1.1. From the Introduction.

My object in writing the following pages has been to supply the growing need of mathematical students in this country for a compact textbook giving the theory of Conic Sections on modern lines. During recent years increasing space has been allowed, in University syllabuses and courses of instruction, to the more powerful and general projective methods, as opposed to the more special methods of what is still known as Geometrical Conics.

The line of cleavage between the two has, however, been sharply maintained, with the result that the already much overworked mathematical student has to learn his theory of Conic Sections three times over: (1) analytically; (2) according to Euclidean methods; (3) according to Projective methods.

The difficulty has been to reconcile the Euclidean and Projective definitions of the curve; in fact to bring in the focal properties into Projective Geometry at a sufficiently early stage. The practice has usually been, in order to pass from the projective to the focal definitions, to introduce the theory of involution. But the latter requires for its fullest and clearest treatment the employment of imaginary elements. It seems undesirable that the more fundamental focal properties of the conics, e.g. the sum or difference of the focal distances and the angles made by these with the tangent and normal, should appear to depend upon properties of imaginary points and lines, even though this might introduce greater rapidity of treatment. The University of London has recognised this, for, while admitting Projective Geometry into its syllabuses for the Final Examination for a Pass Degree, it has excluded involution. Many teachers have felt that this exclusion amounted to a rigorous enforcement of the line of cleavage mentioned above. In the present book the difficulty has been met, it is hoped, successfully. Chapters I-VII cover practically the whole field of Geometry of Conics which is required from the average mathematical student who is not reading for Honours. In these chapters no use has been made of involution. In Chapter VI a proof is given that any conic, projectively defined, can be cut from a real right circular cone. The foci are then obtained from the focal spheres and the rest of the focal properties follow.

In Chapters I-VII the only knowledge presupposed is that of Euclid, Books I-VI and Book XI; also enough of Analytical Geometry to understand the use of signs and coordinates and the meaning of the equations of the straight line and circle, so that the data of the drawing examples should be intelligible to the reader. Of plane Trigonometry the meaning of the sine, cosine and tangent and the formula for the area of a triangle (area = $\large\frac{1}{2}\normalsize ab \sin C$) are all that is required.

Instead of basing the treatment of the subject upon harmonic groups, I have introduced cross-ratio from the very beginning. Although fully appreciating the superior elegance of the former method, insomuch as it enables Projective Geometry to be developed without any appeal to metrical properties, I think it is hardly the one best suited to beginners. For this reason I have used metrical methods whenever their use was obviously indicated, although I hope it will be found that the spirit of the projective methods has been adhered to.

The second part of the book is intended for students reading for Honours or desirous of making themselves familiar with the more advanced parts of the subject. It has been impossible in such a short space to do more than bring the reader to the threshold of the rich treasure house of Modern Geometry and to give him a glimpse of some of its more characteristic methods. No attempt has been made to give a complete account of the theorems obtained in this domain: those given have been chosen to illustrate the methods. Nevertheless it is believed that an Honours student will find there most, if not all, of the fundamental results which he ought to know.

The range of knowledge presupposed on the part of the reader is of course, much wider in the later than in the earlier chapters. Thus the whole theory of imaginaries and of homography has been allowed to rest on an analytical basis. This should present no difficulty, for, by the time the reader reaches these chapters, he will almost certainly have acquired sufficient knowledge of Analytical Plane and Solid Geometry to make his progress easy. On the other hand purely geometrical development of imaginaries would have been too long and laborious for inclusion. But will be found that those results which depend on analytical considerations are in every case broad generalisations, such as those relating to the properties of conjugate imaginaries, to the operations which lead to homographic relations and to the number of points in which curves and surfaces of given degree and order intersect. I have carefully abstained from using analysis to prove particular theorems.

With regard to homography the method of one-one correspondence has been made fundamental. It is true that discrimination has to be used in applying the principle, but this may be said of almost any principle: and a student soon gets to know when a one-one correspondence geometrically given is really algebraic. The notion of homographic involutions, which appears to be a powerful instrument, has been introduced in Chapter XI.

Finally Chapters XIII and XIV deal with geometry of space. Many properties of cones of the second order, of sphero-conics, and of quadrics come most easily from purely geometrical considerations: and it seems a pity that the methods of Pure Geometry are not more frequently employed at this stage.

In the preparation of the book the classical treatises of Cremona and Reye and a more recent but very concise and instructive exposition of the subject by M Ernest Duporcq have been chiefly consulted.

