N. IDELSON. Adjustment Computations by the Method of Least Squares. GIZ. 1927. pp. 192. Price 2 rubles 40 kopecks, binding 15 kopecks.

The method of least squares is such a universal idol and is regarded as such a panacea1 that almost every year, in each of the major

1 Deservedly or undeservedly—this is another question; see, for example, B. P. Weinberg, “On the methodology of averaging,” Journal of Applied Physics, 4, No. 2, 3–24, 1927.

...in the scholarly countries are devoted to somewhat new manuals on the application of this method. It would seem that, in view of the considerable development of the methodology of these applications and the comparatively slow further successes of the subject itself, one might be content in this direction with something like the stereotype used for tables of logarithms; but in fact it turns out otherwise.

Such a striving for novelty of exposition—sometimes extremely insignificant—is a symptom of the fact that the methodology of teaching the method of least squares has been developed to a far lesser degree than the methodology of its applications; and the book by N. I. Idelson, recently awarded the Glavnauka prize, furnishes clear proof of this. One may dispute whether it is rational to begin the exposition “with the method of indeterminate multipliers, called in geodesy the method of correlates” (the first 54 pages of Idelson’s book), pass from there “to the treatment of series of observations of one quantity, studied on the basis of statistical concepts” (the next 47 pages), and then turn “to systems with an excessive number of equations” (56 pages), the solution of which by the method of least squares is usually considered the principal content of such manuals. To me personally, such a distribution of the material, somewhat unusual in a course on the “Method of Least Squares,” seems quite correct from the didactic point of view; but the final judges can only be the consumers—both students of higher educational institutions and those geodesists, astronomers, physicists, and statisticians who, needing to become acquainted with this fundamental method of adjustment, study this book. One thinks that, perhaps, to many of them the exposition will appear somewhat difficult; but in that case the blame will lie not in the, as I have already indicated, wholly rational approach adopted by the author, but in the very manner of exposition (partly, in its style), which is not always distinguished by sufficient simplicity—perhaps because of the author’s overly meticulous striving for rigor and completeness in the proofs of all the details of that purely practical (italics, as throughout, are ours) problem for which “the present course is above all intended.”1 Fully agreeing with the author’s proposition that, in this matter, “not a single detail that might facilitate this work for the practitioner should be omitted,” I nevertheless find that not all details of proofs can be counted among such details, since their presence in places makes it difficult to assimilate the book.

The author rightly believes that “to reduce the exposition of the method of least squares to a few schemes and computational rules would be to take the question in an excessively narrow sense. In the mathematical sense this method acquires its full value only after

of the appearance of P. L. Chebyshev’s memoir on interpolation by the method of least squares, i.e., on the construction of a function that would approximate a given set of its individual values in the best sense from the point of view of the method of least squares.” And one may fully agree with the author’s assertion that “in the last section of the book there is given a rigorous and at the same time elementary exposition of this method (i.e., Chebyshev’s method), organically connected with the preceding parts of the course; moreover, it is shown that everything needed for its application fits conveniently into the usual formulas and schemes of the method of least squares.” This last section, which occupies, despite the inclusion in it of the foundations of the method of correlation and the application of Chebyshev’s method to harmonic analysis, only 33 pages (the last 2 pages are devoted to a very interesting and useful survey of the literature on the method of least squares), is an adornment of the book, and, probably, after its appearance the exposition and construction of Chebyshev’s orthogonal polynomials will become an indispensable part of every serious manual on the application of the method of least squares.

Turning now to the shortcomings of the book, and without dwelling on certain minor defects of exposition which it would hardly be appropriate to mention in a journal review, I shall note that its principal gap is an extreme lack of principle, which, however, represents an inevitable consequence of the lack of principle of the very principle of least squares. But, in calling N. I. Idelson’s exposition extremely lacking in principle, I mean only that he, so scrupulous in matters of strict justification of each of his words, carefully—deliberately or unintentionally—passes over all questions concerning the justification of the method of least squares itself and especially the limits of its applicability, questions which are perhaps no less important for practice and for practitioners than the justification of one or another computational procedure. In order to avoid the reproach of speaking without proof and to acquaint readers with the author’s attitude toward these questions, I shall quote passages concerning the justification of the method of least squares and the most fundamental question concerning the difference between “random” and “systematic” errors (the same could also be done for the “actual” or “true” values of observed quantities, etc.—pp. 14 and 54).

