrown into
perplexity, becomes aware of his ignorance, and felt a desire to know?
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 22 / 140
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Plato: The Meno
Socrates continues his questioning, insistent that he is not teaching,
but rather guiding him.
SOCRATES: Now notice what starting from this state of perplexity, he will discover by
seeking the truth in company with me, though I simply ask him questions without
teaching him. Be ready to catch me if I give him any instruction or explanation instead
of simply interrogating him on his own opinions. Tell me boy, is not this our square of
four feet? You understand?
BOY: Yes.
SOCRATES: Now we can add another equal to it like this?
BOY: Yes
SOCRATES: And a third here, equal to each of the others?
BOY: Yes.
SOCRATES: And then we can fill in this one in the corner?
BOY: Yes
SOCRATES: Then here we have four equal squares?
BOY: Yes
…
SOCRATES: Now does this line going from corner to corner cut each of these squares in
half?
BOY: Yes.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 23 / 140
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Plato: The Meno
The boy discovers the correct solution.
SOCRATES: Now think. How big is this area?
BOY: I don’t understand
SOCRATES: Here are four squares. Has not each line cut off the inner half of
each of them?
BOY: Yes.
SOCRATES: And how many such halves are there in this figure?
BOY: Four.
SOCRATES: And how many in this one?
BOY: Two.
SOCRATES: And what is the relation of four to two?
BOY: Double.
SOCRATES: How big is this figure then?
BOY: Eight feet.
SOCRATES: On what base?
BOY: This one.
…
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 24 / 140
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Plato: The Meno
The boy discovers the correct solution.
SOCRATES: The technical name for it is ‘diagonal’; so if we use that name, it is
your personal opinion that the square on the diagonal of the original square is
double its area.
BOY: That is so Socrates.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 25 / 140
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Plato: The Meno
Socrates and Meno debrief.
SOCRATES: What do you think, Meno? Has he answered with any opinions that
were not his own?
MENO: No, they were all his.
…
SOCRATES: At present these opinions, being newly aroused, have a dream-like
quality. But if the same questions are put to him on many occasions and in
different ways, you can see that in the end he will have knowledge on the subject
as accurate as anybody’s
MENO: Probably.
SOCRATES: This knowledge will not come from teaching but from questioning.
He will recover it for himself.
MENO: Yes.
SOCRATES: And the spontaneous recovery of knowledge that is in him is
recollection, isn’t it
MENO: Yes.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 26 / 140
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Plato: The Meno
Socrates discusses the immortality of the soul.
SOCRATES: Either then he has at some time acquired the knowledge which he
now has, or he has always possessed it. If he always possessed it, he must always
have known; If on the other hand he acquired it at some previous time, it cannot
have been in this life, unless somebody has taught him geometry. He will behave
in the same way with all geometrical knowledge, and every other subject. Has
anyone taught him all these? You ought to know, especially as he has been
brought up in your household.
MENO: Yes, I know that no one has ever taught him.
…
SOCRATES: Then if he did not acquire them in this life, isn’t it immediately clear
that he possessed and had learned them during some other period?
MENO: It seems so.
SOCRATES: When he was in human shape?
MENO: Yes.
SOCRATES: If then there are going to exists in him, both while he is and while
he is not a man, true opinions which can be aroused by questioning and turned
into knowledge, may we say that his soul has been for ever in a state of
knowledge? Clearly he always either is or is not a man.
MENO: Clearly.
SOCRATES: And if the true about reality is always in our soul, the soul must be
immortal, and one must take courage and try to discover–that is–recollect–what
one doesn’t happen to know, or (more correctly) remember, at the moment.
MENO: Somehow I believe you are right.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 27 / 140
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Plato: The Meno
Is Socrates teaching?
