1.1. The title page.

The title page gives the authors as L Wayland Dowling Ph.D., Associate Professor of Mathematics, University of Wisconsin, and F E Turneaure C.E., Dean of the College of Engineering, University of Wisconsin. It gives the publisher as Henry Holt and Company, New York.

1.2. A 1914 advertisement.

On page 87 in The Mathematics Teacher 7 (2) (December, 1914), the book Analytical Geometry is advertised as follows:

Because of its fundamental importance, the concept of functional correspondence and the method of representing such correspondence geometrically has been introduced early in the course. The standard forms of equations of important loci are also developed early, and the properties of these loci discussed later by means of the equations already at hand.

1.3. From the Preface.

In accordance with the general plan of this series of textbooks, the authors of the present volume have had constantly in mind the needs of the student who takes his mathematics primarily with a view to its applications as well as the needs of the student who pursues mathematics as an clement of his education.

The processes of analytical geometry find their application, for the most part, in the scientific laboratory where it is often necessary to study the properties of a function from certain observed values. The fundamental concept is, therefore, that of functional correspondence and the methods of representing such correspondence geometrically. For this reason rather more than usual attention has been given to these subjects.

An intelligent appreciation of functional correspondence requires an intimate knowledge of the relation between an equation and the graphical representation of the functional correspondence determined by the equation. Such a knowledge is most easily obtained by a study of linear equations and equations of the second degree together with their corresponding loci. This knowledge is not only of importance to the student of applied mathematics, but it has a special disciplinary value for the general student.

The standard forms of the equations of a number of important loci are developed early (Chapter IV), and the properties of these loci are discussed in detail later (Chapters VI and VII) by means of the equations already at hand. By this arrangement, it is hoped that some unnecessary repetition has been avoided.

The equations of tangents to the conic sections have been derived by means of the discriminant of the quadratic equation whose roots are the x-coordinates of the points of intersection with a variable secant, rather than by means of the derivative. This course has been adopted, first, because the geometric interpretation of the discriminant is important in itself; and, second, because the use of the derivative ought, logically, to be preceded by a chapter devoted to its definition and the methods for finding it, at least for algebraic functions. Moreover, the use of the derivative for finding the equations of tangents is only one of its many applications. No student should feel that his mathematical education is complete without a knowledge of the calculus, where he will become familiar with the derivative and can appreciate its usefulness in many directions.

The present volume is designed for a four-hour, or a five-hour, course for one semester, but may be shortened to a three-hour course by omitting certain parts of the text. ... The chapters on solid analytic geometry have been added for the benefit of those students who have time only for an outline of the subject matter. No apology is therefore offered for the meagre treatment. The authors desire to express their appreciation to their colleagues of the University of Wisconsin and of the University of Illinois for the assistance and the many helpful suggestions given them during the preparation of the book.

L W Dowling, F E Turneaure.

University of Wisconsin, July, 1914.

1.4. Review by: E J Moulton.
The American Mathematical Monthly 22 (3) (1915), 93-95.

A chief feature of several recent texts on analytics is the emphasis of the general idea of a function and its graph rather than of the theory of conics. This text retains that emphasis to a large extent. After 25 pages devoted to chapters on "Systems of Coordinates" (I) and "Directed Segments and Areas of Plane Figures" (II), we find 65 pages of discussion of "Functions and their Graphic Representation " (III), " Loci and their Equations " (IV), and " Equations, and their Loci" (V), the last chapter including "Transformation of Coordinates." In Chapter III first methods of graphing functions are given with illustrations from algebraic and transcendental functions in rectangular and polar coordinates. The equation of a locus is defined in Chapter IV and the standard equations of straight lines, the conics and Cassinian ovals are derived for both coordinates systems. Chapter V gives methods of discussing an equation with numerous examples. It is not until Chapter VI, "Loci of First Order," beginning on page 98, that we find a systematic treatment of the straight line. After what has preceded, 10 pages suffices. Then in Chapter VII, "Loci of Second Order, Equations in Standard Form," we find, compassed in 34 pages, a fairly complete elementary treatment of the conics, including "Poles and Polars" and "Systems of Conics." The next chapter treats the general equation of second degree.