I have ventured to make certain changes in the recognised nomenclature. Thus what is called by Mr Russell in his Treatise of Pure Geometry the axis and pole of homography I call the cross-axis and cross-centre, as the name seems to bring more vividly before the mind the fundamental property of the thing defined. For a similar reason I have used the term incident to denote two forms such that the elements of one lie in the corresponding elements of the other. The term perspective, which is employed by German writers in this connexion, appears misleading, since it would not then apply to what are universally known as perspective ranges and pencils.

The examples are taken mostly from exercises set to classes at University College, London, and from College and University Examination papers. For permission to use these my thanks are due to the Principal of the University of London and to the Provost of University College, London. The examples contain also many theorems which it has not been found possible to include in the text. A special feature of those on the first seven chapters is that they are divided into two sets. Those marked (A) are theoretical; those marked (B) are drawing exercises. My own experience as a teacher leads me to believe that such actual drawing is of immense value in assisting beginners to understand the subject, as well as intrinsically useful in practice. Considerable stress has therefore been laid upon drawing-board constructions.

1.2. Review by: John J Milne.
The Mathematical Gazette 5 (81) (1909), 127-128.

Dr Filon's book, though bearing a similar title to Cremona's work, deals with the subject in an entirely different manner. Whilst in the latter the law of duality is always kept before the notice of the reader, in the former it occupies a more subordinate position, and this allows Dr Filon to arrange his book more as a continuous treatise, in which the reader has not to stop and, as it were, study each geometrical truth a second time in different words.

Dr Filon gives us a preliminary chapter on projection, and then in chapter ii. proceeds to the theory of cross-ratio of the range and pencil, and harmonic forms. Considering the importance of this chapter, which is the foundation of the subject and contains within the compass of seventeen pages all that the student is supposed to require before dealing with the conic, it appears too condensed, and several of the examples might with advantage to the beginner have been introduced into the text; e.g. all that we are told about two distinct cobasal projective forms is that they are identical if they have more n than two self-corresponding elements, but we are not given any help towards finding the latter. No doubt the author was eager to push on to the more attractive part of the subject relating to the conic, and said with Newton, "Propero ad magis utilia." Here Dr Filon gives us neat proofs of the anharmonic properties of the points and tangents of a conic, followed by a chapter containing the chief properties of poles and polars, which are proved for the circle and then transferred by projection to the conic. We then have the theorems of Pascal and Brianchon, the former proved directly for the conic by cross-ratio, and the latter by reciprocation, and they are employed in the construction of conics subject to different conditions.

These are followed by some elementary propositions which seem somewhat out of place. Although, as Dr Filon remarks, at present a student has in a way to learn his conics three times over, i.e. by the methods of (1) Apollonius, (2) analysis, and (3) projection; on the other hand, considering the importance of the conic in higher geometry, the wisest plan seems to be to reserve the powerful method of projection for the treatment of those advanced parts of the subject for which the other methods are not so suitable. It would hardly be possible to condense into a score of pages an amount of the theory of geometrical conics which would enable a student fully to understand and digest what he might be expected to meet in such a work as the present. Of course it must not be forgotten that projection is only one of the instruments of higher geometry, as Chasles has shown us in his Traité des Sections Coniques, where, after deriving in the first half-dozen pages the conic from the circle, the method of projection is laid aside, and the subject is developed entirely by means of the theory of cross-ratio.

After again dealing with Pascal's theorem and constructions depending on it, we are introduced to imaginaries and homography, treated for the most part analytically, the principle of one-one correspondence being carefully explained and its limitations pointed out. After most interesting sections on involution of the line and conic, and homographic forms of the second order, we come to a fascinating chapter on systems of conics, in which the article on the construction of the common self-conjugate triangle is particularly interesting, and makes us ask why Dr Filon did not go a step further and give us figures showing the pair of real common chords in the different cases to which he refers, viz. when the two conics have only two real intersections, and again when one conic lies entirely inside the other. And here one cannot help saying that the figures do not do the author justice, and are scarcely up to the level of the subject matter. The work concludes with two chapters on the cone and sphere, and quadrics, in which the reader is brought "to the threshold of the rich treasure house of Modern Geometry, and is given a glimpse of some of its more interesting methods."

2.1. Note.

No reviews found.

