THE MAJOR SCHOOLS OF THE CLASSICAL PERIOD 27 such as we have described,has been an enormous and complicated task. Despite the extensive efforts of scholars,there are gaps in our knowledge and some conclusions are arguable.Nevertheless the basic facts are clear 3.The Major Schools of the Classical Period The cream of the classical period's contributions are Euclid's Elements and Apollonius'Conic Sections.Appreciation of these works requires some knowl- edge of the great changes made in the very nature of mathematics and of the problems the Greeks faced and solved.Moreover,these polished works give little indication of the three hundred years of creative activity preceding them or of the issues which became vital in the subsequent history. Classical Greek mathematics developed in several centers that succeeded one another,each building on the work of its predecessors.At each center an informal group of scholars carried on its activities under one or more great leaders.This kind of organization is common in modern times also and its reason for being is understandable.Today,when one great man locates at a particular place -generally a university-other scholars follow,to learn from the master The first of the schools,the Ionian,was founded by Thales (c.640- c.546 B.c.)in Miletus.We do not know the full extent to which Thales may have educated others,but we do know that the philosophers Anaximander (c.610-c.547 B.c.)and Anaximenes (c.550-480 B.c.)were his pupils. Anaxagoras (c.500-c.428 B.c.)belonged to this school,and Pythagoras (c.585-c.500 B.c.)is supposed to have learned mathematics from Thales. Pythagoras then formed his own large school in southern Italy.Toward the end of the sixth century,Xenophanes of Colophon in Ionia migrated to Sicily and founded a center to which the philosophers Parmenides (5th cent.B.c.)and Zeno (5th cent.B.c.)belonged.The latter two resided in ecame mo mte hao he so ive te a half of the fifth century onward,were concentrated mainly in Athens. The most celebrated school is the Academy of Plato in Athens,where Aristotle was a student.The Academy had unparalleled importance for Greek thought.Its pupils and associates were the greatest philosophers, mathematicians,and astronomers of their age;the school retained its pre- eminence in philosophy even after the leadership in mathematics passed to Alexandria.Eudoxus,who learned mathematics chiefly from Archytas of Tarentum (Sicily),founded his own school in Cyzicus,a city of northern Asia Minor.When Aristotle left Plato's Academy he founded another school, the Lyceum,in Athens.The Lyceum is commonly referred to as the Peri- patetic school.Not all of the great mathematicians of the classical period can be identified with a school,but for the sake of coherence we shall occasionally
28 THE CREATION OF CLASSICAL GREEK MATHEMATICS discuss the work of a man in connection with a particular school even though his association with it was not close. 4.The Ionian School The leader and founder of this school was Thales.Though there is no sure knowledge about Thales'life and work,he probably was born and lived in Miletus.He traveled extensively and for a while resided in Egypt,where he carried on business activities and reportedly learned much about Egyptian mathematics.He is,incidentally,supposed to have been a shrewd business- man.During a good season for olive growing,he cornered all the olive presses in Miletus and Chios and rented them out at a high fee.Thales is said to have predicted an eclipse of the sun in 585 B.c.,but this is disputed on the ground that astronomical knowledge was not adequate at that time. He is reputed to have calculated the heights of pyramids by comparing their shadows with the shadow cast by a stick of known height at the same time.By some such use of similar triangles he is supposed to have calculated the distance of a ship from shore.He is also credited with having made mathematics abstract and with having given deductive proofs for some theorems.These last two claims,however,are dubious.Discovery of the attractive power of magnets and of static electricity is also attributed to TThales. The Ionian school warrants only brief mention so far as contributions to mathematics proper are concerned,but its importance for philosophy and the philosophy of science in particular is unparalleled (see Chap.7,sec.2). The school declined in importance when the Persians conquered the area. 5.The Pythagoreans The torch was picked up by Pythagoras who,supposedly having learned from Thales,founded his own school in Croton,a Greek settlement in southern Italy.There are no written works by the Pythagoreans;we know about them through the writings of others,including Plato and Herodotus. In particular we are hazy about the personal life of Pythagoras and his followers;nor can we be sure of what is to be credited to him personally or to his followers.Hence when one speaks of the work of Pythagoras one really refers to the work done by the group between 585 B.c.,the reputed date of his birth,and roughly 400 B.c.Philolaus (5th cent.B.c.)and Archytas (428-347 B.c.)were prominent members of this school. Pythagoras was born on the island of Samos,just off the coast of Asia Minor.After spending some time with Thales in Miletus,he traveled to other places,including Egypt and Babylon,where he may have picked up some mathematics and mystical doctrines.He then settled in Croton.There he
