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Elements of Geometry. 21 



conception of the contiguity of surfaces is not actually 

 required, except perhaps for the measurement of volume. 

 {Cf. § 11.) 



5. The third step places us in a position to introduce 

 measurement and the three fundamental magnitudes, 

 length, angle, and area. For fundamental measurement 

 we need definitions of equality and addition, such that 

 the law of equality and the two laws of addition are 

 true. The choice of unit may be left out of account ; for, 

 with geometric magnitudes, the laws are true whatever 

 unit is selected. The law of equality is Axiom 1; the first 

 law of addition is Axiom 9. Axioms 2-7 are together very 

 nearly equivalent to the second law of addition (which may 

 be stated roughly in the form that the magnitude of a sum 

 depends only on the magnitudes of the parts). Axiom 8 is 

 an attempt to compress the definitions of equality for all 

 three magnitudes into a sinole sentence ; it is better to 

 separate them. Euclid fails to give any definition of 

 addition ; he does not tell us how the " w 7 hole " is to be 

 related to the "parts" in order that it should be greater. 



6. We will now take the magnitudes in turn. For the 

 length of a straight line the necessary definitions are : — 



(1) Two straight lines are equal in length if they can be 

 placed so that when one end of the first is contiguous with 

 one end of the second, the other ends are also contiguous. 



(2) The length of the straight line AB is equal to the sum 

 of the lengths of the straight lines CD, EF, if they can be 

 placed so that C is contiguous with A, F with B, I) with E 

 and with some part of AB between A and B. 



These definitions, like all similar definitions of mag- 

 nitudes, are satisfactory and are subject to the necessary 

 laws of equality and addition only if certain conditions 

 are ful rilled. The conditions are described by saying that 

 the surfaces in w T hich the straight lines lie must be those 

 of rigid bodies. This is a definition of a rigid body: a 

 rigid body is something which (like a perfect balance) is 

 determined by the satisfaction of the conditions for mea- 

 surement *. Rigid bodies according to this test include 

 many of those which satisfy the crude test of § 4, though 

 they include others (e.g., surveyors' tapes used as surveyors 

 use them) which do not satisfy that test. In virtue of the 

 fact that rigid bodies are necessary to measurement, the 



* Cf. H. Dingier, Phys. Zeit. xxi. p. 487 (1920). 



