# Henry Plummer's books

Henry Plummer wrote a number of books on mathematical astronomy, dynamics and probability. We give some information on these books below in the form of extracts from Prefaces and extracts from reviews.

An Introductory Treatise on Dynamical Astronomy (1918)

The Principles of Mechanics (1929)

Probability and Frequency (1940)

Four-Figure Tables with Mathematical Formulae (1941)

**Click on a link below to go to the information about that book.**An Introductory Treatise on Dynamical Astronomy (1918)

The Principles of Mechanics (1929)

Probability and Frequency (1940)

Four-Figure Tables with Mathematical Formulae (1941)

**1. An Introductory Treatise on Dynamical Astronomy (1918), by H C Plummer.**

**1.1. From the Preface.**

This book is intended to provide an introduction to those parts of Astronomy which require dynamical treatment. To cover the whole of this wide subject, even in a preliminary way, within the limits of a single volume of moderate size would be manifestly impossible. Thus the treatment of bodies of definite shape and of deformable bodies is entirely excluded, and hence no reference will be found to problems of geodesy or the many aspects of tidal theory. Already the study of stellar motions is bringing the methods of statistical mechanics into use for astronomical purposes, but this development is both too recent and too distinct in its subject-matter to find a place here.

Nevertheless the book covers a wider range of subject than has been usual in works of this kind. Thereby two advantages may be gained. For the reader is spared the repetition of very much of the same introductory matter which would be necessary if the different branches of the subject were taken up separately. But in the second place, and this is more important, he will see these branches in due relation to one another and will realise better that he is dealing not with several distinct problems but with different parts of what is essentially a single problem. In an introductory work it therefore seemed desirable to make the scope as wide as was compatible with a reasonable unity of method, the more so on account of the almost complete absence of similar works in the English language.

...

Since the main object in view has been to cover a wide extent of ground in a tolerably adequate way rather than to delay over critical details, the absence of mathematical rigour may sometimes be noticed. Very little attention is given to such questions as the convergence of series. It is not to be inferred that these points are unimportant or that the modern astronomer can afford to disregard them. But apart from a few simple cases where the reader will either be able to supply what is necessary for himself, or would not benefit even if a critical discussion were added, such questions are extremely difficult and have not always found a solution as yet. It is precisely one of the aims of this book to increase the number of those who can appreciate this side of the subject and will contribute to its elucidation.

**1.2. List of Contents.**

Contents:-

Chapter I:- The law of gravitation, 1-10;

II:- Introductory propositions, 11-20;

III:- Motion under a central attraction, 21-32;

IV:- Expansions in elliptic motion, 33-48;

V:- Relations between two or more positions in an orbit and the time, 49-64;

VI:- The orbit in space, 65-72;

VII:- Conditions for the determination of an elliptic orbit, 73-84;

VIII:- Determination of an orbit, Method of Gauss, 85-93;

IX:- Determination of parabolic and circular orbits, 94-102;

X:- Orbits of double stars, 103-114;

XI:- Orbits of spectroscopic binaries, 115-128;

XII:- Dynamical principles, 129-141;

XIII:- Variation of elements, 142-157;

XIV:- The disturbing function, 158-176;

XV:- Absolute perturbations, 177-191;

XVI:- Secular perturbations, 192-206;

XVII:- Secular inequalities, Method of Gauss, 207-217;

XVIII:- Special perturbations, 218-235;

XIX:- The restricted problem of three bodies, 236-253;

XX-XXI:- Lunar theory, 254-291;

XXII:- Precession, nutation and time, 292-311;

XXIII:- Libration of the moon, 312-322;

XXIV:- Formulas of numerical calculation, 323-340;

Index, 341-343.

**1.3. Review by: Anon.**

*The American Mathematical Monthly*

**26**(6) (1919), 253-254.

This review contains extracts from the Preface of the book followed by the sentence:

Mr Plummer is professor of astronomy in the University of Dublin and Royal Astronomer of Ireland.

**1.4. Review by: H S J.**

*Science Progress (1916-1919)*

**13**(51) (1919), 492-493.

There is a singular deficiency of works in the English language dealing with the important branches of astronomy which are comprised under the general heading of dynamical astronomy. This is surprising and regrettable when the important contributions which have been made to the subject by English and American astronomers are considered. It is sufficient merely to mention the names of Adams, Darwin, Hill, Newcomb, Cowell, Brown, and Leuschner, amongst others. The publication by Prof Plummer of the volume under review is therefore an event of importance.

