10 MATHEMATICS IN MESOPOTAMIA symbolism came about inadvertently.In some problems two special Sumer- ian words(somewhat modified by Akkadian endings)were used to represent two unknowns that were reciprocals of each other.Moreover,the ancient Sumerian pictorial symbols were used for these words,and since these pictorial symbols were no longer in the current language,the effect was to use special symbols for unknowns.These symbols were used repeatedly and one could identify them without even knowing how they were pronounced in Akkadian. The algebraic problems were solved by describing only the steps re- quired to execute the solution.For example,square 10 obtaining 100; subtract 100 from 1000;this gives 900;and so forth.Since no reasons for steps were given,we can only infer how they knew what to do They summed arithmetic progressions and geometric progressions in concrete problems;for the latter we find,in our notation 1+2+4+.·+29=29+(29-1)=210-1. Also,the sum of the squares of the integers from I to 10 was given as though they had applied the formula 12+22+.+2=（1×写+n×)1+2+3+.+. No derivation accompanied the special cases treated in their texts. Babylonian algebra included a bit of the theory of numbers.Many sets of Pythagorean triples were found,probably by the correct rule;that is,if x三b2- 92,y =2pq,z =p2+92,then x2+y2=z2.They also solved x2+y2=222 in integers. 6.Babylonian Geometry The role of geometry in Babylonia was insignificant.Geometry was not a separate mathematical discipline.Problems involving division of a field or the sizes of bricks needed for some construction were readily converted into algebraic problems.Some computations of area and volume were given in accordance with rules or formulas.However,the figures illustrating geo metric problems were roughly drawn and the formulas may have been in- correct.In the Babylonians'calculations of areas,for examplc,we cannot tell whether their triangles were right triangles or whether their quadri- laterals were squares and,hence,whether their formulas were the right ones for the figures concerned.However,the Pythagorean relationship,the simi larity of triangles,and the proportionality of corresponding sides in similar triangles were known.The area of a circle was obtaincd,apparently by following the rule that d 12wherecis the circumference.Thisrule amounts