In accord with another modern development, we find (included in Chapter IX, "Loci of Higher Order and Other Loci") a twelve page discussion of "Empirical Equations and their Loci." This subject, on account of its importance in applications of mathematics to the sciences, seems destined to become an essential part of a good course in analytics. The presentation of the authors, which includes the use of logarithmic coordinate paper, is excellent.

Following these chapters on plane analytics is a brief treatment (about 50pages) of solid analytics. There is little of novelty in this part of the book.

The book as a whole impresses one very favourably. The general order of presentation is excellently adapted to give the student a real appreciation of the power and beauty of analytic geometry, and also the ability to use it. The tendency to lay a little more stress than usual on the geometrical aspect of the subject, especially in the chapter on "Loci and their Equations," by numerous figures and by various methods of constructing some of the loci, will be welcomed by many teachers.

There are several matters of detail, however, which one may criticise. Exceptional cases of various sorts are, for example, quite generally ignored. Thus in discussing the two-point form of the equation of a straight line, the exceptional cases of lines parallel to either coordinate axis receive no mention. In giving the slope forms, no mention is made of lines parallel to they-axis. And all lines through the origin are ignored in connection with the intercept and normal forms. The determinant form of the equation of a plane through three points is said to be linear without mentioning the exceptional case arising when the points are collinear. Likewise no account is taken of exceptional cases in the discussion of pencils of lines or of conics.

While the general idea of Chapter IV, "Loci and their Equations," appeals to me, the logic seems unsatisfactory in two respects. By definition, "The equation of the locus of a point is an equation in the variables $x$ and $y$ which is satisfied by the coordinates of every point on the locus; and conversely...The "and conversely" is subsequently neglected without comment in deriving equations except in the case of the circle. This, of course, invalidates the proof, for instance, that the locus of every equation of first degree is a straight line. The proof of the "and conversely" for a straight line is as difficult as the direct, and the omission seems hardly excusable. The second criticism is on the expression, "the equation of the locus . . ." without comment on the first "the." There is an infinite number of equations satisfying the prescribed conditions in general, and some remark is apparently necessary before one of them may be designated as "the equation." For example, the following are all equations of the same real locus:

          $x^{2} + y^{2} = 4;  x^{4} + 2x^{2y^{2}}+ y^{4} = 16;  9x^{2} + 9y^{2} = 36;  x^{4}+ x^{2}y^{2} + 4y^{4} = 16.$

At least a footnote of explanation seems desirable.

It is further noted that the proof of the formula for the distance from a line to a point (p. 104) is invalid for some cases (e. g., for Fig. 66, p. 105).

As a minor point for criticism, one is surprised to find that the formula for the distance between two points is not given for rectangular cartesian coordinates as distinct from oblique coordinates. Also no reason for deriving formulas for the mid-point of a line segment by a method different from that required to find the point of division in a given ratio, r, is apparent since the former is a simple corollary of the latter. And although the latter is given in the usual manner, it is open to a certain criticism. A point $P$ is found on $P_{1}P_{2}$ such that $P_{1}P/PP_{2} = r$; its coordinates are

          $x = \Large\frac{1}{1+r}\normalsize (x_{1} + rx_{2}); y = \Large\frac{1}{1+r}\normalsize (y_{1} + ry_{2})$.

There is such a point for all values of $r$ except $r = -1$; and by taking all values of $r$ we get all points on the line $P_{1}P_{2}$ except $P_{2}$. These exceptional cases are avoided and we get simpler formulas, which are more easily derived, if we determine $P$ such that $P_{1}P/P_{1}P_{2} = r$ as follows; namely

          $x = x_{1} + r.\Delta x = x_{1} + r(x_{2} - x_{1})$,
          $y = y_{1} + r.\Delta y = y_{1} + r(y_{2} - y_{1})$.

To sum up, although marred by several inaccuracies in detail, the book as a whole is very good, thoroughly modern, and includes much in a small compass.

1.5. Analytic Geometry Hardcover (27 October 2022), by F E Turneaure and L Wayland Dowling.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilisation as we know it.

This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.

Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

2.1. Title Page.

The title page gives the author as L Wayland Dowling Ph.D., Associate Professor of Mathematics, University of Wisconsin. The book we consulted is the "First Edition, Fourth Impression." The publisher is the "McGraw-Hill Book Company, Inc., New York: 370 Seventh Avenue; London: 6 & 8 Bouverie St, E.C. 4." It is dated 1917.

