**High-School Algebra. Elementary Course (1907), by Herbert Ellsworth Slaught and Nels Johann Lennes.****1.1. Review by: Thomas Emery McKinney.**

*The School Review***16**(3) (1908), 205-207.For some time a revision of the school course in algebra has been an admitted necessity. Many recent studies and discussions have dealt with the essential features of a course adapted to present needs. In this question the authors of the book have been actively interested for several years and are thoroughly familiar with its literature. In the light of, and in conformity with, the exacting requirements of mathematical pedagogy as recently developed in this country and abroad, they propose a course in algebra suited to existing conditions in our high schools and academies. ... In its gain of simplicity by keeping essentials constantly in the fore, in its appeal to the learner by emphasis on important applications and on graphical methods, in its solicitude that he be not forced prematurely into purely abstract, symbolic reasoning, the book embodies the best results of recent studies on the teaching of algebra. It will arouse enthusiasm in the classroom and merits a wide adoption. On textbook and teaching methods it will exert, we believe, a profound influence.

**1.2. Review by: Ernest B Little.**

*Bull. Amer. Math. Soc.***15**(1909), 357-362.The common weakness of our college students in elementary algebra shows a great need for improvement in the teaching of this subject. The appearance of these admirable texts, constructed after a new model, marks a distinct advance in the teaching of algebra in our high schools. ... As stated in the preface, "the main purpose of the Elementary Course is the solution of problems rather than the construction of a purely theoretical doctrine as an end in itself." Throughout the first course the method is inductive. The interests, ability, and needs of the first year high school students seem to demand this selection of the inductive method. In writing elementary texts, mature mathematicians too often let their delight in the subject as an elegant deductive system obscure the ability of the student at this particular stage in his development. The teachableness of this book is greatly increased by the fact that the authors never lose sight of the standpoint of the pupil.

**High-School Algebra. Advanced Course (1908), by Herbert Ellsworth Slaught and Nels Johann Lennes.****2.1. Review by: Walter B Carver.**

*The School Review***17**(2) (1909), 133-135.Anyone who had read the first volume (Elementary Course) of this algebra must have awaited the appearance of the second volume with some interest. The elementary course measured up in a large degree to its avowed purpose, but it was obviously insufficient. Much of what one felt to be lacking in the first volume is contained in the advanced course, and the two combined make a very complete treatment of the subjects covered. The question as to the wisdom of presenting the subject in two distinct courses is one upon which a theoretical opinion would be of little value. It will be settled in due time by practical experience. ... But when one has said the worst that can be said about the book, the fact remains that the defects are neither numerous nor serious; and that in many important respects, it sets a new and higher standard for high-school algebras.

**2.2. Review by: Ernest B Little.**

*Bull. Amer. Math. Soc.***15**(1909), 357-362.The Advanced Course presupposes a knowledge of the principles of elementary algebra, considerable training in deductive reasoning obtained in the preceding year of plane geometry, and greater maturity in the pupils ; hence its point of view is entirely different from that of the Elementary Course. The method is deductive, and instead of principles to be illustrated the algebraic facts are put in the form of theorems to be proved. ... The authors have had the courage to omit entirely topics which they believe have no place in a high school algebra, such as simultaneous equations of more than three unknowns (a single example in four unknowns is given) inequalities, complicated complex fractions.

**Plane Geometry with Problems and Applications (1909), by Herbert Ellsworth Slaught and Nels Johann Lennes.****3.1. Review by: Anon.**

*The Mathematics Teacher***3**(2) (1910), 93.This book is sufficiently different from the numerous editions of "geometry" to deserve special attention. To quote from the preface: " The subject has been enriched by including many applications of special interest to the pupils. ..." Free use is made of certain sources of problems which may be easily comprehend without extended explanations - such problems pertain to decoration, ornamental designs and architectural forms." On closer investigation it appears that these applications are intended merely to furnish a concrete setting for the usually too abstract geometrical theorems. There is no claim made that they are practical in the sense that the study of bookkeeping is practical for one who intends to keep books. The aim is to present the subject so that the pupils may learn to know the essential facts of elementary geometry as properties of the space in which they live and not merely as statements in a book." ... While the logical rigor of the older texts has been fully maintained and in some cases improved (see the treatment of incommensurables), the whole subject has been recast with the avowed purpose of adapting to the needs and powers of the pupil.

**3.2. Review by: Frederick William Owens.**

*Bull. Amer. Math. Soc.***17**(1911), 374-375.This book has several features that distinguish it from the conventional high school text. Among these the most noticeable are the gradual introduction of the severely logical forms, and the introduction, for the purpose of making the subject more attractive, of a large number of applications to geometric forms more or less commonly met with in life. ... The algebraic form of the treatment is a decided improvement. One particularly pleasing feature is the willingness of the authors to introduce assumptions, as such, wherever they deem it desirable. At the same time, they state frankly that the set of axioms given is not complete, and that intuitional inferences concerning such subjects as order and continuity are drawn freely. ... The applications, which occur in large numbers throughout the book, are usually taken from architectural designs and various ornamental designs for decorative purposes, such as tile patterns, parquet floors, grill work, etc. These exercises are for the most part very simple, but by bringing into play a large number of straight lines and circle arcs in a single problem, they are a valuable aid in the development of geometric imagination. These exercises also bring into bold relief the principle of symmetry, which is thereby given the prominence that it deserves. Other applications are made to the problems of finding the distance between two points on opposite sides of a pond, measuring the height of a tree, finding the distance between two inaccessible points, cutting the braces for a roof, etc. All these applications should be a valuable aid in holding the interest of the numerous pupils who care comparatively little for the logical features of the subject.