3.1. Review by: F P W.
Science Progress in the Twentieth Century (1919-1933) 16 (64) (1922), 666-667.

The third edition of Professor Filon's well-known textbook has been seen through the Press by Mr T L Wren, Reader in Geometry in the University of London, and does not differ materially from previous editions. A new chapter has been added on Point-Reciprocation, which is the special case of polar reciprocation with regard to a conic arising when the conic is a circle. Its importance is due to the fact that by it metrical properties may be obtained; also it is considered that the student is most easily familiarised with the general notion of reciprocation by examining this particular case in some detail. Another novelty is introduced in chapter ii, where Menelaus' and Ceva's Theorems are deduced from the proposition that the continued product of ratios of segments of the sides of a triangle is unaltered by projection.

A useful set of Miscellaneous Examples has been added at the end.

In comparison with such a book as Enriques' Geometria Proiettiva the general treatment may seem lacking in breadth, and the book rather overburdened with detail; the author has in mind, of course, the elementary student, and is somewhat circumscribed by the peculiarities of the syllabuses of the University of London, by which involution is postponed to a relatively late stage; but his book remains one of the best introductions to the subject that there is in English, and may be safely recommended to the serious student.

4.1. From the Preface.

During the quarter of a century which has elapsed since the first edition of this book was published, Projective Geometry has found new practical applications. In particular, the uses of photography in Air Surveying, as well as in Astronomy, involve the principles and methods of projection; and it appears probable that, in both cases, the advantages of graphical constructions will be increasingly appreciated. Meanwhile the older applications to Cartography, Geometrical Optics and Engineering Drawing have lost nothing of their importance.

No apology is therefore needed for the insistence on drawing-board constructions, which was a feature of the earlier editions. Indeed this has been emphasised, in the present edition, by the addition, at the end of all the later chapters dealing with the geometry of the plane, of a set of drawing examples marked B, which had previously been restricted to the first seven chapters. From the purely didactic standpoint, actual drawing is even more valuable to clear up difficulties in the more advanced work than it is in the elementary parts of the subject.

It still remains true, however, that the chief interest of projective methods is for the pure mathematician, for whom they provide an instrument of remarkable range and power.

The general scheme of the original edition has remained, save in one important respect, substantially unaltered. In particular I have not modified the lines on which the subject is introduced in Chapter I, though I have tried to remove certain obscurities and have kept the graphical constructions concentrated towards the end of the chapter, so that they may be omitted by those who attach no importance to such constructions. I am aware that this will not satisfy certain critics, but I could not have met their objections without abandoning a conception of the genesis of the subject which I still believe to be the right one.

The chief alteration involving the geometry of the plane has been a rearrangement of order, which brings in Involution before, instead of after, the discussion of foci and focal properties of the conic.

This change was always desirable, for the introduction of foci by means of the focal spheres was never really the natural approach and had the defect of masking the true significance of foci from the projective point of view. The ban on the early introduction of Involution, which used to be imposed by certain University syllabuses, has now been generally abandoned, and the treatment of the whole subject gains thereby in clearness and coherence.

The above modification of plan has necessitated a good many consequential alterations. Chapter VI now deals with ranges and pencils of the second order and self-corresponding elements, and this naturally leads to a discussion of Involution in Chapter VII, followed by the focal properties of the conic in Chapter VIII.

Up to this point the whole treatment, although capable of interpretation in a wider sense, is based upon real elements and constructions actually possible on the drawing-board, as in my view this is essential to give confidence to the beginner.

Chapter IX then introduces imaginary elements and the circular points at infinity. For this, appeal is made, as in the original edition, to algebraic considerations. Although, in strictness, such considerations are outside pure geometry, they are found, in practice, sufficiently convincing to the student, they avoid the usually long-winded arguments based upon a purely geometrical theory of imaginary elements, and it would seem pedantic, at this stage, not to use them. For the same reason I have not hesitated to employ such considerations whenever they lead, as in the treatment of homographic fields or the intersections of loci of various degrees, to general principles most obviously and directly. But I have tried consistently to preserve a geometrical spirit throughout, so far as possible.

Chapters X and XI are devoted to a discussion of homography and reciprocation respectively, in relation to plane fields. In Chapter XI an investigation of Inversion has been added; this is a new feature: although Inversion is not really included under projective methods, it is closely allied to them and usually associated with them in University syllabuses. It is also important to make the student aware of the fact that all one-one point transformations are not necessarily homographic.

Chapters XII-XIV follow the same lines as Chapters XI-XIII in the first edition. The discussion of quadrics, originally limited to one chapter, has appeared inadequate, even at this elementary stage, and has been expanded, so that two chapters, XV and XVI, have now been given to it.