2.2. From the Preface.

The present volume embodies a course of lectures on Projective Geometry given by the author for a number of years at the University of Wisconsin. The synthetic point of view was chosen primarily to develop the power of visualisation and of pure geometric analysis for young men and women preparing to teach geometry in our secondary schools. Such a course should naturally avoid a review of the subject matter of Elementary Geometry and, at the same time, should not be so far removed from familiar concepts as to lose connection with them. ln the second place, the synthetic treatment of loci of the second order and of the second class opens up a new field to the student familiar with analytical processes and bas certain advantages in arousing his enthusiasm for continued work in mathematics.

No especial preparation beyond Elementary Geometry and a slight knowledge of Trigonometry is required in order to read this book with perfect understanding. The reader who knows his Analytic Geometry will often find himself on familiar ground, but no knowledge beyond the use of coordinates is assumed.

The book is frankly patterned after Reye's Geometrie der Lage, with the feeling that the general method of treatment adopted by Professor Reye best serves the purposes outlined above. On the other hand, the author has not failed to consult and profit by other texts on Projective Geometry that occupy important places in recent literature; notably, Veblen and Young, Projective Geometry; Enriques, Geometria Proiettiva: Severi, Complementi di Geometria Proeittiva.

No attempt has been made to set forth a necessary and sufficient set of postulates for Projective Geometry; not that the author fails to recognise the importance of research already completed in this field, but because of the conviction that the student is unfitted to appreciate work of this character until he has assimilated the main body or theorems and their applications based upon concepts familiar to him from the study of Elementary Geometry. This, too, is in accord with the aims set forth above. The existence of ideal elements must be assumed; and the Dedekind postulate, or an equivalent, must be used in order to arrive at continuously projective forms. The treatment of the Dedekind postulate for this purpose is confessedly meagre, and many teachers may feel the need of expanding it, or indeed of restating it, as occasion seems to demand.

No attempt has been made to introduce new or strange terms, the only exception, so far as the author is aware, is the use of the word "confocal" to indicate those elements of a double polarity which are the supports of coinciding involutions of conjugate elements.

While this book has grown out of lectures given to students preparing to teach geometry, the subject matter is by no means of interest to this class of students alone. The engineer and the artisan must of necessity become familiar with the elementary processes of projection and section, and these processes are the same whether they lead to properties of geometrical figures or to methods in mechanical drawing.

L Wayland Dowling
University of Wisconsin
June 1917.

2.3. Review by: F W Owens.
Bulletin of the American Mathematical Society 26 (1) (1919), 39-40.

This book (as the preface tells us) has been developed from a course of lectures given by the author for a number of years at the University of Wisconsin.

In its treatment of the subject it follows closely the main lines of the classical text of Reye in nomenclature and in point of view. It begins with an introductory chapter giving definitions of the elements, projection, section, and ideal elements. The point range, sheaf of lines, sheaf of planes, field of points, field of lines, bundle of lines, bundle of planes, space of points, space of planes, special linear complex, and space of lines are named as the eleven primitive forms. No formal definition of primitive form is given which would enable the reader to judge why exactly these eleven forms are given and others are excluded.

The concept of motion of a line about a point and that of parallelism are made use of in defining an ideal point, and this is followed by the "fundamental assumption: On every straight line there is one and only one ideal or infinitely distant point. This point makes the line continuous from any one point on it to any other point on it in either direction. Through a given point there can be drawn one and only one line parallel to a given line. This parallel intersects the given line in the ideal or infinitely distant point."

This quotation perhaps shows the point of view of the author in basing his projective geometry on a completely developed euclidean geometry. Duality is introduced by means of illustrations followed by a statement of the method of obtaining new theorems from others by interchanging elements and their reciprocals in the statements. Neither the logical bearing of the principle nor the extent of its validity is discussed. The study of the complete quadrangle leads up to harmonic forms and the cross-ratio of a harmonic range is shown to be -1 by the use of ordinary metrics. From harmonic ranges harmonic scales are developed. The harmonic separation theorems are obtained from a free and intuitive use of continuity. A form of the Dedekind postulate is introduced using the concepts of segments and order as modified by the introduction of ideal elements. The proof of the fundamental theorem of von Staudt is then based on continuity. Curves and envelopes of the second order, poles and polars with respect to a curve are then studied in considerable detail, with the use of metrics for some properties. An involution on a form is defined as a cyclic projectivity of order 2 and conjugate imaginary points, lines and planes are introduced from the elliptic involution. An entire chapter is devoted to the focal properties of conies. Collineations, dualities, affinities, and polarities are treated briefly.

