306 THE PRINCIPLES OF SCIENCE. [CHAP. 



revolution or perigon, as it has been called by Mr. Sande- 

 man. 1 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 revolution, but 

 can always refer anew to space itself. Whether we take 

 the whole perigon, its half, or its quarter, is really imma- 

 terial ; Euclid took the right angle, because the Greek geo- 

 meters had never generalised their notions of angular 

 magnitude sufficiently to treat angles of all magnitudes, or 

 of unlimited quantity of revolution. Euclid defines a right 

 angle as half that made by a line with its own continuation, 

 which is of course equal to half a revolution, but which 

 was not treated as an angle by him. In mathematical 

 analysis 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. In 

 this point of view angular magnitude is an abstract ratio, 

 namely, the ratio between the length of arc subtended and 

 the length of the radius. The geometrical unit is then 

 necessarily the angle corresponding to the ratio unity. 

 This angle is equal to about 57, 17', 44" '8, or decimally 

 57'2957795 13... , z It was called by De Morgan the arcual 

 unit, but a more convenient name for common use would 

 be radian, as suggested by Professor Everett. Though this 

 standard angle is naturally employed in mathematical 

 analysis, and any other unit would introduce great com- 

 plexity, we must not look upon it as a distinct unit, since 

 its amount is connected with that of the half perigon, 

 by the natural constant 3-14159 . . . usually denoted by 

 the letter IT. 



When we pass to other species of quantity, the choice 

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

 lutely no mode of defining a length, but by selecting some 

 physical object exhibiting that length between certain 

 obvious points as, for instance, the extremities of a bar, 

 or marks made upon its surface. 



1 Pclicotetics, or the Science of Quantity ; an Elementary Treatise on 

 Algebra, and its groundwork Arithmetic. By Archibald Samleman, 

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



* De Morgan's Trigonometry and Double A Igebra, p. 5. 



