Philosophers and geometers 600 B.C. - A.D. 400

600 B.C.-A.D. 400 SCARA LU. IA. AATIHAEMATILS NND Sc euc® Philosophers and Geometers Fig. 2.1. Tonic capitol and Greek architecture with the potter's wheel and the lathe, and they knew how to solder iron. They used the gnomon (sundial) to tell time. They believed that the pri mary stuff of the physical world consisted of water, earth, air, and fire. During the course of nearly a thousand years, mathematics flourished among the Greeks. We know that they contributed ideas to the theory of 600 B.C.-A.D. 400 T vn science” of the ancient Greeks would scarcely qualify as science ibn Py mo om criteria. Their notions were derived chiefly from specula Some little experimentation, , not from obiec ti ve observations jecti i andi measurements (although they did recogniz e that the earth w h They toye ‘ d with ideas of the natu re of matter and its indestructvctibi ibilility, ty, but b theco ms theory of Democritus was quit . . e different from that of Dalton , t ousand years later. Gree ; k medicine did lean la rgely on experii morph by ap portates, but Plato’s phys ics and physiology were anthropo- + “Aristotle, a keen observer, was more successful i i . . then in physi.cs, but he is chiefly reme? mbered for his logic. m h Physiology . surpassed mecture was dignified and impressive, with a grandeur rarely , Dut like Greek geometry ry, , it lacked d ynamic i qualiity: no flowin i movement, only frozen symmetric beau ty. This can be seen in the Tonic un a the top, or crown, of a supporti ng pillar) at the left in figure 2.1 e e other two types of capitals used by the Greeks were the Doric and the rn Rome. Rom ans used somewhat similar capital designs. Afte r the » these designs were N modified by Gothic architect s, but th classic forms were revived during the Rena issance and have survived to the present lay. The modern Austrian parl iament building at the right in figure Fr or instance, is typical of the archi tecture of ancient Greece 6 a addition to their intellectual, phil osophic, and aesthetic talents, the reeks were fairly skill ful in handicrafts and technics. They 14 were familiar numbers, which they called arithmetike, but were much less interested in the art of computation, which they called logistike. The former was the concern of philosophers and scholars; the latter was relegated to merchants and artisans. The greatest single contribution of Greek mathematics to posterity was, of course, the concept of deductive reasoning, or logical proof, which was probably introduced by Thales (ca. 600 B.c.) and elaborated on by Euclid and his successors. Because of the association of logic with philosophy, the Greeks regarded geometry as an essential part of a liberal education. (In medieval universities the quadrivium consisted of arithmetic, geometry, astronomy, and music, whereas the trivium was composed of grammar, rhetoric, and dialectic; together they formed the seven liberal arts.) We now know that some mathematical knowledge formerly believed to be original with the Greeks must instead be attributed to the Egyptians and to the Babylonians. Nevertheless, the glorious achievements of Greek geometry are not likely to fade even after two thousand years. As G. H. Hardy has so truly said, “The Greeks were the first mathematicians who are still ‘real’ to us today... . Greek mathematics is the real thing.” THE RISE OF GREEK SCIENCE The rise of Greek science began in Ionia, near the Aegean Sea, with Thales and Pythagoras. Of these two men we know very little directly, only

what later writers said of them. It seems that Thales (ca. 600 B.c.) was a man of practical affairs, a shrewd businessman, and a philosopher of sorts, with an interest in astronomy. From his travels in Egypt he brought back some knowledge of Egyptian geometry. He is credited with a few elementary theorems, but it is doubtful if he did more than take the first timid steps toward organizing geometry on a systematic basis. 17 600 B.C.-A.D. 400 1 1 , 8 . . (see fig. 2.2). Even more mystery and legend surround his compatriot Pythagoras (ca. 540 8.c.), for nearly all we know of him has come down through second- and third-hand sources. Without doubt Pytha goras was a mystic and a prophet. In his rather wide travels he gather ed considerable knowledge of mathematics and astronomy as well as religious lore. On settling in Croton, a city in southern Italy, he organized a secret society of kindred spirits interested in philosophy and mathematics. In time this Pythagorean brotherhood became firmly established, with elabor ate rules and rituals, Pythagoras not only taught the members of this “inner circle” but lectured to the general public as well. Credit for each new discovery was generally given to Pythagoras himself, not to individual membe rs of the brotherhood. Much