The figures are fairly well done, the typography is very good, and the volume as a whole is neat and attractive. The large number of exercises scattered throughout the book add much to its utility as a textbook. The little historical notes although very brief are also valuable and stimulating.

In the opinion of the reviewer, such a book as this one, which is avowedly not concerned with the more logical phases of the subject, would be more valuable to many if it gave at least references to such books as that of Veblen and Young, where such treatment could be found. On the whole, however, the book will no doubt be of much service in beginning courses in the subject.

2.4. Review by C J Keyser.
Science, New Series 47 (1222) (1918), 542.

Professor Dowling's "Projective Geometry" is a handsome introduction to the most exquisitely beautiful of mathematical subjects. The treatment, which is in the manner of Rey's classic "Geometrie der Lage," is synthetic as distinguished from algebraic, and presupposes no knowledge beyond ordinary elementary geometry and a very little trigonometry. It dos not aim at the rigour of the postulational method, but is preliminary thereto and admirably qualifies the reader to appreciate the nature and the value of that method.

2.5. Review by: John W Bradshaw.
The American Mathematical Monthly 25 (1) (1918), 15-18.

If long and expectant waiting for the guest's arrival insures a hearty welcome, this little book first sees the light under most auspicious circumstances. For many years those who have had the good fortune to teach projective geometry have been wishing for a text in English that should lay sufficient emphasis on its non-metrical character and at the same time should be adapted to the powers of the average college junior or even the exceptional younger student. Cremona, excellent book though it is, does not satisfy the first condition; Holgate's translation of Reye has long been out of print; and Veblen and Young's masterful treatise has seemed to many too heavy for the purpose. Here we have a book, small and compact, of pleasing external appearance, well printed in good type, with clear and attractive page broken by figures of reasonable size. Its point of view is non-metrical and yet it does not neglect the metrical applications. It merits careful consideration by all who are interested in this beautiful field of mathematical thought and sympathetic trial in many of the institutions where the subject is taught.

Perhaps the first question confronting the author of an elementary text on projective geometry is whether the conic shall be studied first as generated by projective ranges and pencils or as made up of the self-conjugate elements of a polar field. Shall he ally himself with the school of which we may take Reye as an example or with that typified by Enriques? The logical advantages of the latter have been clearly pointed out, its wisdom from a pedagogical stand point is not so clear. It takes much longer to reach what to the student seems worth while.

The teacher finds himself, perhaps, in the position of the guide who must choose between two paths up the mountain. There is on the one hand a long and rugged climb, which leads directly to the top, and on which the wonderful landscape bursts suddenly into view in all its beauty; and on the other is an easier path, which half-way up the mountain affords a fine view, if not the equal of that at the higher level. The most hardy of his party would not stop short of the top in any event. Of those who, exhausted, would fail to finish the longer ascent, some, encouraged by the lesser view, will go on to the greater; others, stopping at the half-way house, will at least have something for their labour. Must it not be agreed that unless the party is made up entirely of hardy climbers the wise guide will choose the second route? But is this a true analogy? We confess doubt.

Our author frankly admits in the preface that he chooses Reye for his pattern, and yet, if we may continue the figure, he does not fail to point the way to the higher level. After viewing the beautiful landscape embracing the conic sections and their properties from the level of generation by projective forms, he concludes the book with chapters on projectively related primitive forms of the second kind and polarities in a plane and in a bundle. The question might be raised whether it would have been possible and desirable in one book to give the teacher his choice of paths. Our author has not attempted this. Much of the book would have to be entirely recast if the teacher wished to make his first approach to the conics through the polar field. In particular the involution on a line could not be derived from the involution on a conic.

