X. On the Theory of Errors of Observation. By Augustus De Morgan, F.R.A.S. 

 of Trinity College, Professor of Mathematics in University College, 

 London. 



[Read Nov. 11, 1861.] 



This paper is an attempt to simplify the mathematical treatment of the subject, mixed 

 with a statement of the grounds on which, in my belief, it ought to rest. I touch only those 

 heads on which I have something to say as to one or other of these points : and I make no 

 remark on preceding writers, except so far as may be inferred from the following preliminary 

 observations. 



In this subject I fancy I have always seen a mixture of modes of thought and modes of 

 treatment which makes it a difficult speculation, though easy of application in practice. 

 Whether this or that be psychological postulate, result of experience, or deduction from one 

 or the other, is often of harder determination than it ought to be : a difficulty sometimes 

 arising from, or augmented by, the very circumstance on which facility of application depends. 

 The peculiar pliability of the function e'"^, which serves our turn be the law of facility of 

 error what it may, is so dexterously used that we hardly know how much of any result is 

 independent of it. This function makes its appearance only as a mathematical instrument. 

 Had any other instrurnent been of more convenient use, it seems as if our results would have 

 had another expression. I shall succeed in shewing that there is no possible choice in this 

 matter, by introducing e'"^ into the representation of results obtained without any reference 

 to it, expressed or implied. 



The theory of probabilities professes to give the way in which belief in elements should 

 affect belief in combinations. The word probability has two different senses, the collision of 

 which is a grand source of confusion : it is used to refer both to the state of the mind, and to 

 the external dispositions which are to regulate the long run of events : to our strength of 

 prediction, and also to the capacity of circumstances to fulfil our prediction. I shall use the 

 words probability and facility, as follows. Head and tail are to our minds of equal proba- 

 bility, so long as we know nothing to distinguish them : but to say they are of equal facility 

 is to make an assertion involving points of symmetry, density, surface, &c. as regards the 

 coin, and we know not what about the habits of the person who is to make the tosses. Ac- 

 cordingly, our theory does not, as many suppose, arrogate to itself a predictive character: it 

 does not prophesy that in six millions of throws with a die, something near to one million 

 will be aces. All it does is to justify to the mind the following alternative, Either very near 

 to one million of aces, or determinate presumption, depending upon the amount of departure, 

 against equal facility in the different faces. Such equality of facility is as likely as a pencil 

 line or a perfectly rigid bar. When we talk of actually applying our theory to observations, 



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