218 Mr. R. Meldrum Stewart on the 



truth. The point which it is desired to emphasize is that, if 

 these two postulates be granted, the whole method of least 

 squares, as properly stated and applied, follows inevitably 

 by rigorous mathematical reasoning. 



It is quite true that in most, if not all, of the present-day 

 text-books on the subject, the reasoning which purports to 

 establish the principles of the method is anything but 

 rigorous; and the method itself is at least in some in- 

 stances grossly misapplied. There is perhaps no subject in 

 the realm of mathematics where clear and careful thinking 

 and rigorous reasoning are more necessary than in the theory 

 of probability and its application to the theory of errors ; 

 and yet it would be difficult to find a subject where con- 

 fusion of thought and loose reasoning are more prevalent : 

 among the most obvious examples of this might perhaps be 

 mentioned the common confusion of the terms " true value " 

 and" most probable value/' "error" and "residual," with the 

 loose reasoning that inevitably arises from such confusion. 



But from the fact that loose reasoning is frequently used 

 in attempts to deduce the principles of the method, it by no 

 means follows that these principles have not been, or cannot 

 be, established by rigorous reasoning. Most, if not all, of 

 the fundamental principles, and many of the details, of the 

 theory have been correctly developed by the classic authors 

 on the subject, as may be readily verified by reference to the 

 original memoirs. It seems worth while to emphasize this 

 fact because, by most of those who have occasion to adjust 

 observations, these classic authors are seldom read, and the 

 prevalent ideas on the method of least squares are those 

 garnered from the more or less faulty text-books of the 

 present day. 



It remains to consider briefly the three t}'pes of problems 

 mentioned by Dr. Campbell. 



The first type is the ordinary case of the arithmetic mean. 

 It may be remarked in passing that the problem before us 

 is not the determination of the " true value/' even with the 

 limitation* imposed by Dr. Campbell. The true value must 

 for ever remain unknown ; all that we can hope to attain is 

 the " most probable value," and this is all that is, or can be, 

 claimed for the arithmetic mean. Further — and here we 

 arrive at the crux of the whole theory of errors — the adoption 

 of the arithmetic mean as the most probable value is a pure 

 assumption ; in fact, if the law of errors is any other than 

 the Graussian taw, the arithmetic mean is not the most 



* Loc. cit. p. 178, footnote. 



