APPENDIX.

—Let the ages at first be x , y , ( x + y ). Now, if a + b = 2 c , then ( a - n ) + ( b - n ) = 2( c - n ), whatever be the value of n . Hence the second relationship, if ever true, was always true. Hence it was true at first. But it cannot be true that x and y are together double of ( x + y ). Hence it must be true of ( x + y ), together with x or y ; and it does not matter which we take. We assume, then, ( x + y ) + x = 2 y ; i.e. y = 2 x . Hence the three ages were, at first, x , 2 x , 3 x ;