12 MATHEMATICS IN MESOPOTAMIA procedural texts and ephemerides,tables of positions of the heavenly bodies at various times.The procedural texts show how to compute the ephemerides. The arithmetic behind the lunar and solar observations shows that the Babylonians calculated first and second differences of successive data,ob- served the constancy of the first or second differences,and extrapolated or interpolated data.Their procedure was equivalent to using the fact that the data can be fit by polynomial functions and enabled them to predict the daily positions of the planets.They knew the periods of the planets with some accuracy,and also used eclipses as a basis for calculation.There was,how- ever,no geometrical scheme of planetary or lunar motion in Babylonian astronomy. The Babylonians of the Seleucid period did have extensive tables on the motions of the sun and moon which gave variable velocities and positions. Also special conjunctions and eclipses of sun and moon were either in the data or readily obtained from them.Astronomers could predict the new moon and eclipses to within a few minutes.Their data indicate that they knew the length of the solar or tropical year(the year of the seasons)to be 12 +22/60 +8/602 months (from new moon to new moon)and the length of the sidereal year(the time for the sun to regain the same position relative to the stars)to within 4 1/2 minutes. The constellations that lent their names to the twelve signs of the zodiac were known earlier but the zodiac itself first appears in a text in 419 B.C. Each sector of the zodiac was a 30 arc.Positions of planets in the sky were fixed by reference to the stars and also by position in the zodiac. Astronomy served many purposes.For one thing,it was needed to keep a calendar,which is determined by the positions of the sun,moon,and stars The year,the month,and the day are astronomical quantities,which had to be obtained accurately to know planting times and religious holidays.In Babylonia,partly because of the connection of the calendar with religious holidays and cere monies and partly because the heavenly bodies were be- lieved to be gods,the priests kept the calendar. The calendar was lunar.The month began when the crescent first appeared after the moon was fully dark (our new moon).The day began in the evening of the first appearance of the crescent and lasted from sunset to sunset.The lunar calendar is difficult to maintain because,although it is convenient to have the month contain an integral number of days,lunar months,reckoned as the time between successive conjunctions of sun and moon (that is,from new moon to new moon),vary from 29 to 30 days. Hence a problem arises in deciding which months are to have 29 and which 30.A more important problem is making the lunar calendar agree with the seasons.The answer is quite complicated because it depends upon the paths and velocities of the moon and sun.The lunar calendar contained extra months intercalated so that 7 such intercalations in each 19 years just about
EVALUATION OF BABYLONIAN MATHEMATICS 13 kept the lunar calendar in time with the solar year.Thus 235 lunar months were equal to 19 solar years.The summer solstice was systematically com- puted,and the winter solstice and the equinoxes were placed at equal intervals.This calendar was used by the Jews and Greeks and by the Romans up to 45 B.c.,the year the Julian calendar was adopted. The division of the circle into 360 units originated in the Babylonian astronomy of the last centuries before the Christian era.It had nothing to do with the earlier use of base 60;however,base 60 was used to divide the degree and the minute into 60 parts.The astronomer Ptolemy (2nd cent. A.D.)followed the Babylonians in this practice. Closely connected with astronomy was astrology.In Babylonia,as in many ancient civilizations,the heavenly bodies were thought to be gods and so were presumed to have influence and even control over the affairs of man. When one takes into account the importance of the sun for light,heat,and the growth of plants,the dread inspired by its eclipses,and such seasonal phenomena as the mating of animals,one can well understand the belief that the heavenly bodies do affect even the daily events in man's life. Pseudoscientific schemes of prediction in ancient civilizations did not always involve astronomy.Numbers themselves had mystic properties and could be used to make predictions.One finds some Babylonian usages in the Book of Daniel and in the writings of the Old and New Testament prophets. The Hebrew "science"of gematria (a form of cabbalistic mysticism)was based on the fact that each letter of the alphabet had a number value because the Hebrews used letters to represent numbers.If the sum of the numerica values of the letters in two words was the same,an important connection between the two ideas or people or events represented by the words was inferred.In the prophecy of Isaiah (21:8),the lion proclaims the fall of Babylon because the letters in the Hebrew word for lion and those in the word for Babylon add up to the same sum. 8.Evaluation of Babylonian Mathematics The Babylonians'use of special terms and symbols for unknowns,their employment of a few operational symbols,and their solution of a few types ofequations involving one or more unknowns,especially quadratic equations, constituted a start in algebra.Their development of a systematic way of writing whole numbers and fractions enabled them to carry arithmetic to a fairly advanced stage and to employ it in many practical situations,especially in astronomy.They possessed some numerical and what we would call algebraic skill in the solution of special equations of high degree,but on the whole their arithmetic and algebra were very elementary.Though they worked with concrete numbers and problems,they evidenced a partial grasp