Apart, however, from alterations of order, a large number of improvements and additions have suggested themselves in the course of revision. Among these may be mentioned a new treatment of the circle of curvature in Chapter V, based upon the perspective transformation, in Chapter Ill, of conics having three-point and four-point contact, and some elementary results on curvature of twisted curves and of quadrics in Chapters XV and XVI; the harmonic envelope and locus of two conics as an illustration of homographic involutions in Chapter XII; an introduction to the general plane cubic and quartic obtained from pencils of conics in Chapter XIII; the focus and directrix property of the sphere-conic in Chapter XIV; a three-dimensional analogue to the complete quadrilateral and quadrangle, and brief discussions of (i) homographic spaces in three dimensions, (ii) inpolar and outpolar quadrics, in Chapters XV and XVI.

Indeed, very few chapters have survived without drastic alteration, and many have been practically rewritten.

A number of new examples have been added; not only have new sets of drawing examples been inserted at the end of Chapters VII, IX-XIII, but a new departure has been to distribute many examples in the text of the chapters, where they serve as illustrations to the articles to which they are appended. In this way the text provides a clue to the solution; conversely the examples help towards the immediate understanding and elaboration of the text. It will be found that the loss of such examples from the sets at the end of the chapters has generally been more than made good, so that in fact the tot al number of examples in the book has been increased from 406 to 893.

Something may be said about the notation. On the whole, experience shows that the notation employed in the earlier editions has proved workable. Certain improvements in nomenclature, however, have been adopted in the present volume. Thus elements not at infinity have been described shortly as "accessible." "Axis of collineation" has been discarded in favour of the now more usual "axis of perspective." The cumbrous terms "harmonically circumscribed to" and "harmonically inscribed in" have been replaced by "outpolar" and "inpolar." The notion of "field" has been used, in preference to that of "figure," in dealing with general transformations. The use of the term "base" has been generally applied to those elements connected with a geometric form which remain constant; thus a flat pencil has two bases, its vertex and its plane, and the word "cobasal" implies that both these bases are the same. In like manner the quadrangle to which a pencil of conics are circumscribed is referred to as the base of the pencil. I have retained the term "equi-anharmonic" to signify forms such that two corresponding sets of four elements have the same cross-ratio; a modern school of thought uses this term to denote a set of four elements such that they are projective with themselves, when any three of them are interchanged cyclically; but a word is required for equi-anharmonic in the old sense, apart from "projective" or "homographic" which, although ultimately equivalent, proceed originally from a different concept.

It must be admitted that, in many respects, the accepted nomenclature of the subject has not always been happy. The word" sheaf," used in the older books for a set of lines and planes passing through a point, does not really convey to the mind a picture of what is intended, and, indeed, would be more appropriately applied to what is known as a regulus. I have adopted the word "star" instead of "sheaf," following a practice which is gradually being introduced. The term "axial pencil" also seems to me unfortunate, and, in fact, in the geometry of the "star," where flat and axial pencil correspond to range and flat pencil respectively in the plane, actually misleading. A new word is needed for a form consisting of planes, e.g. some such word as "fold"; were "fold" used to describe an axial pencil, a "fold" of the second order would denote the set of tangent planes to a cone of the second order, a form for which there is at present no satisfactory short word; conical pencil must obviously denote a cone of lines and corresponds to a range of second order; the cone of planes corresponds to a pencil of second order, but the word "pencil" cannot be used again. Another advantage of the introduction of the word "fold" would be that (leaving systems of conics out of account), a range would always consist of points, a pencil always of straight lines, and a "fold" always of planes. "Axial pencil" is, however, so well entrenched in current practice that I have not ventured to displace it, and I have introduced the word "wrap" to describe, when necessary, the set of tangent planes to a cone.

The term "self-polar," when applied to quadrangles and quadrilaterals, has been changed to "polar"; the nomenclature of the earlier editions appeared unsatisfactory, since such quadrangles and quadrilaterals are not polar figures of themselves. Corresponding changes have been made when dealing with the star and with three-dimensional geometry; also, following Reye, a distinction has been drawn between "polar" and "conjugate" lines with respect to a quadric; in the previous editions the two terms had been used as synonymous.

I have retained the words "pencil of conics (or quadrics)" and "range of conics (or quadrics)," although the latter hardly satisfies me as a description. These terms are by now well-established, and the alternatives would be: either to introduce entirely new words, such as "loop" for "pencil" (suggesting a number of paths through fixed points), and "slide" for "range" (suggesting a deformable curve sliding on fixed guides); or to employ the words "net" and "web," which I have used for linear systems of any grade, to mean, when not accompanied by any qualification, the net and web of the first grade, instead of, as now, those of the second grade. On the whole, however, it seemed that continual changes of notation were to be deprecated. It will be noticed that the word "web" is still used to denote a tangential system, and is correlative to "net." I have not followed a practice sometimes adopted, of using "web" to denote a net of the third grade.

