258 POPULAR SCIENCE MONTHLY. 



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 cur- 

 rency. The annual inundation of the Nile was continually altering the 

 minor features of the country along its course, and washing away land- 

 marks between adjacent properties. Some means of re-establishing 

 these marks and of determining the areas of fields was therefore essen- 

 tial. 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 squar- 

 ing 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 him- 

 self 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 diam- 

 eter 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. 



