THE ARITHMETIC 17 result.Multiplication and division were also reduced to additive processes. To calculate 12 times 12,say,the Egyptians did the following: 2 2 24 4 48 8 96 Each line was derived from the preceding one by doubling.Now since 4.12 =48 and 8.12 =96,adding 48 and 96 gave the value of 12.12.This process was,of course,quite different from multiplying by 10 and by 2and adding.Multiplication by 10 was also performed and consisted of replacing unit symbols by the symbol for 10 and replacing the 10 symbol by the symbol for 100. Division of one whole number by another as carried out by the Egyptians is equally interesting.For example,19 was divided by 8 as follows: 8 16 1/2 1/4 1/8 1 Therefore,the answer was 2+1/4 1/8.The idea was simply to take the number of eights and parts of eight that totaled 19. The denotation of fractions in the Egyptian number system was much more complicated than our own.The symbol,pronounced ro,which originally indicated 1/320 of a bushel,came to denote a fraction.In hieratic writing the oval was replaced by a dot.The or dot was generally placed above the whole number to indicate the fraction.Thus,in hiero- glyphic writing, = 分= 分= A few fractions were denoted by special symbols.Thus the hieroglyph denoted 1/2;T,2/3;and x,1/4. Aside from a few special ones,all fractions were decomposed into what are called unit fractions.Thus Ahmes writes 2/5 as 1/3 +1/15.The plus sign did not appear but was understood.The Rhind papyrus contains a table for expressing fractions with numerator 2 and odd denominators from 5 to 101 as sums of fractions with numerator 1.By means of this table a frac- tion such asour729,which to Ahmes is the integer7 divided by the integer 29,could also be expressed as a sum of unit fractions.Inasmuch as 7 =2+ 2+2+1,he proceeds by converting each 2/29 to a sum of fractions with numerator 1.By combining these results and by further conversion he ends
18 EGYPTIAN MATHEMATICS up with a sum of unit fractions,each with a different denominator.The final expression for 7/29 is 6+24+58+87+232 It so happens that 7/29 can also be expressed as 1/5 +1/29 +1/145,but because Ahmes'2/n table leads to the former expression,this is the one used. The expression of our a/b as a sum of unit fractions was done systematically according to age-old procedures.Using unit fractions,the Egyptians could carry out the four arithmetic operations with fractions.The xtensive and complicated computations with fractions were one reason the Egyptians never developed arithmetic or algebra to an advanced state. The nature of irrational numbers was not recognized in Egyptian arithmetic any more than it was in the Babylonian.The simple square roots that occurred in algebraic problems could be and were expressed in terms of whole numbers and fractions. 3.Algebra and Geometry The papyri contain solutions of problems involving an unknown that are in the main comparable to our linear equations in one unknown.However,the processes were purely arithmetical and did not,in Egyptian minds,amount to a distinct subject,the solution of equations.The problems were stated verbally with bare directions for obtaining the solutions and without ex- planation of why the methods were used or why they worked.For example, problem 31 of the Ahmes papyrus,translated literally,reads:"A quantity, its 2/3,its 1/2,its 1/7,its wholc,amount to 33."This means,for us: 3×+亏++x=33. Simple arithmetic of the Egyptian variety gives the solution in this case. Problem63of the papyrus runs as follows:"Directions for dividing 700 breads among four people,23 for one,1/2for the second,13 for the third, 1/4 for the fourth."For us this means 3*+互x+3*+年x=700 The,as given by Ahm,:“Adn学安子This give1安 Divide1by1专子This gives号年Nowfind吃of70.Tis40” 11 11 In some solutions Ahmes uses the "rule of false position."Thus to determine five numbers in arithmetic progression subject to a further con dition and such that the sum is 100,he first chooses d,the common dif
ALGEBRA AND GEOMETRY 19 ference,to be 5 1/2 times the smallest number.He then picks 1 as the smallest and gets the progression:1,6 1/2,12,17 1/2,23.But these numbers add up to 60 whereas they should add up to 100.He then multiplies each term by 5/3. Only the simplest types of quadratic equations,such as ax2=b,are considered.Even when two unknowns occur,the type is x2+y2=100, so that after eliminating y,the equation in x reduces to the first type.Some concrete problems involving arithmetic and geometric progressions also can be found in the papyri.To infer general rules from all these problems and solutions is not very difficult. The limited Egyptian algebra employed practically no symbolism.In the Ahmes papyrus,addition and subtraction are represented respectively by the legs of a man coming and going,andand the symbol Iis used to denote square root. What of Egyptian geometry?The Egyptians did not separate arith- metic and geometry.We find problems from both fields in the papyri.Like the Babylonians,the Egyptians regarded geometry as a practical tool.One merely applicd arithmctic and algebra to the problems involving areas volumes,and other gcometrical situations.Egyptian geometry is said by Herodotus to have originated in the need created by the annual overflow of the Nile to redetermine the boundaries of the lands owned by the farmers. However,Babylonia did as much in geometry without such a need.The Egyptians had prescriptions for the arcas of rectangles,triangles,and trape- zoids.In the case of the area of a triangle,though they multiplied one number by half another,we cannot be sure that the method is correct because we are not sure from the words used whether the lengths multiplicd stood for base and altitude or just for two sides.Also the figures were so Doorly drawn that one cannot be sure of lust what area or volume were being found.Their calculation of the area of a circle,surprisingly good, followed the formula A =(8d/9)2where d is the diameter.This amounts to using 3.1605 for m. An example may illustrate the "accuracy"of Egyptian formulas for area.On the walls of a temple in Edfu is a list of fields that were gifts to the temple.These fields gencrally have four sides,which we shall denote by a,b,c,d,where a and b and c and d are pairs of opposite sides.The inscrip- tions give the arca of thesc various fields as But some fields 2 are triangles.In this case,d is said to be nothing and the calculation is changed to (a+b)c 2 Even for quadrangles the rule is just a crude approximation
