Download Greek mathematics — part I

Greek mathematics — part I
1
General
After ancient Babylon, beginning about 500 B.C.E., what seems to have been
a completely different kind of ’mathematics’ arose in Greece and the Eastern
Mediterranean (Turkey, Sicily, Egypt, the Middle East). Written in the Greek
language, the work includes some writings which have ever since been thought
of as ‘classics’.
Historians — including me — refer to the civilization which produced these
works as ‘Greek’, and to the people who did them as ‘the Greeks’. We may
need to qualify this language, but it’s reasonable. The people involved include
some mathematical ‘household names’, like Euclid, Archimedes and Ptolemy.
They all, or mostly, worked in the same way, for about a thousand years until
600 C.E., with a high point in the last few centuries B.C.E. What they produced
was not (apparently) useful, it was in general centred on the ideas of geometry
and proof.
There are, then, some obvious questions:
• How, and why. did this kind of mathematics start?
• Did it have any connection to what had gone before — in Egypt or Babylonia?
• Was there any practical mathematics in the ancient Greek world, and if
so, what was its relation to the ‘pure geometrical’ kind? (Discussed by
Asper (on KEATS))
We’ll try to consider these questions, as well as looking at what our sources are
actually like. First, an actual Greek story.
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Origins: the Greeks on ‘Egyptian mathematics’
The king moreover (so they say) divided the country among all the Egyptians
by giving them each an equal square parcel of land, and made this his source
of revenue, appointing the payment of a yearly tax. And any man who was
robbed by the river would come to Sesostris and declare what had befallen him;
then the king would send men to look into it and measure the space by which
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the land was diminished, so that thereafter it should pay in proportion to the
tax originally imposed. From this, to my thinking, the Greeks learned the art
of geometry; the sun-clock and the sundial and the twelve divisions of the day,
came to Hellas not from Egypt but from Babylonia.
From Herodotus History II, 109 (mid fifth century BCE), in F-G extract 1.D4,
p.21.
And second, some doubts.
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An argument against a ‘Babylonian origin’ for
Greek mathematics
Old Babylonian mathematics is inherently metric; all parameters have both
quantity and measure, explicit or implicit, as well as dimension. . . In contrast,
the classical Greek tradition is inherently geometric; parameters have dimension
but no quantity or measure. It is also heavily deductive and axiomatic: the emphasis is on deriving general proofs from explicitly stated theorems and axioms.
So on internal evidence alone we can dismiss Old Babylonian mathematics as
the putative arithmetical precursor to early Greek mathematics, simply because
it was not particularly arithmetical.
(From Robson 2005 (see bibliography, also on KEATS), p.10; a similar argument
would seem to rule out the ‘Egyptian origins’ of Herodotus’ account.)
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From the Meno, by Plato.
This could count as the earliest ‘mathematical text’ from Greece; it comes from
a dialogue by the philosopher Plato (4th century B.C.E.), while the earliest
manuscripts of the dialogue which survive are arounf 1200 years later. In it,
Socrates proposes to his friend Meno’s slave boy the problem of drawing a square
whose size is double that of a ‘four-foot square’ (i.e. one of four square feet —
side two feet). The boy suggests doubling the side. See fig.1 for an illustration
of the argument.
Soc. Mark now the farther development. I shall only ask him, and not teach
him, and he shall share the enquiry with me: and do you watch and see if you
find me telling or explaining anything to him, instead of eliciting his opinion.
Tell me, boy, is not this a square of four feet which I have drawn?
Boy. Yes.
Soc. And now I add another square equal to the former one?
Boy. Yes.
Soc. And a third, which is equal to either of them?
Boy. Yes.
Soc. Suppose that we fill up the vacant corner?
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Boy. Very good.
Soc. Here, then, there are four equal spaces?
Boy. Yes.
Soc. And how many times larger is this space than this other?
Boy. Four times.
Soc. But it ought to have been twice only, as you will remember.
Boy. True.
Soc. And does not this line, reaching from corner to corner, bisect each of these
spaces?
Boy. Yes.
Soc. And are there not here four equal lines which contain this space?
Boy. There are.
Soc. Look and see how much this space is.
Boy. I do not understand.
Soc. Has not each interior line cut off half of the four spaces?
Boy. Yes.
Soc. And how many spaces are there in this section?
Boy. Four.
Soc. And how many in this?
Boy. Two.
Soc. And four is how many times two?
Boy. Twice.
Soc. And this space is of how many feet?
Boy. Of eight feet.
Soc. And from what line do you get this figure?
Boy. From this.
Soc. That is, from the line which extends from corner to corner of the figure of
four feet?
Boy. Yes.
Soc. And that is the line which the learned call the diagonal. And if this is the
proper name, then you, Meno’s slave, are prepared to affirm that the double
space is the square of the diagonal?
