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Augustin-Louis Cauchy

b. 21 August 1789, Paris, France
d. 22 May 1857 in Sceaux, France

The complete works of Cauchy have been published as Oeuvres complètes in two series (Series I - 12 volumes; Series II - 14 volumes) by Gauthier-Villars, Paris from 1882 to 1974. His work on least squares and applications to probability were written during essentially two distinct periods. When a paper is referenced by number, it is to the ordering in the Oeuvres complètes.

The fonts which appear in many of Cauchy's papers are frequently quite poor and used inconsistently. In his astronomical papers, even astronomical symbols are called into use to represent variables.

Minimizing absolute deviations

Boscovich had studied fitting an equation to observations where the criterion of fit is the minimization of absolute deviations. Cauchy's intent is to extend the method of Boscovich to equations having two or more unknown coefficients to be determined. He provides an algorithm to find the linear function of best fit. From the geometric point of view, the feasible set for the problem is represented by an open convex polygon, polyhedron or higher dimensional object. The solution is the lowest point on the surface of such object

The papers are presented here in order of publication, not in order of composition. The 1824 paper is actually an abridgment of the one published in 1831.

Interpolation

A paper which ultimately generated some controversy is the following.

(1835) "Mémoire sur l'interpolation." Lithographed. Printed in Modena, Mem. Soc. Ital. XXI, 1836, pp. 374-389 and in J. Math. Pures Appl. 2, 193-205 (1837). OC II, 2, 5-17.

For a period between 1846 and 1848, Cauchy  presented a sequence of papers in which he demonstrates a convenient method to compute the elements of the orbit of a planet or comet. These papers include a discussion as to how his method of interpolation may be used to advantage. See this page on astronomy.

The controversy between the method of interpolation and the method of least squares

Prompted by a paper by Faye, Cauchy read a paper in which he repeated his method of interpolation. Given a set of n linear equations in m unknowns,  where n>m, he shows how to solve for each unknown, first through successive elimination of each, and then by ascending up from the last to obtain the first to be eliminated. Since the system is overdetermined, there can be no simultaneous solution of all. Consequently, to each equation is associated an error which Cauchy intended to minimize.

(1853) "Mémoire sur l'évaluation d'inconnues déterminées par un grand nombre d'équations approximatives du premier degré." Comptes Rendus Hebd. Séances Acad. Sci. 36, 1114-1122. OC I, 12 (519), 36-46. Read 27 June 1853.

Here we must insert a memoir by Jules Bienaymé. He is concerned that some are using Cauchy's method of interpolation rather than the method of least squares. We'll discover subsequently that Bienaymé is motivated by his fear that Laplace's use of least squares is proven inappropriate and that consequently his research is found to be wrong.

On 4 July 1853, Jules Bienaymé read a paper, "Sur les différences qui distinguent l'interpolation de M. Cauchy de la méthode des moindres carrés, et qui assurent la supériorité de cette méthode." to the Academy of Sciences in which he compared the method of interpolation of Cauchy as presented in his 1835 paper to the method of least squares and in which he further asserted the superiority of the latter. The action of Bienaymé was certainly precipitated by the reading of previous paper (519).

Cauchy insists that the method of least squares is not always appropriate. To this end we have the two memoirs (522 and 523), the first read on the 18th of July and the second on the 25th of July. Memoir 522 contributes little. On the other hand, in Memoir 523, Cauchy observes that the application of least squares is to situations where the number of unknowns in the problem is fixed beforehand. However, there are situations where this is not the case. If one imagines approximating an unknown function from observations and representing this unknown function by a power series, one determines the coefficients of the terms sequentially, beginning with the least power, and stopping when the error  is a small as the error of observation. 

In return, Bienaymé tenders another response at the session of 8 August 1853.

(1853) "Mémoire sur l'interpolation, ou remarques sur les remarques de M. Jules Bienaymé." Comptes Rendus Hebd. Séances Acad. Sci. 37, 64-69. OC I, 12 (522), 63-68. Read 18 July 1853.

(1853) "Sur la nouvelle méthode d'interpolation comparée à la méthode des moindres carrés." Comptes Rendus Hebd. Séances Acad. Sci. 37, 100-109. OC I, 12 (523), 68-79. Read 25 July 1853.

Article 524 consists solely of an announcement that Cauchy presented further to the Academy two papers, one of which was entitled Mémoire sur le calcul des probabilités. and the promise that the results obtained in these two Memoirs will be presented in a later session. The relevant portion, in fact, is included in the text of the previous article (523).

(1853) "Mémoire sur les coefficients limitateurs ou restricteurs." Comptes Rendus Hebd. Séances Acad. Sci. 37, 150-162. OC I, 12 (525), 79-94. Here Cauchy introduces what we call today characteristic functions. That is, those functions with equal 1 on a set and 0 on its complement.

The following Memoir, 526, now incorporates a law of facility of error. Here he introduces the Cauchy distribution. The next, Memoir 527, reminds us that the method of least squares is based on the normal law for facility of error. When another law is applicable, it is no longer appropriate.

(1853) "Sur les résultats moyens d'observations de même nature, et sur les résultats les plus probables." Comptes Rendus Hebd. Séances Acad. Sci. 37, 198-206. OC I, 12 (526), 94-104. Read 8 August 1853.

We have on this same date "Remarques de M. Bienaymé à l'occasion des Notes inserées par M. Cauchy dans les Comptes Rendus de deux des séances précédentes."

(1853) "Sur la probabilité des erreurs qui affectent des résultats moyens d'observations de même nature." Comptes Rendus Hebd. Séances Acad. Sci. 37, 264-272. OC I, 12 (527), 104-114. Read 16 August 1853.

Article 528 consists solely of the following: "M. Cauchy presents also to the Academy a Mémoire sur la probabilité des erreurs qui affectent les résultats moyens d'un grand nombre d'observations." This piece is included at the end of the previous piece, Article 527.

On August 20, Bienaymé read a paper "Considérations à l'appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés" defending Laplace against Cauchy. 

Previous to the publication of this next article, appeared an account of a discussion regarding concerns with the publication of Cauchy's paper read on August 12. This notice appeared in the Comptes Rendus T. XXXVII, No. 9, pp. 324-325. 

The last two Memoirs are somewhat repetitive. The latter, in fact, Memoire 530, is only an abridgment.

(1853) "Sur la plus grande erreur à craindre dans un résultats moyens d'un très-grand nombre des observations." Comptes Rendus Hebd. Séances Acad. Sci. 37, 326-334. OC I, 12 (529), 114-124. Read 29 August 1853.

(1853) "Mémoire sur les résultats moyens d'un très-grand nombre des observations." Comptes Rendus Hebd. Séances Acad. Sci. 37, 381-385. OC I, 12 (530), 125-130. Read 29 August 1853.