22 EGYPTIAN MATHEMATICS knew of the Sothic cycle is open to question.Their calendar was adopted by Julius Caesar in 45 B.c.,but changed to a 365 1/4-day year on the advice of the Alexandrian Greek Sosigenes.Though the Egyptian determination of the year and the calendar were valuable contributions,they did not result from well-developed astronomy,which in fact was crude and far inferior to Babylonian astronomy. The Egyptians combined their knowledge of astronomy and geometry to construct their temples in such a manner that on certain days of the vear the sun would strike them in a particular way.Thus some were built so that on the longest day of the year the sun would shine directly into the temple and illuminate the god at the altar.This orientation of temples is also found to some extent among the Babylonians and Greeks.The pyramids too were oriented to special directions of the heavens,and the Sphinx faces east. While the details of the construction of these works are unimportant for us, it is worth noting that the pyramids represent another application of Egyp- tian geometry.They are the tombs of kings;and because the Egyptians believed in immortality they believed that the proper construction of a tomb was material for the dead person's afterlife.In fact,an entire apartment for the future residence of king and queen was installed in each pyramid.They took great care to make the bases of the pyramids of the correct shape;the relative dimensions of base and hcight were also highly significant.However. one should not overemphasize the complexity or depth of the ideas involved. Egyptian mathematics was simple and crude and no deep principles were involved,contrary to what is often asserted. 5.Summary Let us review the status of mathematics before the Greeks enter the picture. We find in the Babylonian and Egyptian civilizations an arithmetic of integers and fractions,including positional notation,the beginnings of algebra,and some empirical formulas in geometry.There was almost no symbolism,hardly any conscious thought about abstractions,no formulation of general methodology,and no concept of proof or even of plausible argu- ments that might convince one of the correctness of a procedure or formula There was,in fact,no conception of any kind of theoretical science Apart from a few incidental results in Babylonia,mathematics in the two civilizations was not a distinct discipline,nor was it pursued for its owr sake.It was a tool in the form of disconnected,simple rules which answered questions arising in the daily life of the people.Certainly nothing was done in mathematics that altered or affected the way of life.Although Babylonian mathematics was morc advanced than the Egyptian,about the best one can say for both is that they showed some vigor,if not rigor,and more per- severance than brilliance
BIBLIOGRAPHY 23 All evaluation implies some sort of standard.It may be unfair but it is natural to compare the two civilizations with the Greek,which succeeded them.By this standard the Egyptians and Babylonians were crude carpenters, whereas the Greeks were magnificent architects.One does find more favor- able,even laudatory,descriptions of the Babylonian and Egyptian achieve- ments.But these are made by specialists who become,perhaps unconsciously, overimpressed by their own field of interest. Bibliography Boyer,Carl B.:A History of Mathemalics,John Wiley and Sons,1968,Chap.2. Cantor,Moritz:Vorlesungen tiber Geschichte der Mathematik,2nd ed.,B.G.Teubner, 1894,Vol.1,Chap.3. Chace,A.B.,et al.eds.:The Rhind Mathematical Papyrus,2 vols.,Mathematical Association of America,1927-29. Childe,V.Gordon:Man Makes Himself,New American Library,1951. Karpinski,Louis C.:The History f Arithmetic,Rand MeNally,1925. Neugebauer,O.:The Exact Sciences in Antiquity,Princeton University Press,1952, Chap.4. Vorgriechische Mathematik,Julius Springer,1934. Sarton,George:A History f,Harvard University Press,1952,Vol.1, Chap.2 Smith,David Eugene:History of Mathematics,Dover (reprint),1958,Vol.1, Chap.2;Vol.2,Chaps.2 and 4. van der wa crden,B.L.:Science Awakening,P.Noordhoff,1954,Chap.1
3 The Creation of Classical Greek Mathematics This,therefore,is mathematics:she reminds you of the in- visible form of the soul;she gives life to her own discoveries; she awakens the mind and purifies the intellect;she brings light to our intrinsic ideas;she abolishes oblivion and ig- norance which are ours by birth. PROCLUS 1.Background In the history of civilization the Greeks are preeminent,and in the history of mathematics the Greeks are the supreme event.Though they did borrow from the surrounding civilizations,the Greeks built a civilization and culture of their own which is the most impressive of all civilizations,the most in- fluential in the development of modern Western culture,and decisive in founding mathematics as we understand the subject today.One of the great problems of the history of civilization is how to account for the brilliance and creativity of the ancient Greeks. Though our knowledge of their early history is subjcct to correction and amplification as more archeological research is carried on,we now have reason to believe,on the basis of the Iliad and the Odyssey of Homer,the decipherment of ancient languages and scripts,and archeological investiga- tions,that the Greek civilization dates back to 2800 B.c.The Greeks settled in Asia Minor,which may have been their original home,on the mainland of Europe in the area of modern Greece,and in southern Italy,Sicily,Crete, Rhodes,Delos,and North Africa.About 775 B.c.the Greeks replaced various hicroglyphic systems of writing with the Phoenician alphabet(which was also used by the Hebrews).With the adoption of an alphabet the Greeks became more literate,more capable of recording their history and ideas. As the Greeks became established they visited and traded with the Egyptians and Babylonians.There are many references in classical Greek writings to the knowledge of the Egyptians,whom some Greeks erroneously considered the founders of science,particularly surveying,astronomy,and 24
