## Reviews of Sidney Loney's textbooks

We give below reviews of some of Sidney Loney's textbooks. The books proved extremely popular but reviewers were often very critical of Loney's approach. To get a proper flavour of the reviews we have only made some minor shortenings:

**1. Mechanics and Hydrostatics for Beginners (1892), by S L Loney.**

**1.1. Review by: Anon.**

*The Journal of Education*

**38**(3) (928) (1893), 67.

It is interesting to see how differently the English press and publishers presents the text-book from his American neighbour. This is especially noticeable in the case of works on physics, or indeed, and phase of science. This is a purely elementary book adapted to those whose knowledge of geometry and algebra is limited.

**2. Plane Trigonometry (1893), by S L Loney.**

**2.1. Review by: James M Taylor.**

*The School Review*

**2**(4) (1894), 240-241.

This work is divided into two parts; the subject of Part I is Geometrical Trigonometry, and that of Part II Analytic Trigonometry. Part I includes the subjects that are usually presented in elementary trigonometry. The trigonometrical functions of angles are defined as ratios. Before introducing, however, the idea of positive and negative angles and directed lines, the author establishes many of the relations between the functions of acute angles, and employs the natural functions of special angles in the computation of heights and distances, thus introducing the reader to one of the most important and interesting applications of the science. The geometrical proofs of the more common formulas are full, clear, and rigorous. The proving of trigonometrical identities, and the solution of trigonometrical equations receive the attention which their importance warrants. The examples in these subjects as in all others are numerous and well chosen. The properties of triangles and their connected circles are fully discussed. In Part II the author first deduces the common exponential and logarithmic series, and then treats of complex quantities under their trigonometric form. He establishes De Moivre's Theorem and applies it to finding any sort of a complex quantity and to the expansion of cos

*n*π and sin*n*π in terms of the functions of π. From these general formulas are obtained the common expansions of sin π and cos π. Next follows the expansions of the cosine and sine of an angle in terms of the cosines and sines of multiples of that angle, also the expansions of the cosine and sine of a multiple angle in terms of the powers of the cosines and sines of the angle. The author treats with marked clearness the exponential series for complex quantities, the trigonometric functions of complex angles, and the hyperbolic functions defined analytically. It would have added to the interest and clearness of view of the reader to have here compared the geometrical representations of the trigonometric and hyperbolic functions. The discussion of the many-valued logarithms of negative and complex numbers is very satisfactory. Among the other subjects treated the most important are the value of π, summation of series, expansion in series, factoring of mathematical expressions, proportional parts, errors of observation, solution of cubic equations, and geometric representation of complex quantities. The author has succeeded in his purpose to produce "a fairly complete elementary text-book on Plane Trigonometry." The faithful student of this treatise "will have little to unlearn when he commences to read treatises of a more difficult character." The style is clear and simple; even when it is diffuse, the author never hides his thoughts with words either large or small. It is a work that will well repay the reading.**2.2. Review by: Anon.**

*The Journal of Education*

**40**(20) (1003) (1894), 351.

There is little that can be said of this admirable, comprehensive treatment of the subject of plane trigonometry, beyond the announcement of the title and the publishers. Issued by Cambridge University Press, Professor Loney's work has all the advantages of careful preparation, critical editorial approval, and the best possible manufacture.

**3. Coordinate Geometry (1895), by S L Loney.**

**3.1. Review by: Anon.**

*The Journal of Education*

**43**(3) (1061) (1896), 46.

This is a complete and vigorous presentation of Cartesian and Polar coordinates, with the emphasis placed largely upon the straight line and circle. There are more than 1000 examples, and generally of an elementary character. They are carefully graded. The work begins with quadratic equations, and advances from that standpoint. For what it teaches, and for what it leaves unpresented, the teacher who must depend largely upon the author has cause to be grateful.

**4. The Elements of Hydrostatics (1901), by S L Loney.**

**4.1. Review by: Anon.**

*The Mathematical Gazette*

**2**(30) (1901), 123.

The peculiar characteristics of Mr Loney's works are so familiar to most teachers that it is unnecessary to dwell on them here. Suffice it to say that the 'Hydrostatics' is no exception to the rule. The chapter on Centres of Pressure strikes us as more complete than is usually the case in an elementary work. ... The sections dealing with curves of buoyancy and tensions of vessels are as simple as is necessary for ordinary students. We think that some definite reasons should be given why the theory of Hydrostatics here laid down should be even remotely applicable to other than the "perfect liquid," and that, in general, more illustrative matter is advisable referring to the machines in daily use. It is really remarkable how differently a student works at theories which have obvious practical applications. We must be wrong if we neglect any method likely to convince the pupil of the practical value and relevance of his investigations.