4.2. Review by: Walter Buckingham Carver.
The American Mathematical Monthly 44 (8) (1937), 534-536.

In the text under review there is no one chapter, and indeed very few sections, in which one does not find mention of such things as distance, angle, perpendicularity, parallelism, circle, centre, diameter, foci, radius of curvature, cylinder, etc. In other words, there is no part of the text which treats pure projective geometry as distinct from metric specialisations of projective geometry. And the reviewer concurs in the opinion that from such a treatment the reader is unlikely to gain a clear impression of what the word projective implies. Another and more serious objection to this kind of presentation of projective geometry is that it makes for logical confusion and leaves one doubtful as to whether or not anything has been proved. Euclidean geometry and projective geometry are quite different logical structures. An axiom or valid theorem in one may be false in the other. The statement that any two distinct lines in a plane have one and only one point in common is true in projective but not in euclidean geometry; while the statement that through any three distinct points not on a line there is one and only one circle is true in euclidean geometry but is either false or meaningless in projective geometry. When logical use is made of both kinds of statements the argument is, to say the least, not clear. ...

It creates a particularly bothersome situation when the author comes to a statement of his "principle of duality." We quote:

It follows from the transformation by reciprocal polars that to every (the italics are the reviewer's) theorem concerning a figure made up of points and lines there corresponds another theorem concerning a corresponding figure made up of lines and points respectively, so that geometrical theorems appear in pairs. ... It should be noticed, however, ... , that properties of length and angular magnitude (which are termed metrical properties) do not generally reciprocate into like properties. It will be found that the properties to which the principle of duality can be applied successfully are the projective properties.

If the author had previously drawn a clear distinction between projective geometry and other geometries, it would be quite simple to say that to every theorem of projective geometry there corresponds a dual theorem; and one could be sure of applying this principle of duality "successfully" in all cases.

4.3. Review by: Patrick Du Val.
The Mathematical Gazette 20 (237) (1936), 66-67.

We have in the plane a special line, the line infinity; and on this line two special (imaginary) points, the circular points at infinity. A geometrical theorem has either no relation to the special line and points, and it is then descriptive; or it has a relation to them, and it is then metrical.

It is notorious that many timid students have been prevented by the terror inspired by this sentence from ever reading further in the great classic of which it is the somewhat abrupt opening. Those who embark upon Professor Filon's work need fear no such shock; their danger indeed is quite of a contrary kind, less disturbing though perhaps more grave - that, namely, of never discovering at all what the difference is between a metrical and a descriptive theorem. It is indeed never made clear at any stage what Projective Geometry is; an unfortunate omission, if only in view of the title of the book. This curious vagueness is apparent from quite near the beginning, where the metrical definition of a cross-ratio is given, and it is then pointed out, as a rather lucky phenomenon, that this quantity is unaltered by projection; and startles one especially later, where the principle of duality is made to appear as a consequence of the possibility of reciprocating with regard to a conic, i.e., of the properties of poles and polars. All very well; the book is elementary, and this is the familiar elementary approach - yes, but the author is not content to be elementary in the results he reaches. He tells the student, in one way or another, almost all there is to know about conics, as well as an enormous lot about quadrics, metrical properties and all; he even manages to include formulae about radii of curvature and torsion, in a chapter on "Projective methods in three dimensions". He is in fact concerned with giving his reader the projective method (or a method involving as great a proportion of projective machinery as possible) of solving almost any problem that can be set him, rather than with teaching him projective geometry; he has given us a textbook for the training of wranglers, rather than of mathematicians. For this purpose, the book is probably as good as it could be; the omission of all talk about axioms of incidence, continuity, the algebra of casts, and all the fundamentals of projective geometry, while it may damn the student's soul, will doubtless win his heart; and the projective proofs given are on the whole very much more readable than those in the more theologically impeccable classics, such as Reye, or the newer work of Juel.