The amount of ground which is covered is surprising. The contents of the book may be briefly summarised as follows: Chapters I to VI are devoted to preliminary matters mainly concerned with the undisturbed elliptic motion of two bodies. Chapters VII to XI are concerned with the methods for the determination of orbits, including the orbits of double stars and of spectroscopic binaries. Chapters XII to XVIII contain an outline of planetary theory, including general dynamical principles, absolute, secular, and special perturbations, and secular inequalities. Chapter XIX is devoted to the restricted problem of two bodies which serves as an introduction to lunar theory, an outline of which is contained in Chapters XX and XXI. The rotation of the earth and moon is discussed in Chapters XXII and XXIII, whilst the last, Chapter XXIV, is devoted to formulae of numerical calculation.

On reading the book we cannot but feel that it suffers from undue compression, and that an omission of some of the matter such as, e.g., most of the chapter on precession and nutation, an account of which is contained in Newcomb's well-known

*Compendium of Astronomy*, would have been an advantage, enabling the remaining matter to be treated at somewhat greater length. Although completeness is not aimed at, it is to be regretted that an account of Prof Leuschner's admirable method for the determination of an orbit, which enables an accurate approximation to be rapidly made, was not included.

Some doubt may be expressed as to whether it was wise to omit numerical applications altogether in a subject whose ultimate aim is always numerical expression. Not to have done so would have enormously increased the size of the book. For this reason, therefore, the author is justified in taking a via media. The mathematical formulae have, wherever possible, been reduced to a form suitable for numerical application, and, in doing so, the author's skill as a computer has proved itself of value. There is truth in the introductory remarks: "The student who feels the need will have no difficulty in finding forms of computation in other works. At the same the reader who will take the trouble to work out such forms for himself will be rewarded with a much truer mastery of the subject, though he should not disdain what is to be learnt from the tradition of practical computers."

All readers may not agree with the criticisms which we have offered, and much justification can be advanced for the course which Prof Plummer has adopted. The book is written with a freshness of treatment and a careful elucidation of difficulties which will commend themselves to the student. It meets a long-felt want. The elaborate treatises of Laplace, Tisseraud, Poincaré, and other writers on dynamical astronomy are too detailed to enable a student to study them with advantage without previous preparation. Prof Plummers book will make the way easier for him by giving him a general knowledge of the whole subject, which will serve as a stepping-stone to the classical treatises. We are grateful to Prof Plummer for supplying this want in so admirable a manner.

**1.5. Review by: Robert Grant Aitken.**

*Publications of the Astronomical Society of the Pacific*

**31**(179) (1919), 61-62.

No attempt will be made in this note to present a "review" of Professor Plummer's book; for, in the first place, an adequate review could be written only by a specialist in dynamical astronomy, and, in the second place, such a review would of necessity be too technical in its phrasing to be suited to the pages of these Publications. The note is intended simply to call attention to an excellent book which fills a gap in English astronomical literature. The only other book known to me, which is written in English and which is readily accessible to students, that aims to present a general introduction of the theory of motion under the law of gravitation is Professor Moulton's An Introduction to Celestial Mechanics, and in content as well as in form of presentation the two books differ widely.

The first six of the twenty-four chapters of Professor Plummer's book deal with "preliminary matters," the fundamental propositions and relations that lead to the discussion of methods of orbit determination, to which the next five chapters are devoted. In his Preface the author says that the latter section "is in no sense complete and is not intended to replace those works which are entirely devoted to this subject. Otherwise it would have been necessary to describe in detail such admirably effective methods as Professor Leuschner's. ..." Notwithstanding this explanation, regret must be expressed that these methods have been omitted, for by actual use in the computation of orbits of comets, asteroids and satellites they have proved their great value and wide applicability. On the other hand, it is a pleasure to note that two of the five chapters in this section present an excellent discussion of the theory of orbits of visual and spectroscopic binary star systems.

