42 Colorado College Studies. 



here. In the newspaper communication referred to in the fo^ot- 

 note, the first problem and its answer are stated as follows: 

 x'- + y- = a" = z~ + ir = D 

 and X- — iv' = z" — y' = □ 



Answer, It a — 7585, then x — 7400. y =1665. z = 6273, 

 IV = 4264. 



The answers are, apparently, expected to be in integers. If 

 a general solution is to be given, then the above problem is not 

 so very easy, but Diophantists care but little for general solu- 

 tions. Assuming a particular number for a and then finding 

 one set of values for an answer gives them perfect satisfaction 

 How many other values, if any, really exist, and what they are, 

 does not concern them much. 



Among the writers in the Mcdhemaiical Companion inter- 

 ested in Diophantine Analysis, appear the names of Dr. 

 O'Eiordon, Wm. Wright, T. Beverly, and Cunliffe. 



One of the best Diophantine scholars in this country was 

 Charles Gill, editor of the Maihemaiical Miscellany and pro- 

 fessor of mathematics at the St. Paul's Collegiate Institute at 

 Flushing, Long Island. He published, in 1848, a book of 90 

 pages, entitled, " Application of the Angular Analysis to the 

 solution of Indeterminate Problems of the Second Degree." * 

 This is a charming little work. Of all publications on Indeter- 

 minate Analysis that we have seen, this least deserves the 

 adverse criticisms we shall make on the work done in this 

 country, taken as a whole. Here an effort is made to introduce 

 method and to exhibit general solutions. The novelty of the 

 book consists in the application of the notation of Trigonometry 

 to Diophantine Analysis. So far as we know Prof. Gill is the 

 first one to make this application. If the sine and cosine of 

 an angle can be expressed as rational numbers, we may assume 



sin A = "^ ^ ^ and cos A = °^ , ^ , , and then all other functions 

 m-+n- m-+n- 



* A copy of this now rare book was lent the writer by Dr. Artemas Martin, of 

 Washington. 



