Jime 20, 1872] 



NATURE 



141 



when I was examining all the investigations I could find on the 

 subject, after looking through Lagiange's memoir (and reading 

 carefully Todhunter's ;<■.(;;/«.' of it), 1 came to the conclusion that 

 it contained nothing that conld, properly speaking, be regarded 

 as an anticipation of the later investigations of Gauss, Laplace, 

 &c., and I contented myself therefore with merely a passing 

 reference. 



Lagrange's paper, as its title implies, gives a mathematical 

 justification of the choice of the mean of a series of discordant 

 observations, and a determination of the chance that the result- 

 ing error lies between certain limits, with developments, &c. ; 

 but the method of Least Squares may be described as an exten- 

 sion of the piinciple of the arithmetic mean to ihe combination 

 of linear equat or.s, involving more than one unknown ; the 

 prolikm being to obtain the best values of the unknowns from a 

 series of discordant Uiu\ir si?>iitlftifttOiis I't^natious, 



The meihod of Leas-t Squares was first proposed in print by 

 Lfgerdre in his " Orbiles des Coraetes " (Paris, 1S05), as a con- 

 venient way of treating observations without leference to the 

 Theory of Chance. Legendre's words are "la methode qui me 

 paroit la plus simple et la plus generate, consiste h. lendre 

 minimum la somme des quarres des erreurs . . . et quej'appelle 

 Methode des moindres quarres. The method, regarded from a 

 practical point of view, is a very natural one ; we shall clearly 

 get a good result by determining the quantities to be found so as 

 to make the sum of lhe2;/ih powers of the errors a minimum, and 

 in order that the resulting equations may be linear (and there- 

 fore manageable), we must take 11 equal to unity. 



Though first published by Legendre, the lule was applied 

 by Gauss, as he himself states, as early as 1795, and the method 

 is explained and the usual law of facility for the first time found 

 in the " Theoria Motus Corporum Ccdestium, Hamburgh, iSog 

 (not iSoS, as in Prof Hall's letter). The principle on which 

 Gauss proceeds hiay fairly, I think, be stated as follows: — If 

 there are given a number of discordant observations l\, I\, 

 &c., of a quantity .\-, so that we have the equations 

 ,v — I\ = o,j: - /'„, = o, &.C., then it is known that a very 

 good result is obtained by giving to .v the arithmetic mean of its 



observed values, and writing -r = — (P' + . . . + J'„) ; and 



it is required to find an equally good rule for deteimining x, y, c, 

 cVc, fiom a number of discordant equations of the form 

 "i-v -t- I'O' + <-!=+... = l\, ".X + 1>«J' + <-.,: -1- ... = r„, &c. 

 Assume therefore that x = ^ (V^ + , . . ]\) is the most pro- 

 bable value of X derived from the first system of equations, and 

 find the law of facility of error that this may be the ca.se ; then, 

 having this law, the most probable values of .v, y, c, &c., can be 

 found for the second system. 



The law of facility Gauss finds to be represented by — — e- ''--^'ifx, 



viz., this is the chance of an error of magnitude mtermediate to 

 .V and X + dx ; and thence it follows that the most probable 

 values of x, y, c, &c., are found by making (a^x + li^y + c^z H- 

 ... - }\) - + (a„x + l<.,y + c.z + . . . - r„) - + , &c., a 

 minimum. Gauss then pioceeds to determine /; in the manner 

 still generally adopted. 



Subsequent writers, Laplace, Poisson, &c., have in conse- 

 quence investigated how far the arithmetic mean is the most 

 probable result, &c. , and in one sense Lagrange (and a fortiori 

 .Simpson) may be said to have very slightly anticipated a portion 

 of the analysis required in these researches, although, as far as 

 the method of Least Squares is concerned, there is no anticipa- 

 tion. A slight examination will show how greatly superior 

 Laplace's analysis is to Lagrange's on the same subject. 



With reference to the independent discovery of the method of 

 Least Squares by Dr. Adrain of New Brunswick, U..S. (see Prof. 

