1. No category

Euclid of Alexandria
Euclid was a Greek mathematician best
known for his treatise on geometry: The
Elements. This book influenced the
development of Western mathematics for
more than 2000 years.
History of Mathematics
The lecturer of this course
Dr Vasos Pavlika
[email protected]
[email protected]
[email protected]
[email protected]
(Vas for short)
Course Lecturer
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Dr Vasos Pavlika, Subject Lecturer at SOAS, University of
London.
Subject Lecturer and online Tutor in Mathematical Economics at
SOAS, University of London.
Senior Teaching Fellow, SOAS, University of London
Lecturer for the Department for Continuing Education, University
of Oxford.
Associate Lecturer: New College, Oxford
Saturday School Lecturer: The London School of Economics
and Political Science.
Associate Tutor: St George’s Medical School, University of
London.
Consultant Mathematician.
Previously Senior Lecturer at the University of Westminster.
Field Chair at the University of Gloucestershire
Portfolio Exercises
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There will be a portfolio exercise for the
course.
This will be an extended essay of a topic
discussed on the course.
This essay will count towards to 10
CATS points.
Euclid
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Little is known about Euclid's actual life.
He lived in Alexandria about 300 B.C.E.
based on a passage in Proclus' Commentary
on the First Book of Euclid's Elements.
Indeed, much of what is known or
conjectured is based on what Proclus says.
After mentioning two students of Plato,
Proclus writes:
Euclid
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All those who have written histories bring to
this point their account of the development of
this science.
Not long after these men came Euclid, who
brought together the Elements, systematizing
many of the theorems of Eudoxus, perfecting
many of those of Theatetus, and putting in
irrefutable demonstrable form propositions
that had been rather loosely established by
his predecessors.
Euclid
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He lived in the time of Ptolemy the First,
for Archimedes, who lived after the time
of the first Ptolemy, mentions Euclid.
It is also reported that Ptolemy once
asked Euclid if there was not a shorter
road to geometry that through the
Elements, and Euclid replied that
“There is no royal road to geometry”.
Euclid
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It is apparent that Proclus had no direct evidence for
when Euclid lived, but managed to place him between
Plato's students and Archimedes, putting him, very
roughly, about 300 B.C.E.
Proclus lived about 800 years later, in the fifth century
C.E.
There are a few other historical comments about Euclid.
The most important being Pappus' (fourth century C.E.)
comment that Apollonius (third century B.C.E.) studied
"with the students of Euclid at Alexandria."
Euclid of Alexandria
(about 325 BC - about 265 BC)
Some quotes by Euclid
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In reply to King Ptolemy Euclid said
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A youth who had begun to read geometry
with Euclid, when he had learnt the first
proposition, inquired, "What do I get by
learning these things?" So Euclid called a
slave and said:
"Give him threepence, since he must make a
gain out of what he learns."
Stobaeus, Extracts
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• “There is no royal road to geometry”.
The parallel postulate
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That, if a straight line falling on two straight lines makes
the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely,
meet on that side on which the angles are less than two
right angles. (see next slide)
[the 5th postulate]
This is the famous parallel postulate, which caused so
many problems for mathematicians in the late 17th and
18th century.
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Led to the development of so-called non-Euclidean
geometries.
This is a consistent geometry that contradicts (or does away
with) the 5th or parallel postulate
The Parallel Postulate
If a straight line falling on two
straight lines makes the interior
angles on the same side less
than two right angles, the two
straight lines, if produced
indefinitely, meet on that side on
which the angles are less than
two right angles.
What are postulates
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Results that can be taken as being true
without proof.
It has been said (E.T. Bell and Heath)
that Euclid’s genius lay in the fact that he
was aware that the five postulates that
he chose were the essential ones
required to derive his theorems.
The parallel postulate
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This postulate influenced the creation of
non-Euclidean geometry at the hands of
• Gauss (the Prince of Mathematics)
• Riemann (Ph.D., examined by Gauss)
• Bolyai (friend of Gauss)
• Lobachevsky
Euclid’s postulates
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A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a
straight line.
3. Given any straight line segment, a circle can be drawn
having the segment as radius and one endpoint as centre.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough. This postulate is
equivalent to what is known as the parallel postulate.
Euclid’s Postulates
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Euclid's fifth postulate cannot be proven as a theorem, although
this was attempted by many people.
Euclid himself used only the first four postulates ("absolute
geometry") for the first 28 propositions of the Elements, but was
forced to invoke the parallel postulate on the 29th.
In 1823, Janos Bolyai and Nicolai Lobachevsky independently
realized that entirely self-consistent "non-Euclidean geometries"
could be created in which the parallel postulate did not hold.
(Gauss had also discovered but suppressed the existence of
non-Euclidean geometries.)
J.C.F. Gauss
Johann Carl Friedrich Gauss
 1777 - 1855
Known as the Prince of
Mathematics.
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Quotes by Gauss
Pauca sed matura (few but ripe)
I confess that Fermat's Theorem as an isolated proposition has
very little interest for me, because I could easily lay down a
multitude of such propositions, which one could neither prove
nor dispose of.
Sums an arithmetic progression
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Gauss at 7 sums the first 100 integers in
class.
His teacher immediately realised his
potential.
Gauss was supported by the Prince of
Brunswick, enabled him to study
At 17 proves the fundamental theorem of
algebra, bit too advanced to show.
Gauss
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Gauss worked in a wide variety of fields in both
mathematics and physics including:
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Number Theory: Higher Arithmetic, his favorite past time
Analysis
Differential geometry: geodesics
Geodesy: map making (similar to Leibniz)
Magnetism
Astronomy and optics.
Ordinary Least squares. Show this
His work has had an immense influence in many
areas. There are few areas of mathematics where
his name does not appear
Georg Friedrich Bernhard Riemann
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1826 – 1866 (student of Gauss) (died
