Math and Sciencemath.fau.edu/schonbek/HistMath/Alexandria2015.pdf · Nile, in Egypt, he modestly called Alexandria. After his death, when his empire was divvied up among his generals,

Math and Science

My Sources Things I learned and read (a lot forgotten) over many

years.

A fascinating book: The Forgotten Revolution (La Rivoluzione Dimenticata) by Lucio Russo, English Translation, Springer Verlag 2004.

The repository of all knowledge, Wikipedia.

Alexander

Colin Farrell as Alexander

Roman copy of a statue that was made during Alexander’s lifetime (Louvre)

Born: 356 BCE King of Macedonia: 336 BCE Battles: Granicus-334 BCE Issus-333 BCE Gaugamela-331 BCE Persian Gate-330 BCE Hydaspes (Pakistan)-326 BCE Died: 323 BCE, not quite 33 years old.

Alexandria In 331 BCE Alexander founded a city on the delta of the

Nile, in Egypt, he modestly called Alexandria. After his death, when his empire was divvied up among his generals, it became the capital of Egypt.

The Ptolemies The general of Alexander who wrangled Egypt was called Ptolemy. He

crowned himself king and started a dynasty that lasted for some three hundred years. All succeeding kings where called Ptolemy, daughters were called mostly Berenice or Cleopatra. All in all, there were 15 Ptolemies. The last one was the son of the famous Cleopatra, who was the last real ruler of Egypt before the Romans took over.

Ptolemy 1, the savior (Louvre)

Bust of Cleopatra (Altes Museum, Berlin)

Elizabeth Taylor as Cleopatra

Ptolemy I, Soter Perhaps one of the most important things the first

Ptolemy did was to build a center of learning, a temple to the muses that naturally was called the Museum. In it was the library.

The Library Over the centuries, the Library of Alexandria became the greatest

repository of knowledge in the world. The Internet of antiquity. They say that when a ship laid anchor in Alexandria, library employees went aboard and confiscated all books. Copies were made and the copies were given back to the ship captain. The originals went to the Library.

The last Ptolemy

The last Ptolemy, Ptolemy XV Philopator Philometor Caesar, known as Caesarion, was the son of Cleopatra and possibly Julius Caesar. He co-reigned with his mother, meaning he never really got to reign. Marc Antony declared him the real son of Caesar, a thing that angered Caesar's adopted son Octavian. And so Octavian fought with Marc Antony, defeated him in the battle of Actium (31 BCE). Marc Antony and Cleopatra committed suicide; Octavian had the 17 year old Caesarion assassinated and then declared himself emperor changing his name to Augustus.

Caesarion

Enter the Romans In 30 BCE Egypt becomes a Roman province.

Alexandria begins a slow decline.

Gaius Julius Caesar (100-44 BCE)

Caesar Augustus (63 BCE- 14 CE)

Toward the end Diophantus (c. 200-284) Number Theory.

Pappus (290-350) Important work in geometry.

Hypatia (c. 360-415) First female mathematician of whom we know the name.

Rachel Weisz as Hypatia in the film Agora

The end of the Library Sometime between 48 BCE and the year 1, a fire caused great

havoc in the library. It may or may not have been Julius Caesar’s fault, who conquered Egypt around 48 BCE.

In AD 391 Christianity became the official religion of the empire. Pagan temples were to be closed. The Library was in a pagan temple.

The absolute coup de grace came in 642 when the Muslims took over Egypt. “If a book agrees with the Koran we don’t need it; if it disagrees it has to be destroyed, “ said the caliph. Or this could be an invention.

What happened in Alexandria? Alexandria is perhaps the first true scientific

revolution. It is a revolution that eventually failed, because the Romans were less interested in science and matters of the mind, than in daily pleasures. But it didn’t fail completely, we still recognize some of the work done there as among the finest the human mind has achieved.

The Antikythera find A lot of ideas about the ancient Greeks had to be

revised by a discovery made in the early 20th century by divers off the coast of a small island called Antikythera. It was the wreck of a ship containing objects dated to 80-50 BCE. The most impressive was a strange mechanism.

A seven minute interlude From You Tube.

The Antikythera Mechanism

In the June/July issue of the Notices of the AMS, there is a review of a

new book by Jo Marchant: Decoding the Heavens: A 2,000-Year-Old Computer—and the Century-Long Search to Discover its Secrets.

Alexandrian Science One of the important contributions was the appearance of the scientific method. The classical Greeks introduced the notion of proof into mathematics. In Alexandria a similar approach, a mathematical approach was applied to the study of nature. That method is still the method used in what are called the hard sciences. Here are a few Alexandrian scientific contributions, before we look at two of the greatest of the mathematicians of the time.

Medicine Greek medicine starts with the semi-mythical

Hippocrates of Cos (~ 500 BCE).