Boy. Certainly, Socrates.
Soc. What do you say of him, Meno? Were not all these answers given out of
his own head?
Men. Yes, they were all his own.
Soc. And yet, as we were just now saying, he did not know?
Men. True.
Soc. But still he had in him those notions of his — had he not?
Men. Yes.
Soc. Then he who does not know may still have true notions of that which he
does not know?
Men. He has.
Soc. And at present these notions have just been stirred up in him, as in a
dream; but if he were frequently asked the same questions, in different forms,
he would know as well as any one at last?
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Men. I dare say.
Soc. Without any one teaching him he will recover his knowledge for himself, if
he is only asked questions?
Men. Yes.
Soc. And this spontaneous recovery of knowledge in him is recollection?
Men. True.
Soc. And this knowledge which he now has must he not either have acquired or
always possessed?
Men. Yes.
Soc. But if he always possessed this knowledge he would always have known;
or if he has acquired the knowledge he could not have acquired it in this life,
unless he has been taught geometry; for he may be made to do the same with all
geometry and every other branch of knowledge. Now, has any one ever taught
him all this? You must know about him, if, as you say, he was born and bred
in your house.
Men. And I am certain that no one ever did teach him.
Soc. And yet he has the knowledge?
Men. The fact, Socrates, is undeniable.
Soc. But if he did not acquire the knowledge in this life, then he must have had
and learned it at some other time?
Men. Clearly he must.
Criticism. First, Socrates is being dishonest when he says the slave boy ‘found
out’ the result without any prompting — he’s being prompted all the time.
Second, as Michel Serres points out (1995),
Socrates knows, but the boy doesn’t,
√
that the side of the eight-foot square, 8, isn’t a rational number and so there’s
no way of guessing it.
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Definitions, etc. from Euclid’s book I.
About 150 years after Plato — dates are vague — a book called ‘Euclid’s Elements of Mathematics’ was written. There’s no hard evidence on who ‘Euclid’
was; but in any case, the book has been a long-term classic, even though it
is (in part) quite difficult. It’s divided into thirteen ‘books’ which follow from
one another. Often each book begins with a set of definitions, from which the
following results are deduced; in this the ‘Elements’ has been a model for almost
all later mathematics books.
But the definitions themselves are sometimes rather strange. Here is the opening
section of Book I.
Definitions
Definition 1. A point is that which has no part.
Definition 2. A line is breadthless length.
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Definition 3. The ends of a line are points.
Definition 4. A straight line is a line which lies evenly with the points on itself.
Definition 5. A surface is that which has length and breadth only.
Definition 6. The edges of a surface are lines.
Definition 7. A plane surface is a surface which lies evenly with the straight
lines on itself.
Definition 8. A plane angle is the inclination to one another of two lines in a
plane which meet one another and do not lie in a straight line.
Definition 9. And when the lines containing the angle are straight, the angle is
called rectilinear.
Definition 10. When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the
straight line standing on the other is called a perpendicular to that on which it
stands.
Definition 11. An obtuse angle is an angle greater than a right angle.
Definition 12. An acute angle is an angle less than a right angle.
Definition 13. A boundary is that which is an extremity of anything.
Definition 14. A figure is that which is contained by any boundary or boundaries.
Definition 15. A circle is a plane figure contained by one line such that all the
straight lines falling upon it from one point among those lying within the figure
equal one another.
Definition 16. And the point is called the center of the circle.
Definition 17. A diameter of the circle is any straight line drawn through the
center and terminated in both directions by the circumference of the circle, and
such a straight line also bisects the circle.
Definition 18. A semicircle is the figure contained by the diameter and the
circumference cut off by it. And the center of the semicircle is the same as that
of the circle.
Definition 19. Rectilinear figures are those which are contained by straight lines,
trilateral figures being those contained by three, quadrilateral those contained
by four, and multilateral those contained by more than four straight lines.
Definition 20. Of trilateral figures, an equilateral triangle is that which has
its three sides equal, an isosceles triangle that which has two of its sides alone
equal, and a scalene triangle that which has its three sides unequal.
Definition 21. Further, of trilateral figures, a right-angled triangle is that which
has a right angle, an obtuse-angled triangle that which has an obtuse angle, and
an acute-angled triangle that which has its three angles acute.
Definition 22. Of quadrilateral figures, a square is that which is both equilateral
and right-angled; an oblong that which is right-angled but not equilateral; a
rhombus that which is equilateral but not right-angled; and a rhomboid that
which has its opposite sides and angles equal to one another but is neither
equilateral nor right-angled. And let quadrilaterals other than these be called
trapezia.
Definition 23 Parallel straight lines are straight lines which, being in the same
plane and being produced indefinitely in both directions, do not meet one another in either direction.
