.182 ON THE EARLY 



7th. To divide the root of one number by that of another : divide one 

 by the other, the root of the quotient is the answer. 



8th. To find a perfect number : that is a number which is equal to the 

 sum of its aliquot parts, (Euclid, book 7, def. 22.) The rule is that 

 delivered by Euclid, book 9, prop. 36. 



9th. To find a square in a given ratio to its root: divide the first num- 

 ber of the ratio by the second ; the square of the quotient is the square 

 required. 



loth. If any number is multiplied and divided by another, the product 

 multiplied by the quotient is the square of the first number. 



11th. The difference of two squares is equal to the product of the sum 

 and difference of the roots. 



12th. If two numbers are divided by each other, and the quotients 

 multiplied together, the result is always one. 



Book tenth, contains nine examples, all of which are capable of solu- 

 tion by simple equations, position, or retracing the steps of the operation, 

 and some of there by simple proportion ; so that it is needless to specify 

 them. 



The conclusion, which marks the limits of algebraical knowledge in the 

 age of the writer, I shall give entire, in the author's words. " Conclu- 

 sion. There are many questions in this science which learned men have 

 to this time in vain attempted to solve ; and they have stated some of 

 these questions in their writings, to prove that this science contains diffi- 

 culties, to silence those who pretend they find nothing in it above their 

 ability^ to warn arithmeticians against undertaking to answer every ques« 



