PROGRESS OF SCIENCE TO 1450 A.D. 179 



Leonardo inherits Egyptian as well as Greek traditions, for ex- 

 ample, the type of fraction just mentioned, square and cube root, 

 progressions, the method of false assumption. It would appear 

 that when the Arabs conquered Alexandria some of the old 

 Egyptian culture was preserved. The rule of three, partner- 

 ship, powers and roots, and the solution of equations are also 

 included. 



In 1225 the emperor, impressed by the accounts of Pisano's 

 mathematical power, arranged a mathematical tournament of 

 which the challenge questions are preserved : 



' To find a number of which the square, when either increased or 

 diminished by 5, would remain a square. 



'To find by the methods used in the tenth book of Euclid a line 

 whose length x should satisfy the equation x 3 -f- 2x 2 + Wx = 20. 



' Three men, A, B, C, possess a sum of money u, their shares being 

 in the ratio 3:2:1. A takes away x, keeps half of it, and deposits 

 the remainder with D ; B takes away y, keeps f of it, and deposits 

 the remainder with D ; C takes away all that is left, namely z, keeps 

 f of it, and deposits the remainder with D. This deposit is found 

 to belong to A, B, and C in equal proportions. Find u, x, y and z.' 

 Leonardo gave a correct solution of the first and third, also a root 

 of the cubic equation correct to nine decimals. Ball. 



Jordanus Nemorarius wrote important Latin works on arith- 

 metic, geometry, and astronomy. His De Triangulis the most 

 important of these consists of four books dealing not only 

 with triangles, but with polygons and circles. He generally uses 

 Arabic numerals, and denotes quantities known or unknown by 

 letters. He solves the problem of finding two numbers having a 

 given sum and product, by a method equivalent to our elemen- 

 tary algebra. This is practically the first European syncopated 

 algebra, but seems to have become too little known to have far- 

 reaching results in a time not yet ripe for this invention. A book 

 on Weights contains elements of mechanics. 



Albertus Magnus, born near the end of the twelfth century, 

 became an ardent champion of the newly discovered but pro- 

 scribed works of Aristotle. In particular he interpreted the Milky 