Critical assumptions of Plato
▶ Socrates is not teaching
▶ The only way to learn something is to be taught
Socrates asking very leading questions
Alternative to the knowing/not knowing disjunction: the role of
reasoning
The process of mathematical insight
At one point Plato says the claim that all learning is recollection is
extends to “…all geometrical knowledge, and every other subject”. Do
you think this thesis could be applied to other subjects as well?
What mathematical knowledge is innate? Propositions? Concepts?
Abilities?
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 28 / 140
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Plato’s Meno and beyond
The process of mathematical insight
Modern mathematics education: Inquiry based learning
▶ George Pólya How to Solve It (1945): Process of discovery in
mathematics
▶ Imre Lakatos Proofs and Refutations (1976): Euler’s characteristic for
Polyhedra
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 29 / 140
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Three great geometrical problems
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Three great geometrical problems
Extremely influential on the
development of geometry
Constraint: Straight edge
and compass
Engaged many scholars over
centuries
▶ Hippocrates of Chios
▶ Anaxagoras
▶ Hippias (ca. 400 BCE)
Emergence of the idea of
demonstration/proof in
mathematics
19th century
mathematicians proved the
impossiblity proofs these
three constructions
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 31 / 140
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Squaring the circle: Egypt
Egyptian problem
Rhind Mathematical Papyrus
(1650 BCE ?) bought by
Henry Rhind in 1858.
6m long 1/3m wide
Ahmes an Egyptian Scribe
Papyrus contains around 87
problems
Problem 48: “Cut off 1/9th
of the diameter and
construct a square on the
remainder”
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 32 / 140
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Squaring the circle: Egypt
Specific example, not a general rule
Take away one nineth of it, namely 1; the remainder is 8. Make
the multiplication 8 times 8; becomes it 64; the amount of it,
this is, in area 64 setat.
Circle has diameter d = 9.
Construct square with base d = 9 less a nineth, i.e., 9 − 1 = 8.
Area of square is 82 = 64.
A = (d −
d
9
)2
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 33 / 140
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Squaring the circle: Egypt
More generally:
Cut off 1/9th of the diameter of a circle and construct a square on the
remainder:
A = (d −
d
9
)2
How good is this method?
πr 2 = (d − d9)
2
πd 2
4
=
64
81
d 2
π = 3.16049
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 34 / 140
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Squaring the circle: India
Sacred texts Śulbasūtras (ca.
800BCE)
Rope and marking stick
If it is desired to transform a square into a
circle, a cord of length half the diagonal of
the square is stretched out from the center
to the east (a part of it lying outside the
easternside of the square); with one third (of
the part lying outside) added to the
remainder (of the half diagonal, the
(required) circle is drawn.
[Baudhāyana-Śulbasūtra 2.9]
How good is this method?
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 35 / 140
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Squaring the circle: India
If it is desired to transform a square into a
circle, a cord of length half the diagonal of
the square is stretched out from the center
to the east (a part of it lying outside the
easternside of the square); with one third (of
the part lying outside) added to the
remainder (of the half diagonal), the
(required) circle is drawn.
d is half-diagonal of square, s half-side of
square, then d 2 = 2s 2 or d =
√
2s.
r = s + 13(d − s)
= s + 13s(
√
2 − 1)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 36 / 140
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Squaring the circle: India
How good is this method?
r = s + 13(d − s)
= s + 13s(
√
2 − 1)
= s3(2 +
√
2)
so that
πr 2 = 4s 2
π
(
s
3(2 +
√
2)
)2
= 4 s2
π ≈ 3.08831…
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 37 / 140
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Squaring the circle: Greece
Antiphon (5th century BCE?): Inscribe a square in a circle, then a
regular polygon with 8 sides, then with 16 sides, etc…
Hippias (ca. 400 BCE) quadratrix
Hippocrates: Lunes
Archimedes investigation of the spiral may have been motivated by
this problem
Archimedes invention of a mechanical ruler led to study of concoids
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 38 / 140
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Squaring the circle: Greece
Hippias (ca. 400 BCE) a new drawing instrument: the quadratrix
Curve generated by motion
Can convert a quarter circle into square of the same area. In this way,
a square with double the side length has the same area as a complete
circle. (Dinostratus 250 BCE)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 39 / 140
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Squaring the circle: Greece
Hippocrates (ca. 5th century BCE) and his lunes.
