The third part of the paper relates to Gauss's second method. 

 The relation in -which it stands to that of Laplace is distinctly- 

 pointed out ; the difference between them arising merely from this, 

 that whereas Laplace considered the importance of a positive or ne- 

 gative error (that is, of an error in excess or defect) to be propor- 

 tional to its arithmetical magnitude, Gauss assumes the square of the 

 magnitude of the error as the measure of its importance, alleging 

 that this importance not heing a magnitude, does not strictly admit 

 of numerical evaluation ; that some assumption is therefore requisite, 

 and that that which he proposes is not more arbitrary than Laplace's, 

 while, from the absence of discontinuity, it leads to far simpler and 

 more satisfactory calculations. 



The writer then shows that neither Laplace's investigations, nor 

 that of Gauss in the Theoria Combinationis Observationum, tends to 

 prove, that the results of the method of least squares are the most 

 probable of all possible results. This point, with regard to which 

 there has occasionally been some degree of confusion, seems to be 

 essential to a just apprehension of the nature of the subject. It may 

 be remarked, with reference to it, that Laplace uniformly speaks of 

 the method of least squares as the most advantageous method of com- 

 bining discordant observations, or as that which gives the most ad- 

 vantageous results, and never as a method by which the most pro- 

 bable results are to be obtained. 



Lastly, the writer proceeds to consider three demonstrations of 

 the method of least squares, given by Mr. Ivory in the Philosophical 

 Magazine. These demonstrations are independent of the theory 

 of probabilities. The first is founded upon an assumed analogy 

 between the equilibrium of weights on a lever and the combination 

 of discordant observations ; the second upon another unsatisfactory 

 analogy ; and the third upon the principle that the error committed 

 at one observation is independent of that committed at any other. 

 None of these demonstrations appear to the writer to be at all con- 

 clusive, but they seemed to deserve consideration, not only from the 

 high reputation of their author, but also from the terms in which they 

 have been mentioned in a recent work on the theory of probabilities. 



On Divergent Series, and various Points of Analysis connected 

 with them. By Augustus De Morgan, Esq. 



The author states that he does not pretend to have perfect confi- 

 dence even in convergent series. It is the main object of his paper 

 to show that the continental analysts are not justified in their rejec- 

 tion of some classes of divergency, and retention of others, by any- 

 thing but experience ; that they have underrated the character of 

 most which they reject, and overrated that of all they receive. 



Divergent series are either of infinite divergence, such as 1 + 2 + 3 

 + 4 + &c, in which summation of terms may give any sum, however 

 great ; or of finite divergence, such as cos 9 + cos 2 + . . ., in which 

 no number of terms can give more than a certain quantity. The 

 former are rejected by most modern continental writers, the latter 

 are retained. 