young)
Riemann's ideas concerning geometry of
space had a profound effect on the
development of modern theoretical physics.
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General Relativity
He clarified the notion of the integral by
defining what we now call the Riemann
integral.
Riemann integral
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Riemann integrals
The Riemann hypothesis
• The holy grail of mathematics now that
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Fermat has been dispensed with
Goldbach conjecture also very old conjecture
Known more formally as the Euler-Goldbach
conjecture
• Any even integer can be expressed as the sum of
two primes.
The Riemann Integral
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Let f(x) be a non-negative real-valued function
of the interval [a,b], and let
S = {(x,y) | 0 < y < f(x)} be the region of the
plane under the function f(x) and above the
interval [a,b] (see the figure on the next slide).
We are interested in measuring the area of S.
Once we have measured it, we will denote the
area by:
The Area S
The Riemann Integral
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b
S   f ( x)dx
a
The Riemann Integral
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The basic idea of the Riemann integral is to
use very simple approximations for the area of
S.
By taking better and better approximations, we
can say that "in the limit" we get exactly the
area of S under the curve.
Note that where f can be both positive and
negative, the integral corresponds to a signed
area; that is, the area above the x-axis minus
the area below the x-axis.
http://en.wikipedia.org/wiki/Riemann_integral
G.B Riemann 1826 – 1866
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G.B Riemann
Riemann Integration
The Riemann Hypothesis
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When studying the
distribution of prime
numbers Riemann
extended Euler's
zeta function
(defined just for s
with real part greater
than one)
The Riemann Hypothesis
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to the entire complex plane (with simple pole at
s = 1).
Riemann noted that his zeta function had trivial
zeros at -2, -4, -6, ... and that all nontrivial
zeros that he could calculate were symmetric
about the line Re(s) = ½
The Riemann hypothesis is that all
nontrivial zeros are on this line.
Proving the Riemann Hypothesis would allow
us to greatly sharpen many number theoretical
results.
The Prime number theorem
The Prime number theorem
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The prime number
theorem gives an
asymptotic form for the
prime counting function,
which counts the number
of primes less than some
integer.
Legendre (1808)
suggested that for large
n,
The Prime number theorem
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with B=-1.08366 (where B is sometimes
called Legendre’s constant).
See also
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http://mathworld.wolfram.com/PrimeNumberTheorem.html
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Nikolai Ivanovich Lobachevsky
1792 - 1856
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In 1829 Lobachevsky published his
non-Euclidean geometry, the first
account of the subject to appear in print,
contradicting the 5th
postulate of Euclid.
Euclid of Alexandria
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Euclid of Alexandria is the most prominent
mathematician of antiquity best known for his
treatise on mathematics
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The Elements
The long lasting nature of The Elements must
make Euclid the leading mathematics teacher
of all time.
However little is known of Euclid's life except
that he taught at Alexandria in Egypt.
Proclus, the last major Greek philosopher, who
lived around 450 AD wrote about Euclid:-
Proclus Diadochus
Proclus Diadochus
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Proclus was a Greek philosopher who
became head of Plato's Academy and is
important mathematically for his
commentaries on the work of other
mathematicians.
He was more of a promoter of Greek
thought.
Euclid according to Proclus
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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.
Euclid
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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”.
Euclid
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The geometry Applet
http://aleph0.clarku.edu/~djoyce/java/ele
ments/usingApplet.html
Eudoxus of Cnidus (410 or 408 BC – 355 or 347 BC)
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An algebraic curve (the Kampyle of Eudoxus)
is named after him
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a2x4 = b4(x2 + y2).
Eudoxus was a Greek mathematician and
astronomer who contributed to Euclid's
Elements.
He mapped the stars and compiled a map of
the known world.
His philosophy influenced Aristotle.
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Influenced Alexander the Great
Kampyle of Eudoxus
a2x4 = b4(x2 + y2)
Euclid
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There is other information about Euclid given by
certain authors but it is not thought to be reliable.
Two different types of this extra information exists.
The first type of extra information is that given by
Arabian authors who state that Euclid was the son of
Naucrates (not much literature is available on him)
and that he was born in Tyre (Lebanon).
It is believed by historians of mathematics that this is
entirely fictitious and was merely invented by the
authors.
Archimedes
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Born about 287 BC in Syracuse, Sicily.
At the time Syracuse was an independent Greek
city-state with a 500-year history.
Died 212 or 211 BC in Syracuse when it was being
sacked/attacked by a Roman army.
He was killed by a Roman soldier who did not know
who he was.
Archimedes was in the middle of doing some work
when the soldier asked him to leave the building,
Archimedes asked for more time to complete his
work and as result of this suggestion he was killed.
Archimedes
Archimedes
Burning mirrors
Mirrors
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When Marcellus withdrew them [his ships] a
bow-shot, the old man [Archimedes]
constructed a kind of hexagonal mirror, and at
an interval proportionate to the size of the
mirror he set similar small mirrors with four
edges, moved by links and by a form of hinge,
and made it the centre of the sun's beams--its
noon-tide beam, whether in summer or in midwinter.
Mirrors
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At last in an incredible manner he [Archimedes] burned up the
whole Roman fleet.
For by tilting a kind of mirror toward the sun he concentrated
the sun's beam upon it; and owing to the thickness and
smoothness of the mirror he ignited the air from this beam and
kindled a great flame, the whole of which he directed upon the
ships that lay at anchor in the path of the fire, until he
consumed them all.
The above passage is from
DIO'S ROMAN HISTORY
Translated by Earnest Cary,
Loeb Classical Library, Harvard University Press, Cambridge,
1914, Volume II, Page 171
Mirrors
Archimedes
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Afterwards, when the beams were reflected in the mirror, a