Source: Mactutor History of Mathematics
Finding the area of figures with curved sides
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 40 / 140
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Squaring the circle: Greece
… he proceeded to show in what way it was
possible to square a lune the outer circumference
of which is that of a semicircle. This he affected
by circumscribing a semicircle about an isosceles
right-angled triangle and a segment of a circle
similar to those cut off by the sides. Then, since
the segment about the base is equal to the sum
of those about the sides, it follows that, when
the part of the triangle above the segment about
the base is added to both alike, the lune will be
equal to the triangle. Therefore the lune, having
been proved equal to the triangle, can be
squared. (–Eudemus)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 41 / 140
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Square: ABCD, centre O.
Circle through A,B,C,D and centre O
Circle through A and C with centre D
Segment 1 (on AB) subtends a right angle on circle
whose centre is O (angle AOB)
Segment 2 (on AC) subtends a right angle on circle
whose center is D (angle ADC)
Therefore segments are similar: seg 1
seg 2
=
AB2
AC2
=
1
2
.
(AB2 + BC2 = AC2 and AB = BC so that AC2 = 2AB2).
Therefore segment 2 is twice segment 1 → segment 2 is
the sum of the two segments 1.
Semicircle ABC with two segments 1 removed is the
triangle ABC.
A square equal to a triangle already established.
Furthermore take segment 2 from semicircle ABC (lune
in second diagram).
Therefore you can square a lune!
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 42 / 140
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Three great geometrical problems: Doubling the Cube
Legends abound as to the origin of this problem. One of them is as follows:
Eratosthenes, in his work entitled Platonicus
relates that, when the god proclaimed to the
Delians through the oracle that, in order to get rid
of a plague, they should construct an altar double
that of the existing one, their craftsmen fell into
great perplexity in their efforts to discover how a
solid could be made the double of a similar solid;
they therefore went to ask Plato about it, and he
replied that the oracle meant, not that the god
wanted an altar of double the size, but that he
wished, in setting them the task, to shame the
Greeks for their neglect of mathematics and their
contempt of geometry. [Theon of Smyrna quoting
a work of Eratosthenes]
Problem reduces down to finding two mean proportionals x, y, such that
a : x = x : y = y : b
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Three great geometrical problems: Trisecting the angle
Provenance of this problem is unknown
Trisection of arbitrary angles
▶ Can do 90◦ and 27◦
Hippocrates (5th century BCE)
Hippias (ca. 400 BCE) quadratrix
Mechanical solution by Archimedes
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Three great geometrical problems: Trisecting the angle
Hippocrates’ method:
Given an arbitrary angle CAB,
construct CD perpendicular to
AB and label meeting point
D. Draw a rectangle ADCF on
AD. Extend FC to E and let
AE be drawn to cut CD at H.
Choose E such that HE =
2AC. Angle EAB will be a
third of angle CAB.
Source: Mactutor History of Mathematics
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 45 / 140
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Three great geometrical problems: Trisecting the angle
Let G be the midpoint of HE so that HG = GE = AC
Since ECH is right, CG = HG = GE
angle EAB = angle CEA = angle ECG
And since AC = CG, angle CAG = angle CGA
But angle CGA = angle GEC + angle ECG
This is equal to 2 x CEG = 2 x EAB as required.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 46 / 140
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Systems of Numeration in Ancient Greece
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 47 / 140
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Greek systems of Numeration
Two distinct forms of notation
Both based on base-ten
Not positional/place value
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 48 / 140
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Greek systems of Numeration: Attic
Acrophonic
pente = 5, deka = 10, hekaton = 100, chilioi = 1000, myriad =
10,000
Based on a simple iterative scheme found in Egypt
Precursor to Roman numerals
Source: Cuomo (2001) p. 11
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Greek systems of Numeration: Ionian
Source: Cuomo (2001) p. 11
Emerged mid-fith century (maybe even earlier)
Based on 24 letters of Greek alphabet plus three additional ones
(stigma, koppa, sampi)
Not place value
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 50 / 140
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Ionian numbers in action: Your turn!
