linear Valuation of Surd Forms. 213 



same semicircle, AZB is the circle re- 

 quired. But if A Z B be less than a semi- 

 circle, as in the figure, we may first reject 

 the consideration of all the points con- 

 tained between the arc A B and its chord. 

 We must then find 0', 0", &c, the centres 

 of the circles passing through A, B, C ; 

 A, B, D, &c. : these will all lie in the same 

 straight line O f 0" 0. Selecting the one the nearest to O, say 

 0", we describe the corresponding circle, in which A C will now 

 take the place of A B in the former circle. If the points A, B, C 

 are not contained in less than a semicircle, i. e. if A B C is an 

 acute-angled or right-angled triangle, A B C is the circle re- 

 quired ; but if they do lie within the same semicircle so that 

 ABC forms an obtuse angle, B will now have to be rejected, and 

 we must find a new centre as before, and so on continually. By 

 this process we must inevitably at last exhaust all the given 

 points ; and the final circle so obtained will be the circle sought, 

 unless the three points through which it has been drawn are 

 distributed over the same semicircle, in which case the circle 

 required is that described upon the chord joining the two ex- 

 treme points as its diameter. The- solution will evidently be 

 unique, and (as already hinted at) merely require the construction 

 upon the sphere either of a circle passing through a certain 

 set of three out of all the given points, or else passing through 

 only two of them, so as to be perpendicular to the radius bi- 

 secting their joining line. 



If we imagine an india-rubber band (similar, we may suppose, 

 in form to a i ' parlour quoit " but more elastic) having the faculty 

 of maintaining its figure always circular, or which is more simple 

 in the case before us, capable of maintaining itself in the same 

 plane, and imagine this sufficiently stretched over the surface of 

 the sphere to contain all the given points (represented by very 

 minute pins 5 heads given upon it), this band will by its contrac- 

 tion upon the surface of the sphere, however originally placed, 

 imitate the steps of Prof. Peirce's method of solution ; and after 

 (it may be) passing through and quitting successive sets of three 

 points, come to a position of geometrical equilibrium, either when 

 its circumference contains a triad of the given points lying at 

 the angles of an acute-angled triangle, or a duad at the extremi- 

 ties of one of its diameters *. 



* The annexed is a more complete and, I think, a correct account of what 

 would happen to the band under the supposed conditions. It will begin 

 to move parallel to its own plane, and continue so to do until it comes in 

 contact with one of the physical points (call it A) upon the surface of the 

 sphere. Supposing that the position of equilibrium is not then attained 



