32 A SHORT HISTORY OF SCIENCE 



and side 10 is said to have as its area i X 10 = 20, the actual area 

 being of course f X \/ 100 4(= 19.6 approximately). It is 

 interesting that this and similar crude methods continued in use 

 by surveyors for many centuries, even after Euclid had given geo- 

 metrical science its modern form. Another problem amounts tc 

 finding two squares having a given total area and their sides in 

 a given ratio, being thus equivalent to solving the equations 



x z + y* = 100 x : y = 1 : f 



By trial x = 1, y = f , give x 2 + y 2 = (f) 2 . 



Since 100 = (f) 2 X 8 2 , the trial values must be multiplied by 8, 



so that x = 8 and y = 6. 



The classical problem of " squaring the circle " is attempted, the 

 result being equivalent to the approximation TT = ^- = 3.16, as 

 against the actual 3.14 -- an excellent result for the time. 



Other computations deal with the capacity of storehouses of 

 unknown shape for grain. A remarkable group of problems 

 deals with a certain geometrical ratio in pyramids equivalent to 

 a modern cosine or cotangent, and of interest in connection with 

 the uniform slope of the great pyramids. 



EGYPTIAN LAND MEASUREMENT. Greek writers emphasize 

 the methods of land measurement of the Egyptians consequent on 

 the obliteration of boundaries by floods of the Nile. Herodotus 

 relates that Sesostris had so divided the land among all Egyptians 

 that each received a rectangle of the same size, and was taxed ac- 

 cordingly. Whoever lost any of his land by the action of the river 

 must report to the king, who would then send an overseer to meas- 

 ure the loss, and make a proportionate abatement of the tax. 

 Thus arose geometry (geometria = earth measurement). Dio- 

 dorus, for example, says: "The Egyptians claim to have intro- 

 duced alphabetical writing and the observation of the stars, like- 

 wise the theorems of geometry, and most of the arts and sciences." 

 The priests "occupy themselves busily with geometry and arith- 

 metic, for as the river annually changes the land, it causes many 

 controversies as to boundaries between neighbors. These cannot 

 be easily adjusted unless a geometer ascertains the real facts 