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The legacy of Aristotle
For just as in geometry it is useful to have been trained in the
elements, and in arithmetic to have a ready knowledge of the
multiplication table up to ten times helps much to the
recognition of other numbers which are the result of
multiplication, so too in arguments it is important to be prompt
about first principles and to know your premisses by heart
Aristotle (Topics 163b24–29)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 52 / 140
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Numerical Tables
Papyrus Michigan 146 (Knorr, 1982)
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Numerical Tables
Papyrus Michigan 146 (Knorr, 1982)
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Numerical Tables
Unit fraction tables
ninths
of the 1 the 9 is 9
[of the 6000] the 9 is 666 3
of the 2 [the 9 is] 6 18
of the 3 [the 9 is] 3
of the 4 [the 9 is] 3 9
of the 5 [the 9 is] 2 18
of the 6 [the 9 is] 3
of the 7 [the 9 is] 3 9
of the 8 [the 9 is] 2 3 18
of the 9 [the 9 is] 1
of the 10 [the 9 is] 1 9
of the 20 [the 9 is] 2 6 18
…
of the 90 [the 9 is] 10
of the 100 [the 9 is] 11 9
of the 200 [the 9 is] 22 6 18
…
of the 900 [the 9 is] 100
of the 1000 [the 9 is] 111 9
of the 2000 [the 9 is] 222 6 18
…
of the 9000 [the 9 is] 1000
of the 1 0000 [the 9 is] 1111 9
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Greek systems of Numeration: Zero
Second century CE papyrus
Small circle with a bar over the top
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Zero in Greek papyrus
Source: Jones (1999) p. 62
Tied to sexagesimal notation
Always in astronomical contexts
Papyrus dates: 100BCE-500CE
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Representation of big numbers!
Archimedes (ca. 287–212
BCE)
Psammites ‘Sand-Reckoner’
Developing a system for
expressing very BIG numbers
“Again there are some who,
without regarding it as
infinite, yet think that no
number has been named
which is great enough to
exceed its multitude”
Curiosity about very big
quantities that can be
imagined
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 58 / 140
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Archimedes Sand Reckoner
Scenario of Archimedes:
Fill the entire universe with
grains of sand.
Grain of sand the smallest thing
you can imagine
Universe the largest thing you
can imagine
How many grains?
Need to invent a system of really
big numbers
Using this system compute and
state and upper bound for this
number of grains.
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 59 / 140
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Archimedes Sand Reckoner
Indeed, it happens that the names of numbers up to a myriad
have been given to us by tradition and we can recognize numbers
greater than a myriad by stating the number of myriads up until
a myriad myriads. Let us call these so-called numbers up to a
myriad myriads “first numbers”.
What are the first numbers?
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Archimedes Sand Reckoner
Indeed, it happens that the names of numbers up to a myriad
have been given to us by tradition and we can recognize numbers
greater than a myriad by stating the number of myriads up until
a myriad myriads. Let us call these so-called numbers up to a
myriad myriads “first numbers”.
First numbers: 1 to a myriad myriad (108)
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Archimedes Sand Reckoner
Let the myriad myriads of the first numbers be called the “units”
of the second numbers and let us enumerate the units of the
second numbers and from tens of units and hundreds and
thousands and myriads up to a myriad myriads.
What are the second numbers?
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 62 / 140
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Archimedes Sand Reckoner
Let the myriad myriads of the first numbers be called the “units”
of the second numbers and let us enumerate the units of the
second numbers and from tens of units and hundreds and
thousands and myriads up to a myriad myriads.
