1.1. Review by: Francis S Macaulay.
The Mathematical Gazette 1 (11) (1897), 120.

This is a small volume of 100 pages, in which the chief properties of the simpler solids are clearly explained and demonstrated. The elementary geometry of spherical triangles is dealt with very thoroughly; and we specially like the theorem in which the various cases of the identical equality of two spherical triangles are given in order in one long enunciation.

2.1. From the Preface.

This book is divided into two sections the first deals with those parts of the Infinitesimal Calculus which have been recently introduced into the syllabus of some examinations for higher-school certificates, while the two sections taken together correspond fairly closely to the curriculum of students reading for the first part of an honours course in mathematics or for the ordinary degree in arts, science or engineering.

Believing that there is no royal road which leads smoothly and directly to the Infinitesimal Calculus, the author has made no attempt to evade all the difficulties which at the outset face the student in this subject. The road has, however, been laid in the first section so as to pass through those domains of number and function with which the student is  probably already acquainted, while the functions which are likely to be unfamiliar to him have been reserved for the second section.

To assist the student in mastering the fundamental conception of a differential coefficient, two ideas which are usually reserved for books of a more advanced character have been introduced at the beginning and used throughout the book, namely, range and sequence, and the ordinary symbolism in connection with them has been varied. The symbols to express open and closed ranges have, however, been changed so slightly that the alteration amounts to little more than a typographical modification but a more important change has been made in the arrow notation, which is now so generally used to express the limits of a sequence. The author is indebted to his former colleague, Dr James Mercer, for the beautiful suggestion of the arrow with a single barb, either upper or lower. This notation has been successfully tested in class work, but has not previously appeared in print. Teachers who have recognised the great advantages which have resulted from the introduction of the fully barbed arrow may perhaps be willing to try the experiment of arrows with a single barb.

No attempt has been made in the first section to deal with the definite integral, nor has the usual notation for the indefinite integral been introduced until a comparatively advanced stage. It need hardly be said that it is no part of the authors plan to exclude such an important and universally adopted symbol his plea for the postponement of its use is based upon the impossibility of justifying the symbol $\int f(x) dx$ as a representation of inverse differentiation until the nature of a definite integral has been explained. Something also may be gained by introducing students at an early stage to a differential equation, even though it is of the simple type

          $\Large\frac{dy}{dx}\normalsize = f(x)$.

The book is not written for any particular group of students; it is designed for those who wish to use the Infinitesimal Calculus as an instrument in the attainment of further knowledge. It is therefore essentially a book of practical mathematics. With this end in view, fundamental ideas are explained at great length for the easiest and quickest way to master this subject is to acquire a firm grasp of the conceptions upon which it is based. The student is advised to return again and again to the earlier chapters it is only gradually that the matter contained in them can be assimilated, and the knowledge which the student acquires as he progresses will sometimes furnish the key to open some doors that may be closed to him at a first attempt. The Infinitesimal Calculus cannot be learnt either by memory or by mimicry it needs judgment and reflection but if studied in the right spirit it may, by strengthening these qualities of mind, fulfil one of the most valuable aims of education.

2.2. Review by: Anon.
The Mathematics Teacher 10 (2) (1917), 121-122.

[Review of Section 1] This book is written "for those who wish to use the infinitesimal calculus as an instrument in the attainment of further knowledge." There are some things which the American student will find in this elementary text which may strike him as quite different from corresponding American texts. Besides a few modifications of notations he will find an early introduction of the ideas of range and sequence and the late introduction of the usual symbol for an indefinite integral. Besides elementary integrations it takes up areas, volumes and moments.

2.3. Review by: Anon.
The Mathematics Teacher 10 (4) (1918), 211.

[Review of Section 2] The first section of this work, which was reviewed in an earlier issue, closed with Chapter XI and this section begins with Chapter XII on exponential and hyperbolic functions, inverse circular and hyperbolic functions. Chapter XIII. treats of the motion of a particle along an axis and Chapter XIV of definite integrals and some general theorems. The six remaining chapters deal with polar coordinates, curvature, partial differentiation, double integration, expansion in power series, curve tracing, envelopes, evolutes, roulettes, differential equations, graphics and nomography.