 Abbe's note in the Atiiirican yotirtial of Science, June 1871), I 

 may remark that if for distinction we call the introduction of the 

 merely practical use of the rule its " invention," and its philoso- 

 phical deduction by tl-.e Theory of Probabilities its " discovery " 

 (so that Legendre invented the method and Gauss discovered it), 

 then Dr. Adrain can only be credited with the independent in- 

 vention of the rule, viz. , he only did what Legendre had done 

 two years previously. This is woith noticing, as from the 

 occurence of the function e - -^- in Dr. Adrain's paper, it might 

 be supposed that it contained some anticipation of Gauss' in\es- 

 tigation ; but such is not the case, and Dr. Adrain's reasons for 

 the adoption of the law are of so trivial a nature that it is in- 

 credible that any mathematician should have been led to the 



discovery of the method by means of them. I imagine that he 

 had noticed the practical convenience of the rule, and subse- 

 quently endeavoured to justify it analytically ; it may be noted 

 that it is possible that Dr. Adrain may have seen or heard of 

 Legendre's memoir published two years before ; his silence on 

 the matter, however, renders it unlikely that this was so. 

 On Ihe whole, by far the greater part of the merit of the intro- 

 duction of the method is due to Gauss ; while the credit of the 

 first suggestion of the practical rule must be assigned to Legendre, 

 Dr. Adrain having, in all probability independently, also suggested 

 the same rule subsequently. It is necessary to be thus ]>articular, 

 as Gauss' publication having taken place in 1S09 and Adrain's 

 in iSoS, it might be thought that the latter had anticipated the 

 former to some extent, which is in no wise the case. 



In writing the history of the Theory of Errors or the Theory 

 of the Treatment of Observations, there are several memoirs 

 anterior to Legendre's that would have to be included, and 

 notably Thomas Simpson's "Miscellaneous Tr.icis," 1757 

 (which is the work Prof. Hall doubtless refers to), Daniel 

 Bernoulli's " Dijudicatio maxime probabilis pluiium observa- 

 lionum discrepantium," &c. Acta. Petrop. 1777, Trembley's 

 paper in the "Berlin Memoirs," 1801, " Observations sur la 

 methode de prendre le milieu entre les observations," iSic. For 

 the above references I was indebted to Todhunter's " History of 

 the M.ithematical Theory of Probability from the time of Pascal 

 to that of Laplace" (London, 1S65), which contains a notice of 

 every work or memoir on the subject to the commencement 

 of the present century (there is a resume of Lagrange's memoir 

 occupying 13 pages), so that no one need have any fear of passing 

 over any writings published previously to iSoo. Having had 

 occasion to make much use of the work, I may be permitted to 

 say that its value, both as regards accuracy and completeness, 

 cannot be over-estimated. J. W. L. Glaisher 



Trinity College, Cambridge, June 8 



Solar Halos 

 A BEAUTIFUL combination of solar halos w,as visible here 

 during 1 lie morning of March 2. At 10-45 the sun having an 

 altitude of aI>out 40° was surrounded by a complete rainbow- 

 tinted circle of some iS° or 20° radius, red inside and blue out- 

 side. An Drc of a larger circle coloured in the same way touched 

 the complete circle at its highest point, rendering the point of 

 contact dazzlingly bright. A short arc touched the lowest point 



of the circle in the same manner. A white halo passed through 

 the sun's position parallel to the horizon, and two fainter white 

 arcs intcrstcttd it obliquely in the point opposite to the sun, 

 forming a conspicuous sun-dog. There were also two rainbow- 

 arcs having their convexities towaid the sun. These were blue 

 inside and red outside, and their centres appeared to be about 

 90" fiom the sun, and some 15" below the horizon. Later an arc 

 concentric with that touching the complete circle appeared above 

 it, having the colours reversed, namely, blue inside and red out- 

 side. Tliese spi earances lasted about an hour and a half before 

 beginning to fade away. W. W. J. 



Gambler, Ohio, March 5 



The Volcanoes of Central France 

 The Auvergne volcanoes threaten to be as periodic a subject of 

 controversy as the authorship of the letters ot Junius. It is only 

 seven years since the last eruption of letters. At that time I con- 

 ti ibutcd apaper to the Gcolo^'iea/ A/agazine (vol. ii. p. 24 1 ), in which 

 I collected, printed, and translated all that I could find on the 

 subject, and came to the conclusion that it was very probable 

 there had been some local outbreak of volcanic action. Thus I 

 agree with Mr. Garbett, but it appears to me that he has not 