Second numbers: A myriad myriad (108) to a myriad myriad of these
(1016)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 63 / 140
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Archimedes Sand Reckoner
Again, let the myriad myriads of the second numbers be called
the unit of the third numbers and let us enumerate the units of
the third numbers and from tens of units and hundreds and
thousands and myriads up to a myriad myriad of these units.
What are the third numbers?
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 64 / 140
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Archimedes Sand Reckoner
Again, let the myriad myriads of the second numbers be called
the unit of the third numbers and let us enumerate the units of
the third numbers and from tens of units and hundreds and
thousands and myriads up to a myriad myriad of these units.
Third numbers: (1016) to a myriad myriad of these (1024)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 65 / 140
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Archimedes Sand Reckoner
In the same way, let the myriad myriad of the third numbers be
called the unit of the fourth numbers, and a myriad myriad of
the fourth numbers, the unit of the fifth numbers, and
continuing in this very way, numbers designated up to the myriad
myriadth of a myriad myriads numbers.
What is this number?
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 66 / 140
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Archimedes Sand Reckoner
In the same way, let the myriad myriad of the third numbers be
called the unit of the fourth numbers, and a myriad myriad of
the fourth numbers, the unit of the fifth numbers, and
continuing in this very way, numbers designated up to the myriad
myriadth of a myriad myriads numbers.
WOW! A myriad myriadth of a myriad myriads is
(100, 000, 000)100,000,000
(That is, 1 followed by 8 hundred million zeros!)
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 67 / 140
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Archimedes Sand Reckoner: A quick comparison
The number of elementary particles in the universe is a mere 1080
(i.e., 1 with 80 zeros after it)
A modern day googol is a bit larger: 10100:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000
But 1 followed with 8 hundred million zeros?
And Archimedes is only getting started!!
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 68 / 140
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Archimedes Sand Reckoner
Numbers named in this way would indeed be enough, but it is possible
to progress even further. For let the previously mentioned numbers be
called the “first period”, and the last number of the first period be
called the “units” of first numbers of the second period. Again, let the
myriad myriad of first numbers of the second period be called the unit
of second numbers of the second period. Likewise, the last number be
called the unit of the third numbers of the second period. And
progressing thus, the numbers proceeding through the numbers of the
second period are named up to the myriad myriadth of myriad myriad
numbers. Again, let the last number of the second period be called the
unit of the first numbers of the third period, and continuing thus up to
a myriad myriad units of myriad myriadth numbers of the myriad
myriadth period.
Unbelievable! This last number, a myriad myriad units of myriad myriadth
numbers of the myriad myriadth period, is 1 followed by 80,000 million
millions of zeros!
Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 69 / 140
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Archimedes Sand Reckoner
Now, to put this system to use. Two pieces of empirical information:
Diameter of the universe < 10,000,000,000 stadia Sphere of diameter 1 finger breadth = 640,000,000 grains of sand Archimedes works through carefully to produce number of grains to fill the universe: 1051 There is a larger sphere out there—the sphere of the size of the fixed stars: Upper bound for the number of grains of sand to fill up the sphere of fixed stars is: 1063 Our modern day vigintillion! Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 70 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of big numbers: Jainism Religious tradition of Jainism Big numbers formed basis of spiritual contemplation Contemplate issues that transcended everyday life Cycles of time Curiosity about very big quantities that can be imagined Developed sophisticated notions of infinity Unenumerable (asaṃkhyāta in Sanskrit) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 71 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The first unenumerable number: Jambū-Island How many mustard seeds can be pilled up in a space with Jambū Island at the center? Jambū island is a circular island Its diameter is fixed at 100,000 yojanas Surrounding Jambū island is a series of concentric rings of alternating bands of sea and land. The widths of these rings double in length each time we move outwards Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 72 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The first unenumerable number: Jambū-Island Now the filling up and emptying begins! Firstly, mustard seeds are to be dropped onto Jambū island and this continues until not a single seed more can be contained without slipping off. Jain thinkers imagined the over-full shape that resulted from this piling would be something we might call a cylindrical pit. Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 73 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The first unenumerable number: Jambū-Island The procedure was as follows: 1 Over-fill Jambū island whose radius is r0 with s0 seeds. 2 Empty this cylindrical pit and drop each of these s0 seeds, one by one, onto various successive rings. You will cover s0 rings. 3 Consider the resulting disk whose radius r1 will include the first s0 rings. 4 Over-fill this disk whose radius isr1 with s1 seeds. 5 Empty this cylindrical pit and drop each of these s1 seeds, one by one, onto various successive rings. You will cover the next s1 rings. 