2.4. Review by: Philip E B Jourdain.
Science Progress (1916-1919) 13 (49) (1918), 148.

This book is one of "Longmans' Modern Mathematical Series," and the first section "deals with those parts of the Infinitesimal Calculus which have been recently introduced into the syllabus of some examinations for higher school certificates, while the two sections taken together correspond fairly closely to the curriculum of students reading for the first part of an honours course in mathematics or for the ordinary degree in arts, science or engineering". The first section passes through "those domains of number and function with which the student is probably already acquainted, while the functions which are likely to be unfamiliar to him have been reserved for the second section". In symbolism there are some good innovations in this book: there is a convenient notation for open and closed ranges of real variables by square and round brackets, and arrows with a single (upper or lower) barb are used to express convergence of the general term of a sequence down or up to a limit. This notation is a decided advance on the fully-barbed arrow now so generally used to replace signs often involving the notion of "equality to infinity." The most striking difference of this book from others is that "no attempt has been made in the first section to deal with the definite integral, nor has the usual notation for the indefinite integral been introduced until a comparatively advanced stage", because of the impossibility of justifying the use of the usual symbol for an indefinite integral as a "representation of inverse differentiation until the nature of a definite integral has been explained."

The book is not written for any particular group of students; it is designed for "those who wish to use the Infinitesimal Calculus as an instrument in the attainment of further knowledge"; and yet more attention is paid to logic than in the usual textbook: in fact, if we describe, as it seems that we may, the object of any textbook to be to give, for teaching purposes, a judicious compromise between history and logic, this book must be accused of being logical. Though it is on "infinitesimal" calculus, infinitesimals are banished. This is a good book, and it is pleasing to see a treatment of integrals giving mean values - an important notion in the theory of functions.

2.5. Review by: William P Milne.
The Mathematical Gazette 9 (139) (1919), 327-328.

The above text-book consists of some 360 pages and is issued in two volumes. Chapters I and II deal with the fundamental notion of Number, Function, Graph, Limit, and Continuity. Chapters III, IV, V, VI, VII treat of such subjects as Differential Coefficient and its elementary processes and applications relative to the simpler functions, i.e. powers of $x$, polynomials, trigonometrical functions, etc. In Chapter VIII. the second differential coefficient is introduced and its connection with radii of curvature, acceleration, etc., explained. Chapters IX, X, XI discuss Inverse Differentiation (leading at once to the necessity for introducing the Logarithm in dealing with $D^{-1}x^{-1})$ and the practical subjects of Areas, Volumes and Moments of Inertia. Chapter XII disposes of Exponential and Hyperbolic Functions, Inverse Circular and Hyperbolic Functions. In Chapter XIII the motion of a particle along an axis is discussed. In Chapter XIV we find general theorems such as the Mean Value Theorem, as well as the Definite Integral. Polar Coordinates and further geometrical properties are treated of in Chapter XV, and Partial Differentiation, together with Double Integration, in Chapter XVI. Expansions in Power Series with applications to Curve-Tracing appear in Chapter XVII, and more geometry - Envelopes, Evolutes, Roulettes - in Chapter XVIII. Chapter XIX is occupied with the easier forms of Differential Equations. The last chapter contains a distinct innovation in the form of the more elementary principles of Graphics and Nomography.

The book is not a mere engineering manual, but is intended mainly for students who desire to obtain a sound academic knowledge of the subject. It gives plenty of practical applications on which the student can exercise himself, and it does not "hedge" the fundamental principles of Number, Continuity, etc., on which the modern conception of the Calculus is based; but wherever these abstract motions are introduced, Prof Carey attempts by concrete illustration to vivify and vitalise as far as possible those nebulous entities which in many modern treatises and lectures leave the ordinary student vague and miserable.
...
It is impossible at this stage in the development of the teaching of the Calculus to appraise with accuracy and justice the true value and practicability of Prof Carey's presentation of the subject. What is certain is, that he has written a most valuable and noteworthy book, and and one which will well repay reading. On the other hand, modern pedagogy is almost unanimous in thinking that the best plan for any student is to proceed in his first reading fast and far into the depths of the subject which he is studying, using only intuition as far as possible and taking much for granted. After thus getting a bird's-eye view of the mathematical territory to be explored, he is in a position to re-traverse the ground more carefully, studying the philosophical bases on which the subject rests and trying to discover the minimum number of axioms. Chrystal states this pedagogical principle very clearly in the Preface to his Algebra, and we have recently had another explicit statement on the same subject by Sir Ronald Ross in his presidential lecture to the Science Masters' Association. It will probably be found best in in the long-run, therefore, to make one's first perusal of the Calculus graphical and intuitional, paying great heed to mechanical processes and practical applications such as Mechanics and Geometry. The student is then in a position to pursue a later course of study, in which he pays little heed to process and applications, but devotes all his attention to concepts of "Number" and "Continuity," rigorous theorems on "Limits," and so forth, - in fact, the modern courses on the so-called subject of "Epsilonology" as set forth in treatises on "Analysis," "Sets of Points," etc. Prof Carey's book is not based on this view; he aims at a high standard of rigour from the outset, but quot homines, tot sententiae, and the two volumes will well repay reading by all whom it may concern.

