Page:A History of Mathematics (1893).djvu/34

 For example, it is required to increase $$\scriptstyle { 1 \over 4 } { 1 \over 8 } { 1 \over 10 } { 1 \over 30 } { 1 \over 45 }$$ to 1. The common denominator taken appears to be 45, for the numbers are stated as $$\scriptstyle 11 { 1 \over 4 }$$, $$\scriptstyle 5 { 1 \over 2 } { 1 \over 8 }$$, $$\scriptstyle 4 { 1 \over 2 }$$, $$\scriptstyle 1 { 1 \over 2 }$$, 1. The sum of these is $$\scriptstyle 23 { 1 \over 2 } { 1 \over 4 } { 1 \over 8 }$$ forty-fifths. Add to this $$\scriptstyle {1 \over 9} {1 \over 40}$$ and the sum is $$\scriptstyle { 2 \over 3 }$$. Add $$\scriptstyle { 1 \over 3 }$$, and we have 1. Hence the quantity to be added to the given fraction is $$\scriptstyle {1 \over 3} {1 \over 9} {1 \over 40}$$.

Having finished the subject of fractions, Ahmes proceeds to the solution of equations of one unknown quantity. The unknown quantity is called 'hau' or heap. Thus the problem, "heap, its $$\scriptstyle { 1 \over 7 }$$, its whole, it makes 19," i.e. $$\scriptstyle {x \over 7}+x=19$$. In this case, the solution is as follows: $$\scriptstyle {8x \over 7}=19$$; $$\scriptstyle {x \over 7}=2 {1 \over 4} {1 \over 8}$$; $$\scriptstyle x=16 {1 \over 2} {1 \over 8}$$. But in other problems, the solutions are effected by various other methods. It thus appears that the beginnings of algebra are as ancient as those of geometry.

The principal defect of Egyptian arithmetic was the lack of a simple, comprehensive symbolism—a defect which not even the Greeks were able to remove.

The Ahmes papyrus doubtless represents the most advanced attainments of the Egyptians in arithmetic and geometry. It is remarkable that they should have reached so great proficiency in mathematics at so remote a period of antiquity. But strange, indeed, is the fact that, during the next two thousand years, they should have made no progress whatsoever in it. The conclusion forces itself upon us, that they resemble the Chinese in the stationary character, not only of their government, but also of their learning. All the knowledge of geometry which they possessed when Greek scholars visited them, six centuries B.C., was doubtless known to them two thousand years earlier, when they built those stupendous and gigantic structures—the pyramids. An explanation for this stagnation of learning has been sought in the fact that their early discoveries in mathematics and medicine had the misfortune of