104? Prof. Gauss on the Representation of the Parts 



This mineral was analysed some years ago by Vauquelin 

 and Dolomieu ; but the numbers which they have mentioned, 

 owing to the insufficient mode of analysis employed at that 

 time, are not entitled to any confidence. — [Journal des Mines 

 ix. 778.) 



XVI. General Solution of the Problem : to represent the Parts 

 of a given Surface on another given Surface, so that the 

 smallest Parts of the Representation shall be similar to the cor- 

 responding Parts of the Surface represented. By C. F. Gauss. 

 Answer to the Prize Question proposed by the Royal Society 

 of Sciences at Copenhagen*. 



Ab his via sternitur ad majora. 



r T , HE author of this paper believes that he must consider 

 -*- the repeated selection by the Royal Society of the question 

 which forms the subjects of it, as a proof of the importance 

 which the Royal Society attaches to it ; and has thereby been 

 induced to submit a solution found by him some consider- 

 able time since, as the lateness of the time at which he was in- 

 formed of the prize question would otherwise have prevented 

 him from sending an answer. He regrets that the latter cir- 

 cumstance has obliged him to limit his inquiry to the essen- 

 tial part only, besides hinting some obvious applications to 

 the projection of maps and the higher branches of geodetics. 

 Had it not been for the near approach of the term fixed by 

 the Society, he would have followed up several inquiries, and 

 have detailed numerous applications of the subject to geo- 

 detical operations ; all which he must now reserve to himself 

 for another moment and another place. 

 December 1822. 



1. The nature of a curve surface is determined by an equa- 

 tion between the coordinates belonging to every point of the 

 same x, y, z. In consequence of this equation, every one of 

 these three variable quantities may be considered as a function 

 of the two others. It is still more general to introduce two 

 new variable quantities t, u, and to represent each of the quan- 

 tities x, y, z as a function of t and u, by which at least generally 

 speaking, determinate values of t and u always belong to every 

 determinate point of the surface, and vice versa. 



2. Let X, Y, Z, T, U have the same signification for a se- 

 cond surface, which x, y, z, t, u had in reference to the first. 



3. To represent the former surface on the second means to 



* From Prof. Schumacher's Astronomische Abhandlungen, No. 3. 



establish 