With what simple means and how quickly are we led to the conics! Only four chapters, 36 pages, are necessary to introduce the student to the fundamental notions, the primitive forms, the theorem of Desargues, the principle of duality, the theorem of perspective quadrangles, the harmonic set, as a preparation for projective one-dimensional primitive forms, the generation of the conic, the theorems of Pascal and Brianchon, and the polar theory of the conic. These are treated in four chapters covering 57 pages. To be sure, the involution is not yet known, nor those properties that depend on it. Three chapters intervene, Chapter IX on the Diameters, Axes, and Algebraic Equations of Curves of the Second Order, Chapter X on Ruled Surfaces of the Second Order, and Chapter XI on Projectively Related Elementary Forms, before in Chapter XII we have the involution defined as a cyclic projectivity of order two. This chapter deals with the theory of the involution and imaginary elements, and the following Chapter XIII with the foci and focal properties. This closes the discussion of one-dimensional forms, the remaining fifty pages being occupied with two-dimensional forms, the two chapters mentioned above.

In general outline these first thirteen chapters handle the subject in about the manner that one familiar with Reye would expect. A few features deserve special mention.

In considering the fundamental theorem the author constructs a harmonic scale, calls attention to the fact that harmonic constructions from three points can never yield all the points of a line, though "theoretically we may arrive at a point-row whose points are everywhere dense," and gives a formulation of the Dedekind postulate as here applied. He confesses that the treatment is meagre, but it furnishes a starting point for the teacher who considers it advisable to lay stress on the logical foundation of the fundamental theorem.

Defining the involution on a conic as a cyclic projectivity of order two furnishes the author with an occasion for a short discussion of cyclic projectivities in general. The definition of imaginary elements and the solution of certain problems involving their use follow closely the latest edition of Reye's first volume.

Though nearly a quarter of the book is devoted to metrical matters the double ratio receives scant attention.

A more detailed outline of the last two chapters may be in place, since in the selection of material here there is a wider range of possibilities. The definition and determination of perspective and projective transformations of two-dimensional forms are followed by the plane perspectivity, affinity, similitude, and congruence, with a short discussion of the double elements of a collineation. In the last chapter we have the construction and classification of polarities in the plane and bundle; orthogonal and absolute polarity and antipolarity; two polarities in the same plane or bundle with resulting collineation and with application to the cyclic planes and focal axes of cones; quadratic transformations, inversion, circular transformations.

In handling these two chapters the teacher is afforded ample opportunity to leave his impress upon his course. He will wish to amplify the treatment at many points and readjust the emphasis, in order to bring out the advantages of the polar field approach to the conic. One must delay long enough on the summit for the mists to roll away. He will find it necessary even to correct misleading statements such as,' "If two planes are collinearly related and have a self-corresponding line, they are in perspective position, or else they are superposed and have in common a sheaf of rays." Evidently what the author means is a line of self-corresponding points rather than a self-corresponding line. Again, in the demonstration of the theorem,2 "In affinately related planes, the ratio between the areas of corresponding figures is constant," it is implied that corresponding segments bear to each other a ratio that is constant throughout the plane.

Scattered through the book there are some fifty sets of exercises averaging six exercises to the set. For the formulation of these the teacher will be grateful. We have noticed two of them that are faulty in statement: "Show that in the configuration of Desargues any line may be taken as an axis of perspectivity. The two triangles and the centre of perspectivity will then be uniquely determined," is not clear; "If a hexagon whose vertices are not coplanar nor its three diagonals concurrent is projected from any point on a line which meets all three of the diagonals, show that the lines projecting the vertices are rays of a cone of the second order," draws the conclusion of Pascal from the hypothesis of Brianchon.

The historical notes are few and do not attempt to sketch the development of the subject. "Chasles, Geom6trie Superieure, 1880," may be thought misleading, since the date is not that of the first edition.

While the author, as he himself says, has patterned after Reye, he has by no means given us a mere translation. The language and style are the author's, not Reye's. Some things we think might be better said in the interest of clearness and accuracy. For example, "Like primitive forms are each composed of the same kind of elements," appears in its connection to be a definition. The breaks in the argument for the purpose of dualizing are sometimes disturbingly frequent. For some things we should use other names; throw of points or lines where the author uses range and pencil, two-dimensional forms for forms of the second kind, quadratic transformation for quadric transformation. We see no reason for banishing the diagonal point. In passing may we express the hope that some steps will soon be taken to standardise the nomenclature of this subject. It is certainly not an advantage pedagogically to have several different names for the same thing.

