6o Algebra. 



lem really works out all right is that it was made to order ; that is, 

 the number 2944 was especially selected so that the problem 

 would ' ' work out " . If we wish this problem stated in words so 

 that it is more nearly identical with its expression in the form of 

 an equation, we must throw out the word "digits" as follows : 



There are two numbers whose sum is ten. If ten times the 

 first plus the second is multiplied by ten times the second plus 

 the first, the product will be 2944. Find the numbers. 



This is nearly as general as the algebraic equation. It permits 

 of either positive or negative, integral or fractional, commensur- 

 able or incommensurable, results, and indeed as the word num- 

 ber is often used it would permit of imaginary results. This prob- 

 lem can be made identical with the original by adding at its close 

 some such caution as this : 



Do not obtain a fractional, a negative, nor an incommensur- 

 able result, nor any result greater than 9. 



It is such conditions as these that we fail to incorporate into an 

 algebraic equation. The algebraic statement, as far as the un- 

 known is concerned, is always the most general possible and con- 

 tains in it no restriction of the unknown to any particular class of 

 numbers, and for this reavSon the algebraic statement of a problem is 

 often more general than the problem itself. This fact should be re- 

 membered, as it will help to explain many apparent difficulties 

 which arise in some problems. These non-algebraic conditions in 

 a problem must be ignored until after the solution is had, and 

 then if a result is obtained like a fractional number of live sheep 

 or a negative price per head, it must be cast out, not because the 

 mathematics is unreliable, but because the problem is cramped 

 and does not fill up the full measure of generality which algebraic 

 methods provide for. 



The greatest breadth and elegance of algebraic analysis would 

 be observed in the treatment of problems in geometry, mechanics 

 and physics, but since we cannot presume any considerable famil- 

 iarity with these, only problems involving the simplest geomet- 

 rical principles have been inserted. While the elegance of alge- 

 braic methods is best seen in the solution and discussion of 

 problems of equal generality with their algebraic statement, yet 

 those we give are not entirely of this class. 