4.4. Review by: D W B.
Science Progress (1933-) 31 (121) (1936), 158.

Prof Filon' s book was first published twenty-seven years ago and the issue of a new edition is a tribute to its continued popularity. No great alterations have been made in order and content. A chapter on inversion in the plane has been added to illustrate transformations which are not projective. Involution has rightly been given an earlier place in the development of the subject, but, contrary to modern usage, imaginary elements, and in particular the circular points at infinity, are not mentioned until after the chapters on foci and focal properties of the conic. The reason for this is the author's insistence on the value of graphical construction to the beginner. In the earlier part of the book little is said that cannot be reproduced on the drawing board, and many graphical exercises are given: the author has anticipated criticism of this approach to the subject by printing these exercises separately from the rest, so that they can be omitted by those who prefer to do so.

We think it is a fair criticism to say that the author's subject is the projective treatment of Euclidean geometry rather than projective geometry in its generality; in this connection it is significant that although coordinates are introduced no mention is made of homogeneous coordinates. From a formal point of view we should prefer to begin the book with a more general discussion in which homography and involution could take their place - they come now as late as Chapters X and VII respectively - and then to show how a metric can be introduced with the help of absolute elements. This is perhaps a counsel of perfection and one more likely to appeal to the advanced geometer than to the beginner.

The last quarter of the book deals with three-dimensional geometry and includes some elementary differential geometry of curves and surfaces.

Anyone who works through this book carefully - and there are nearly 900 examples on which he can test his knowledge - can hardly fail to feel the fascination of his subject and to be stimulated to further reading.

4.5. Review by: Anon.
Nature 137 (1936), 297.

The new edition of this valuable work differs so widely from the older ones, with many changes and additions, and more than twice the original number of examples, that it is almost a new book. Noteworthy features are the treatment of inversion, the circle of curvature, three-point and four-point contact, the harmonic envelope and locus of two conics, the plane cubic and quartic, and the focus-directrix property of the sphero-conic. The three-dimensional portions include homographic spaces, inpolar and outpolar quadrics, analogues of the complete quadrilateral and quadrangle, and the curvature of quadrics and twisted curves.

5.1. Review by: R V South.
The Mathematical Gazette 16 (220) (1932), 277-279.

From their published papers one conjectures that Professor Filon's special interest has been in the underlying physics of photo-elastic phenomena, Professor Coker's in their application to practical problems; but in this book the dovetailing is so neatly done that guessing becomes a hazardous occupation. Chapter I gives an account of physical optics, based on the classical electromagnetic theory of light; Chapter II contains what is needed of the theory of elasticity for an understanding of photo-elastic applications; Chapter III is an historical summary of knowledge relating to the principles of photo-elasticity; in the remaining chapters (IV-VIII) the authors discuss various problems of engineering practice which they have had occasion to study by photo-elastic methods.

The book contains 720 pages, hundreds of line illustrations, sixteen plates in colour; the quality of its printing is what we have come to expect in the well-known "dark blue" series of the Cambridge University Press; it is written by men who have made its subject peculiarly their own. Of such a work, and more especially when the reviewer has found no leisure for adequate study, criticism is presumptuous, complaint ungrateful; all that is known of photo-elasticity is contained within its pages, and a reader can ask no more. But a scientific treatise must be judged in relation to the needs of those who will study it; and this reviewer confesses that he has found it difficult to decide what is the class of readers which the authors have had specially in mind. There is of course, first of all, the reader who is interested in everything that pertains to photo-elasticity, whether in theory or in practical application; who will study systematically both the fundamental theory of optics and of elasticity and the engineering problems treated in the concluding chapter. His needs are fully met, and he will find little to criticize beyond the weight of the volume which he has to hold: surely two volumes were indicated, both by the bulk and arrangement of the subject-matter.

But readers of this kind, one would suppose, are few in comparison with the many who are interested in photo-elasticity as a tool for research, and whose desire is to acquire such knowledge of its technique as will make them able to set about experiments, should occasion arise. Such readers will need some account of the undulatory theory of light, for they must have some mental picture of the phenomena of polarisation; but it may be questioned whether, in order to acquire that picture, they should be asked to follow in detail the highly mathematical investigations which have given us a more or less complete theory of optics. Their interest is in phenomena-above all, in the birefringent property of strained glass or xylonite; if theory can account for those phenomena, well and good: if not, so much (from their standpoint) the worse for theory. And when they read that mathematical conclusions, even in regard to the phenomena of polarisation, are sometimes contradicted by experiment, will they not slacken in their resolve to understand photo-elasticity at such cost in mental effort ?