fearful kindling of fire was raised in the ships, and at the
distance of a bow-shot he turned them into ashes.
In this way did the old man prevail over Marcellus with his
weapons.
• This is most likely apocryphal
The previous passage is from
GREEK MATHEMATICAL WORKS
Translated by Ivor Thomas, Loeb Classical Library, Harvard
University Press, Cambridge, 1941, Volume II, Page 19
Archimedes and Pi
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Archimedes calculated that
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220/71<π<22/7
3.14< π <3.157
A little bit of trivia 14th March is Einstein’s
birthday
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A better approximation was not obtained for
another 2000 years.
Archimedes and pi
Pi: continued fractions
What about this constant π
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What is π ?
It is the ratio of the circumference of a circle to
its diameter, the Greeks were aware that C/D
was the same for every circle
Is this ratio the same as A/r2?
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How did Archimedes obtain his estimate?
By a form of integral calculus, known as the method of
exhaustion
Euclid
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The second type of information is that Euclid was
born at Megara.
This is due to an error on the part of the authors who
first gave this information.
In fact there was a Euclid of Megara, who was a
philosopher who lived about 100 years before the
mathematician Euclid of Alexandria.
It is not quite the coincidence that it might seem that
there were two learned men called Euclid.
Euclid
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In fact Euclid was a very common name
around this period and this is one further
complication that makes it difficult to
discover information concerning Euclid
of Alexandria since there are references
to numerous men called Euclid in the
literature of this period.
Euclid
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Returning to the quotation from Proclus given above,
the first point to make is that there is nothing
inconsistent in the dating given.
However, although we do not know for certain
exactly what reference to Euclid in Archimedes' work
Proclus is referring to, in what has come down to us
there is only one reference to Euclid and this occurs
in On the sphere and the cylinder.
The obvious conclusion, therefore, is that all is well
with the argument of Proclus and this was assumed
until challenged by Hjelmslev.
Euclid
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He argued that the reference to Euclid was
added to Archimedes book at a later stage,
and indeed it is a rather surprising reference.
It was not the tradition of the time to give
such references, moreover there are many
other places in Archimedes where it would be
appropriate to refer to Euclid and there is no
such reference.
Despite Hjelmslev's claims that the passage
has been added later, Bulmer-Thomas writes
in:-
Euclid
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Although it is no longer possible to rely
on this reference, a general
consideration of Euclid's works ... still
shows that he must have written after
such pupils of Plato, Eudoxus and before
Archimedes.
Plato
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(427-347BCE)
The son of wealthy and influential Athenian
parents, Plato began his philosophical career
as a student of Socrates.
When the master died, Plato traveled to Egypt
and Italy, studied with students of Pythagoras,
and spent several years advising the ruling
family of Syracuse.
Eventually, he returned to Athens and
established his own school of philosophy at the
Academy.
Plato
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Plato (429–347 B.C.E.) is, by any
reckoning, one of the most dazzling
writers in the Western literary tradition
and one of the most penetrating, wideranging, and influential authors in the
history of philosophy.
Plato
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An Athenian citizen of high status, he displays in his
works his absorption in the political events and
intellectual movements of his time, but the questions
he raises are so profound and the strategies he uses
for tackling them so richly suggestive and
provocative that educated readers of nearly every
period have in some way been influenced by him,
and in practically every age there have been
philosophers who count themselves Platonists in
some important respects.
Plato
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He was not the first thinker or writer to whom the word
“philosopher” should be applied.
But he was so self-conscious about how philosophy should be
conceived, and what its scope and ambitions properly are, and
he so transformed the intellectual currents with which he
grappled, that the subject of philosophy, as it is often
conceived—a rigorous and systematic examination of ethical,
political, metaphysical, and epistemological issues, armed with
a distinctive method—can be called his invention.
Few other authors in the history of Western philosophy
approximate him in depth and range: perhaps only Aristotle
(who studied with him), Aquinas, and Kant would be generally
agreed to be of the same rank.
Plato
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For students enrolled there, Plato tried
both to pass on the heritage of a
Socratic style of thinking and to guide
their progress through mathematical
learning to the achievement of abstract
philosophical truth.
The written dialogues on which his
enduring reputation rests also serve both
of these aims.
Plato
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In his earliest literary efforts, Plato tried to
convey the spirit of Socrates's teaching by
presenting accurate reports of the master’s
conversational interactions, for which these
dialogues are our primary source of
information.
Early dialogues are typically devoted to
investigation of a single issue, about which a
conclusive result is rarely achieved.
Pythagoras of Samos
about 569 BC - about 475 BC
Euclid
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This is far from an end to the arguments
about Euclid the mathematician.
The situation is best summed up by Itard
who gives three possible hypotheses.
Euclid
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(i) Euclid was an historical character who wrote the
Elements and the other works attributed to him.
(ii) Euclid was the leader of a team of mathematicians
working at Alexandria. They all contributed to writing the
'complete works of Euclid', even continuing to write books
under Euclid's name after his death.
(iii) Euclid was not an historical character. The 'complete
works of Euclid' were written by a team of
mathematicians at Alexandria who took the name Euclid
from the historical character Euclid of Megara who had
lived about 100 years earlier.
Euclid
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It is worth remarking that Itard, who accepts
Hjelmslev's claims that the passage about
Euclid was added to Archimedes, favors the
second of the three possibilities that was listed
on the last slide (i.e. of a team)
We should, however, make some comments
on the three possibilities which, it is fair to say,
sum up pretty well all possible current theories.
Euclid
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There is some strong evidence to accept (i). (Euclid