THE PYTHAGOREANS 29 founded a religious,scientific,and philosophical brotherhood.It was a formal school,in that membership was limited and members learned from leaders.The teachings of the group were kept secret by the members,though the secrecy as to mathematics and physics is denied by some historians.The Pythagoreans were supposed to have mixed in politics;they allied themselves with the aristocratic faction and were driven out by the popular or democratic party.Pythagoras fled to nearby Metapontum and was murdered there about 497 B.c.His followers spread to other Greek centers and continued his teachings. One of the great Greek contributions to the very concept of mathe- matics was the conscious recognition and emphasis of the fact that mathe- matical entities,numbers,and geometrical figures are abstractions,ideas entertained by the mind and sharply distinguished from physical objects or pictures.It is true that even some primitive civilizations and certainly the Egyptians and Babylonians had learned to think about numbers as divorced from physical objects.Yet there is some question as to how much they were consciously aware of the abstract nature of such thinking.Moreover,geo metrical thinking in all pre-Greek civilizations was definitely tied to matter. To the Egyptians,for example,a line was no more than either a stretched rope or the edge of a field and a rectangle was the boundary of a field. The recognition that mathematics deals with abstractions may with some confidence be attributed to the Pythagoreans.However,this may not have been true at the outset of their work.Aristotle declared that the Pythag oreans regarded numbers as the ultimate components of real,material objects.1 Numbers did not have a detached existence apart from objects of sense.When the early Pythagoreans said that all objects were composed of (whole)numbers or that numbers were the essence of the universe,they meant it literally,because numbers to them were like atoms are to us.It is also believed that the sixth-and fifth-century Pythagoreans did not really distinguish numbers from geometrical dots.Geometrically,then,a number was an extended point or a very small sphere.However,Eudemus,as reported by Proclus,says that Pythagoras rose to higher principles(than had the Egyptians and Babylonians)and considered abstract problems for the pure intelligence.Eudemus adds that Pythagoras was the creator of pure mathematics,which he made into a liberal art. The Pythagoreans usually depicted numbers as dots in sand or as pebbles. They classified the numbers according to the shapes made by the arrange- ments of the dots or pebbles.Thus the numbers 1,3,6,and 10 were called triangular because the corresponding dots could be arranged as triangles (Fig.3.1).The fourth triangular number,10,especially fascinated the Pythagoreans because it was a prized number for them,and had 4 dots on 1.Metaphys.I,v,986a and 986a 21,Loeb Classical Library ed
30 THE CREATION OF CLASSICAL GREEK MATHEMATICS Figure 3.1 Figure 3.2. 。 。 each side,4 being another favorite number.They realized that the sums 1,1 +2,1 +2 +3,and so forth gave the triangular numbers and that 1+2+.+n=(n/2)(n+1). The numbers 1,4,9,16,.were called square numbers because as dots they could be arranged as squares (Fig.3.2).Composite (nonprime) numbers which were not perfect squares were called oblong. From the geometrical arrangements certain properties of the whole numbers became evident.Introducing the slash,as in the third illustration of Figure 3.2,shows that the sum of two consecutive triangular numbers is a square number.This is true generally,for as we can see,in modern notation, a+1)+"支'a+2)=a+1 That the Pythagoreans could prove this general conclusion,however,is doubtful. To pass from one square number to the next one,the Pythagoreans had the scheme shown in Figure 3.3.The dots to the right of and below the lines in the figure formed what they called a gnomon.Symbolically,what they saw here was that n2 +(2n +1)=(n +1)2.Further,if we start with 1 and Figure 3.3
32 THE CREATION OF CLASSICAL GREEK MATHEMATICS Figure 3.6.Hexagonal numbers this rule gives only some sets of such triples.Any set of three integers which can be the sides of a right triangle is now called a Pythagorean triple. The Pythagoreans studied prime numbers progressions,and those ratios and proportions they regarded as beautiful.Thus if p and g are two numbers,the arithmetic mean A is(p+g)/2,the geometric mean G is vpg and the harmonic mean H,which is the reciprocal of the arithmetic mean of I/p and 1/g,is 2pg/(p +q).Now G is seen to be the geometric mean of A and H.The proportion A/G=G/H was called the perfect proportion and the proportion (p+g)/2 =2pg/(p+g):g was called the musical proportion Numbers to the Pythagoreans meant whole numbers only.A ratio of two whole numbers was not a fraction and therefore another kind of number, as it is in modern times.Actual fractions,expressing parts of a monetary unit or a measure,were employed in commerce,but such commercial uses of arithmetic were outside the pale of Greek mathematics proper.Hence the Pythagoreans were startled and disturbed by the discovery that some ratios- for example,the ratio of the hypotenuse of an isosceles right triangle to an arm or the ratio of a diagonal to a side of a square-cannot be expressed by whole numbers.Since the Pythagoreans had concerned themselves with whole-number triples that could be the sides of a right triangle,it is most likely that they discovered these new ratios in this work.They called ratios expressed by whole numbers commensurable ratios,which means that the two quantities are measured by a common unit,and they called ratios not so expressible,incommensurable ratios.Thus what we express as v2/2 is an incommensurable ratio.The ratio of incommensurable magnitudes was called aloyos (alogos,inexpressible).The term appnros (arratos,not having a ratio)was also used.The discovery of incommensurable ratios is attributed to Hippasus of Metapontum (5th cent.B.c.).The Pythagoreans were sup- posed to have been at sea at the time and to have thrown Hippasus over- board for having produced an element in the universe which denied the Pythagorean doctrine that all phenomena in the universe can be reduced to whole numbers or their ratios