2.6. Review by: E J Moulton.
Amer. Math. Monthly 27 (12) (1920), 470-472.

How far should one go in introducing new ideas at the beginning of a first course in the calculus? And how much time can one afford to spend in explaining those ideas? It is in answering these two questions that the text before us differs most radically from our standard texts. Most writers seem to agree that while in an introductory course the notions of variable, function and limit must be discussed, a very brief discussion is best, introducing as little notation and as few new terms as possible, with the thought presumably that those notions will become most rapidly and truly understood through use.

The attitude of the author is given in the preface. Believing that there is no royal road which leads smoothly and directly to the Infinitesimal Calculus, the author has made no attempt to evade all the difficulties which at the outset face the student in this subject. The road has, however, been laid in the first section so as to pass through those domains of number and function with which the student is probably already acquainted ... . To assist the student in mastering the fundamental conception of a differential coefficient, two ideas which are usually reserved for books of a more advanced character have been introduced at the beginning and used throughout the book, namely, range and sequence, and the ordinary symbolism in connection with them has been varied ... . The book ... is essentially a book of practical mathematics. With this end in view, fundamental ideas are explained at great length; for the easiest and quickest way to master this subject is to acquire a firm grasp of the conceptions upon which it is based. The student is advised to return again and again to the earlier chapters; it is only gradually that the matter contained in them can be assimilated. ...

In accord with these statements the author has taken more than twice as much time for his introductory material as is common in other introductory texts, and the notions range and sequence are brought into unusual prominence. Most of us will, I believe, doubt the necessity, and hence the desirability also, of directing so much energy toward the mastery of these concepts so early in the student's career. But if we accept the author's point of view, and do not try to quarrel with him over what should be the purpose of his book, we must compliment him upon the skill with which he has carried out his plan. Anyone agreeing with his point of view will be delighted with his full and elementary presentation.
...
The American reader is impressed with the amount of geometry that is given. The chapters on curve tracing, envelopes, evolutes, roulettes, and graphics and nomography have much in them that is not included in our introductory texts. As there are no counterbalancing omissions, we would find it necessary to lengthen our course to include it all. The book furnishes an excellent reference work for this material. The last chapter, on "Graphics, nomography," gives in ten pages a good idea of the possibilities of the subject, and should, as the author states, "enable the reader to follow the complicated nomograms which are largely used in France and other countries."

2.7. Review by: F M Morgan.
Bull. Amer. Math. Soc. 25 (1919), 772-473.

This text is bound in two separate sections, the first containing sufficient material for a good elementary course, while the two sections together cover the topics usually presented in a longer elementary course. Chapter one reminds one of function theory, for the author treats of such topics as sequence of numbers, the arithmetical continuum, closed and open ranges, etc. The reviewer doubts if a beginner can grasp such concepts. Chapter two, on "Limits," contains many interesting examples to illustrate what a limit is, but nowhere is there to be found a concise definition of the word. The question of left hand and right hand limit, the question of the limit of a sum, product and quotient of two functions, is very thoroughly discussed. In the third chapter the rules for the derivative of a sum, a product and quotient of two functions are derived, but the last two derivations are very blind. The next chapter "The sign of the differential coefficient" treats of maxima and minima, and many fine examples are to be found among the exercises. This is followed by a chapter on algebraic functions. The remaining topics treated in the short elementary course are: "The inverse of a function," "Function of a function," "Tangent and normal," "Parametric equations," "Point of inflexion," "Circle of curvature," "Order of magnitude," "Inverse differentiation," "Logarithmic functions," "Areas," "Volume," "Parabolic approximation," "Simpson's rule," "Moments" and "Centre of gravity." No definite integrals are used, in fact the symbol $\int$ is not introduced.