was the sole author).
It was accepted without question by everyone for
over 2000 years and there is little evidence which is
inconsistent with this hypothesis.
It is true that there are differences in style between
some of the books of the Elements yet many authors
vary their style.
Again the fact that Euclid undoubtedly based the
Elements on previous works means that it would be
rather remarkable if no trace of the style of the
original author(s) remained.
Euclid
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Even if we accept (i) then there is little doubt
that Euclid built up a vigorous school of
mathematics at Alexandria.
He therefore would have had some able pupils
who may have helped out in writing the books.
However hypothesis (ii) (team) goes much
further than this and would suggest that
different books were written by different
mathematicians.
Euclid
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Other than the differences in style referred to above,
there is little direct evidence of this.
Although on the face of it (iii) might seem the most
fanciful of the three suggestions, nevertheless the
20th century example of Bourbaki shows that it is far
from impossible.
Henri Cartan, Andre Weil, Jean Dieudonne, Claude
Chevalley, and Alexander Grothendieck wrote
collectively under the name of Bourbaki and
Boubaki's Eléments de mathématiques contains
more than 30 volumes.
Euclid
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Of course if (iii) were the correct
hypothesis then Apollonius, who studied
with the pupils of Euclid in Alexandria,
must have known there was no person
'Euclid' but the fact that he wrote:-
Euclid
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.... Euclid did not work out the syntheses of the
locus with respect to three and four lines, but
only a chance portion of it ...
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certainly does not prove that Euclid was an
historical character since there are many
similar references to Bourbaki by
mathematicians who knew perfectly well that
Bourbaki was fictitious.
Euclid
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Nevertheless the mathematicians who
made up the Bourbaki team are all well
known in their own right and this may be
the greatest argument against
hypothesis (iii) in that the 'Euclid team'
would have had to had consisted of
outstanding mathematicians.
So who were they?
Euclid
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We shall assume that hypothesis (i) is
true but, having no knowledge of Euclid,
we must concentrate on his works after
making a few comments on possible
historical events.
Euclid must have studied in Plato’s
Academy in Athens to have learnt of the
geometry of Eudoxus and Theaetetus of
which he was so familiar.
Euclid
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None of Euclid's works have a preface,
at least none has come down to us so it
is highly unlikely that any ever existed,
so we cannot see any of his character,
as we can of some other Greek
mathematicians, from the nature of their
prefaces.
Pappus writes that Euclid was:-
Euclid
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... most fair and well disposed towards
all who were able in any measure to
advance mathematics, careful in no way
to give offence, and although an exact
scholar not vaunting himself.
Euclid
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Some claim these words have been added to
Pappus, and certainly the point of the
passage (in a continuation which we have
not quoted) is to speak harshly (and almost
certainly unfairly) of Apollonius.
The picture of Euclid drawn by Pappus is,
however, certainly in line with the evidence
from his mathematical texts.
Another story told by Stobaeus is the
following:-
Euclid
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Euclid's most famous work is his treatise on
mathematics
•
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The Elements.
The book was a compilation of knowledge that
became the centre of mathematical teaching
for 2000 years.
Probably no results in The Elements were first
proved by Euclid but the organisation of the
material and its exposition are certainly due to
him.
Euclid
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In fact there is ample evidence that
Euclid is using earlier textbooks as he
writes the Elements since he introduces
quite a number of definitions which are
never used such as that of an oblong, a
rhombus, and a rhomboid.
Euclid’ postulates again
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The fourth and fifth postulates are of a different nature.
Postulate four states that all right angles are equal.
This may seem "obvious" but it actually assumes that
space in homogeneous - by this we mean that a figure
will be independent of the position in space in which it is
placed.
The famous fifth, or parallel, postulate states that one and
only one line can be drawn through a point parallel to a
given line.
Euclid's decision to make this a postulate led to Euclidean
geometry.
It was not until the 19th century that this postulate was
dropped and non-Euclidean geometries were studied.
Euclid
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There are also axioms which Euclid calls
'common notions'.
These are not specific geometrical properties
but rather general assumptions which allow
mathematics to proceed as a deductive
science. For example:Things which are equal to the same thing are
equal to each other.
Zeno
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Zeno of Sidon, about 250 years after Euclid wrote
the Elements, seems to have been the first to show
that Euclid's propositions were not deduced from the
postulates and axioms alone, and Euclid does make
other subtle assumptions.
The Elements is divided into 13 books.
Books one to six deal with plane geometry.
In particular books one and two set out basic
properties of triangles, parallels, parallelograms,
rectangles and squares.
Achilles and the tortoise
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
“In a race, the quickest runner can never
overtake the slowest, since the pursuer
must first reach the point whence the
pursued started, so that the slower must
always hold a lead.”
—Aristotle, Physics, VI:9, 239b15
Achilles and the tortoise
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
In the paradox of Achilles and the Tortoise,
Achilles is in a footrace with the tortoise.
Achilles allows the tortoise a head start of 100
feet.
If we suppose that each racer starts running at
some constant speed (one very fast and one
very slow), then after some finite time, Achilles
will have run 100 feet, bringing him to the
tortoise's starting point.
Achilles and the tortoise
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During this time, the tortoise has run a much shorter
distance, for example 10 feet.
It will then take Achilles some further time to run that
distance, in which time the tortoise will have advanced
farther; and then more time still to reach this third point,
while the tortoise moves ahead.
Thus, whenever Achilles reaches somewhere the tortoise
has been, he still has farther to go.
Therefore, because there are an infinite number of points
Achilles must reach where the tortoise has already been-he can never overtake the tortoise
Euclid


