25^ Mr. De Morgan on the relative Signs of Coordinates. 



fixed axes, with given directions of sign, a straight Hue oblique 

 to the axes need not be considered as having either sign, un- 

 less in conjunction with an hypothesis relatively to the signs 

 of lines perpendicular to it; in whicli case the law of signs 

 which is absolutely necessary for continuity is the one just 

 laid down. And by continuity in this case is simply meant 

 the condition, that whenever a moving line comes momentarily 

 to coincide with a given line, its relations of sign are the 

 same as those in the given line. This being assumed, the 

 assignment of the relations of sign in any two given perpen- 

 diculars determines a relation between the signs of any two 

 other perpendiculars. The indeterminateness of equation (1.) 

 simply implies that it was obtained without reference to any 

 such relation. Those who are used to this subject will see 

 how the expression of a length by means of the tangent of an 

 angle instead of its sine and cosine introduces this ambiguity. 

 When the two latter are both given, the line on one side of a 

 point is distinguished from the other, which is not the case 

 when the tangent is given. 



T take this opportunity to append a few remarks upon the 

 answer given by Mr. Graves in this Magazine of last April, 

 to the objections to his view of the logarithms of unity con- 

 tained in my Calculus of Functions. That I did not make a 

 forma] reply before this time arose from my considering both 

 sides of the argument as brought to a sufficiently distinct 

 point by Mr. Graves in his communication, and particularly 

 in the following sentence. 



" If I am told that logarithm ought to be so defined that 

 X ought to be called an a-log., or logarithm to base a only 

 of the arithmetical value of «', I must say that I should 

 not approve such a restriction. It would, if the proposed 

 theorems be correct, arbitrarily exclude from the name of 

 logarithms orders of functions which enjoy the same funda- 

 mental and characteristic jaro/'^r//(?5 as those that are favoured 

 with that name." 



It is then proposed b}'^ Mr. Graves to extend the name of 

 logarithm so as to include, not only what I should call the 

 logarithms to the arithmetical form of the base, but also those 

 to every other form yet considered in algebra. That is to 

 say, writing s in its most general form g i+Stottv'— i and j/ in 

 its form s log^+2 « tt V-i, where log j/ is the arithmetical lo- 

 garithm, he defines x, the general logarithm of j/, by the equa- 

 tion 



c (l+2TO7r'/^^),c = J logy+2n 7r\^— i. 



