nrents. It is also shown that the terms which denote the two ele- 

 ments of this antithesis cannot in any case be applied absolutely and 

 exclusively. We cannot say, this is a fact and not a theory, or this 

 is a theory and not a fact ; for a true theory is a fact ; a fact is a fa- 

 miliar theory. It was further observed, that the antithesis being 

 inseparable, one element seems, and is asserted, in different systems 

 of philosophy, to be derived from the other ; ideas from experience, 

 or experience from ideas. But we must always have both elements : 

 thus in mechanics, and in our experience, we have necessary princi- 

 ples, such as that every event must have a cause ; and in chemistry 

 also other necessary principles, as that the chemical composition of 

 a body determines its kind and properties. 



March 4, 1S44. 



On the Method of Least Squares. By R. L. Ellis, Esq. 



The aim of this paper is to give a succinct exposition of the differ- 

 ent demonstrations by which it has been proposed to establish the 

 validity of the rule known as the method of least squares. The first 

 demonstration of this celebrated rule (which had been previously 

 proposed by Legendre) is that given by Gauss in the Theoria Mottis. 

 The next appears to be that of Laplace, which has been followed, 

 without variation of principle, by Poisson and other French writers. 

 The demonstration of Gauss is based upon the assumption, that the 

 arithmetical mean is the most probable result to be derived from a 

 series of direct observations of an unknown magnitude. This as- 

 sumption is alleged by Laplace to be altogether precarious ; and it 

 appears that Gauss acquiesced in this remark, as he subsequently, 

 in the Theoria Combinationis Observationum, produced another de- 

 monstration, which is independent of this assumption. As the first 

 method of Gauss has been followed by later writers, of whom Encke 

 is one, it seemed desirable to endeavour to ascertain if the objection 

 of Laplace be well-founded ; and this the writer has attempted to 

 do in the first part of the present communication. His conclusion 

 is, that although the practice of adopting the arithmetical mean as 

 an approximation to the true value of the unknown magnitude ob- 

 served, is founded on just principles, yet that we are not entitled to 

 say that it leads to the most probable result ; and consequently that 

 the demonstration in question is invalid. 



The writer then proceeds to consider Laplace's demonstration. 

 This involves no precarious assumption, but the mathematical part 

 of the investigation is of very considerable difficulty, and cannot be 

 said to be altogether free from doubt. For Laplace's analysis, an- 

 other, founded on a theorem which was first made use of by Fourier 

 in his researches on heat, is substituted ; and by this change the 

 mathematical difficulties of the subject are very much diminished. 

 An attempt is also made to test the accuracy of Laplace's methods 

 by reference to a particular case. 