Book three studies properties of the
circle while book four deals with
problems about circles and is thought
largely to set out work of the followers of
Pythagoras.
Book five lays out the work of Eudoxus
on proportion applied to commensurable
and incommensurable magnitudes.
Heath says:
Euclid
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Greek mathematics can boast no finer discovery than this
theory, which put on a sound footing so much of
geometry as depended on the use of proportion.
Book six looks at applications of the results of book five to
plane geometry.
Books seven to nine deal with number theory.
In particular book seven is a self-contained introduction to
number theory and contains the Euclidean Algorithm for
finding the greatest common divisor of two numbers.
Book eight looks at numbers in geometrical progression
but van der Waerden writes that it contains:-
Euclid
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... cumbersome enunciations, needless
repetitions, and even logical fallacies.
Apparently Euclid's exposition excelled only in
those parts in which he had excellent sources
at his disposal.
Book ten deals with the theory of irrational
numbers and is mainly the work of Theaetetus.
Euclid changed the proofs of several theorems
in this book so that they fitted the new
definition of proportion given by Eudoxus.
Euclid
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Books eleven to thirteen deal with three-dimensional
geometry.
In book eleven the basic definitions needed for the
three books together are given.
The theorems then follow a fairly similar pattern to
the two-dimensional analogues previously given in
books one and four.
The main results of book twelve are that circles are
to one another as the squares of their diameters and
that spheres are to each other as the cubes of their
diameters.
These results are certainly due to Eudoxus.
Euclid
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