Lastly there is the student to be considered, and the research worker in engineering science: for them, at a first reading, clear-cut statement is essential, devoid of scientific niceties. Thus we are led to a constructive suggestion, offered here in the conviction that the authors, by adopting it, could add greatly to the usefulness of their work, and without much cost in additional labour. What now is wanted, to supplement the treatise which lies before us, is a short manual for the laboratory, severely practical, which in itself would enable the student to set up his own apparatus, but at the same time, by copious references to the larger work, would stimulate him to closer study of the phenomena which he employs. Whether by the authors or by some other, this manual will be written, for it is sorely needed; if it is left to others, it may be badly done. So, though the authors may fairly claim to have given us already more than we had any right to expect, we can only hope that they, who can draw from such unequalled store of knowledge, may find the time to put us still further.

5.2. Review by: J T W.
Journal of the Royal Society of Arts 80 (4169) (1932), 1081-1082.

The authors of this book have, for the past twenty years or more, been universally recognised as the pioneers and the chief authorities in the application of photo-elasticity to the solution of mechanical problems. It will therefore be a cause for general satisfaction that they have found the time to prepare a work of such completeness as that which has just appeared. It is true that the reader, when opening a book of this size (it weighs nearly 5 lbs) may be excused some surprise when he reads in the opening lines of the preface that "this account of photo-elastic science arose from the need felt by us both for a handbook, which would give, in a convenient compass, not only the essential mathematical and physical data found necessary in our investigations in this branch of science, but also a connected account of the principal technical developments" (the italics are the reviewer's). On reading further, however, he is bound to rejoice that the authors "were ultimately compelled to diverge from their primary object in favour of a larger and more comprehensive work, in the endeavour to cover the field in an adequate manner."
...
There can be no doubt that this book will be for a long time the classic on the subject, and it is clearly part of the essential equipment of anyone who has to study the behaviour of engineering materials and structures under strain.

6.1. Review by: Peter P Benham.
Science Progress (1933-) 46 (182) (1958), 365.

This is the second edition of a work first published twenty-six years ago. The reviews accorded to it at that time, some of which are reproduced on the dust cover of the present edition, overwhelmingly acclaim its timeliness and academic merit. Since the text remains unaltered except for the correction of two minor errors and misprints, the latter quality still holds good. With regard to the former, that of timeliness, this may now be only of concern to someone wishing to obtain a work that has been out of print for some time. The question remains, however, whether this treatise is now obsolete due to the advances that have been made in photo-elasticity during the last two decades. It should be made clear at this point that Coker and Filon's colleague, Col. H T Jessop, has contributed to this edition a preface, a discussion of the developments that have occurred since the first edition and a most useful bibliography of the more recent important contributions to the field.

It may well be considered that this is not the type of book which could or should be brought up to date. Although some of the practical techniques have now been discarded it is essentially a manual on the fundamental principles and engineering applications of the science of photo-elasticity and its complement, the theory of elasticity, and as such forms the corner stone of the literature on the subject. It is of equal value and importance to the university graduate or the industrial research laboratory and at a time when experimental stress analysis plays such a great and essential part in engineering it is gratifying that this work is once more made available.

6.2. Review by: B Sugarman.
Physics Bulletin 8 (12) (1957), 396.

The reappearance of the comprehensive volume on photo-elasticity by Coker and Filon as a second edition will surely be widely acclaimed not only by current workers in the field but also by the many industrial laboratories exploring the method. As the reviser comments "Nothing, in fact, has been published which could replace Coker and Filon as the most comprehensive and authoritative work on the theory of photo-elasticity, and on the principles of its application to stress-analysis."

6.3. Review by: Donald A Wiegand.
Physics Today 11 (10) (1958), 38-39.

A republication of this excellent book which first appeared in 1931 can be justified on several grounds. In the first place the theory necessary to fully understand photo-elastic phenomena is thoroughly and concisely developed; second, a wealth of information concerning technique is presented. As an added attraction, the book contains considerable material of historical interest in the field of optics, elasticity, as well as photo-elasticity. The editor has, however, made changes pertaining only to errors in the first edition. While this has resulted in the return to print of a familiar volume, it is the opinion of the reviewer that several items could have been brought up to date with little modification to the original text. In the introduction, H T Jessop does provide a short review of photo-elasticity since the first edition and gives references to important works during this period.

7.1. From the Preface.

The present little book is not intended to supersede in any way its larger predecessor, the Treatise on Photo-Elasticity by Professor Coker and myself. It has a more limited and modest aim, namely to give, to a practical investigator who wishes to use photo-elastic methods in order to explore stress-distributions occurring in any problem in which he is interested, a brief account sufficiently complete and explicit to enable him to set up his apparatus and to use it in the best possible manner.

