46 Colorado College Studies. 



III. Estimate of the Work Done. 



In the examination of the more difficult Diopliantine prob- 

 lems solved by the men above referred to, we are often struck 

 by the great ingenuity displayed. In following the intricate 

 paths, we often Avonder how the patience of the solver endured 

 to accomplish it. We are sometimes tempted to exclaim, 

 " Here we have indeed a true disciple of Diophantus," or " This 

 solution is after Diojihantus's own heart." Some of the work is, 

 in point of difficulty, in advance of anything we have of Dio- 

 phantus himself. And indeed it ought to be, for the ancient 

 algebraist worked under the enormous disadvantage of having 

 but few algebraical symbols and of not possessing the Arabic 

 notation of numbers. And yet, our ingenious solutions, however 

 dazzling, fail to satisfy us. They fail to teach us general 

 methods. The solution of one problem may give us no clue what- 

 ever as to how the next one should be solved. Our Diophantists 

 are dexterous and indefatigable; they lead us by intricate and 

 laborious processes to an answer which may consist of rational 

 fractions with numerators and denominators of 30 to 50 digits, 

 or which may have integral values of upward to 150 digits. 

 But little or no effort is made to determine whether all possible 

 solutions have been found, and to show how the remaining 

 answer can be obtained. The problems are, after all, treated 

 very superficially. Their inner secrets remain, to a great extent, 

 unraveled. The lack of general methods by which full and 

 complete solutions can be obtained, renders the work done of 

 little or no scientfic value. That the discovery of powerful 

 methods is more valuable than the obtaining of an answer to 

 some particular problem will be conceded by all. To illustrate : 

 AVhen Nathaniel Bowditch, in the Analyst of Robert Adrain, 

 succeeded in adjusting discordant observations obtained from 

 a survey of a certain piece of land, so as to obtain the most 

 probable values in that particular case, he displayed cleverness; 

 but when Robert Adrain worked out a general method by which 