Euclid proves these theorems using the
“method of exhaustion" as invented by
Eudoxus.
The Elements ends with book thirteen which
discusses the properties of the five regular
polyhedra and gives a proof that there are
precisely five.
This book appears to be based largely on an
earlier treatise by Theaetutus.
Euclid
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Euclid's Elements is remarkable for the clarity with
which the theorems are stated and proved.
The standard of rigor was to become a goal for the
inventors of the calculus centuries later.
As Heath writes:
This wonderful book, with all its imperfections, which
are indeed slight enough when account is taken of
the date it appeared, is and will doubtless remain the
greatest mathematical textbook of all time. ... Even in
Greek times the most accomplished mathematicians
occupied themselves with it: Heron, Pappus,
Porphyry, Proclus and Simplicius wrote
commentaries; Theon of Alexandria re-edited it,
altering the language here and there, mostly with a
view to greater clearness and consistency...
Euclid
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

It is a fascinating story how the Elements has
survived from Euclid's time and this is told
well by Fowler.
He describes the earliest material relating to
the Elements which has survived:Our earliest glimpse of Euclidean material
will be the most remarkable for a thousand
years, six fragmentary ostraca containing text
and a figure ... found on Elephantine Island in
1906/07 and 1907/08...
Euclid

These texts are early, though still more than 100 years
after the death of Plato (they are dated on palaeographic
grounds to the third quarter of the third century BC);
advanced (they deal with the results found in the
"Elements" [book thirteen] ... on the pentagon, hexagon,
decagon, and icosahedron); and they do not follow the
text of the Elements. ... So they give evidence of
someone in the third century BC, located more than 500
miles south of Alexandria, working through this difficult
material... this may be an attempt to understand the
mathematics, and not a slavish copying ...
Euclid
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
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The next fragment that we have dates from 75 - 125
AD and again appears to be notes by someone
trying to understand the material of the Elements.
More than one thousand editions of The Elements
have been published since it was first printed in
1482.
Heath discusses many of the editions and describes
the likely changes to the text over the years.
B L van der Waerden assesses the importance of
the Elements:-
Euclid
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Almost from the time of its writing and lasting
almost to the present, the Elements has
exerted a continuous and major influence on
human affairs.
It was the primary source of geometric
reasoning, theorems, and methods at least
until the advent of non-Euclidean geometry in
the 19th century. It is sometimes said that,
next to the Bible, the "Elements" may be the
most translated, published, and studied of all
the books produced in the Western world.
Euclid
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Euclid also wrote the following books which have
survived:
Data (with 94 propositions), which looks at what
properties of figures can be deduced when other
properties are given;
On Divisions which looks at constructions to divide a
figure into two parts with areas of given ratio;
Optics which is the first Greek work on perspective; and
Phaenomena which is an elementary introduction to
mathematical astronomy and gives results on the times
stars in certain positions will rise and set.
Euclid

Euclid's following books have all been
lost:
• Surface Loci (two books),

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Porisms
Conics (four books),
Book of Fallacies and Elements of
Music. The Book of Fallacies is
described by Proclus:-
Euclid
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Since many things seem to conform with the truth
and to follow from scientific principles, but lead
astray from the principles and deceive the more
superficial, [Euclid] has handed down methods for
the clear-sighted understanding of these matters
also ...
The treatise in which he gave this machinery to us is
entitled Fallacies, enumerating in order the various
kinds, exercising our intelligence in each case by
theorems of all sorts, setting the true side by side
with the false, and combining the refutation of the
error with practical illustration.
Euclid


Elements of Music is a work which is
attributed to Euclid by Proclus.
We have two treatises on music which
have survived, and have by some
authors attributed to Euclid, but it is now
thought that they are not the work on
music referred to by Proclus.
Euclid


Euclid may not have been a first class mathematician but
the long lasting nature of The Elements must make him
the leading mathematics teacher of antiquity or perhaps
of all time.
As a final personal note let me add that my own
introduction to mathematics at school did not include in
the late 1970s-80s any edition or part of Euclid's
Elements and thus the concept of proof was lacking as it
is in the mathematics teaching in schools today.
Euclid
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
Book 1 of The Elements begins with
numerous definitions followed by the famous
five postulates.
Then, before Euclid starts to prove theorems,
he gives a list of common notions. The first
few definitions are:
Def. 1.1. A point is that which has no part.
Def. 1.2. A line is a breadthless length.
Def. 1.3. The extremities of lines are points.
Def. 1.4. A straight line lies equally with
respect to the points on itself.
Euclid
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The postulates are ones of construction such
as:
One can draw a straight line from any point to
any point.
The common notions are axioms such as:
Things equal to the same thing are also equal
to one another.
•