Section two starts with an excellent chapter on exponential and hyperbolic functions. The results of several integration formulas are expressed in terms of these functions. This is followed by a discussion of the motion of a particle along an axis. The definite integral is now introduced and many of the elementary properties which we usually assume in an elementary course are proved in full detail. This is followed by a chapter on polar coordinates in which pedal curves and intrinsic equations are discussed together with the usual material to be found in such a chapter. Work on partial differentiation, double integration, triple integration, expansion in power series, curve tracing, singular points, Newton's method of ascertaining the form of a curve at the origin and at infinity, envelopes, involutes, roulettes and planimeters, finish the work in the calculus. This is followed by a short course in differential equations and a short but very interesting chapter on nomography.

The book is well written and the typographical errors are few, there being more, however, in the second section than in the first.

3.1. Form the Introduction.

This book is a systematic treatise on the fundamental principles of the science of Mechanics: it also provides material which will enable students to become proficient in the application of these principles. It has been written in conformity with certain rules, the most distinctive of which is that concepts and principles are introduced one at a time, and that the introduction of any particular concept or principle is delayed until further progress without it would be inconvenient. In this way, the authors believe that a student will learn to appreciate the logical structure of the subject - an appreciation which is of prime importance, not only in the full understanding of the theory, but also in the solution of problems.

The second rule followed consists in proceeding from the concrete to the abstract, and from the more familiar to that which is less familiar. In pursuance of this rule, the gravitational measure of force is used before the absolute measure is introduced; and consequently the concept of mass is not discussed until a relatively late stage. This has the effect of making portions of the book conform to the practice in engineering instruction.

Statics and Dynamics are developed side by side with as much similarity as possible. The internal forces of a body in equilibrium are treated on the same lines as those adopted in the discussion of moving bodies. This is a prominent feature of the book, and is an extension of Newton's methods to Statics. It is, of course, the practice of books in which the principles of Statics are deduced from those of Dynamics, but in such books little attention is usually given to Statics; books in which Statics is dealt with as a separate section generally follow the pre-Newtonian tradition.

In the appeal to mechanical phenomena, familiar experience and a posteriori evidence are quoted more often that the results of specially designed laboratory experiments. This is in accord with the logical nature of the book, but the authors recognise the high value of work in a laboratory for a student of Mechanics.

3.2. Review by: Harry Bateman.
Amer. Math. Monthly 33 (4) (1926), 224.

This book by a professor emeritus in the University of Liverpool and a young professor who has made notable advances in the study of the tides is a good example of what can be accomplished when the experience of a great teacher is combined with the brilliance of an ardent investigator. The subject is well presented, one good feature being the frequent use of vectors. Many illustrations are given to assist the reader and numerous examples are worked out at the end of the book to help along any student who is reading the subject for the first time. Though the work is quite elementary, room is found for a useful chapter on frameworks and an elementary treatment of hydrostatics is included. We fully agree that instruction on these subjects should be given in a course on mechanics.

3.3. Review by: David E Smith.
The Mathematics Teacher 18 (7) (1925), 438-439.

This is the most recent volume in the Longmans' Modern Mathematical Series, a collection of books that has done much to set forth the modern purposes and methods in the teaching of pure and applied mathematics. It is written by Professor Emeritus Carey and Professor Proudman, both of the University of Liverpool. Professor Carey has long been known for his contributions to the study of the calculus and mechanics and for the high standard set by him when directing the work in these subjects in his university, and his collaborator has been prominent in the same field. The book represents, therefore, an author ship that assures a scholarly treatment of the subject.

The work is intended for use in the first year of the university, but it represents a standard that is hardly reached as early as this in our American colleges. It does not require a preliminary study of the calculus, but it presupposes what we do not generally have in this country, namely, a good course in mechanics in the secondary school.

While the chapter headings naturally represent the classical topics, the presentation of these topics represents a considerable departure from classical methods. The theory in general follows a statement of the problem to which it applies, proceeding from simple facts well known to the reader to the mathematical explanation of these facts. This is seen in the case of speed and its measurement, of velocity-acceleration, of gravity, and so on through the list. Much attention is also paid to simple laboratory practice, a feature which British teachers have been able to carry out with much more attention to mathematics than is usually the case with us.