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Postulates Let the following be postulated:
Postulate 1. To draw a straight line from any point to any point.
Postulate 2. To produce a finite straight line continuously in a straight line.
Postulate 3. To describe a circle with any center and radius.
Postulate 4. That all right angles equal one another.
Postulate 5. 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.
Common Notions
Common notion 1. Things which equal the same thing also equal one another.
Common notion 2. If equals are added to equals, then the wholes are equal.
Common notion 3. If equals are subtracted from equals, then the remainders
are equal.
Common notion 4. Things which coincide with one another equal one another.
Common notion 5. The whole is greater than the part.
Exercise What, if any, is the difference between definitions, postulates and
common notions? Which are obvious and which are not?
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Two propositions from Euclid’s Elements..
Proposition I.35. Parallelograms which are on the same base and in the same
parallels equal one another.
Let ABCD and EBCF be parallelograms on the same base BC and in the
same parallels AF and BC.
I say that ABCD equals the parallelogram EBCF . [Euclid’s diagram is given
in Fig. 2.]
[The proof then continues:]
Since ABCD is a parallelogram, therefore AD equals BC. (Proposition I.34.)
For the same reason EF equals BC, so that AD also equals EF . And DE is
common, therefore the whole AE equals the whole DF . (Common Notions 1,
Common Notions 2.)
But AB also equals DC. Therefore the two sides EA and AB equal the two
sides F D and DC respectively, and the angle F DC equals the angle EAB, the
exterior equals the interior. Therefore the base EB equals the base F C, and
the triangle EAB equals the triangle F DC. (Propositions I.34, I.29, I.4.) [So
at this point, we have proved the triangles EAB and F DC are what is usually
called ‘congruent’ — all sides and angles are the same. This implies that they
have equal areas, of course.]
Subtract DGE from each. Then the trapezium ABGD which remains equals
the trapezium EGCF which remains. (Common Notions 3.)
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Add the triangle GBC to each. Then the whole parallelogram ABCD equals
the whole parallelogram EBCF . (Common Notions 2.)
Therefore parallelograms which are on the same base and in the same parallels
equal one another. Q.E.D.
Notes. The proposition asserts that two parallelograms have the same area —
in Euclid’s terms this is simply called ‘equal’. The method is by adding and
subtracting congruent triangles. As you can see, it depends on DC and EB
meeting in the point G, which is not defined and which (in some cases) is not
there. This is a typical piece of ‘carelessness’, if not an important one, in Euclid.
Proposition XII.2. Circles are to one another as the squares on their diameters.
[This proposition has been chosen since, although it is ‘difficult’, (a) it shows
what, in Euclid, replaced the standard statement about areas of circles (area
= (some number) times (diameter)2 ) which was in use among the Babylonians,
Egyptians etc, and which we use, calling the number π/4; (b) it is an example
of the well-named ‘method of exhaustion’, where you show that two curved
areas are equal by approximating them by polygons as closely as you need. In
Proposition XII.1, it has been shown that the areas of regular polygons are in
the ratio of the squares on their sides — not difficult. This is what is now used.
See fig.3 for the diagram.]
Let ABCD and EF GH be circles, and let BD and F H be their diameters.
I say that the circle ABCD is to the circle EF GH as the square on BD is to
the square on F H.
For, if the square on BD is not to the square on F H as the circle ABCD is to
the circle EF GH, then as the square on BD is to the square on F H, the circle
ABCD is either to some less area than the circle EFGH, or to a greater area.
First, let it be in that ratio to a less area S.
Inscribe the square EF GH in the circle EF GH. Then the inscribed square is
greater than the half of the circle EF GH, for if through the points E, F , G,
and H we draw tangents to the circle, then the square EF GH is half the square
circumscribed about the circle, and the circle is less than the circumscribed
square, hence the inscribed square EF GH is greater than the half of the circle
EF GH.
(IV.6, III.17)
Bisect the circumferences EF , F G, GH, and HE at the points K, L, M , and
N . Join EK, KF , F L, LG, GM , M H, HN , and N E.
Therefore each of the triangles EKF , F LG, GM H, and HN E is also greater
than the half of the segment of the circle about it, for if through the points K,
L, M , and N we draw tangents to the circle and complete the parallelograms
on the straight lines EF , F G, GH, and HE, then each of the triangles EKF ,
F LG, GM H, and HN E is half of the parallelogram about it, while the segment
about it is less than the parallelogram, hence each of the triangles EKF , F LG,
GM H, and HN E is greater than the half of the segment of the circle about it.
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(III.17)
Thus, by bisecting the remaining circumferences and joining straight lines, and
by doing this repeatedly, we shall leave some segments of the circle which will
be less than the excess by which the circle EF GH exceeds the area S.
