270 



Mr DE morgan, ON THE ROOT OF AN ALGEBRAIC EQUATION. 



joined with an odd number of obtuse angles, except only three acute sides joined with three 

 obtuse angles. This may be remembered by the following mnemonic diagram. 



12 3 



Obtuse Sides. 



Obtuse Angles. 



Acute Angles. 



Acute Sides. 



or two Ns with an inverted N between them. The numerals, from left to right, describe the 

 numbers of obtuse sides and of obtuse angles ; from right to left, acute sides and acute angles ; 

 the lines connecting numerals assert the possibility of the combination. Thus, with two 

 obtuse sides may coexist either one, two, or three obtuse angles ; with two acute angles may 

 coexist either one, two, or three acute sides. 



The base of the triangle being given, and one of the hemispheres formed by its plane, the 

 regions which are the loci of the vertices, for different classes of triangles, may be described as 

 follows. Let the hemisphere be orthographically projected upon the plane of the base, and 



let orthographic projections be used for originals in description. Let AB be the base when 

 acute, BA' when obtuse. Let aa, jSjS' be perpendicular to AA', BB'. Draw an hyperbola 

 AB, A'B', having aa, (ifi', for asymptotes. Then AA', BB', aa, /3/3', represent secondaries 

 o[ ABA'B', and AB,A'B', represent intersections of the sphere with an hyperbolic cylinder. 

 The hemisphere is divided into ten regions ; and each region is the locus of the vertices of one 

 class of triangles for the base AB, and of one class for the base BA'. A point on the hyper- 

 bola, on either branch, is the vertex of a right-angled triangle, be the base either AB or BA'. 

 Thus, if the vertex C be taken within the region BOa, then, the base being AB, all the sides 

 are acute, and the one angle ABC obtuse ; or the triangle may be symbolised as 01, no obtuse 

 side and one obtuse angle. But with reference to BA', we have 21, two obtuse sides A'B, 

 A'C, and one obtuse angle A'CB. These symbols are entered in the compartment BOa, and 

 all the other compartments are treated accordingly. 



A. De M. 

 Dec. 19, 1857. 



