to the Case of an Autotomic Plane Circuit, 453 



and magnitude as at starting. Here, of course,, the extremities 

 of the line trace out independent circuits, the algebraic sum of 

 whose areas is equal to the area described by the line. The 

 consideration of a particular case of this., viz. that in which the 

 area of one of the circuits is zero, leads to a rule for finding the 

 area of any given circuit ; but this rule, though considered rather 

 elegant (" concinniorem methodum ") by its author and therefore 

 deserving of systematic proof, is cumbrous and troublesome as 

 compared with De Morgan's. On this account it is not repro- 

 duced here. 



The next section of the subject is 



Descriptio figurarum per motum recta circular em. 



The introduction starts with the theorem, " Any triangle 

 is equal to the algebraic sum of the triangles whose common 

 vertex is any point in the plane of the given triangle and 

 whose bases are the given sides." This leads to the more 

 general proposition, "The area of any plane figure whatever 

 (including, of course, those whose perimeters are autotomic) 

 is equal to the algebraic sum of the triangles having a com- 

 mon vertex in the plane of the figure and so constructed that 

 their bases make up the given perimeter;" it being pointed out 

 that in the case of figures whose perimeter is curvilineal, the 

 simplest set of triangles is obtained by drawing from the given 

 point in the plane all possible tangents to the perimeter. And 

 now thus prepared we are invited to consider in succession the 

 various cases of the description of figures by a straight line re- 

 volving now in one way now in another round a fixed point 

 situated in the direction of its own length. These cases are six 

 in number, and may, for the sake of condensation, be logically 

 arranged as follows, the author's elucidations being omitted as 

 superfluous : — 



A. When the describing line preserves the same sign, 



{ 



a> and is constant in length. 

 bj and is variable in length. 



fl, centre in one extremity of line. 

 (a) (b) < 2, centre between extremities. 



[ 3, centre in line produced. 



It may be mentioned that many of the theorems in this section 

 are stated alternatively in the language of projections, much cir- 

 cumlocution being thus obviated; e. g. the result of (A, a, 3) is 

 put thus : — " In whatever plane a pyramid, however truncated, 

 may be projected, the pro'ection of the bases equals the projec- 

 tion of the sides." 