Seven chapters are devoted to the problems of perturbations, the first one dealing "exclusively with abstract dynamical principles which are subsequently employed." One chapter is given to the restricted problem of three bodies, and two to lunar theory. The three closing chapters treat respectively of precession, nutation and time; libration of the Moon; and formulae of numerical application.

A somewhat careful reading of the section devoted to orbit determinations shows that the book will appeal primarily to the student of astronomical theory rather than to the practical computer; but Professor Plummer is himself a skilful computer and has succeeded in his effort "to leave no formulae in a shape unsuitable for translation into numbers."

**1.6. Review by: Harold Jeffreys.**

*Nature*

**102**(1918), 322.

There has long been a need for a general book on celestial mechanics on a smaller scale and at a more accessible price than the standard work of Tisserand, and Prof Plummer's recent publication is a very successful effort to satisfy that need. It is so concisely written that a most remarkable amount of material is made available, within a limited space, with only occasional loss of clearness. Halphen's theorem, that if the acceleration of a particle is a function of its position alone, and all the trajectories are plane curves, then the acceleration is always directed towards a fixed point, forms the commencement of the problem of two bodies. This constitutes a most welcome innovation, as Kepler's second law now becomes a consequence of the first, and its truth a confirmation of it. The methods of determining from observation the orbits of planets, comets, and visual and spectroscopic binaries are treated in detail, and then the author passes on to the treatment of perturbations, which is dealt with much as usual; but it is pleasant to see that chap, xvi., on secular perturbations, includes a table of numerical results. Very useful features are the account of recent work on methods of numerical interpolation and integration, and the description of Cowell and Crommelin's method of computing special perturbations. The lunar theory and the theories of precession, nutation, and lunar libration are also discussed in some detail. Misprints are few and the index is good. The present writer would like to suggest, however, that in a future edition tables of the best available values of the elements of the solar system, in astronomical and metrical units, should be included; for these are the fundamental data of dynamical astronomy, and are not usually presented in any convenient form.

**2. The Principles of Mechanics (1929), by H C Plummer.**

**2.1. Review by: Louis Melville Milne-Thomson.**

*Nature*

**124**(1929), 331.

For the student beginning theoretical mechanics a treatment based entirely on Newton's laws of motion seems to be not only desirable but also absolutely necessary if a sound basis on which further development may rest is to be assured. At the outset the reduction to mathematical form of the problems of mechanics has to be faced, and this proves as a rule to be the greatest stumbling-block; the subsequent mathematical treatment appears in the main to offer much less difficulty than the original formulation. The reason for this difficulty in formulation is generally traceable to two causes (1) an insufficient appreciation of the simplifying assumptions in the external conditions which must be made in order to render the problem manageable; (2) an inadequate grasp of the mechanical principles contained in Newton's laws. To overcome (1) there must be a careful specification of the conditions under which the solution is sought and a minimum use of conventional language. The remedy for (2) is a proper explanation and exemplification of principles. This, it is Prof Plummer's object to supply.

The book is divided into five parts, the first dealing with the geometry of motion. In the second, Newton's laws are introduced and applied to the dynamics of translation. This part would have been improved by a somewhat fuller treatment, with more examples, of impulsive motion, for it is here that mechanical principles appear in their most applicable form without mathematical elaboration. In Part III. statical considerations are explained. Part IV. returns to dynamics, the two-dimensional motion of a rigid body. Part V. gives elementary principles of the elastic behaviour of bodies under stress. This is a welcome addition to the ordinary treatment of elementary mechanics, as it gives a glimpse of the deviations of actual solid bodies from rigidity.

Elementary notions of the infinitesimal calculus are used throughout. This is quite as it should be; in fact the beginner profits doubly by a concurrent use of the two subjects, while to exclude the calculus is deliberately to obscure and restrict the possibilities of elementary mechanics. Good introductory text-books on mechanics are rare. Where a teacher is available the book is not the most important matter. For a student reading by himself a good book is absolutely essential. This one can be recommended.

**2.2. Review by: C O Tuckey.**

*The Mathematical Gazette*

**14**(204) (1929), 584-585.

This book professes to be a first stage book for those "with only a slight equipment of mathematical knowledge," "suitable for those who will ultimately go further." The reader is "warned that the beginnings of the Calculus will still lie before him."

But it is difficult to envisage the type of beginner capable of tackling the book, especially if ignorant of Calculus. ...