“Such an attempt was in fact made by Laplace, who intended to justify the whole theory of errors and of adjustment computations on the principle \([(\lambda)]\)—minimum. However, for reasons on which we shall not dwell, this idea remained mathematically fruitless” (p. 18). “Thus, the best solution in the sense of ‘\([\varepsilon \varepsilon]\)—minimum’ is at the same time the best solution in the sense of ‘weight of the unknowns—minimum,’ and in this circumstance lies the justification of the method of least squares as a general method of adjustment” (140).

“We suppose that all observations have been made with equal accuracy and are not subject to so-called systematic errors of instrumental or personal character. Experience shows that,

Bibliography

that, owing to random errors (NB: the author’s first use of this term!) of the observations, the measurement data \(l_1, l_2 \ldots l_n\) will differ somewhat from one another, fluctuating to either side of a certain mean value. We take into account in the simplest way all these random deviations if, as the best determination of the measured quantity \(x\) on the basis of the given series, we take the arithmetic mean of all the obtained \(l\)’s” (16–17). “We make the basic assumption that the random errors with which each of the observations is associated are grouped, according to their magnitude, in accordance with a certain quite definite law, common to all and every series of observations, provided only that these observations are not distorted by constant or systematic errors” (54). “If there is reason to assert that all measurements of a given series are free from a constant or systematic error, i.e. from an error varying according to a definite, though perhaps unknown, law, then discrepancies between the results of individual observations can be explained only by the influence of random errors” (70). “Such are the fundamental formulas of the theory of errors. We draw attention once more to the fact that they follow as a consequence of the very definition of a ‘random’ distribution, which finds its mathematical expression in Gauss’s law. One may say that the theory of errors recognizes as random precisely those series of errors to whose distribution Gauss’s law is applicable; and if, with respect to a given series, it is possible to establish a noticeable deviation from this law, we shall no longer consider such a series to be free from constant or systematic error” (72–73). “However, one can imagine cases where any system of corrections gives a very small improvement of the solution (the author’s spacing). This will show that the observed quantities have become inaccurate and contradict one another, and that the method of least squares cannot introduce any appreciable improvement into the matter” (111).

I allow myself to doubt that the reader, even if he finds and compares all these “definitions” of systematic and random errors, will understand what the difference between them consists in and—if he even agrees “to consider it obligatory to investigate every series of observations for a ‘normal distribution’” (75)—will know how to do this. Should one use for this purpose at least those three criteria which N. I. Idelson cites (64), or simply compare the observed distribution of deviations from the mean with the theoretical one, as the author does? If the reader prefers the latter, he will have to find himself at an impasse before the question of what agreement is to be considered “in general excellent” (61), what “satisfactory” (62 and 77 on the last page all the discrepancies are of one sign!), what “very close” (96), and what “cannot be called quite satisfactory” (127). He will encounter no fewer difficulties in the former case as well, for the only indication of how to proceed will be that, in cases where “the extreme errors do not satisfy the conditions of the normal distribution, ... the corresponding observations should not be taken into account with full weight in processing the series” (76).

Bibliography

I have deliberately cited everything that can be drawn, in this respect, from the book under review, in order to show what difficulties are encountered by someone wishing to make conscious use of the method of least squares—even with such an exemplary exposition of it at hand as that provided by Idelson’s book. The fault here lies not with the author, but with the fact that there are not only no manuals, but also no monographs—and even few separate works—devoted to the general questions of the theory and practice of the processing of observations; the need for them is often painfully felt by the physicist, the statistician, the astronomer and the biologist, the engineer of a factory laboratory, and the exploration geologist.

In any case, gratitude is due both to the author, who has compiled this excellent manual, and to the State Publishing House, which has printed this book very well and issued it at a comparatively inexpensive price.

B. P. Weinberg.

1. The author himself, apparently without noticing it, often departs from such a practical framing of questions, setting aside, for example, in the choice of weights (p. 89), all “questions of expediency, which stand apart from mathematical investigation.” ↩