If a=b and b=c then a=c
We should note certain things.
Euclid
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Euclid seems to define a point twice
(definitions 1 and 3) and a line twice
(definitions 2 and 4). This is rather strange.
Euclid never makes use of the definitions
and never refers to them in the rest of the
text.
Some concepts are never defined.
For example there is no notion of ordering
the points on a line, so the idea that one
point is between two others is never
defined, but of course it is used.
Euclid
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
As we noted in the real numbers, Pythagoras
to Stevin, Book V of The Elements considers
magnitudes and the theory of proportion of
magnitudes.
However Euclid leaves the concept of
magnitude undefined and this appears to
modern readers as though Euclid has failed
to set up magnitudes with the rigor for which
he is famed.
Euclid
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
When Euclid introduces magnitudes and numbers he
gives some definitions but no postulates or common
notions. For example one might expect Euclid to
postulate a + b = b + a, (a + b) + c = a + (b + c), etc.,
but he does not.
When Euclid introduces numbers in Book VII he
does make a definition rather similar to the basic
ones at the beginning of Book I: A unit is that by
virtue of which each of the things that exist are called
one.
Euclid
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
Some historians have suggested that the difference
between the way that basic definitions occur at the
beginning of Book I and of Book V is not because
Euclid was less rigorous in Book V, rather they
suggest that Euclid always left his basic concepts
undefined and the definitions at the beginning of
Book I are later additions.
What is the evidence for this?
The first comment would be that this would explain
why Euclid never refers to the basic definitions. If
they were not in the text that Euclid wrote then of
course he could not refer to them.
Euclid
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
The next point to note is that they are very similar to
the work which is ascribed to Heron called
Definitions of terms in geometry.
This contains 133 definitions of geometrical terms
beginning with points, lines etc. which are very close
to those given by Euclid.
 Knorr argues convincingly that this work is in fact
due to Diophantus.
The point here is the following. Is Definitions of terms
in geometry based on Euclid's Elements or have the
basic definitions from this work been inserted into
later versions of The Elements?
Diophantus of Alexandria
about 200 - about 284


Diophantus was a Greek mathematician
sometimes known as 'the father of
algebra' who is best known for his
Arithmetica.
This had an enormous influence on the
development of number theory.
• Pierre de Fermat
Diophantus: Fermat’s Last
theorem

"Cubum autem in duos cubos, aut
quadrato-quadratum in duos quadratoquadratos, et generaliter nullam in
infinitum ultra quadratum potestatem in
duos eiusdem nominis fas est dividere
cuius rei demonstrationem mirabilem
sane detexi. Hanc marginis exiguitas
non caperet"
Heron of Alexandria
about 10 - about 75


Heron or Hero of Alexandria was an
important geometer and worker in
mechanics who invented many
machines including a steam turbine.
His best known mathematical work is the
formula for the area of a triangle in terms
of the lengths of its sides.
Euclid
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We have to consider what Sextus Empiricus says about
definitions.
First note that Sextus wrote about 200 AD and it was
believed until comparatively recently that Heron lived later
than this.
Were this the case, then of course Sextus could not have
referred to anything written by Heron.
However more recently Heron has been dated to the first
century AD and this tells us that Sextus wrote after
Heron.
The other part of the puzzle we have to consider here is
the earliest version of Euclid’d Elements to be found.
When Vesuvius erupted in 79 AD,
Euclid
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
Herculaneum together with Pompeii and
Stabiae, were destroyed.
Herculaneum was buried by a compact
mass of material about 16 m deep which
preserved the city until excavations
began in the 18th century.
Euclid
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

Special conditions of humidity of the ground
conserved wood, cloth, food, and in particular
papyri which give us important information.
One papyrus found there contains fragments of
The Elements and was clearly written before
79 AD.
Since Philodemus, a student of Zeno of Sidon,
took his library of papyri there some time soon
after 75 BC the version of The Elements is
likely to be of around that date.
Zeno of Sidon

about 150 BC - about 70 BC
Zeno


Zeno of Sidon was a Greek philosopher
who became head of the Epicurean
school.
He criticised some of the axioms that
Euclid set out in The Elements.
Euclid
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Let us go back to Sextus who writes about
"mathematicians describing geometrical entities" and
it is interesting that the word "describing" is not used
in The Elements but is used by Heron in Definitions
of terms in geometry.
Again the descriptions he gives are closer to the
exact words appearing in Heron than those of Euclid.
When Sextus gives "the definition of a circle" he
uses the word "definition" which is that of Euclid.
Sextus quotes the precise definition of a circle which
appears in the Herculaneum fragment.
Euclid

