358 THE PRINCIPLES OF SCIENCE. 



Angular magnitude is the second case in which we 

 have a natural and almost necessary unit of reference, 

 namely, the whole revolution or perigon, as it has been 

 called by Mr. Sandeman a . 



It is a necessary result of the uniform properties of 

 space, that all complete revolutions are equal to each 

 other, so that we need not select any one, and can always 

 refer anew to space itself. Whether we take the whole 

 perigon, its half, or its quarter, is really immaterial ; 

 Euclid took the right angle, because the Greek geome- 

 ters had never generalized their notions of angular 

 magnitude sufficiently to conceive clearly angles of all 

 magnitude, or of unlimited quantity of revolution. But 

 Euclid defines a right angle as half that made by a line 

 with its own continuation, not called by him an angle, and 

 which is of course equal to half a revolution. In mathe- 

 matical analysis, again, a different fraction of the perigon 

 is taken, namely, such a fraction that the arc or portion 

 of the circumference included within it is equal to the 

 radius of the circle-. This angle, called by De Morgan the 

 arcual unit, is equal to about 57, 17', 44"'8, or decimally 

 57'2957795i3 , and is such that the half revolu- 

 tion contains 3' 14159265.... such units b . Though this 

 standard angle is naturally employed in mathematical 

 analysis, and any other unit would introduce needless 

 complexity, we must not look upon it as a distinct unit, 

 since its amount is connected with that of the half peri- 

 gon, by a natural constant 3*14159 usually signified 



by the letter TT. 



When we pass to other species of quantity, the choice 

 of unit is found to be entirely arbitrary. There is abso- 



a ' Pelicotetics, or the Science of Quantity ; an Elementary Treatise on 

 Algebra, and its groundwork Arithmetic.' By Archibald Sandeman, M.A. 

 Cambridge, (Deighton, Bell, and Co.) 1868, p. 304. 



b De Morgan's ' Trigonometry and Double Algebra,' p. 5. 



