256 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE 



property belongs to Cos os. Consequently according to this view it is not true that Sin os and Cos « 

 have each a single value, or any finite set of values definitely ; but they each have all possible 

 values from — 1 to + 1 in such a manner and sense that not one of these values is pre-eminent 

 above another, and no one has a claim to be put forward above its fellows, but all stand in exactly 

 the same relation to the function Sin eo (or Cos eo) so that at one and the same moment Sin os 

 (or Cos os) is equal to every one of them but not more properly equal to any one than any other 

 of them. From this reasoning and kindred reasons of an equally general character, I satisfied 

 myself that Sin 05 and Cos os cannot be replaced by zero, unless under some special hypothesis, 

 and that when taken in a general sense they cannot justly be s\ipposed to have definite values at 

 all. I shall now proceed to some considerations which are preliminary to a more formal proof that 

 they have not the value zero, even when considered as the limits of more general forms. 



In conducting my inquiry into the values of the symbols Sin os, Cos os, I am unavoidably 

 brought upon the confines of the much controverted subject of divergent series. In a certain sense 

 which will be explained, I agree with Professor De Morgan that all non-convergent series stand on 

 the same basis, though I cannot subscribe to the train of reasoning by which this is usually main- 

 tained, involving as it appears to me some disputable positions. Much of the obscurity which 

 attaches itself to the subject of divergent series may be traced to the discordant and strange 

 significations applied to the symbol =, when used in connection with infinite series. The pre- 

 sumption is that when this symbol stands between two quantities it indicates, that either may be 

 used for the other in algebraical processes. A very eminent author states that it " may be rendered 

 by the phrase gives as its result, when it is placed between two expressions, one of which is the 

 result of an operation which in the other is indicated and not performed ;" — an explanation which 

 agrees exactly witli what Woodhouse states in his Principles of Analytical Calculations, who insists 

 upon this definition of it at intervals through his work with an earnestness which indicates the 

 confidence with which he regarded it as true. Now if this definition be closely examined it cannot 

 be understood to denote that the expressions connected by = differ in any thing but form ; for 

 one side denotes that an operation is to be performed, and the other is the result of the actual 

 operation : if then the operation has been correctly and completely performed, there is no difference 

 except in form between quantities connected by =. But an examination of the Principles of Ana- 

 lytical Calculations, will not fail to satisfy us, that in giving this definition the author must have 

 understood it in some modified sense whicii he has not expressed in the definition itself. For when 

 it is said that " = is a symbol which serves merely to connect an involved expression and the result 

 of an operation," it is evident that "numerical equality" could not then be, what the author affirms 

 it is, a contingent result. But whatever was the sense which the author mentally attached to the 

 symbol, it involved a principle which necessitated the making distinctions where by ordinary minds 

 the difference cannot easily be grasped : for it was found impossible to be consistent without demand- 

 ing a license to consider ^ and (as also and 1 as essentially distinct. Now 



what difference is there between 2 and 1 -I- 1, except in form ? Is not 1 -t- 1 an expression in which 

 an operation is to be performed the result of which is rightly denoted by 2 ? and if so, then by his 

 own definition 2 and 1 -t- 1 are algebraically equivalent. I must confess that I cannot consent to 

 such distinctions as are here demanded without being satisfied that there is no means of avoiding 

 them ; and I cannot but suspect that in the present case there is no other necessity for them, than 

 what arises from a misapplication of the definition which the author has given of the symbol =. 

 For if this symbol serve merely to connect an involved expression and the result of an opera- 

 tion, it is clearly a misuse of it to employ it in connecting an involved expression and a part 

 only of the result of an operation. Let me explain by an example. Professor Woodhouse writes 



= 1 - .T + ,T^ - Now the operation denoted on the left-hand is the division of 1 by 1 + ,t, 



1 + a- '^ ■' 