THE GENERAL SOURCES 25 arithmetic.Many Greeks went to Egypt to travel and study.Others visited Babvlonia and learned mathematics and scicnce there. The influence of the Egyptians and Babylonians was almost surely felt in Miletus,a city of Ionia in Asia Minor and the birthplace of Greek phil- osophy,mathematics,and science.Miletus was a great and wealthy trading city on the Mediterrancan.Ships from the Greek mainland,Phoenicia,and Egypt came to its harbors;Babylonia was connected by caravan routes leading eastward.Ionia fell to Persia about 540 B.c.,though Miletus was allowed some independence.After an Ionian revolt against Persia in 494 B.c. was crushed,Ionia declined in importance.It became Greek again in 479 B.c. when Greece defeated Persia,but by then cultural activity had shifted to the mainland of Grecce with Athens as its center. Though the ancient Greek civilization lasted until about A.D.600,from the standpoint of the history of mathematics it is desirable to distinguish two periods,the classical,which lasted from 600 to 300 B.c.,and the Alexandrian or Hellenistic,from 300 B.c.to A.D.600.The adoption of the alphabet, already mentioned,and the fact that papyrus became available in Greece during the seventh century B.c.may account for the blossoming of cultural activity about 600 B.c.The availability of this writing paper undoubtedly helped the spread of ideas. 2.The General Sources The sources of our knowledge of Greek mathematics are,peculiarly,less authentic and less reliable than our sources for the much older Babylonian and Egyptian mathematics,because no original manuscripts of the important Greek mathematicians are extant.One reason is that papyrus is perishable; though the Egyptians also used papyrus,by luck a few of their mathematical documents did survive.Some of the voluminous Greek writings might still be available to us if their great libraries had not been destroyed Our chief sources for the Greek mathematical works are Byzantine Greek codices (manuscript books)written from 500 to 1500 years after the Greek works were originally composed.These codices are not literal repro- ductions but critical editions,so that we cannot be sure what changes may have been made by the editors.We also have Arabic translations of the Greek works and Latin versions derived from Arabic works.Hcre again we do not know what changes the translators may have made or how well they under. stood the original texts.Moreover,even the Greek texts used by the Arabic and Byzantine authors were questionable.For example,though we do not have the Alexandrian Greck Heron's manuscript,we know that he made a number of changes in Euclid's Elements.He gave different proofs and added new cases of the theorems and converses.Likewise Theon of Alexandria (end of 4th cent.A.D.)tells us that he altered sections of the Elements in his edition
26 THE CREATION OF CLASSICAL GREEK MATHEMATICS The Greek and Arabic versions we have may come from such versions of the originals.However,in one or another of these forms we do have the works of Euclid,Apollonius,Archimedes,Ptolemy,Diophantus,and other Greek authors.Many Greek texts written during the classical and Alexandrian periods did not come down to us because even in Greek times they were superseded by the writings of these men The Greeks wrote some histories of mathematics and science.Eudemus (4th cent.B.c.),a member of Aristotle's school,wrote a history of arithmetic, a history of geometry,and a history of astronomy.Except for fragments quoted by later writers,these histories are lost.The history of geometry dealt with the period preceding Euclid's and would be invaluable were it available Theophrastus (c.372-c.287 B.c.),another disciple of Aristotle,wrote a history of physics,and this,too,except for a few fragments,is lost. In addition to the above,we have two important commentaries. Pappus (end of 3rd cent.A.D.)wrote the Synagoge or Mathematical Collection; almost the whole of it is extant in a twelfth-century copy.This is an account of much of the work of the classical and Alexandrian Greeks from Euclid to Ptolemy,supplemented by a number of lemmas and theorems that Pappus added as an aid to understanding.Pappus had also written the Treasury of ceeorsAwo公cambewtmreksmhid works themselves.This book is lost,but in The second important commentator is Proclus (A.D.410-485),a prolific writer.Proclus drew material from the texts of the Greek mathematicians and from prior commentarics.Of his surviving works,the Commentary,which treats Book I of Euclid's Elements,is the most valuable.Proclus apparently intended to discuss more of the Elements,but there is no evidence that he ever did so.The Commentary contains one of the three quotations traditionally credited to Eudemus'history of geometry (see sec.10)but probably taken from a later modification.This particular extract,the longest of the three,is referred to as the Eudemian summary.Proclus also tells us something about Pappus'work.Thus,besides the later editions and versions of some of the Greek classics themselves,Pappus'Mathematical Collection and Proclus'Com mentary are the two main sources of the history of Greek mathematics. Of original wordings (though not the manuscripts)we have only a fragment concerning the lunes of Hippocrates,quoted by Simplicius (first half of 6th cent.A.D.)and taken from Eudemus'lost History of Geometry,and a fragment of Archytas on the duplication of the cube.And of original manu- scripts we have some papyri written in Alexandrian Greek times.Related sources on Greek mathematics are also immensely valuable.For example, the Greek philosophers,especially Plato and Aristotle,had much to say about mathematics and their writings have survivcd somewhat in the same way as have the mathematical works. The reconstruction of the history of Greck mathematics,based on sources