of what was once regarded as contribution s of Pythagoras, including the so-called Pythagorean theorem itself, had actually been known earlier by the Babylonians. It is even possible that the Chinese may have been familiar with the Pythagorean relation (see Was Pythagoras Chinese? by Frank J. Swetz and T. I. Kao [University Park, Pa.: University and NCTM, 19771). Pennsylvania State Yet the Pythagorean school gave mathematics in Greec e an entirely new complexion. Mathematics was henceforth regarded as closely related to philosophy and wisdom in general. And arithm etic (number theory, not computation) was regarded by the Pythagoreans as even more Significant than geometry. Pythagoras is supposed to have said, In fact, the Pythagoreans were among the first “Number rules all.” Triangular Numbers 1, 3, 6,10... | Square Numbers 1, 4, 9, 16,... Pentagonal Numbers 1,5,12,22,... Fig. 2.2 of i edge he aac Of interest also is the five-pointed star, or pentagram ane “ een a badge of the brotherhood and which ee wc ction, is i length of the entre eine see ee arte such that the ratio of the sem ais to the length of the greater part as the greater part is to the B or C) on ary aller. art (fig. 2.3). Either intersection point (eg golden section. the into diagonal of a regular pentagon divides that diagonal A Aly D > T ne to use geometry to express relations between quantities and numbers. They developed a theory of Proportion that served them well until they encountered numbers that could not be expressed as the ratio of two whole numbers, such as V2 and V40. In any discussion of the Pythagoreans, it is difficult to separate legend from fact. However, the most notable aspect of the Pythagorean influence is the belief it fostered in the universal importance of numbers, Clinging to number mysticism and worship, they assigned all sorts of characteristics to numbers. Odd numbers had male attributes; even numbers, female traits. The number two was the first female numbe r; three was the first male number. Four was the number of justice; five represented marriage; and so on. This sort of idolatry led to numerology and astrology, which still enjoy popularity today. Of far greater significance mathematically was the IS, S r-s AD . AC AC CD or sr rer Fig. 2.3 itself. ne We come now to the famous right-triangle theorem the | | an se ° At s. oras and his theorem have been honored on stamp Pen ase oe me er 2.4 is an artistic representation of the meo i le;; the stamp at the right depicts a :4: i t triang goras presumably land ofSamos birthplace of Pythagoras, showing Pytha hows a geometric consulting an oracle; in the center another stamp

600 B.C.-A.D. 400 19 PAAAL| Fig. 2.6. Fig. 2.4. Pythagorean theorem mera of the Pythagorean relation. A statement of the theorem, A? + = C*, is given on the stamp in figure 2.5, which nicely illustra tes À NICARAGUA Greek architecture once more. as the intellectual center of the Greek world. Philosophers, mathematicians, and astronomers gathered in Athens, among them Democritus, Plato, and Aristotle. Democritus struggled with the problem of dividing a solid such as a cone or a cylinder into many thin slices, an idea that anticipated the development of the calculus many centuries later. His thinking suggested the doctrine of atomism. According to this notion, all physical phenomena could be explained in terms of extremely small but finite, hard atoms, incapable of subdivision and moving about in empty space. He believed these atoms were unchangeable and indestructible. The atoms differed from one another in size and form and properties, which accounted for the variation in the properties of different substances. Democritus is honored on the stamp at the right in figure 2.7, which also bears the contemporary symbol of atomic structure; it was issued to commemorate the inauguration of the Democritus Nuclear Research Center at Aghia Paraskevi, in Greece. Democritus’s concept was remarkably similar to modern theories of the structure of matter, but whereas contemporary physics and chemistry are based on quantitative experiments and CENTAVOS LAS 16 FORMULAS MATEMATICAS QUE CAMBIARON LA FAZ DE LA TIERRA Fig. 2.5. Law of Pythagoras have nor clear how the Pythagoreans justified the theorem, which, as we we ady said, was known to the early Babylonians and Egyptians. at “geometric proof,” if any, Pythagoras gave is not known, but Euclid and others gave several proofs. Since the days of Greek geomet hundred mathematical proofs of the theorem have been given, ove some of them geometric, others largely algebraic. Two are suggested in fi ure 2.6; readers might like to see if they can figure them out by themselves | one we vom ane Stage of Greek mathematics known as the tn ad ingens faa eed over the fifth and fourth centuries B.c. pened the way, layi i geometry and theoretical arithmetic. Now the city of Athens was thriving Fig. 2.7. Aristotle and Democritus

