The Method of Least Squares is in constant use among engineers, for the adjustment of observations, but a simple and yet perfectly satisfactory proof of its main principle is still a desideratum. In this article, I wish to bring to notice a demonstration which has not yet found a place in English books, and which is far more simple and conclusive than the one usually given. I shall attempt, in writing it, to everywhere simplify and illustrate the reasoning of the proof, and to give enough of the fundamental principles of Probability to render the whole argument plain to those entirely unacquainted with the subject.
The proof alluded to is that of Dr. Hagen, who now, at a ripe old age, enjoys the honor of being one of the greatest hydraulic engineers of the century, and was first published at Berlin, in 1837, in his book entitled " Grundziige der Wahrscheinlichkeitsreehnung." Although often used in subsequent German and French works, it received no attention from English mathematicians until 1865, when Prof. Tait re-discovered it in a greatly modified and much less satisfactory form, a;nd published it in Yol. xxiv of the .Edinburgh Transactions. This, and a paper by Mr. Kummell, of the U. S. Lake Survey, in Tile Analyst, 1876, Vol. iii, p. 133, are the only sources of information concerning it in the English language, i Mr. Kummell's discussion, although very abbreviated, and requiring in its readers a previous knowledge of the subject, was very welcome to mathematicians, and it contains one or two modifications of the German method of presentation, which considerably shorten the algebraic work.
Hagen's proof has, I think, but one difficulty, and that lies in the fundamental axiom or hypothesis upon which it is based. This difficulty is very slight compared with those in Gauss's or Laplace's proofs, while the mathematical work is vastly simpler. For the bei See also, Price's [~ttl~fitesl~(~l Calc!dus (London, 18~5), VoI. ii, pp. ;17~;