The general impression made by the style of presentation is that the book is suitable for the Mathematical Scholarship candidate in his second year of specialisation in Mathematics, or perhaps for the first year of the University course.

This ensures that the reader will not be a beginner in Mechanics, except in very rare cases.

In fact, it seems better to regard the book as a revision course for those who have already studied one of those books in which "a certain range of elementary problems is contemplated without much regard for the need of a coherent foundation."

From this point of view there is much to commend in it.

The bookwork is set out with considerable care; the worked-out examples are judiciously chosen and their solutions are given to a suitable degree of detail; and each of the nine chapters concludes with a set of 25 to 30 examples of "scholarship standard," in the usual sense of that somewhat elastic phrase.

The arrangement of the chapters is as follows:

The first three deal with Kinematics, Chaps. IV, V with Dynamics of Translation, Chaps. VI, VII with Statics, Chap. VIII with Dynamics of a Rigid Body in two dimensions and Chap. IX with Elasticity.

In common with many writers the author uses g either for the acceleration due to gravity or for its numerical value. This is very convenient, but so careful a writer might have been expected to say beforehand that he proposed to do so.

The author takes the view that Newton's second law is merely a definition of force. He gives an illustration in which pressure of gas balances either a weight or the pressure of a hand and explains that while we know the pressure of the gas as rate of change of momentum, we really know nothing of the opposing forces in spite of their apparent familiarity.

It is the reviewer's opinion that if Newton's second law were merely a definition, then these three celebrated laws would have proved much less adequate as a foundation of Dynamics than they have proved, and that if Newton had intended his second law merely as a definition of "vis impressa," his knowledge of Latin was quite sufficient to enable him to make his intention clear - but probably this is personal prejudice.

In any case the discussion given of Newton's laws is interesting and so is the discussion on the limitations of the principle of Conservation of Energy in Dynamics.

In the Statics an unusual arrangement is adopted.

The parallelogram law gives the rule for the balancing of two couples, and from this all needed results as to parallel forces are derived. On the whole these chapters seem less interesting than those on Dynamics.

The chapter on Rigid Dynamics is largely concerned with moments of inertia and with pendulums, the latter being dealt with at some length.

The chapter on Elasticity has nothing to do with collisions, but deals with elastic strings, plates under stress, bending of rods and such like.

The book gives too short a treatment to be suitable as sole book at any stage, e.g. a chapter of 30 pages with 30 examples has to suffice for the whole of Statics, except parallel forces and centre of gravity.

But the careful discussions of the principles make it an interesting revision book suitable for able pupils in their last year or two at school or first year at the University.

**3. Probability and Frequency (1940), by H C Plummer.**

**3.1. Review by: Maurice George Kendall.**

*Journal of the Royal Statistical Society*

**103**(1) (1940), 97.

This book is deceptively entitled. One opens it (apprehensively) expecting to find that another Fellow of the Royal Society has joined in the scrimmage round the question whether probability and relative frequency are the same thing. Actually Professor Plummer is not concerned with this question at all, what little he says about the foundations of the theory of probability being severely orthodox. His main object is to give "an approach to probability and statistics from a mathematical point of view," which amounts, in his modest treatment, to an account of the mathematics required in probability, the theory of errors and the elementary statistical theory of frequency distributions and correlation.

Chapters 1 and 2 deal with the probabilities of discontinuous and continuous populations, and as Professor Plummer himself indicates, are such as might be found in any serious textbooks of algebra or integral calculus. They are nearly all occupied with direct probabilities, and we meet the usual problems whose solutions depend on counting the number of ways in which events can happen the probability that two knights placed at random on a chessboard will attack each other; the chance that if a stick is broken in two places the resulting pieces will form a triangle; the St Petersburg paradox; Buffon's problem; and so on. Professor Plummer obtains Bernoulli's theorem and Bayes's theorem in the orthodox way; but he does not venture on a complete discussion of their use in statistical inference.

In his Chapter 3 the author deals with the normal law of error and the concomitant topics of precision, rejection of observations and least squares. The fourth and fifth chapters treat of the normal and Poisson laws, the Pearson family of curves, and correlation. There is nothing much in this book which has not appeared elsewhere, but students of statistics may find it useful to have some of the basic mathematics assembled under one cover.