6 Consider the resulting disk whose radius r2 will include the first s0 + s1 rings. 7 Over-fill this disk whose radius is r2 with s2 seeds. 8 Empty this cylindrical pit and drop each of these s2 seeds, one by one, onto various successive rings. You will cover the next s2 rings. 9 Consider the resulting disk whose radius r3 will include the first s0 + s1 + s2 rings. 10 Over-fill this disk whose radius is r3 with s3 seeds. 11 Empty this cylindrical pit… And so on. This process is to be repeated a staggering s30 times. Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 74 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The first unenumerable number: Jambū-Island This process is to be repeated a staggering s30 times. s30 = 7.9656x10 135 Can you even fathom the resulting number of mustard seeds? The Jains could, and gave this number a name: jaghanya-parīta-asaṁkhyāta “unenumerable of low enhanced order” Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 75 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Really big numbers: A modern example Jorge Luis Borges The Library of Babel This amazing library contains all books written from 25-symbol alphabet and are exactly 1,312,000 characters long. The number of books in this library? 251,312,000 Physicists have protested that the size of this library would have to be larger than our observable universe! Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 76 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EUCLID’S ELEMENTS Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 77 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aristippus on the Rhodian Shores Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 78 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aristippus on the Rhodian Shores Bene speremus, Hominum enim vestigio video Take heart! For I see the traces of man Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 79 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid’s Elements Euclid (ca. 300 BCE) Elements 13 books Plane and solid geometry and number theory Strict format: proposition and demonstration Deductive structure Long lasting legacy More than 1000 printed editions (first in 1482) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 80 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid’s legacy In England the text-book of Geometry consists of the Element of Euclid; for nearly every official programme of instruction or examination explicitly includes some portion of this work. Numerous attempts have been made to find an appropriate substitute for the Elements of Euclid; but such attempts, fortunately, have hitherato been made in vain. The advantages attending to a common standard of reference in such an important subject, can hardly be overestimated; and it is extremely improbable, if Euclid were once abandoned, that any agreement would exist as to the author who should replace him. from the preface of Todhunter’s Euclid London 1882 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 81 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 82 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 83 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 84 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 85 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 86 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 87 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 88 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editions of Euclid’s Elements Online Version D J Joyce with hyperlinks and interactive applets Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 89 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid’s legacy Translations made into Latin, English, French, German, Russian, Spanish, Italian, Dutch, Modern Greek... Abraham Lincoln: “He studied and nearly mastered the six books of Euclid since he was a member of Congress” Until early years of 20C in the UK...the geometry curriculum was determined exclusively by Euclid�s Elements Doug French (2004) What is it about Euclid that explains his lasting impact for so long? Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 90 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What do we know about Euclid? Not much younger than these [pupils of Plato] is Euclid, who put together the “Elements”, arranging in order many of Eudoxus’s theorems, perfecting many of Theaetetus’s, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger than Plato’s circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole “Elements” the construction of the so-called Platonic figures. Proclus (450 CE) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 91 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The definitions Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 92 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The definitions Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 93 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Postulates Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 94 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Postulates Dynamic language: ‘to draw’, ‘to produce’ Parallel postulate apeiron: indefinite or infinite? Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 95 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . apeiron: indefinite or infinite? IX.20 Euclid very careful about avoiding ‘infinite’ Proposition is carefully worded Given a quantity of primes, another one can be found This is quite different from ‘the number of primes is infinite’ Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 96 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . apeiron: indefinite or infinite? I.12 D is on the other side of the line Proposition is carefully worded Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 97 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition one Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 98 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition one Proclus (5th century CE) Commentator on the parts of a Euclidean proposition: protasis: enunciation ekthesis: setting out kataskeuē : construction diorismos: definition of goal apodeixis: demonstration sumperasma: conclusion Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 99 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) Your turn! protasis: enunciation ekthesis: setting out kataskeuē : construction diorismos: definition of goal apodeixis: demonstration sumperasma: conclusion Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 100 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46): Your turn! Takes 46 propositions to get to square construction deductive sequence Role of the diagram Lettered points Is the diagram indispensible? Or part of the demonstration? Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 101 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructing a square: Greece Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 102 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructing a square: Book I Proposition 46 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 103 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructing a square: Book I Proposition 47 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 104 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) protasis: enunciation ekthesis: setting out kataskeuē : construction diorismos: definition of goal apodeixis: demonstration sumperasma: conclusion Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 105 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 106 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) To describe a square on a given straight-line. Let AB be the given straight-line. So it is required to describe a square on the straight-line AB. Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 107 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) Let AC have been drawn at right-angles to the straight-line AB from the point A on it. [To draw right-angles, we need proposition I.11] Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 108 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) and let AD have been made equal to AB. Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 109 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) And let DE have been drawn through point D parallel to AB. And let BE have been drawn through point B parallel to AD. [To draw parallel lines we need proposition I.31, which in turn requires proposition I.23] Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 110 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) And let DE have been drawn through point D parallel to AB. And let BE have been drawn through point B parallel to AD. [How do we find E before we have both parallel lines intersecting?] Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 111 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) Thus ADEB is a parallelogram. [A parallelogram? We require a square! Euclid isn’t finished yet.] Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 112 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Constructing a square (I.46) Therefore, AB is equal to DE, and AD to BE [Prop. 1.34]. But, AB is equal to AD. Thus, the four (sides) BA, AD, DE, and EB are equal to one another. Thus, the parallelogram ADEB is equilateral. So I say that (it is) also right-angled. For since the straight-line AD falls across the parallels AB and DE, the (sum of the) angles BAD and ADE is equal to two right-angles [Prop. 1.29]. But BAD (is a) right-angle. Thus, ADE (is) also a right-angle. And for parallelogrammic figures, the opposite sides and angles are equal to one another [Prop. 1.34]. Thus, each of the opposite angles ABE and BED (are) also right-angles. Thus, ADEB is right-angled. And it was also shown (to be) equilateral. Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 113 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Construction and Diagram is different Instructions connect with figure using lettered points No reference to tools All mathematical objects required have already been definied (parallel lines, a line at right angles) Deductive sequence Diagram and construction are quite different What then is the role of the diagram in the text? Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 114 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid’s Elements: The role of the diagrams Is it a construction or a accompanying figure? Reviel Netz: Greek Mathematical Practice (1999) ▶ Determination of objects is achieved through the diagram ▶ Study on Elements XIII for determined figures ▶ 838 letters: 370 determined, 247 underdetermined, remaining mixed. Greek geometry is based on diagrams in an essential way The diagram is a metonym of the proposition Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 115 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid’s Elements: Concluding thoughts Emergence of deductive structure and axiomatic method What role do numbers play in his geometry? Emphasis on demonstration/proof New significance to the nature of geometric truth Euclid’s audience Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 116 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aristippus on the Rhodian Shores Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 117 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Geometrical Algebra Debate Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 118 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.1 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 119 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.1 a(b+c+d+...) = ab+ac+ad+... Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 120 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.2 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 121 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.2 ab + ac = a2 if a = b + c Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 122 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.3 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 123 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.3 (a + b)a = ab + a2 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 124 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.4 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 125 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclid: Proposition II.4 (a + b)2 = a2 + 2ab + b2 Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 126 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebra? or Geometry? Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 127 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra Interpretation of Euclid’s Elements as ‘geometric algebra’ began almost a century ago Very colourful interaction between scholars Questioning the very heart of essence of mathematics and its operations Two sides: P Tannery A Szabo B Van Der Waerden S Unguru O Neugebauer D Rowe A Weil Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 128 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Proponents Van Der Waerden: When one opens book II of the Elements, one finds a sequence of propositions which are nothing but geometric formulations of algebraic rule...We have here so to speak, the start of an algebraic textbook, dressed up in geometric form. Andre Weil: The greeks must have been in possession of powerful algebraic tools. Hans Freudenthal: Throughout Greek mathematics, one finds numerous applications of this algebra. The line of thought is always algebraic, the formulation geometric. Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 129 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Opponents (1975) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 130 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Opponents Sabetai Unguru: I believe such a view is offensive, naive and historically untenable...in more cases than not it is non history...[these individuals are] largely unable to relinquish and discard their laboriously acquired mathematical competence when dealing with periods in history during which such competence is historically irrelevant Sabetai Unguru: The crucial historiographical point is that in this transfer process one does irreparable violence and inflicts unrectifiable damage to the unique perculiar suigeneris traists of Greek geometry which are not, let me stat this emphatically, reducible to something ‘simpler’, less ‘clumsy’ ‘awkward’ ‘cumbersome, and so forth about Greek mathematics when it is not taken out of its own context Arpad Szabo: If the history of mathematics is to be rewritten, I would like it to be done by somebody who shows higher esteem for ancient mathematics Sabetai Unguru:The attempt to see algebra lurking behind the propositions of greek geometry can only be done at the cost of overlooking the intrinsic geometrical setting that persistently motivates greek mathematics Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 131 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Proponents (1976) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 132 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Proponents Proponents: ...their failure to see the general significance of these quadratic formulas...and shortsightedness in failing to extend the number concept so as to include the negative and the ordinary complex numbers [Euclid could have been improved] by paying explicit attention to the subject of mensuration and to other possible applications Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 133 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Opponents Opponents: this is an historically and philosophically faulty procedure, distorts of the true character of ancient mathematics If scholars...neglect the peculiar specificities of a given mathematical culture„, then by definition their work is ahistorical Greek mathematics must be understood in its own right. this can be done by refusing to apply to its analysis foreign, anachronistic criteria Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 134 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Proponents (1978) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 135 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Proponents: [...the data is] a textbook on solving equations...given certain magnitudes a, b, c, and a relation f(a, b, c, x) then x too is given... Opponents: Greek geometry contain no equations...indeed had euclid at his disposal frendenthals’s functional notation it is rather easy to infer that he would not have needed ninety-four propositions to get his point across. Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 136 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fight over geometric algebra: Proponents: Algebra is the art of handling algebraic expressions like (a + b)2 and of solving equations like x2 + ax = b.. If this definition is applied to any text, it is unimportant what symbolism the text uses Opponents: There are no equations of the type (a + b)2 in Euclid! Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 137 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Issues with geometric algebra: Absense of philological category-there are no letters or symbols in Euclid Differences between algebraic and geometric ways of thinking Incongruence with gradual historical development of algebra Much information is lost when you convert the geometric operations into algebraic ones Many of the algebraic equivalents are trivial; geometric content is superfluous in algebraic form No concrete definitions of operations (multiplication; division, restrictions on the operations of addition and subtraction) Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 138 / 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is algebra? Unguru’s definition Operational symbolism: using symbolism to abstract a structure of a mathematical problem Preoccupation with relationships over objects Abstractness What do you think? Dr Clemency Montelle (UC, NZ) History of Math July 26, 2020 139 / 140