| II. Ancient Mathematics
The earliest records of advanced, organized mathematics date back to the ancient Mesopotamian country of Babylonia and to Egypt of the 3rd millennium scaps bc. There mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry and with no trace of later mathematical concepts such as axioms or proofs.
The earliest Egyptian texts, composed about 1800 scaps bc, reveal a decimal numeration system with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as in the system used by the Romans. Numbers were represented by writing down the symbol for 1, 10, 100, and so on as many times as the unit was in a given number. For example, the symbol for 1 was written five times to represent the number 5, the symbol for 10 was written six times to represent the number 60, and the symbol for 100 was written three times to represent the number 300. Together, these symbols represented the number 365. Addition was done by totaling separately the units¡ª10s, 100s, and so forth¡ªin the numbers to be added. Multiplication was based on successive doublings, and division was based on the inverse of this process.
The Egyptians used sums of unit fractions ( 1/n), supplemented by the fraction 2/3, to express all other fractions. For example, the fraction 2/7 was the sum of the fractions ? and 1/28. Using this system, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. In geometry, the Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the volumes of figures such as bricks, cylinders, and pyramids. To find the area of a circle, the Egyptians used the square on 8/9 of the diameter of the circle, a value of about 3.16¡ªclose to the value of the ratio known as pi, which is about 3.14.
The Babylonian system of numeration was quite different from the Egyptian system. In the Babylonian system¡ªwhich, when using clay tablets, consisted of various wedge-shaped marks¡ªa single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics. The number 60, however, was represented by the same symbol as 1, and from this point on a positional symbol was used. That is, the value of one of the first 59 numerals depended henceforth on its position in the total numeral. For example, a numeral consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 ¡Á 602 + 27 ¡Á 60 + 10. This principle was extended to the representation of fractions as well, so that the above sequence of numbers could equally well represent 2 ¡Á 60 + 27 + 10 ¡Á ( 1/60), or 2 + 27 ¡Á ( 1/60) + 10 ¡Á ( 1/60-2). With this sexagesimal system (base 60), as it is called, the Babylonians had as convenient a numerical system as the 10-based system.
The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation (see Equation). They could even find the roots of certain cubic equations. The Babylonians had a variety of tables, including tables for multiplication and division, tables of squares, and tables of compound interest. They could solve complicated problems using the Pythagorean theorem; one of their tables contains integer solutions to the Pythagorean equation, a2 + b2 = c2, arranged so that c2/a2 decreases steadily from 2 to about 4/3. The Babylonians were able to sum arithmetic and some geometric progressions, as well as sequences of squares. They also arrived at a good approximation for . In geometry, they calculated the areas of rectangles, triangles, and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders. However, the Babylonians did not arrive at the correct formula for the volume of a pyramid.