HISTORY OF ALGEBRA. 181 



and add the root of the sum to half the co-efficient of the Shai. This is 

 the unknown quantity. For example. What number is that which being 

 subtracted from its square, and the remainder added to the square, is 10? 

 Subtract Shai from Mdl and go on with the operation, 2 Mdl all but Shai 

 is equal to 10 ; and after Jebr and Radd, Mdl is equal to 5 and \ of Shai. 

 The square of half the coefficient of the Shai and 5, is 5 and half an 

 eighth, and its root is 'i%i To this add £, the result is sf, which is 

 the number required. 



Book ninth, contains twelve rules regarding the properties of numbers, 



Viz. 



1st. To find the sum of the products of a number multiplied into itself 

 and into all numbers below it: add one to the number, and multiply the 

 sum by the square of the number ; half the product is the number 

 required. 



2d. To add the odd numbers in their regular order: add one to the 

 last number and take the square of half the sum. 



3d. To add even numbers from two upwards : multiply half the last 

 even number by a number greater by one than that half. 



4th. To add the squares of the numbers in order: add one to twice 

 the last number, and multiply a third of the sum by the sum of the num- 

 bers. 



5th. To find the sum of the cubes in succession : take the square of 

 the sum of the numbers. 



6th. To find the product of the roots of two numbers : multiply one 

 by the other, and the root of the product is the answer. 



X 2 