It is a pleasure to the teacher to be able to point out to his class the excellences of good figures. This pleasure is in measure denied to the user of the book we are discussing; for while the figures are in general clear and easily read, the teacher, if he calls attention to them at all, will be forced to remark on the carelessness exhibited in their construction. The inconsistency in the use of small circles surrounding designated points and in the use of dotted lines is astonishing. In each of two figures three lines that should pass through a point form a triangle of considerable size. Very unfortunate is Fig. 120, one of the most pretentious in the book. The crude approximations to ellipses here shown are an offence to even the slightly trained eye. What a pity that a book must carry a blemish of this sort, which might so easily have been avoided!

Let us conclude with the hope that this book will find wide acceptance and accomplish much in bringing the subject to the attention of a larger body of students.

2.6. Modern Mathematical Texts; Projective Geometry Paperback – 10 March 2016, by L Wayland Dowling (Author).

Publisher's information.

Projective Geometry, a branch of mathematics that transcends traditional Euclidean concepts, is eloquently presented in "Modern Mathematical Texts: Projective Geometry" by L Wayland Dowling. Originally published in 1917, this classic text is a vital resource for those delving into the intricate relationships between points, lines, and planes through the lens of projective geometry.

This book is structured to enhance the reader's visualisation and understanding of geometric principles by employing a synthetic approach rather than an analytic one. Dowling's explanations, backed by rigorous theorems such as Desargues' and Pascal's, make this text indispensable for both students and educators alike, particularly those preparing to teach mathematics at the secondary school level.

The volume offers comprehensive coverage on fundamental concepts like central projection, the principle of duality, harmonic ranges, projectively related forms, and the generation of curves and envelopes of the second order. Dowling expertly guides readers through complex topics such as elementary forms, conic sections, and involutions, offering both theoretical frameworks and practical applications.

Perfect for students with a background in elementary geometry and trigonometry, this volume assumes no advanced knowledge of analytic geometry, making it accessible while challenging the intellect. As part of the "Leopold Classic Library," this edition has undergone meticulous quality control to ensure a superior reading experience.

Whether you are revisiting classical geometric concepts or exploring projective geometry for the first time, this work remains a significant mathematical text, bridging traditional geometric ideas with more abstract, modern approaches.

3.1. From the Preface.

The present volume has grown out of lectures on the mathematics of life insurance given at the University of Wisconsin for a number of years, and is intended primarily as a first course for those young men and women who wish to become trained actuaries; or as a final course for other students who, for one reason or another, desire an elementary knowledge of the fundamental mathematical principles underlying a vast and growing business in our modern world.

In this book, with the exception of the few remarks in the introductory chapter, a knowledge of elementary algebra and its application to the theory of annuities-certain is assumed.

The elementary theory of probability is developed in chapter II with sufficient fullness to make its application to the theory of contingent functions clear.

A considerable symbolism has grown up in connection with actuarial theory and practice. The attempt is here made to keep this symbolism as simple as possible while retaining its classic form.

The tables herein included are for illustrative purposes only. Many of the commutation symbols, as well as the Fackler valuation symbols, were recomputed with the aid of a five-place table of logarithms and are not, therefore, accurate enough where large sums of money are involved. Glover's tables entitled Tables of Applied Mathematics in Finance, Insurance, Statistics, is particularly valuable, not only for the actuary, but for all who have to deal with financial operations or with the application of statistics to science in general.

While this volume presupposes only a knowledge of elementary mathematics, it is perhaps hardly necessary to remark that the mathematical equipment of the actuary should include a knowledge of the calculus. Some examples of the use of this subject have been added in the Conclusion. Others will be found in advanced treatises on actuarial science or on statistics.

These treatises were freely consulted in the preparation of the lectures out of which this volume has grown.

The author desires to express his gratitude to Dean Charles S Slichter for his aid in seeing the book through the press.

L Wayland Dowling
University of Wisconsin
July 1925.

