in the Elemcntari/ Investigations of Geometry. 103 



anoles will be mere numbers, and of course C as a number 

 may be legitimately determined by 



It is here strangely forgotten that the equation right angle = 1 

 is merely symbolical or abbreviatory, and can hold only when 

 angles are compared together : whereas C in the foregoing 

 equation is not determined to be a part of the unit xvhich re- 

 presents R, but of the abstract unit which measures the ratio 



— . To show the difficulties to which the French geometers 

 ^ .... 



seem reduced in their attempts to gloss over this curious piece 



of reasoning, let us quote one sentence from Maurice's dis- 

 sertation. After speaking of the trigonometrical equation 



6: £« 



1 + ^--^ 



COS C = 



2- 

 as derived from ^^ ^ ^ ^^^^ ^_^ ^) 



he says : " Now this example is not, as might at first sight be 

 supposed, at variance with die principles here laid down. If 

 the angular unit, though not affecting all the terms, has dis- 

 appeared, the reason is, that the proposed equation turns out 

 to establish tliat a certain function of the sides of a triangle is 

 a function of one of its angles : and as this latter function may 

 be transcendental, and consequently equal to the cosine of 

 the arc which corresponds to that angle, it follows that just 



as we have the general relation angle — -;^^^, so also we 

 have a certain cosine (and every cosine must be an abstract 

 number) equal to a certain algebraical function of the ratios 

 between the sides." This paragraph has certainly the merit 

 of mystifying the subject fconsiderably. Ui)on analysing it, 

 however, we find it composed of two assertions. Maurice 

 first states that the functional e(|uation may give a tniiiscen- 

 dental function of the angle ccjual to a function of the ratios 

 of the sides; and that this transcendental function may be a 

 number, or, as it is, the cosine. This is singularly at variance 

 with another part of his paper, in which he rightly asserts that 

 no unit can disappear, excei)ting by division, by itself. Here, 

 however, an angular unit disappears without division ; and 

 Maurice accouiUs for it merely by slating that it does so ! 



These remarks, brief as they are, will, 1 hope, sufficiently 

 show die highly objectionable character of the nianncr in 

 which one ptirty in the tonlroviisv would distinguish betwixt 



till' 