For it was proved in the first theorem of the tenth book that if two unequal
magnitudes are set out, and if from the greater there is subtracted a magnitude
greater than the half, and from that which is left a greater than the half, and
if this is done repeatedly, then there will be left some magnitude which is less
than the lesser magnitude set out.
(X.1)
Let segments be left such as described, and let the segments of the circle EF GH
on EK, KF , F L, LG, GM , M H, HN , and N E be less than the excess by which
the circle EFGH exceeds the area S.
Therefore the remainder, the polygon EKF LGM HN , is greater than the area
S.
Now inscribe in the circle ABCD the polygon AOBP CQDR similar to the
polygon EKF LGM HN .
Therefore the square on BD is to the square on F H as the polygon AOBP CQDR
is to the polygon EKF LGM HN .
(XII.1)
But the square on BD is to the square on F H as the circle ABCD to the area
S, therefore the circle ABCD is to the area S as the polygon AOBP CQDR
is to the polygon EKF LGM HN . Therefore, alternately the circle ABCD is
to the polygon inscribed in it as the area S is to the polygon EKF LGM HN .
(V.11, V.16)
But the circle ABCD is greater than the polygon inscribed in it, therefore the
area S is also greater than the polygon EKF LGM HN . But it is also less,
which is impossible.
Therefore the square on BD is to the square on F H not as the circle ABCD is
to any area less than the circle EF GH.
Similarly we can prove that the circle EF GH is to any area less than the circle
ABCD not as the square on F H is to the square on BD.
I say next that neither is the circle ABCD to any area greater than the circle
EF GH as the square on BD is to the square on F H.
For, if possible, let it be in that ratio to a greater area S. Therefore, inversely
the square on F H is to the square on DB as the area S is to the circle ABCD.
But the area S is to the circle ABCD as the circle EF GH is to some area less
than the circle ABCD, therefore the square on F H is to the square on BD as
the circle EF GH is to some area less than the circle ABCD, which was proved
impossible. Therefore the square on BD is to the square on F H not as the
circle ABCD to any area greater than the circle EF GH.
(V.11)
And it was proved that neither is it in that ratio to any area less than the circle
EF GH, therefore the square on BD is to the square on F H as the circle ABCD
is to the circle EF GH.
Q.E.D.
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Bibliography.
Benno Artmann, Euclid: The Creation of Mathematics, Springer-Verlag, 1999.
Walter Burkert, Lore and Science in Ancient Pythagoreanism tr. E.L.Minar,
jr., Cambridge, Mass. 1972.
Euclid, The Thirteen Books of Euclid’s Elements, ed, and intro. Sir Thomas
Heath, New York: Dover, 1967. The Elements can also be found on the Internet,
particularly at aleph0.clarku.edu/ djoyce/java/elements.
David Fowler The Mathematics of Plato’s Academy (Second Edition), Oxford,
O.U.P., 1999.
T.L.Heath, A History of Greek Mathematics, Oxford: O.U.P., 1921; reprinted
New York: Dover, 1981.
Wilbur Knorr, The Origin of Euclid’s Elements, Dordrecht: Reidel, 1975.
—, The Ancient Tradition of Geometric Problems, Boston: Birkhäuser 1986.
G.E.R.Lloyd, Magic, Reason and Experience, Cambridge: C.U.P., 1979.
Reviel Netz, The Shaping of Deduction in Greek mathematics, Cambridge: Cambridge University Press, 1999.
Otto Neugebauer, The Exact Sciences in Antiquity, Princeton: Princeton University Press, 1952.
Proclus, A commentary on the first book of Euclid’s Elements, translated, with
introduction and notes, by Glen R. Morrow, Princeton: Princeton University
Press, 1970.
Eleanor Robson, ‘Influence, ignorance, or indifference? Rethinking the relationship between Babylonian and Greek mathematics.’ Bulletin of the BSHM 4
(2005), pp. 1-17.
Richard Seaford, Money and the Early Greek Mind: Homer, Philosophy, Tragedy,
Cambridge: Cambridge University Press, 2004.
Michel Serres, ‘Gnomon’, in Serres ed., A History of Scientific Thought, Oxford:
Blackwell, 1995.
Alfred Sohn-Rethel, Intellectual and Manual Labour: Critique of Epistemology,
London: Macmillan, 1978.
B.L. van der Waerden,Science Awakening, New York: Oxford University Press,
1961.
Finally, Eleanor Robson and Jacqueline Stedall eds. The Oxford Handbook of
the History of Mathematics (Oxford: OUP, 2009), in the general course reading
list, contains four contributions on or around Greek mathematics from different
viewpoints; I’d particularly recommend G. E. R. Lloyd (comparing Greek and
Chinese approaches), pp. 1-25.
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Fig.1. The diagram from the ‘Meno’
Fig.3. The figure for Euclid XII.2.
Fig.2. The figure for Euclid I.35.
G