3.2. Beginning of Section on Probability.

Fundamental Principle of Probability. Originating at the gaming table in the middle of the seventeenth century and developed by the brilliant school of French mathematicians that flourished for the next 150 years, the theory of probability has come to occupy a place of fundamental importance in very diverse fields of human inquiry. Today the physicist, the chemist, the statistician, the biologist, the actuary, depend in ever-increasing measure upon the results of this theory for the further development of their respective sciences.

The theory of probability is based upon the following fundamental principle, or definition:

If an event can happen in $a$ ways and fail to happen in $b$ ways, all being equally likely, the probability that it will happen in any one trial is
          $p = \Large\frac{a}{a+b}\normalsize$          (1)
and the probability that it will fail to happen in any one trial is
          $q = \Large\frac{b}{a+b}\normalsize$          (2)
Whether the events are all equally likely or not can be determined, in general only by experiment. For example, in throwing a cubical die, the presumption is that the six possible ways in which the die can fall are all equally likely. Since the ace can fall uppermost in one way and can fail to fall uppermost in five ways, the probability of getting ace in a single throw is one in six, and the probability of not getting ace is five in six. But if, in a large number of trials, the ace should fail to fall uppermost in approximately one-sixth the number of trials, the presumption would be that the six possible ways in which the die can fall are not all equally likely and that the die is either imperfect or else the method of throwing is not a matter of pure chance.

Many games of chance are like the example just quoted; that is, it is possible to determine a priori, or beforehand, in how many ways a given event can happen and in how many ways it can fail to happen.

3.3. Review by: W P E.
Journal of the Institute of Actuaries (1886-1994) 57 (1) (1926), 80.

Professor Dowling's book is very elementary: it does once more what has been done by other writers as well or better.

3.4. Review by: Rainard B Robbins.
Journal of the American Statistical Association 21 (154) (1926), 237-239.

The first eight pages of this book review briefly the theory of compound interest, developing many of the usual formulae in connection with annuities. The author's methods in these pages should prove stimulating to the student who is already familiar with the subject matter, but would doubtless prove very difficult for a beginner. The author points out in the preface that a knowledge of the applications of algebra to the theory of annuities is assumed. On page 3 under the topic "Annuities" we find the following:

An annuity is a sum of money payable at stated intervals of time and for a period of years. Usually an annuity is payable annually and at the end of each year. If the annuity is payable at the beginning of each year, it is called an annuity-due.

It is felt that the definition given in the first sentence can be improved upon. The accuracy of the second sentence is questioned. The third sentence appears as a definition of an annuity-due and limits it to one payable annually. The expression "contingent annuity," although it may not be generally used in American texts, serves a real need and its use or the use of some other words with the same definition should be encouraged. It is not clear that the expression "term annuity" fills a real need.

After Chapter II on the "Theory of Probability," life insurance proper is introduced in Chapter III with a discussion of the mortality table. The first sentence of the chapter seems unfortunate: "The probability of living, or dying, at a given age is measured by a mortality table." It would seem rather that the mortality table is the result of a statistical effort to find out what the actual probability of living or dying is at different ages. Of course, for practical purposes, in conducting a life insurance business, it is necessary in many calculations to assume that deaths will be related in a definite manner to those expected by a chosen mortality table. The second sentence of this chapter reads as follows:

The fundamental column in a mortality table exhibits the number of people left alive at each age out of an assumed number alive at a given early age until all are dead.

In the minds of many there is little doubt that the fundamental column of a mortality table is the column showing the probability of death at each age. However, this is not a matter of great importance since either column can be obtained from the other.
...
This book covers in seventy-nine pages most of the material which is essential as a basis for ordinary actuarial work. The reviewer feels that the book would be far more useful as a text if it covered twice as many pages. Most students will make more rapid progress with a lucid text written with the thought of anticipating the student's difficulties and clearing up as many of them as possible in the text. It seems that the author has, in places, deliberately chosen the hardest method of development for a student to see instead of the most natural method. For instance, the formula for the single premium for a whole life insurance is developed by a method which is probably very difficult for most students going through the subject for the first time, and then the more natural development is given on the following page. Throughout the book the author loses no chance to use a summation sign. This makes for compactness, and probably for disgust on the part of the student.