Aristotle (384-322 BCE), the great sage, on the brain (De partibus animalium): For there are many who think that the brain itself consists

of marrow, and that it forms the commencement of that substance, because they see that the spinal marrow is continuous with it. In reality the two may be said to be utterly opposite to each other in character. For of all the parts of the body there is none so cold as the brain; whereas the marrow is of a hot nature, as is plainly shown by its fat and unctuous character. Indeed this is the very reason why the brain and spinal marrow are continuous with each other. For, wherever the action of any part is in excess, nature so contrives as to set by it another part with an excess of contrary action, so that the excesses of the two may counterbalance each other. Now that the marrow is hot is clearly shown by many indications.

Aristotle goes on: The coldness of the brain is also manifest enough. For in the first place it is cold

even to the touch; and, secondly, of all the fluid parts of the body it is the driest and the one that has the least blood; for in fact it has no blood at all in its proper substance. This brain is not residual matter, nor yet is it one of the parts which are anatomically continuous with each other; but it has a character peculiar to itself, as might indeed be expected. That it has no continuity with the organs of sense is plain from simple inspection, and is still more clearly shown by the fact, that, when it is touched, no sensation is produced; in which respect it resembles the blood of animals and their excrement. The purpose of its presence in animals is no less than the preservation of the whole body. For some writers assert that the soul is fire or some such force. This, however, is but a rough and inaccurate assertion; and it would perhaps be better to say that the soul is incorporate in some substance of a fiery character. The reason for this being so is that of all substances there is none so suitable for ministering to the operations of the soul as that which is possessed of heat.

Aristotle still.

For nutrition and the imparting of motion are offices of the soul, and it is by heat that these are most readily effected. To say then that the soul is fire is much the same thing as to confound the auger or the saw with the carpenter or his craft, simply because the work is wrought by the two in conjunction. So far then this much is plain, that all animals must necessarily have a certain amount of heat. But as all influences require to be counterbalanced, so that they may be reduced to moderation and brought to the mean (for in the mean, and not in either extreme, lies the true and rational position), nature has contrived the brain as a counterpoise to the region of the heart with its contained heat, and has given it to animals to moderate the latter, combining in it the properties of earth and water. For this reason it is, that every sanguineous animal has a brain; whereas no bloodless creature has such an organ, unless indeed it be, as the octopus, by analogy. For where there is no blood, there in consequence but little heat. The brain, then, tempers the heat and seething of the heart.

Alexandrian Medicine Herophilus (335-280 BCE) describes the anatomy of the

brain, discovers nerves and distinguishes between sensory and motor nerves. Describes the optic, oculomotor, trigeminal, facial, auditory, and hypoglossal nerves. Describes the digestive and the circulatory systems. Has a special water clock constructed to measure heart rate in patients. Stresses the importance of prevention in medicine. And one could go on. The change is night to day.

Botany and Zoology Many advances, thanks to the vastness of new

territories opened by Alexander’s conquests. In this area, Aristotle already had made some good observations. Teophrastus (c. 371-287 BCE), who was Aristotle’s favorite pupil wrote botany treatises discussing mutations.

Astronomy Aristarchus of Samos (c. 310-230 BCE) postulates a

heliocentric theory of the universe. He also calculates the distance from earth to moon and sun, and the relative sizes of these bodies. Archimedes (of whom much more later) writes:

Archimedes on Aristarchus: You (King Gelon) are aware the 'universe' is the name given by most

astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the Floor, and that the sphere of the fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface (in The sand reckoner)

Other astronomers Hipparchus (190-120 BCE). Developed trigonometry,

invented astronomical instruments.

Ptolemy (90-168) (No relation to the kings) His Mathematical Treatise, better known by its Arabic name of Almagest, became the most influential astronomy textbook for many centuries. It places the earth at the center of the universe and then uses very ingenious though eventually unsustainable theories to explain the motion of the planets.

Names that have to be mentioned: Eratosthenes of Cyrene (276-195 BCE) Measures quite

accurately the circumference of the earth; his sieve is a methof for finding prime numbers.

Hero of Alexandria (c. 10-70) Mathematician (formula for area of a triangle) and inventor (a steam engine).

Apollonius of Perga (262-190 BCE) Treatise on conics.

A Glimpse into Alexandrian Mathematics (With an analysis bias)

Euclid (~ 300 BCE)

Euclid-Some Facts Nothing is known of his personal life.

Wrote at least ten books, of which five survived. That is, no original copy remains. We have copies, of copies, of copies…

His most famous work is the Elements. The elements seem to be the most widely published (and translated) book after the Bible. The first printed version is from 1482.

The oldest extant version is from the tenth century.

The Elements Thirteen Books, 465 Propositions cover ALL of the mathematics known at the time, not only geometry. Books I-IV are geometry. Book V is the Theory of Proportions. Some two centuries before the Pythagoreans had caused a havoc in mathematics with their discovery of irrational magnitudes. The problem was brilliantly solved by Eudoxus of Cnidus (c. 400-350 BCE); the main part of Book V is a masterful exposition of the Eudoxian theory. Book VI is more geometry. Books VII-IX contain number theory. Book X deal with irrational quantities. Books XI-XIII are solid geometry.