Such an account is perhaps not very readily extracted from the larger volume of some 700 pages, in which the information needed is inevitably rather scattered, and a number of enquiries have convinced both Professor Coker and myself of the desirability of writing a short manual for workers in this field.

The book, as its title implies, is intended for engineers; those who are desirous of studying the subject from the point of view of the physicist will find that side more fully developed in Chapter III of the Treatise; only enough of it is dealt with here to allow the reader to understand the phenomena and to guard against important sources of error.

To use any scientific method successfully one must grasp the fundamental principles involved; some explanation of the properties of polarised light which are relevant to the applications in view has therefore been given in Chapter I, for the benefit of those engineers whose memories of Physical Optics may need refreshing; those to whom that subject is familiar may well skip most of this chapter. On the other hand, the treatment of the theory of stress and strain, with which engineers are well acquainted, has been less elaborate.

Generally speaking, the mathematics have been reduced to a minimum and I have been content to quote a number of results of which the proofs are complicated and difficult. Those curious of such things will find the proofs in question in the Treatise. Such proofs are in no way essential to the intelligent use of the practical methods and would have made the present volume unnecessarily long and tedious.

In like manner the discussion of the very numerous engineering problems to which photo-elastic methods have been, or can be, applied, a discussion which bulks very largely in the longer work, has been entirely omitted, My object was not to produce evidence of the value of photo-elastic observations of stress by showing what has been accomplished by such means and how it accords with Elastic Theory. I have assumed that, by now, the value of such methods is generally recognised. Here, also, the reader who desires further information is referred to the Treatise.

Similarly all references to sources, original papers, etc., have been left out. They would, owing to the restricted scope of the present work, have been incomplete from the nature of the case; and there seemed no point in duplicating the bibliography and ample references of the Treatise, which are available if required.

By kind permission of the Syndics of the University Press, six of the illustrations have been reproduced from the Treatise and the descriptions of certain instruments have been borrowed largely from its text.

It must not be supposed, however, that the following pages are merely an abridgement of the previous work. They contain a considerable amount of new matter, in particular discussions of certain errors and adjustments (for example of the circular polariscope) and of methods of observation such as those dealing with frameworks, and with the thermal and membrane measurements of P+Q, also descriptions of instruments like the projection apparatus, the portable polariscope and the triple-reflection polariser in Chapter V.

My chief regret is that my old friend and colleague, Professor E G Coker, who collaborated with me for many years in the preparation of the larger Treatise, has unfortunately been prevented by illness from continuing his collaboration in this case. Indeed his great authority as one of the most distinguished living scientific engineers, no less than his unique position in respect of this particular subject, indicated him clearly as the one of us two who should have been responsible for giving this short account of photo-elastic practice to his fellow-engineers. I can only plead that I have taken his place by necessity and that, although by training a mathematician and physicist, yet, by long and close contact with engineers and their work, I have tried to attain at least some sympathetic understanding of their problems, which I hope will help to make the present effort acceptable to engineering readers.

7.2. Review by: J C.
Science Progress (1933-) 31 (124) (1937), 751.

According to the preface, this little book is intended "to give, to a practical investigator who wishes to use photo-elastic methods in order to explore stress-distributions occurring in any problem in which he is interested, a brief account sufficiently complete and explicit to enable him to set up his apparatus and to use it in the best possible manner."

The value of the methods of photo-elasticity for instructional purposes is, perhaps, not sufficiently appreciated in this country, and, for this reason alone, a cheap book giving a clear exposition of the theoretical basis of the method is particularly welcome. Prof Filon's book certainly fulfils that function; all the necessary theory is there, whilst there is nothing superfluous for a proper understanding of the optical and elastic principles involved. But it seems questionable whether, with only this book to guide him, a research worker or teacher, using the method for the first time, and with no previous knowledge of the apparatus, would be able "to set up his apparatus and to use it in the best possible manner." The gulf between this little book and the large book by Profs Filon and Coker seems too great; we should have liked to see more precise laboratory instructions, the methods of drawing the isoclinics and lines of principal stress, and the calculation of the stresses worked out from the start for at least one example. If one or two illustrations of the practical value of the method had been included, the book would have had more propaganda value, it would have done more to encourage the hesitant chief designer or professor of mechanics to install the apparatus; such encouragement is necessary, for, at least in the opinion of the reviewer, a far more widespread use of the method of photo-elasticity is highly desirable, particularly as a medium of instruction.