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This does not include a definition of
"circumference" although Euclid does
use the notion of circumference of a
circle.
The later versions of The Elements
which have come down to us include a
definition of "circumference" within the
definition of a circle.
Euclid
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None of the above proves whether the basic
definitions of geometric objects have been added to
The Elements later.
They do show fairly convincingly that the definition of
a circle has been extended to include the definition
of circumference in later editions of the book.
The hypothesis is that Sextus has The Elements and
Definitions of terms in geometry in front of him when
he is writing and he uses the word "describe" when
he refers to Heron and "define" when he refers to
Euclid.
Euclid
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Even it this is correct it still doesn't prove that
the version of The Elements sitting in front of
Sextus does not contain basic definitions of
geometric objects but it does make such a
possibility at least worth debating.
What do you think?
Euclid
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One last point to think about.
We quoted above:
Def. 1.4. A straight line lies equally with respect to
the points on itself.
What does this mean? It does seem a strange
description for Euclid to give, for it appears to be
meaningless. Compare it with the definition of a
straight line in Definitions of terms in geometry:
A straight line is a line that equally with respect to all
points on itself lies straight and maximally taut
between its extremities.
Euclid
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Again we ask: do you think that the
definition appearing in The Elements is a
corruption of Heron’s definition and so
was added later, or do you think that
Euclid gave a rather poor definition
which was improved by Heron?
Why do neither use the definition of a
straight line as the shortest distance
between two points?
Euclid
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
He is therefore younger than Platro’s circle, but
older than Erarosthenes and Archimedes; for
these were contemporaries, as Erarosthenes
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.
Bertrand Russell on Euclid
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Bertrand Russell wrote an article The Teaching of Euclid
in which he was highly critical of the Euclid's axiomatic
approach.
Although this article is very interesting, it seems
extremely harsh to criticize Euclid in the way that Russell
does.
As someone once said, Euclid's main fault in Russell's
eyes is that he hadn't read the work of Russell.
The article appeared in The Mathematical Gazette in
1902. Its full reference is B Russell, The Teaching of
Euclid, The Mathematical Gazette 2 (33) (1902), 165-167.
We give below our version of Russell's article.
The Teaching of Euclid
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It has been customary when Euclid, considered as a
text-book, is attacked for his verbosity or his
obscurity or his pedantry, to defend him on the
ground that his logical excellence is transcendent,
and affords an invaluable training to the youthful
powers of reasoning.
This claim, however, vanishes on a close inspection.
His definitions do not always define, his axioms are
not always indemonstrable, his demonstrations
require many axioms of which he is quite
unconscious.
The Teaching of Euclid
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

A valid proof retains its demonstrative force when no
figure is drawn, but very many of Euclid's earlier
proofs fail before this test.
The first proposition assumes that the circles used in
the construction intersect - an assumption not
noticed by Euclid because of the dangerous habit of
using a figure.
We require as a lemma, before the construction can
be known to succeed, the following:
Euclid

If A and B be any two given points, there
is at least one point C whose distances
from A and B are both equal to AB.
Euclid
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
This lemma may be derived from an axiom of continuity.
The fact that in elliptic space it is not always possible to
construct an equilateral triangle on a given base, shows
also that Euclid has assumed the straight line to be not a
closed curve - an assumption which certainly is not made
explicit.
When these facts are taken account of, it will be found
that the first proposition has a rather long proof, and
presupposes the fourth.
We require the axiom: on any straight line there is at least
one point whose distance from a given point on or off the
line exceeds a given distance.
Euclid
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
Euclid's Elements form one of the most beautiful and
influential works of science in the history of
humankind.
Its beauty lies in its logical development of geometry
and other branches of mathematics.
It has influenced all branches of science but none so
much as mathematics and the exact sciences.
The Elements have been studied 24 centuries in
many languages starting, of course, in the original
Greek, then in Arabic, Latin, and many modern
languages.
Euclid
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
I am discussing Euclid's Elements for a couple
of reasons.
The main one is to rekindle an interest in the
Elements, and the web is a great way to do
that. Using Java applets we can illustrate
geometry. That also helps to bring the
Elements alive.
The text of all 13 Books is complete, and all of
the figures are illustrated using the Geometry
Applet, even those in the last three books on
solid geometry that are three-dimensional.
Euclid

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
The Geometry Applet is used to illustrate the figures in the Elements.
With the help of this applet, you can manipulate the figures by
dragging points.
In order to take advantage of this applet, be sure that you have
enabled Java on your browser. If you disable Java, or if your browser
is not Java-capable, then the illustrations in the elements will still
appear, but as plain images.
If you click on a point in the figure, you can usually move it in some
way. The free points, usually colored red, can be freely dragged about,
and as they move, the rest of the diagram (except the other free
points) will adjust appropriately. Sliding points, usually colored orange,
can be dragged about like the free points, except their motion is limited
to either a straight line, a circle, a plane, or a sphere, depending on the
point. Other points can be dragged to translate the entire diagram. But
if a pivot point appears, usually colored green, then the diagram will be
rotated and scaled around that pivot point.
Euclid
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
Take, for example, the figure below showing the relation
between a tetrahedron and a cube inscribed in a sphere. The
diameter of the sphere has length AB, and you can drag the
endpoints A and B to change the size of the sphere. The side of
the cube has length BD, and the side of the tetrahedron has
length AD. The cube is drawn with red edges while the
tetrahedron is shaded light blue and drawn with blue edges.
The center of the sphere is the red dot, and you can drag it to
move the sphere around. The point E can be dragged
anywhere on the surface of the sphere.
The point F has to be at length BD from E on the surface of the
sphere, and so it drags along a certain circle on the sphere.
The rest of the cube and tetrahedron are then determined. (See
proposition XIII.15 for background on the mathematics.)