**First Principles of Algebra (1911), by Herbert Ellsworth Slaught and Nels Johann Lennes.****4.1. Review by: Anon.**

*The Mathematics Teacher***5**(1) (1912), 37.The authors in writing this book have kept two aims before them: (1) To provide a gradual and natural introduction to the symbols and processes of algebra. (2) To give vital purpose to the study of algebra by using it to do interesting and valuable things. The equation is introduced and developed early, and the principles are codified in a few short rules. It is an interesting and carefully written book.

**Solid Geometry: With Problems and Applications (1911), by Herbert Ellsworth Slaught and Nels Johann Lennes.****5.1. Review by: Frederick William Owens.**

*Bull. Amer. Math. Soc.***18**(1912), 198-199.This book follows the plane geometry of the textbook series of the authors. It is divided into seven chapters entitled lines and planes in space, prisms and cylinders, pyramids and cones, regular and similar polyhedrons, the sphere, variable geometric magnitudes, and theory of limits. The logical phase of the development of solid geometry, as here treated, is a great improvement over that usually found in our textbooks. Many of the more fundamental principles are formally stated as axioms. The first striking example of this is Axiom III: "If two planes have a point in common, then they have at least another point in common." This fundamental theorem of three-dimensional geometry has usually been kept as74 obscure as possible. In all, ten axioms are thus stated. ... The book is very teachable, and taken altogether is a marked improvement over the usual text.

**Source Book of Problems for Geometry (1912), by Mabel Sykes, Herbert Ellsworth Slaught and Nels Johann Lennes.****6.1. Review by: Herbert E Cobb.**

*Amer. Math. Monthly***20**(5) (1913), 164-165.The present volume is a welcome contribution to the endeavour to make the mathematical work in our schools more practical and tangible. The belief that mathematics is an excellent discipline has for many years removed the mathematics of the secondary schools farther and farther from the daily life and experience of the pupils. The text-books have emphasized the logical side, and have contained a large number of exercises and problems for drill and a few problems in applied mathematics which are for the most part spurious. ... As a supplementary book in the high school course in geometry this text will, in the writer's opinion, prove of great value. Geometrical forms occur so frequently in tiles, mosaics, needlework, jewellery, iron grills, steel ceilings, tracery of windows and other architectural forms that a study of these forms will add greatly to the interest of geometry. Moreover it will be a pleasure to recognize known geometrical forms in architectural details which meet the eyes daily but have failed to attract attention.

**Plane Trigonometry and Applications (1914), by Ernest Julius Wilczynski and Herbert Ellsworth Slaught.****7.1. Review by: Derrick Norman Lehmer.**

*Amer. Math. Monthly***21**(10) (1914), 329-330.Every friend of mathematics and every friend of mathematical teaching must rejoice when scholars of national and international reputation give their time to the compiling of text-books in the more elementary branches of the science. It is a serious mistake to suppose that high school teachers are best equipped to write high school texts. It is particularly fortunate when the author is not only a scholar of wide outlook, but also a man with a keen instinct for teaching. When, moreover, the work of such a man has passed under the searching eye of an editor who is also an acknowledged leader in mathematical teaching, it is natural to expect much in advance of the book. ... Perhaps the most unusual feature of the book is the insertion of the chapter on the Theory of Wave Motion which closes the book. It is a subject of unusual interest and importance, and the chapter is one of the most beautiful in the book. We doubt, however, whether many teachers will venture to include it in a course where students are apt to have as little mathematical maturity as most students of trigonometry do. ... Much of interest in historical matter has been introduced "not in the form of detached historical notes, but organically connected with the topic under discussion." This appears to the reviewer to be a decided improvement on the usual method, but he cherishes the belief that the effect would be much better if the historical matter were collected into a separate chapter at the end of the book, and the subject discussed as a whole, - but each one to his taste in such matters

**Logarithmic and Trigonometric Tables (1914), by Ernest Julius Wilczynski and Herbert Ellsworth Slaught.****8.1. Review by: Derrick Norman Lehmer.**

*Amer. Math. Monthly***21**(10) (1914), 329-330.**Elementary Algebra (1915), by Herbert Ellsworth Slaught and Nels Johann Lennes.****9.1. From the Preface.**The Elementary Algebra is planned to cover the work of the first year in this subject. It is in no sense a revision of the authors' First Principles of Algebra, but a new book, designed to meet the most exacting requirements of college entrance or other examination boards and the syllabi of various states. The presentation of topics, therefore, follows the traditional order. The new Algebra contains numerous attractive features, all aiming to make the subject more simple and interesting and therefore more valuable to first year pupils. Among these features the most distinctive are perhaps the following four: (i) The presentation of the subject is as simple as it can be made. (ii) The book is equipped with an unusually full and complete set of exercises and problems. (iii) Vital purpose is given to the study of algebra by using it to do interesting and useful things. (iv) Emphasis is given to the human interest of algebra.