The four pages covering the whole subject of preliminary term valuation seem to be absolutely hopeless from the standpoint of all but the very exceptional student and even the best student must supplement from some other source what is in the text before he can hope to grasp preliminary term valuation in such a way that it will be of any practical use to him.

There is little danger of writing a text on this subject which will be too simple for the good of the student. A text written from the standpoint of the student to whom the subject is new and who is having his difficulties in grasping even those points which will seem so simple later, such a text, designed to inspire the student rather than to discourage him, is of many times the value of a text written in a compact form, with every sentence expressed as generally as possible with the apparent intention of making it a challenge for the student to understand what it is all about. The compact text costs less but if a student's time is worth anything the lucid text is worth many times as much as the compact.

3.5. Review by: Robert Henderson.
The American Mathematical Monthly 33 (2) (1926), 102-103.

This book is intended primarily as a first course for those young men and women who wish to become trained actuaries or as a final course for other students who desire an elementary knowledge of the fundamental principles underlying life insurance.

It is well designed as to scope and the execution is on the whole admirable, the explanations being notable for lucidity. This makes it all the more regrettable that it at the same time presents a few rather serious defects. These will be mentioned in the order in which they appear in the book rather than in order of importance.

The subject of probability is taken up from the a priori or subjective standpoint and while the author has some good company in this respect it has always seemed to the present writer a peculiarly inappropriate method of presentation as an introduction to life insurance where the probabilities are based upon experience.

In Article 21 it is said, with respect to the "life curve" which is a graphic representation of the "number living" column of the life table, that it should be regarded as "the lower limit of a fluctuating curve representing actual mortality." I suppose the author has in mind the fact that, due to the improvement in mortality rates during the past half century the mortality tables in general use have come to represent a considerably higher rate of mortality than the average experience of recent years. That is, however, an accidental condition and even then does not justify the statement quoted. If the mortality table represents the actual experience its "life curve" corresponds to the mean position of the fluctuating curve rather than a lower limit.

Probably the gravest defect is however the use of the official international symbol for the annual premium for an endowment insurance as the symbol for the annual premium for a term insurance. This requires the invention of a new and inconvenient symbol for the former when it is needed. The accepted notation provides symbols for both these functions and no new ones were necessary.

It only remains to point out that on page 109 there is a clerical error apparently due to some confusion between mean algebraic deviation and mean absolute deviation. The integral there given is the proper one for the former, which is equal to zero, but it is equated to the expression for the latter.

3.6. Review by: W Palin Elderton.
The Mathematical Gazette 13 (181) (1926), 92.

This book is of the elementary kind; it is intended "as a course for those young men and women who wish to become trained actuaries; or as a final course " for certain other students. The author appreciates that, though his book assumes only an elementary knowledge of algebra, a considerably greater mathematical equipment is essential to an actuary, and we think it was a mistake to assume that the mathematics were not available for the first course. In England the mathematical work comes first and study of life contingencies afterwards; and this arrangement seems the one. If, however, an author is determined to assume only elementary ledge of algebra it is inconsistent to write a concluding chapter on the curve, Stirling's formula, standard deviations, etc., in which integrals used (sometimes where they might have been avoided), or to define Makeham's "Law of Mortality" in terms of a function (force of mortality) which involves differential equation, when it can be defined just as easily in terms of a function (the logarithm of the probability of surviving a year).

3.7. Mathematics Of Life Insurance: Modern Mathematical Texts Hardcover (23 June 2012), by L Wayland Dowling.

Publisher's information.

The Mathematics of Life Insurance is a modern mathematical textbook written by L. Wayland Dowling. The book is designed to provide an in-depth understanding of the mathematical principles behind life insurance. The author covers a wide range of topics, including probability theory, life tables, actuarial notation, and the calculation of premiums and reserves. The book is aimed at students and professionals in the field of actuarial science, as well as anyone interested in the mathematics of life insurance. The author uses clear and concise language, making complex mathematical concepts accessible to readers with varying levels of mathematical knowledge. The book includes numerous examples and exercises to help readers develop their understanding of the subject matter. Overall, The Mathematics of Life Insurance is an essential resource for anyone seeking a comprehensive understanding of the mathematical principles behind life insurance. This scarce antiquarian book is a facsimile reprint of the old original and may contain some imperfections such as library marks and notations. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions, that are true to their original work.