冬 MATHEMATICS IN MESOPOTAMIA of abstract mathematics in their recognition that some procedures were typical of certain classes of equations. The question arises as to what extent the Babylonians employed mathematical proof.They did solve by correct systematic procedures rather complicated equations involving unknowns.However,they gave verbal in- structions only on the steps to be made and offered no justification of the steps.Almost surely,the arithmetic and algebraic processes and the geomet- rical rules were the end result of physical evidence,trial and error,and in- sight.That the methods worked was sufficient justification to the Babylonians for their continued use.The concept of proof,the notion of a logical structure based on principles warranting acceptance on one ground or another.and the consideration of such questions as under what conditions solutions to problems can exist,are not found in Babylonian mathematics. Bibliography Bell,E.T.:The Development of Mathematics,2nd ed.,McGraw-Hill,Chaps.1-2. Boyer,Carl B.:A History of Mathematics,John Wiley and Sons,1968,Chap.3. Cantor,Moritz:Vorlesungen iber Geschichte der Mathematik,2nd ed.,B.G.Teubner, ”1894,Vol.l,Chap- 1 Chiera,E.:They Wrote on Clay,Chicago University Press,1938. Childe,V.Gordon:Man Makes Himself,New American Library,1951,Chaps.6-8. Dantzig,Tobias:Number:The Language of Science,4th ed.,Macmillan,1954, Chaps.1-2. Karpinski,Louis C.:The History f Arithmetic,Rand McNally,1925. Menninger,K.:Number Words and Number Symbols:A Cultural History of Numbers, Massachusetts Institute of Technology Press,1969. Neugebauer,Otto:The Exact Sciences in Antiquity,Princeton University Press,1952, Chaps.1-3 and 5. Vorgriechische Mathematik,Julius Spr ger,1934,Chaps.1-3 and 5 Sarton,George:A Hisr Harvard University Pres,92,Vol 1,Chap. -The Study of the History of Mathematics and the History of Sciene,Dover (reprint),1954. Smith,David Eugene:History of Mathematics,Dover (reprint),1958,Vol.1, Chap.I. Struik,Dirk J.:A Concise History of Mathematics,3rd cd.,Dover,1967,Chaps.1-2. van der Waerden,B.L.:Science Awakening,P.Noordhoff,1954,Chaps.2-3
2 Egyptian Mathematics All science,logic and mathematics included,is a function of the epoch- all science,in its ideals as well as in its achievements. E.H.MOORE 1.Background While Mesopotamia experienced many changes in its ruling peoples,with resulting new cultural influences,the Egyptian civilization developed un- affected by foreign influences.The origins of the civilization are unknown but it surely existed even before 4000 B.c.Egypt,as the Greek historian Hero- dotus says,is a gift of the Nile.Once a year this river,drawing its water from the south,floods the territory all along its banks and leaves behind rich soil. Most of the people did and still do make their living by tilling this soil.The rest of the country is desert. There were two kingdoms,one in the north and one in the south of what is present-day Egypt.Some time between 3500 and 3000 B.c.,the ruler, Mena,or Menes,unified upper and lower Egypt.From this time on the major periods of Egyptian history are referred to in terms of the ruling dynasties,Menes having been the founder of the first dynasty.The height of Egyptian culture occurred during the third dynasty (about 2500 B.c.), during which period the rulers built the pyramids.The civilization went its own way until Alexander the Great conquered it in 332 B.c.Thereafter, until about A.D.600,its history and mathematics belong to the Greek civilization.Thus,apart from one minor invasion by the Hyksos(1700- 1600 B.c.)and slight contact with the Babylonian civilization(inferred from the discovery in the Nile valley of the cuneiform Tell al-Amarna tablets of about 1500 B.c.),Egyptian civilization was the product of its native people The ancient Egyptians developed their own systems of writing.One system,hieroglyphics,was pictorial,that is,each symbol was a picture of some object.Hieroglyphics were used on monuments until about the time of Christ.From about 2500 B.c.the Egyptians used for daily purposes what is called hieratic writing.This system employed conventional symbols,which at first were merely simplifications of the hieroglyphics.Hieratic writing is
16 EGYPTIAN MATHEMATICS syllabic;each syllable is represented by an ideogram and an entire word is a collection of ideograms.The meaning of the word is not tied to the separate ideograms. The writing was done with ink on papyrus,sheets made by pressing the pith of a plant and then slicing it.Sincc papyrus dries up and crumbles, very few documents of ancient Egypt have survived,apart from the hiero glyphic inscriptions on stone. The main surviving mathematical documents are two sizable papyri: the Moscow papyrus,which is in Moscow,and the Rhind papyrus,dis- covered in 1858 by a Britisher,Henry Rhind,and now in the British Museum. The Rhind papyrus is also known as the Ahmes papyrus after its author,who opens with the words"Directions for Obtaining the Knowledge of All Dark Things."Both papyri date from about 1700 B.c.There are also fragments of other papyri written at this time and later.The mathematical papyri were written by scribes who were workers in the Egyptian state and church administrations. The papyri contain problems and their solutions-85 in the Rhind papyrus and 25 in the Moscow papyrus.Presumably such problems occurred in the work of the scribes and they were expected to know how to solve them. It is most likely that the problems in the two major papyri were intended as examples of typical problems and solutions.Though these papyri date from about 1700 B.c.,the mathematics in them was known to the Egyptians as far back as 3500 B.c.,and little was added from that time to the Grcek conquest. 2.The Arithmetic The hieroglyphic number symbols used by the Egyptians were I for 1, nfor 10,and for 100,for 1000,for 10,000,and other symbols for larger units.These symbols were combined to form intermediate numbers.The direction of the writing was from right to left,so that represented 24.This system of writing numbers uses the base 10 but is not positional. Egyptian hieratic whole numbers are illustrated by the following symbols: 1 2345678 0 10 The arithmetic was essentially additive.For ordinary additions and sub- tractions they could simply combine or take away symbols to reach the proper