**3.2. Review by: Henry Thomas Herbert Piaggio.**

*The Mathematical Gazette*

**24**(258) (1940), 63-65.

As recently as 20th June, 1939, Dr John Wishart, in an address to the Royal Statistical Society (reported in full in their Journal, vol. 102, part iv, 1939), regretted that "there appears to be no standard treatise on the theory of statistics treated from the mathematical point of view". He declared that "if the teaching of the subject, at least in relation to its mathematical foundations, is to be seriously discussed, I submit that the question of the text-book must be put right in the forefront. ..." "... It should begin with a study of probability theory, a section which may well prove to be the most difficult of all to write.... Thereafter, there is a fairly clear road through distributional theory as an application of probability; the study of distribution functions and their parameters; the theory of estimation (an important section); theories of fitting; tests of goodness of fit; sampling distributions of statistics, leading to tests of significance ... profusely illustrated by examples; inferences respecting population parameters to be deduced from samples, involving a close examination of fiducial probability; ... through problems in a single variable to those for many variables, and from methods of dealing with homogeneous data to those concerned with data of a multiply-connected character, the important method known as the analysis of variance occupying a central and commanding place at this stage."

The specifications above of the ideal mathematical treatise on statistics are not easy to comply with, and indeed Dr Wishart himself considers that it is probably beyond the capacity of any one man to deal with them. In these circumstances, it is the duty of a reviewer to draw particular attention to any book which gives even a part of what is required. There is the last chapter of Uspensky's

*Introduction to Mathematical Probability*, but only a very small fraction of this work bears on statistics. There is also a "planigraphed" reproduction of a course of lectures by Dr Wilks, but it is difficult to obtain it in England. Thus Professor Plummer's book, which he describes as a modest attempt to supply an approach to various topics of probability and statistics from a simple mathematical point of view, will be welcomed. In comparing the contents of his book with the ideal specifications above, it is only fair to remember that the author, in his preface, mentions the omission of several aspects, and claims only to provide an introduction to the general ideas of the subject.

The first chapter deals with probability, introduced at the outset from two points of view, the "equally likely" definition of Laplace and the frequency definition of Chrystal. Professor Plummer says that "it should be considered an important aim of the subject to reconcile these two points of view as far as possible", but the rest of the chapter, which "is such as might be found in any serious textbook of Algebra", is concerned exclusively with the application of Laplace's definition. There is no mention of any of the modern developments of the frequency definition, due to von Mises and his school.

The second chapter deals with continuous probability: "it owes more than a little to the suggestive chapter contributed by W M Crofton to Williamson's

*Integral Calculus*". For the convenience of students it contains a proof of the elementary properties of Beta and Gamma functions, sufficient for the use of these functions in probability problems.

The third chapter deals with the theory of errors. There is an interesting discussion of the normal law from different points of view. This is followed by an account of the method of least squares.

The titles of the fourth and fifth chapters are somewhat misleading. The fourth chapter is headed "statistical distributions", but really deals chiefly with curve fitting. After a short treatment of Poisson's sequence, and of practical Fourier analysis, there is an elaborate account of the method of moments, culminating in Pearson's set of generalised probability curves and the methods of fitting empirical data to them.

The fifth and last chapter, headed "correlation", contains a short discussion of that subject and of the cognate subject regression, but it deals chiefly with statistical distributions. In the reviewer's opinion, it is the most valuable part of the book. It contains the proofs of various theorems due to "Student" (W S Gosset), Karl Pearson, and R A Fisher. It starts with the extension of the normal law to two, three, and n dimensions, and deduces the distribution of "chi-squared", with applications to tests of goodness of fit. Then follows an introduction to the theory of small samples drawn from a normal population.

...

**3.3. Review by: Alexander Craig Aitken.**

*Nature*

**146**(1940), 114.

The approach to probability adopted in this book is conventional, but cautious. The two definitions, the one a priori, based on equally likely aspects of a system, the second a posteriori, based on relative frequency in repeated trials, are stated and criticised, the first being preferred, but with qualifications. In the first two long chapters sixty illustrative examples of the most varied kind, divided equally between discrete and continuous problems, are posed, solved and commented upon. The reader who has worked through these will have learned a great deal about cards, dice, the Problem of Points, the St. Petersburg Paradox, Laplace's Law of Succession, Buffon's needle and many ingenious and interesting problems in geometrical probability; he should indeed be able to attack almost any reasonable problem of the kind.