**9.2. Review by: Anon.**

*The Mathematics Teacher***8**(1) (1915), 60-61.This book is planned to cover the first year in the subject. It gives a long course for this time, for it includes all the required topics for Elementary and Intermediate Algebra except the Progressions. Like previous books by these authors there is great emphasis on simple presentation and easy gradation in each topic, and on the side of concrete applications. There is an abundance of exercises and problems through the text, and the last thirty-five pages is given up to review exercises on the various chapters. While it is true that the large part of the problems are concerned with concrete things, rather than having a practical application in themselves, there are many interesting ones, some of which correlate this subject with other school subjects to good purpose. The historical notes are well chosen and are attractively arranged. The table of contents and the index are particularly usable, and the book is at tractive in its general makeup.

**Intermediate Algebra (1916), by Herbert Ellsworth Slaught and Nels Johann Lennes.****10.1. From the Preface.**The Intermediate Algebra is designed to follow the authors' Elementary Algebra, It meets the most exacting requirements of college entrance and other examination boards and of the syllabi of various states. The presentation of topics, there- fore, follows the traditional order. While recognizing the increased maturity of the pupils, the authors nevertheless maintain in this present text that simple and interesting form of presentation which characterizes the earlier book. For example, while the axioms and fundamental laws are used in the proofs of theorems, yet these are stated and the proofs are given in an informal manner that at once attracts and holds the pupils' attention and interest.

**10.1. Review by: Anon.**

*The Mathematics Teacher***9**(1) (1916), 68.This is a continuation of the elementary algebra by the same authors, and follows the same general methods as the earlier book. It contains an excellent chapter on the use of determinants in solving equations, and one on the use of logarithms. The book covers the usual requirements in intermediate algebra, and seems to be well written and teachable.

**College Algebra with Applications (1916), by Ernest Julius Wilczynski and Herbert Ellsworth Slaught.****11.1. Review by: Derrick Norman Lehmer.**

Amer. Math. Monthly 24 (5) (1917), 230-231."The material included in this book," says the author in his suggestions to the instructor, "probably contains everything ever given under the title College Algebra in any American college." The problem of making a selection that will suit the needs of all teachers of algebra is one that might well daunt the most ingenious maker of textbooks.

**Plane Geometry, Revised Edition (1918), by Herbert Ellsworth Slaught and Nels Johann Lennes.****12.1. Review by: Anon.**

*The Mathematics Teacher***10**(3) (1918), 164.In this edition the authors retain the main features of their former book, the human interest side being represented by historical notes, diagrams of applied geometry, etc. The number of theorems has been somewhat reduced, and the book has a more attractive interior.

**Solid Geometry with Problems and Applications (1919), by Herbert Ellsworth Slaught and Nels Johann Lennes.****13.1. From the Preface.**

*Amer. Math. Monthly***27**(10) (1920), 373.In re-writing the Solid Geometry the authors have consistently carried out the distinctive features described in the preface of the Plane Geometry. ... Owing to the greater maturity of the pupils it has been possible to make the logical structure of the Solid Geometry more prominent than in the Plane Geometry. The axioms are stated and applied at the precise points where they are to be used. Theorems are no longer quoted in the proofs but are only referred to by paragraph numbers; while with increasing frequency the student is left to his own devices in supplying the reasons and even in filling in the logical steps of the argument. For convenience of reference the axioms and theorems of plane geometry which are used in the Solid Geometry are collected in the Introduction. In order to put the essential principles of solid geometry, together with a reasonable number of applications, within limited bounds, certain topics have been placed in an Appendix. This was done in order to provide a minimum course in convenient form for class use and not because these topics, Similarity of Solids and Applications of Projection, are regarded as of minor importance. In fact, some of the examples under these topics are among the most interesting and concrete in the text. ... The treatment of incommensurables throughout the body of this text, both Plane and Solid, is believed to be sane and sensible. In each case, a frank assumption is made as to the existence of the concept in question (length of a curve, area of a surface, volume of a solid) and of its realization for all practical purposes by the approximation process. Then, for theoretical completeness, rigorous proofs of these theorems are given in Appendix III, where the theory of limits is presented in far simpler terminology than is found in current text-books and in such a way as to leave nothing to be unlearned or com- promised in later mathematical work.

**13.2. Review by: Anon.**

*The Mathematics Teacher***12**(1) (1919), 35.In this book the authors have carried out again the features that distinguish their former books on geometry. There is an increasing tendency to leave some scope for the pupils' originality in the propositions, and a good amount of exercises are included. Excellent features are the summaries at the end of each book, and the table of the axioms and theorems from plane geometry that are used in the text.