ditionalism made it an act of sacrilege to alter what had become part of the sacred writings.
When we consider the conditions of life in Egypt we can easily see why this particular kind of geometric knowledge so early gained currency. The annual inundation of the Nile was continually altering the minor features of the country along its course, and washing away landmarks between adjacent properties. Some means of re-establishing these marks and of determining the areas of fields was therefore essential. To meet this demand the surveyors devised the rules which Ahines has given us. The further necessity of ascertaining the contents of a barn of given shape and dimensions likewise gave rise to the rules for determining volumes.
We learn also that the Egyptians were acquainted with the truth of the Pythagorean theorem, that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, for they applied this knowledge practically by means of a triangle whose sides were 3, 4 and 5 respectively, in laying down right angles. This general truth was derived in all probability by deduction from a large number of individual cases. The Egyptian rule for the area of a circle was remarkably accurate for such an early date. It consisted in squaring eight-ninths of the diameter. This gives to n the value 3.1605.
It is generally supposed that the Greeks had their attention drawn to geometry through intercourse with the Egyptians. It was but a step, however, for them to pass beyond the latter, and with them we find the birth of the true mathematical spirit which refuses to accept anything upon authority, but requires a logical demonstration. It is well known what an important place was held by geometry in Greek philosophy. The Pythagorean school contributed much that was important along with a great deal that was fanciful and of little value. Pythagoras himself was the first to prove the theorem referred to above, which goes by his name. The Greeks for the most part pursued the study of geometry as a purely intellectual exercise. Anything in the nature of practical applications of the subject was repugnant to them, and hence but little attention was paid to theorems of mensuration. This reminds one of the story told of a professor of mathematics in modern times who, in beginning a course of lectures, made the remark: "Gentlemen, 'to my mind the most interesting thing about this subject is that I do not see how under any circumstances it can ever be put to any practical use." Euclid in his 'Elements' does not mention the theorem that the area of a triangle is equal to half the product of its base and altitude, nor does he enter into any discussion of the ratio of the circumference to the diameter of a circle. This last, however, was a problem which as early as the time of Pythagoras had attracted much attention. 'Squaring the circle' was a stumbling block to the Greeks and has been ever since.
