SECT. 35.] Theory of the Average. 499 



able accordance with the law of error. The result of these 

 two assumptions is that if we collect a very large number of 

 measurements of the different parts and proportions of some 

 ancient building, say an Egyptian temple, whilst no as 

 signable length is likely to be permanently unrepresented, 

 yet we find a marked tendency for the measurements to 

 cluster about certain determinate points in our own, or any 

 other standard scale of measurement. These points mark the 

 length of the standard, or of some multiple or submultiple of 

 the standard, employed by the old builders. It need hardly 

 be said that there are a multitude of practical considerations 

 to be taken into account before this method can be expected 

 to give trustworthy results, but the leading principles upon 

 which it rests are comparatively simple. 



35. The case just considered is really nothing else 

 than the recurrence, under a different application, of one 

 which occupied our attention at a very early stage. We 

 noticed (Chap. II.) the possibility of a curve of facility which 

 instead of having a single vertex like that corresponding to 

 the common law of error, should display two humps or 

 vertices. It can readily be shown that this problem of the 

 measurements of ancient buildings, is nothing more than the 

 reopening of the same question, in a slightly more complex 

 form, in reference to the question of the functions of an 

 average. 



Take a simple example. Suppose an instance in which 

 great errors, of a certain approximate magnitude, are dis 

 tinctly more likely to be committed than small ones, so that 

 the curve of facility, instead of rising into one peak towards 

 the centre, as in that of the familiar law of error, shows a 

 depression or valley there. Imagine, in fact, two binomial 

 curves, with a short interval between their centres. Now if 

 we were to calculate the result of taking averages here we 



