PEINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 247 



true that there is a remarkable (but not complete) parallelism 

 between purely numerical and ordinary algebraic operations and 

 their geometrical interpretation; that the limits to the parallelism 

 have to be determined independently of the arithmetic or algebra, 

 is evidence of their real or essential independence. 1 



In algebra we have a conceptually loose method of treating 

 infinitesimals as zeros, 2 because as a rule, that procedure does not 

 lead to practical error; but there is a real difference between an 

 infinitesimal and an absolute zero which ought not to be disre- 

 garded when we are dealing with the fundamental conceptions of 

 algebra or of 'analytical,' i.e. algebraic, geometry. Ha and b are 

 the terminals of a straight line, we may say that from any point 

 c not in the line, but opposite say its centre, that the lines ca and 

 cb approach more and more closely to the one straight line as ab 

 is increased in length; but it surely is not correct to say that 

 when a and b are each infinitely distant, ac, cb are one and the 

 same straight line : they differ by a small but real angle ; that is 

 to say the angle bac is infinitesimal, but the angle bab is nought. 



Our knowledge that two straight lines can intersect but once, 3 

 does not depend upon spatial or physical experiment; and the fact 



1 This may be simply illustrated as follows: — If a, b, and c are ordinary 

 numbers, and algebraically ab—c, we can find a number d, such that d- 

 +/c, if c be positive. If a alone be an abstract number, 6 X the number of 

 units on a line x, then ab x = c x , and dd x = c x , nevertheless there is no such 

 thing as the square root of a line : therefore the operation Vc x is really 

 impossible. The result that the unit d x repeated d times is equal to c x , 

 is an interpretation through which an essentially impossible operation 

 may nevertheless be assigned an arbitrary and yet consistent meaning. 

 The symbols a x b y = d x d y = c xy may be thought to be unambiguous, but it 

 may easily be verified that in geometry ( - a x ) ( - b y ) is not the same 

 surface as (+a x ) ( + b y ), therefore the product of two negatives, is not always 

 the same as that of two positive numbers. 



2 The zero in +a-a = 0, may be called an absolute zero or nought : the 

 plus sign signifies the positing of the quantity a, the minus its removal 

 again, hence nothing is left. Infinity multiplied by this nought or zero 

 yields nothing, i.e. co x = 0. We call also a-f-oo , zero, this is really an 

 infinitesimal and not nought ; i.e. if a is an angle, line, or space, it is an 

 infinitesmal angle, line, or space. In this case we have oo x = a instead 

 of 0. It is because zeros do not always mean the same thing that 0/0 

 and x etc. are indeterminate. 



3 Chrystal states that 'Experience does not settle the question/ Proc. 

 Roy. Soc, Edin. x., p. 641. For reasons which are later given it may also 

 be said that it would be idle to attempt to settle it by experiment. 



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