In the long chapter on the Theory of Errors the normal law is derived first by Herschel's considerations, then by other sets of postulates, the conclusion being that it is "an empirical rule which is supported by experience in general, but which in the last resort must be verified by the same test". The principle of Least Squares is derived from the assumptions of normal distribution of error, and maximum probability; the other derivation, by minimum variance of consistent linear estimate, is mentioned, but as of secondary importance. (The present reviewer cannot agree with this assignment, nor with the statement at the end of \168, p. 162, that "the basis of probability" is abandoned in this second approach; and incidentally it is not necessary for its validity that the law of error should be symmetrical.) The treatment and notation follow classical tradition.

The statistical distributions in one variable treated in Chapter iv are those of Poisson, the Type A Series with the cubic correction term, and Pearson's system, introduced by way of the limiting case of the difference equation of the hypergeometric distribution. The point of view of curve-fitting is empirical. "The results have only to be justified by success."

...

The book is a welcome addition to the rather exiguous list of textbooks in English on the subject. Lack of space is doubtless the cause of the omission of certain important topics, such as the theory of estimation of parameters in a probability function. In a few places more precise statements could have been made. ... And lastly the reviewer, holding as he does that to keep an open mind is to assign no probability whatever, must demur to the suggestion that "ignorance may be an element conducing to equal likeliness." These, however, are mere obiter dicta; the book is at all times interesting, and punctuated with dry common sense.

**3.4. Review by: K B Madhava.**

*Current Science*

**9**(5) (1940), 245-246.

The world is rapidly coming to realise the immense role played by "chance" in all matters commercial, social, political or scientific. In other words, "indeterminism" seems to have come to stay as an organic part of our civilisation and culture. Though as Bertrand Russell put it "chance" may be negation of all laws, mathematical acumen has been sharp enough to penetrate beneath the apparent lawlessness or randomness and evolve a working set of rules for our guidance. Theories of Probability are the result of mathematical studies of "chance events". Many textbooks on Probability and Statistics have been written with specific applications to economics, business statistics, medicine, biology, and education, in order to cater to the needs of non-mathematicians interested in these fields. They cannot therefore afford to put in sufficient mathematics. Here we have a book by an eminent astronomer boldly coming forward to give the necessary mathematical background to the subject without any fear of persons unsophisticated mathematically. The author is however rather modest in regard to his aim. His ambition is only to provide an introduction to the general ideas of the subject from a simple mathematical point of view, and refers to his work as too "slight" to need a bibliography. Perhaps the consciousness of the author that he is not making a novel contribution has affected the exposition which is very much condensed and not quite easy to follow, explanations in worked examples being of the tabloid form.

The treatment of probability is designedly of the usual textbook variety based on the Laplacian doctrine of exclusive and exhaustive alternatives of equal probability and therefore makes no mention either of modern theories or criticisms of the classical school like those of Richard von Mises, H Cramer, A Kolmogoroff or Maurice Frechet. The author believes, in J M Keynes' manner, that what he calls Bernoulli's Capital theorem which is deducible from the Laplacian definitions gives a precise meaning to the relative frequency definition originally due to Leslie Ellis but here attributed to Chrystal. The first two chapters deal with probability and introduce the reader to almost all the simple and noteworthy problems and theorems due to Huygens, Montmort, Tschebyscheff, Bernoulli, Bertrand, Burnside, Poincaré, Crofton, Buffon, and others. They contain also some incidental and necessary pure mathematics, Stirling's approximation to factorial n, Euler's Gamma and Beta functions, and Dirichlet's integral. The introduction to a posteriori probability and the theorem of Bayes is remarkably clear and leads on to a suggestion of a sort of fiducial probability, the range to be attributed to the probability to be inferred when the frequency is given. The discussion is enlivened by such humorous touches as the following:

"Not even the most primitive mind contemplates a Twilight of the Gods, when the sun will rise on weekdays only; it is doubtful if the Eskimo thinks of the play of chance when the sun remains above or below the horizon."

