g94 THIRD REPORT 1833. 



Co + flfi cosfii + a2COs(9, +S^) + , . «„-, cos (3, + fi2 + . • fi«-i) = (1.) 



aisinfli + ff2sin(fii + fl2) + . . a„_, sin (9i + 92 + . . 9„-i) = (2.) 



91 + ^2+ ... 9„_, . = (« - 2 r) TT (3.) 



The first two of these equations may be called equations of 

 figure, and the last the equation of angles, and all of them must 

 be satisfied in order that the lines in question may be capable 

 of being formed into a figure, along the sides of which if a point 

 be moved it will circulate continually. If the values of flj, 

 ^2 ~ ^1' ^3 ~ ^2 • • ^«-i "" ^n-2 ^^ ^11 positive, aud if r = 1, then 

 the equation of angles will correspond to those rectilineal figures 

 to which the corollaries to the thirty-second proposition of the 

 first book of Euclid are applicable, and which are contemplated 

 by the ordinary definitions of rectilineal figures in geometry. 

 If we should suppose r = 2 or 3 or any other whole number 

 different from 1, the equation would correspond to stellated 

 figures, where the sum of the exterior angles shall be 8, 12, or 

 4 r right angles. The properties of such *^t?Z/a^e(/ figures were 

 first noticed by Poinsot in the fourth volume of the Journal 

 de FEcole Polytechnique, in a very interesting memoir on the 

 Geometry of Situation*. 



All equal and parallel lines drawn or estimated in the same 

 direction are expressed by the same symbol aflfected by the 

 same sign, whatever it may be : and it is this infinity of lines, 

 geometrically different from each other, which have the same 

 algebraical representation, which renders it necessary to con- 

 sider the position of lines, not merely with respect to each other, 

 but also with respect to fixed lines or axes, through the medium 

 of the equations of their generating points. In other words, it 

 is not possible to supersede even rectilineal geometry by means 

 of affected symbols only. We are thus led to the consideration 

 of a new branch of analytical science, which is specifically de- 

 nominated the Application of Algebra to Geometry, and which 

 enables us to consider every relation of points in space and the 

 laws of their connexion with each other, whatever those laws 

 may be. It is not our intention, however, to enter upon the 

 discussion of the general principles of this science, or to notice 

 its present state or recent progress. 



A great number of elementary works on trigonometry have 

 been published of late years in this counti'y, many of which are 

 remarkable for the great simplicity of form to which they have 

 reduced the investigation of the fundamental formulae. Such 

 works are admirably calculated to promote the extension of 



• See also Peacock's Algebra, p. 448. 