20 EGYPTIAN MATHEMATICS The Egyptians also had rules for the volume of a cube,box,cylinder, and other figures.Some of the rules were correct and others only approxi- mations.The papyri give as the volume of a truncated conical clepsydra (water clock),in our notation, r=(传0+： h 13 where h is the height and (D+d)/2 is the mean circumference.This formula amounts to using 3 for m. The most striking rule of Egyptian geometry is the one for the volume of a truncated pyramid of square base,which in modern notation is V=(@2+b+6, where h is the height and a and b are sides of top and bottom.The formula is surprising because it is correct and because it is symmetrically expressed(but of course not in our notation).It is given only for concrete numbers.How- ever,we do not know whether the pyramid was square-based or not because the figure in the papyrus is not carefully drawn. Neither do we know whether the Egyptians recognized the Pythagorean theorem.We know there were rope-stretchers,that is,surveyors,but the story that they used a rope knotted at points to divide the total length into parts of ratios 3 to 4 to 5,which could then be used to form a right triangle, is not confirmed in any document. The rules were not expressed in symbols.The Egyptians stated the problems verbally;and their procedure in solving them was essentially what we do when we calculate according to a formula.Thus an almost literal translation of the geometrical problem of finding the volume of the frustum of a pyramid reads:"Ifyou are told:a truncated pyramid offor the verti- cal height by 4 on the base,by 2 on the top.You are to square this 4,result 16.You are to double,result 8.You are to square 2,result 4.You are to add the 16,and 8,and the 4,result 28.You are to take one-third of 6,result 2. You are to take 28 twice,result 56.See,it is 56.You will find it right." Did the Egyptians know proofs or justifications of their procedures and formulas?One belief is that the Ahmes papyrus was written in the style of a textbook for students of that day and hence,even though no general rules or principles for solving types of equations were formulated by Ahmes,it is very likely that he knew them but wanted the student to formulate them himself or have a teacher formulate them for him.Under this view the Ahmes papyrus is a rather advanced arithmetic text.Others say it is the notebook of a pupil.In either case,the papyri almost surely recorded the types of problems that had to be solved by business and administrative clerks,and the methods of solution were just practical rules known by ex- perience to work.No one believes that the Egyptians had a deductive
EGYPTIAN USES OF MATHEMATICS structure based on sound axioms that established the correctness of their rules. 4.Egyptian Uses of Mathematics The Egyptians used mathematics in the administration of the affairs of the state and church,to determine wages paid to laborers,to find the volumes of granaries and the areas of fields.to collect taxes assessed according to the land area,to convert from one system of measures to another,and to calcu- late the number of bricks needed for the construction of buildings and ramps The papyri also contain problems dealing with the amounts of corn needed to make given quantities of beer and the amount of corn of one quality needed to give the same result as corn of another quality whose strength relative to the first is known. As in Babylonia a major use of mathematics was in astronomy,which dates from the first dynasty.Astronomical knowledge was essential.To the Egyptian the Nile was his life's blood.He made his living by tilling the soil which the Nile covered with rich silt in its annual overflow.However,he had to be well prepared for the dangerous aspects of the flood;his home,equip- ment,and cattle had to be temporarily removed from the area and arrange ments made for sowing immediately afterwards.Hence the coming of the flood had to be predicted,which was done by learning what heavenly events preceded it. Astronomy also made the calendar possible.Beyond the need for a calendar in commerce was the need to predict religious holidays.It was believed essential,to ensure the goodwill of the gods,that holidays be cele- brated at the proper time.As in Babylonia,keeping the calendar was largely the task of the priests The Egyptians arrived at their estimate of the length of the solar year by observing the star Sirius.On one day in the summer this star became visible on the horizon just before sunrise.On succeeding days it was visible for a longer time before the sun's growing light blotted it out.The first day on which it was visible just before sunrise was known as the heliacal rising of Sirius,and the interval betwcen two such days was about 365 1/4 days;so the Egyptians adopted,supposcdly in 4241 B.c.,a civil calendar of 365 days for the year.The concentration on Sirius is undoubtedly accounted for by the fact that the waters of the Nile began to rise on that day,which was chosen as the first day of the vear. The 365-day year was divided into 12 months of 30 days,plus 5 extra days at the end.Because the Egyptians did not intercalate the additional day every four years,the civil calendar lost all relation to the seasons.It takes 1460 years for the calendar to set itself right again;this interval is known as the Sothic cycle,from the Egyptian name for Sirius.Whether the Egyptians