**5. An Elementary Treatise on the Dynamics of a Particle and of Rigid Bodies (1909), by S L Loney.**

**5.1. Review by: Anon.**

*The Mathematical Gazette*

**5**(87) (1910), 315.

Mr Loney's name is too well known as a writer of text-books for it to be necessary to do more than call attention to the scope of the class-book he now publishes upon Particle and Rigid Dynamics. The student is supposed to have read some course of elementary Dynamics, to have a fair working knowledge of the Differential and Integral Calculus, and, with the aid of the Differential Equations solved in the text and appendix, he will have all the equipment necessary for his purposes. The ground covered is that required for a Science or Engineering Degree in Applied Mathematics, or for Junior Students for an Honours Degree in Mathematics. The work is clearly that of a skilful and experienced teacher, and candidates for the examinations mentioned will probably find in it all that they require in order to cut a respectable figure when their time of trial arrives. On the other hand, we cannot look upon the book as doing much more than take the student through a subject for examination purposes, however excellently this may be done. As a treatise on cut and dried lines, it has the qualities that have made its more elementary predecessors of value in the past. But we think the author would have made his book far more inspiring and attractive had he remembered that the age at which students are fit to begin this stage of their reading is also the age at which it is well to cultivate their imaginative capacity. In another couple of years every boy who reaches this stage will have had his ideas developed largely through the experimental course, which is everywhere forming an essential part of the preliminary training in applied mathematics. This then is the time for a more critical examination of the familiar concepts, for a glance, or even more than a glance, at the history of the subject and the development of principles. From the point of view of the student's mental growth, it does not seem to us to be right that at 18 or 19 he should be left entirely to himself to discover the existence of a working hypothesis where he has hitherto met the infallible rigour of a law. We cannot, however, blame the author for not transgressing the limits he has deliberately laid down for himself in this otherwise excellent manual, and, of course, we are open to challenge as to our view of the propriety of allowing a student to suppose that the problems dealt with in a mathematical text-book are exactly those we find in the world around us.

**6. An Elementary Treatise on Statics (1912), by S L Loney.**

**6.1. Review by: Anon.**

*The Mathematical Gazette*

**7**(105) (1913), 125.

This treatise will be found useful for scholarship candidates, and it covers the course required for Science and Engineering degrees and for junior students for Mathematical Honours. The reader is assumed to have some knowledge of the methods of the calculus and elementary solid geometry. Sections are devoted to Shearing Stresses, Forces in Three Dimensions, Wrenches, Nul Lines and Planes, Strings and Chains, Attractions and Potential, and Slightly Elastic Beams. It contains a large collection of examples of every degree of difficulty. The quality of Prof Loney's work is too well known to need further comment. This volume is intended to be a companion to the author's 'Dynamics of a Particle and of Rigid Bodies'. It is sure to find an audience, for there are very few text-books covering precisely the same ground from which the student can make his selection.

**6.2. Review by: Anon.**

*The Mathematics Teacher*

**4**(4) (1912), 173.

This work presupposes a knowledge of elementary calculus and solid geometry as well as the more fundamental notions of statics. Besides others it has chapters on Work, Centre of Gravity, Machines, Attraction and Potential, and Graphic Solutions. The book seems to be written in Professor Loney's usual clear style and contains much to be commended. There are many examples covering a wide range of application and of varying degrees of difficulty.

**7. The Elements of Coordinate Geometry. Part II. Trilinear Coordinates, etc. (1923), by S L Loney.**

**7.1. Review by: A. R.**

*The Mathematical Gazette*

**12**(169) (1924), 67-68.

The first chapter of this volume deals with Cross Ratio, Homographic Ranges and Pencils, and Involution, and these subjects are continued in Chapter IV. Reciprocation and Projection occupy Chapters VI and VII. A text-book containing a blend of Pure and Analytical methods accords well with modern ideas, but in the present volume the treatment of Pure Geometry is half-hearted. The postponement of Projection to so late a stage makes it unlikely that the student will appreciate the relevance of cross-ratio methods; nor is he likely to be impressed by their power when he finds even such results as the harmonic property of the quadrilateral, the "cross-join" property of homographic ranges, and the involution theorem of Desargues and Sturm proved by analysis. No hint is given that this last theorem leads to many of the principal properties of the conic. If it is intended that Pure Geometry should be learnt from an independent treatise, then the space devoted to this part of the subject seems to be excessive. Chapter II introduces Trilinear coordinates and equations of the first degree. Results such as the formulae for the distance between two points, for the angle between two lines, and for the length of the perpendicular are proved by transference to Cartesian axes. Some of this work could be made simpler and more attractive by the use of elementary vector theory. Reasons exist for preferring Areals as the primary system, and other Homogeneous Coordinates deserve some mention. Equations of the second degree are considered in the following chapter. ... After the properties of the general equation have been investigated, the equations to various circles and conics specially related to the triangle of reference are obtained; envelope as well as locus equations are given, but their importance is not sufficiently emphasised. It is true that Chapter V is devoted to Tangential Coordinates, but even at this stage the "principle of duality" has not been enunciated, and the treatment of the subject suffers accordingly. There is no need for details of metrical reciprocation, but the student must be made familiar with the idea of duality before he can be expected to acquire a grasp of envelope methods. ... Some attention is paid to parametric representation, but the ordinary method of arriving at parametric equations is not given. ... In the closing chapter there is an account of the invariants of two conics which does not, however, include consideration of Tangentials. As in most text-books, there is no serious attempt to justify the use of complex numbers as coordinates, and the treatment of points at infinity is unsatisfactory. ... The number of examples exceeds 400, and solutions or hints are given for about 100 of them, with answers at the end of the book.

**7.2. Review by: F. P. W.**

*Science Progress in the Twentieth Century (1919-1933)*

**18**(72) (1924), 643-644.

Prof Loney' s elementary textbook on analytical geometry, published in 1895, is a well-known and excellent introduction to the subject and is still widely used. He has now written a sequel which will no doubt prove popular as a good elementary exposition of the old-fashioned kind, with numerous and well-chosen examples. Most of the matter, homographic ranges, trilinear coordinates, tangential coordinates, and so on, is of course to be found in Salmon; the last chapter, on invariants and covariants of systems of conics, contains, however, a good deal, both in the text and in the examples, which is not there. We are glad that Prof Loney has included this chapter; the subject is too often omitted in elementary books. But the treatment is old-fashioned. The whole point of homogeneous coordinates is to develop projective as distinct from metrical properties; the specification of the particular coordinates, trilinear or areal, is of secondary importance, and formulae for lengths of perpendiculars and so on might well have been omitted. So, too, we could wish that the principle of duality had not been introduced as though it necessarily depended upon or had anything to do with reciprocation in regard to a conic.