In support of the urgency of Bayes' rule, it is asserted that any justification must be found in the absence of any alternative way of grafting a few isolated facts on the tree of ignorance. Every meteorologist knows of occasions when an answer must be found and the best, if not the ideal answer, must be given, provided it shall not violate the established principles of probability. It is probably not too much to affirm that nowhere has a student of probability been warned against the misuse or abuse of the ideas of continuous probability in relation to a posteriori probability as in para 122 of this book. The real obstacle to the application of the theory lies in the uncertainty regarding the a priori probability. When this law of probability is known, the a posteriori probability can be calculated in the light of observed facts on principles which are sound enough. The result is a modified law of probability, which virtually represents the probability of a probability or probability at second hand, as it were. As a measure of frequency this inverse probability or probability of causes must be regarded as inferior to a more direct determination based on fuller knowledge. But only too commonly a knowledge of the a priori probability is entirely absent. It is then only possible to make some hypothesis to supply the deficiency. The assumption should be as reasonable as may be, and should at least cover all conceivable possibilities. It may then be fairly hoped that the result will represent a tolerable approximation to the truth.

The third chapter deals with the theory of errors in which the astronomer feels naturally at home. Prof Plummer does not admit the validity of the deduction of the normal law from the assumption that the errors of observations are the result of unlimited independent sources. On the other hand, in such typical cases as the astronomical observations of position, the law represents the errors of a good observer in favourable circumstances. While the law admits the possibility of errors larger than can occur, it tends to underestimate the probability of errors which do occur. However, the astronomer has adopted it on account of the practical advantages it offers. After studying the error functions in some detail, the author proceeds to expound the mathematics of the Method of Least Squares deriving formulae for the standard deviations of the unknown. A note of warning is sounded that when everything (mathematically) possible has been done, it must not be regarded as surprising when the probable error of a determination is proved to be exceeded several times by the result of a later investigation. The mathematical method gives a measure of the self-consistency of the solution rather than that of the true accuracy of the observations. We are told that astronomers' troubles lie in other directions than mathematical, such as inadequacy of the number of observations, choice of appropriate weights to be attached in the combination of observations from different sources and the presence of systematic error.

The last two chapters, (four and five) deal with Statistics proper. Chapter four treats of curve- fitting by the methods of Fourier analysis and Pearson's differential equations. Hardy's method of summation is expounded for the calculation of moments of different orders. Chapter five is headed "correlation" which properly applies only to the first half, giving an account of the well-known Bravais' method for the normal distribution of points in two and three dimensions. We must be grateful to the author for devoting the rest of the chapter to the proofs of results which are usually merely enunciated in statistics text-books on account of the mathematical difficulties involved. But they are fully faced here and we get a mathematically satisfying discussion of Pearson's root-mean-square contingency, and the frequency distributions of chi-square, standard deviation, Student's ratio and Pearson's correlation coefficient. The dispersion of the correlation coefficient is also included. In regard to these difficult distributions, it may be remarked that a precedent had already been set up in their inclusion in books on probability by H Levy and L Roth and J V Uspensky However, the chi-square distribution makes its first public appearance in full mathematical robes only in the present text.

On the whole, Prof Plummer's book will appeal greatly to all persons whose mathematical equipment is more or less of the Honours standard of an Indian University. We have no hesitation in recommending it to every mathematical aspirant, student or teacher, who wants to enter into the new discipline which is coming to have an ever expanding field of applications.

**4. Four-Figure Tables with Mathematical Formulae (1941), by H C Plummer.**

**4.1. Note.**

Consists of a rearrangement and expansion of a short pamphlet containing a collection of mathematical formulae, by C E Wright, combined with "the well-known tables of the late Frank Castle".

**4.2. Review by: B M Brown.**

*The Mathematical Gazette*

**25**(265) (1941), 187.

This book is made up of two separate parts bound together into one volume. The first part consists of the familiar four-figure tables of Castle with certain modifications. The most interesting of these are the inclusion of an extra table to overcome the difficulty of the large mean differences in the early part of the table of logarithms, and the omission of tables of antilogarithms and square roots. Positive characteristics are used for logarithms of numbers less than unity.

The second part, which is a collection of formulae, occupies over forty pages and appears to cover every conceivable branch of elementary mathematics. There are in addition three pages of constants, including a large number of special interest to astronomers and physicists.

Last Updated June 2021