rigorous mathematical analysis, the doctrine of Democ ritus grew out of intuition and philosophic speculation. Although his atomism was decried by Socrates and others, his views never died out completely. They were revived, so to speak, about 1800 by John Dalton, the English chemist, who built his theory on a century and a half of chemical experimentation. A generation or so after Democritus, two other philosophers helped to mold Greek thought—Plato (ca. 380 B.c.) and Aristotle (ca. 340 8.c.). Plato was more deeply interested in moral philosophy than in natural philosophy (i.e., physical science), which he regar ded as somehow inferior. He viewed mathematics as a more or less lofty, abstract form of thought, remote from the mundane affairs of everyday life. The single exception that Plato conceded was the relation of mathematic s to astronomy. In his view, the heavens reflected the perfection of abstract mathematics, and this conviction prevailed for centuries until the time of Johannes Kepler (1570-1630). Aristotle is honored on the left-hand stamp in figure 2.7. A great philosopher and an able biologist, he is also know n for his systematic treatment of deductive logic—that is, reasoning from accepted statements to necessary conclusions. This elaborate Aristotelian verbal logic exerted profound influence until the nineteenth century, when the modern mathematician George Boole introduced the study of symbolic logic to matheatomistic speculations of Democritus. Aristotle’s Philosophy in his own day matics. Aristotle tended toward exper imental science; he rejected the was not as influential as that of Plato, but the Arabs Jater “rediscovered” him, and for centuries thereafter Europe regarded as the last word. the authority of Aristotle THE GOLDEN AGE The period from 300 to 200 B.c. is often called the golden age of Greek mathematics. During this time creativity reach ed its zenith. Three men towered high above their contemporaries as well as above those who had preceded 21 600 B.C.-A.D. 400 university, namely, Archimedes and Apollonius, probably the greatest s of antiquity. msthematician uclid’s greatest contribution was his work end ds lemen ij . i of thirteen boo ks. In this he collected and sarr reality i a series 1, 2, Sn (a bout 300 B.c.). Books i i s known at the time the mathematic deals ( due chiefly to Pythagoras); Boo fi figures ane and plane l i h lines deal with propo onan of theory the to is devoted 5 s Book s); (Hippocrate ippocrates); ci with ith circles ry 7, 8, and d 9, 9, Boc ; Books fig figures; imilar doxus); Book 6, areas and similar te: Book quantities; irrational 10, Book ); of numbers (mostly Pythagoras ‚Pp solid geometry; Book 12, the method of rete and Boo u | and constructions for the five regular so Mi Survived vor l monumenta Despite certain logical weaknesses, this io lids (Plato). nchanged for more than two thousan d| years. Indeed, Was so revered that in England many centuries later, instead orn geometry one simply “read Euclid.” The logical eee N postulate ate.. |paralle s he so-called icularly the concern the postulates, particu hiefly i nine These shortcomings were eventually remedied in the middle of the teenth century, but that story must wait until a later chapter. | Archimedes i a gout pe ere test Although Archimedes (ca. 250 B.C.) was without Ae uns the first also achiev ne also times, he ( ician o f ancient ienti and mathematatici scientist devices. al mechanic ingenious and ies “practical” di that illed byby its usefulnessa th lever a nd was so thrilled of the law of thelaw i the explain toei ean onan’ stand to place a me “Give d, exclaime have he is Supposed to move the world.” Actually, in recognizing that the weights ane © am aEnd years, quantitat antitatiive measuremen i i n, he applied : are iniinverse proportio ea: thousand two two nearly by s mechanic of science icipatiing the tions, anticipat 8:a es is honored on the two stamps shown in fig i Archimed J artist y th-centur seventeen the by painting a right is a reproduction of them or who were to follow: Euclid, Archi medes, and Apollonius. Euclid Pite F2*2 With the founding of the city of Alexandria matical activity shifted from Greece to Egypt about 330 B.C., mathe- LEY DE ARQUIMEDES . Although Alex ander the Great did not live to see it, the metropolis that he founded at the mouth of the Nile blossomed into a renowned center of learning to which both Greek and Oriental scholars flocked in great numbers. In the course of lime, the university at Alexandria boasted more than 700 000 volumes. One of the first teachers at the university was Euclid. His influence was felt a generation later in the brilliant achievemen ts of two other scholars at the Fig. 2.8. Archimedes and his law