numerators are 2 to a sum of fractions each of whose numerators is 

 1. Thus gg is stated to be the sum of ^, g&, ^74 and ^- 

 Whether Ahmes has any general rule for this kind of reduction, or 

 whether the solutions given represent the accumulative experience 

 of previous writers, it is hard to determine. Having finished the 

 subject of fractions, Ahmes next proceeds to the solution of pro- 

 blems which are nothing more nor less than equations of one un- 

 known quantity 1 . The problem, in its translation, is generally 

 stated thus, Heap, its % its whole, it makes 19, or as it would be 

 put to-day #+f =19. Ahmes gives the answer as 16 + i + i, still 

 following the Egyptian custom illustrated above of reducing all 

 fractions to the sum of those whose numerators are 1. Algebra 

 then, in its beginnings, would appear to be as ancient as geometry. 

 The document also devotes itself to attempting to find the areas 

 of isosceles triangles and of isosceles trapezoids. The areas of 

 circles are also found with a very fair approximation to correctness. 

 It is evident then that mathematics, more especially of a 

 practical kind, had reached a relatively high degree of attainment 

 among the Egyptians at a very early period. But the same 

 appears true, though perhaps in a somewhat less degree of the 

 ancient Babylonians. Two tablets have been discovered and the 

 cuneiform system of writing deciphered. The most interesting 

 information gleaned from the first tablet is that the Babylonians 

 must have used not only the decimal system but also a sexagesimal 

 one. For instance, the tablet contains a table of square numbers 

 up to GO 2 . The square of 7 is given as 49, that of 8 as 1.4, that of 9 

 as 1.21 and so on. The only intelligible solution for this is that 

 they used the sexagesimal scale. Cantor has suggested that the 

 reason the Babylonians used this system is because they at first 

 reckoned the year at 360 days. The circle was then divided into 

 360 degrees, each degree being analogous to the amount of the 

 supposed daily movement of the sun around the earth. If it be 

 assumed that they were familiar with the fact that the radius may 

 be applied as a chord to the circumference six times and that each 

 of the resulting chords subtends an arc, measuring exactly 60, we 

 have a probable basis for their using the sexagesimal system. The 



1 Cf. Cajori, History of Mathematics, P. 15. 



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