The Elements-Highlights Book I begins with a number of definitions and then

has 5 postulates (axioms) from which all plane geometry will follow. The first four are as axioms should be, reasonable and obvious. But the fifth, the so called parallel postulate caused quite a stir. For centuries people believed that it was a theorem, not an axiom. The matter was finally settled in the nineteenth century, over two thousand years later, by the discovery of non-euclidean geometries by Gauss, Bolyai, Lobachevsky, and Riemann.

The Elements-Highlights The parallel postulate states 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 are the angles less than the two right angles.

An easier, more manageable, equivalent version is: Given a line and a point not on it, there is a unique line parallel to the given line, through the point.

The Elements-Highlights Proposition 47 of Book I is the famous Theorem of Pythagoras. Euclid’s proof is not the easiest, but it is very pretty. He proves that square ACEF has twice the area of triangle AFB, triangle AFB is congruent to triangle ACI, and triangle ACI has half the area of rectangle AIKL. Thus square ACEF has the same area as rectangle AIKL. Similarly, square CBGH has the same area as rectangle BLKJ. In conclusion the square on the hypotenuse has an area equal to the sum of the areas of the squares on the legs.

The Elements-Highlights Book VII begins with what today is somewhat

anachronistically called the Euclidean algorithm.

It is an efficient process for finding the greatest common divisor of two numbers.

The Elements-Highlights Book IX, Proposition 14, is the Fundamental Theorem

of Arithmetic:

Every positive integer greater than 1 can be decomposed into a product of primes. This decomposition is unique up to the order of appearance of the factors.

This establishes prime numbers as the basic building blocks of mathematics.

The Elements-Highlights Questions about prime numbers have fascinated

mathematicians, professionals and amateurs alike, for millennia. In Book IX, Proposition 20, Euclid gives a first important result. He proves that there is no end to the primes; there is an infinity of primes. No matter how large a number you have, there is always a prime larger than that number. People have called Euclid’s proof beautiful in its simplicity.

The Elements-Highlights

Book XIII is dedicated to the five platonic solids: Regular polyhedra. Euclid shows how to construct them and then proves there cannot be any regular polyhedron other than the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron.

The Theory of Proportions What would mathematics have been if the Greeks had

had digital computers?

Maybe they would have not been so shocked by the discovery of irrational numbers. But they didn’t and dealing with irrationals became a necessity. Eudoxus Theory of Proportions was the answer.

In my opinion, one of Eudoxus’ great insights was that for numbers (or proportions) a, b it doesn’t really make sense to ask if a = b. One defines a = b as meaning that both a > b and b> a are false. So is 0.99999… = 1? Yes, because it is clearly false that 0.99999… >1, and that 1 > 0.99999… Our next mathematician used this idea masterfully.

And now for the greatest Alexandrian scientist, one of all time greats: Archimedes. Having talked about him already, I’ll only mention one of his results.

Archimedean Works On Plane Equilibriums (two books)

Quadrature of the Parabola

On the Sphere and the Cylinder (two books)

On Spirals

On Conoids and Spheroids

On Floating Bodies (two books)

Measurement of the Circle

The Sandreckoner

Squaring the parabola By the time of Archimedes people had figured out that

for a circle the ratio of the area to the square of its radius equaled the ratio of its circumference to its diameter, a ratio that some two thousand years later would be known as π

But little was known of π. In fact, Archimedes was the first to invent an algorithm that would allow the calculation of π to an arbitrary precision.

But that’s another story.

Parabolic Segments A parabolic segment is the region bounded by a

parabola and a straight line intersecting the parabola in two points

Parabolic Segments

A parabolic segment is the region bounded by a parabola and a straight line intersecting the parabola in two points

The Vertex of a Parabolic Segment

M is the midpoint of the segment AB. Through M draw a line parallel to the axis of the parabola. It intersects the parabola at V. That is the vertex. Incidentally, the tangent to the parabola at V is parallel to AB.

A First Lemma

The Fundamental Lemma

How did Archimedes do it?

Archimedes set out to evaluate the area of a sector of a parabola. I want to give a few details on how he proceeded.

Squaring the parabola Archimedes style

The Theorem

The Proof Begins

The Proof Continues

How the Proof Does NOT End

And so, ad infinitum, as we’d say nowadays. The triangles exhaust the parabolic sector and by the formula for sum of a geometric series, The area is and we are done. But Archimedes was very precise.

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Archimedes, the real epsilon delta man Showing an understanding of real numbers and limits

that perhaps was only recovered with the work of Richard Dedekind in the nineteenth century, Archimedes now proves another

Lemma. If A0, A1, . . ., An form a finite sequence of magnitudes, each one of which is four times the next, then

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End of the Proof

Example

Parabola y = px2

The triangle has area By Archimedes, the area of the parabolic segment is In Calculus 1 we compute this area by saying it is the area of the rectangle containing the sector minus the area beneath the curve, namely:

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