In the matter of arrangement in presentation the authors have adopted a plan that is worthy of attention. In the chapters there is printed a general treatment of the topic; then the contents refers to "Worked Examples" and "Examples" in the second half of the book. In this way they combine a text of theory and a book of exercises. Whether this is as good as our custom of combining the examples with the text is probably answerable only by considering the habits of the users.

Perhaps the most striking feature of the treatment is to be found in the use of vectors and in the well-arranged set of exercises on the subject.

The English writers have for a long time excelled in their exercises in applied mathematics. This fact strikes an American reader as he looks over the work under review. There are more than a hundred pages devoted to applications. Since the type is small, they represent the equivalent of 150 to 200 pages of our ordinary textbooks. Such a supply, including a large amount of new material, will prove very helpful to American students.

The entire work is worthy of careful reading and it seems not improbable that it will find a place as a textbook in some of our colleges.

3.4. Review by: Charles R MacInnes.
Bull. Amer. Math. Soc. 32 (2) (1926), 172.

The book starts with kinematics treated from a geometric point of view; a point of view which is carried so far that what they call speed-acceleration is defined as the slope of the time-speed graph, and velocity-acceleration as a velocity on the hodograph. Along this same line, in illustrating speed as the slope of a curve, the authors make the curious statement that "it is important that the student realize that in the concept of speed, we divide a distance by a time." This should give pause even to one who would accept the corresponding statement concerning the measure of a speed.

The early chapters cover the ideas of velocity, acceleration, projectiles, relative motion, and general kinematics. Then comes a chapter on vector addition and subtraction; multiplication is left to a later chapter, in fact to a chapter which comes after some of the ideas have been used in getting the moment of a force. Statics and dynamics of a particle and of a set of particles are developed side by side. Then comes the idea of moments, followed by centres of gravity and hydrostatics. The authors have quite justly left to this late date the difficult idea of mass. After momentum and impact are taken up, come work and energy. The book closes with a little celestial mechanics and a few interesting historical notes.

The book is not intended for use in a first course; on the other hand, no use is made of the calculus. The authors have gathered a large collection of problems, about a thousand, and many of them quite substantial enough to test the ingenuity of the best students. They are all gathered at the end, even the illustrative ones, arranged by chapters. Answers are furnished. Graphical methods are used a good deal, including Maxwell diagrams for the solution of truss problems.

4.1. Review by: Eric H Neville.
The Mathematical Gazette 14 (201) (1929), 468-569.

The pages of the tables designed at Liverpool have an elaborate appearance. The decimal figures are from one fount when the entry is in excess of the true value, from another when it is in defect, and from a third in the few cases in which it is exact. The first two founts are used also for the difference columns; these are conveniently placed in the centre of the page, and a simple device makes them in effect columns for five instead of for ten of the main columns. I do not know to what extent the practical man prefers refinements in four figures to five figures used mechanically. Economy of space is not to be despised, for half the labour of working with such a volume as Chambers is in turning over the pages. In this direction the limit was doubtless attained by old Oliver Byrne, who reduced the entries of the seven-figure logarithms of the first ten million numbers to ten folio double-pages, not by small print, but by a variety of typographical ingenuities. Mr Grace and the late Prof Carey have been quite successful in their less ambitious plan of making the most of four figures. And as they say, if their devices are ignored, there remain the ordinary tables which schoolboys can use. At the same time, a healthy boy wants to understand gadgets if they are put into his hands, and the mathematical master has more important things to explain than forced decimals.

This book is distinguished from the others also by the inclusion of exponential and hyperbolic functions and of natural logarithms, a difference which to my mind is more important for class use than an improvement in the order of accuracy obtainable. Practice in numerical integration which does not frequently proceed to the bitter end is an absurdity: any teacher who questions this assertion may try the experiment of setting a few examples to a class which he thinks will find them child's play. The absurdity is perhaps tolerable if suitable tables are not available. It is true that the necessary calculations can all be performed without difficulty by means of ordinary logarithms. This is not an excuse to be urged by compilers who print separate sine and cosine tables, for they at least can hardly plead that they are taking intelligence and industry for granted. But if subsidiary calculations have to be performed and corrected, the teacher may reasonably object to numerical examples on the ground that three-quarters of the time spent on them is spent irrelevantly. Any serviceable set of tables which removes this objection should be welcomed widely.