## B F Finkel's Mathematical Solution Book

A mathematical solution book containing systematic solutions of many of the most difficult problems.
The First Edition of the work was completed in 1888. However, the printer went out of business in the middle of printing the book and so the book was not published until 1893. We give first a version of the Preface from this First Edition (written in 1888): |

**Preface**

This work is the outgrowth of eight years' experience in teaching in the Public Schools, during which time I have observed that a work presenting a systematic treatment of solutions of problems would be serviceable to both teachers and pupils.

It is not intended to serve as a key to any work on mathematics; but the object of its appearance is to present, for use in the schoolroom, such an accurate and logical method of solving problems as will best awaken the latent energies of pupils, and teach them to be original investigators in the various branches of science.

It will not be denied by any intelligent educator that the so-called "Short=Cuts" and "Lightning Methods" are positively injurious to beginners in mathematics. All the "whys" are cut out by these methods and the student robbed of the very object for which he is studying mathematics; viz., the development of the reasoning faculty and the power to express his thoughts in a forcible and logical manner. By pursuing these methods, mathematics is made a mere memory drill and when the memory fails, all is lost; whereas, it should be presented in such a way as to develop memory, the imagination, and the reasoning faculty. By following out the method pursued in this book, the mind will be strengthened in these three powers, besides a taste fr neatness and a love of the beautiful will be cultivated.

Anyone who can write out systematic solutions of problems can resort to "Short Cuts" at pleasure; but, on the other hand, let a student who has done all his work in mathematics by formulae, "Short Cuts," and "Lightning Methods" attempt to write out a systematic solution - one in which the work explains itself - and he will soon convince one of his inability to express his thoughts in a logical manner. These so-called "Short-Cuts" should not be used at all in the schoolroom. After pupils and students have been drilled on the systematic method of solving problems, they will be able to solve more problems by short methods, than they could by having been instructed in all the "Short Cuts" and "Lightning Methods" extant.

It cannot be denied that more time is given to, and more time wasted in the study of arithmetic in the public schools than in any other branch of study; and yet, as a rule, no better results are obtained in this branch than in any other. The reason od this, to my mind, is apparent. Pupils are allowed to combine the numbers in such a way as "to get the answer" and that is all that is required. They are not required to tell why they do this, or why they do that, but, "did you get the answer?" is the question. The art of "cyphering" is thus developed at the expense of the reasoning faculty.

The method of solving problems pursued in this book is often called the "Step Method." But we might, with equal propriety, call any orderly manner of doing any thing, the "Step Method." There are only two method of solving problems - a right method and a wrong method. That is, the right method which take up, in logical order, link by link, the chair of reasoning and arrives at the correct result. Any other method is wrong and hurtful when pursued by those who are beginners in mathematics.

One solution, thoroughly analysed and criticised by a class, is worth more than a dozen solutions the difficulties of which are seen through a cloud of obscurities.

This book can be used to a great advantage in the classroom - the problems at the end of each chapter affording ample exercise for supplementary work.

Many of the Formulae in Mensuration have been obtained by the aid of the Calculus, the operations alone being indicated. This feature of the work will not detract from its merits for those persons who do not understand the Calculus; for those who do understand the Calculus it will afford an excellent drill to work out the steps taken in obtaining the formulae. Many of the formulae can be obtained by elementary geometry and algebra. But the Calculus has been used for the sake of presenting the beauty and accuracy of that powerful instrument of mathematics.

In cases where the formulae lead to series, as in the case of the circumference of the ellipse, the rule is given for a near approximation.

It has been the aim to give a solution of every problem presenting anything peculiar, and those which go the rounds of the country. Any which have been omitted will receive space in future editions of this work. The limits of this book have compelled me to omit much curious and valuable matter in Higher Mathematics.

...

Hoping that the work will, in a measure, meet the object for which it is written, I respectfully submit it to the use of my fellow teachers and co-labourers in the field of mathematics.

We give below a version of the Preface to the Third Edition of 1899: |

**Preface to Third Edition**

The hearty reception accorded this book, as is attested by the fact that two editions of 1,200 copies each have already been sold, encouraged me to bring out this third edition.

In doing so, I have availed myself of the opportunity of making some important corrections, and such changes and improvements as experience and the suggestions of teachers using the book have dictated. The very favorable comments on the work by some of the most eminent mathematicians in this country confirm the opinion that the book is a safe one to put into the hands of teachers and students.

While mathematics is the exact science, yet not every book that is written upon it treats of it as though it were such. Indeed, until quite recently, there were very few books on Arithmetic, Algebra, Geometry or Calculus that were not mere copies of the works written a century ago, and in this way the method, the spirit, the errors and the solecisms of the past two hundred years were preserved and handed down to the present generation. At the present time the writers on these subjects are breaking away from the beaten paths of tradition, and the result, though not wholly apparent, is a healthier and more vigorous mathematical philosophy. Within the last twenty-five years there has set in, in America, a reaction against the spirit and the method of previous generations, so that C A Laisant, in his 'La Mathematique Philosophie Enseignement', Paris, 1898, says, "No country has made greater progress in mathematics during the past twenty-five years than the United States. The most of the text-books on Arithmetic, Algebra, Geometry, and the Calculus, written within the last five years, are evidence of this progress."

The reaction spoken of was brought about, to some extent, by the introduction into our higher institutions of learning of courses of study in mathematics bearing on the wonderful researches of Abel, Cauchy, Galois, Riemann, Weierstrass, and others. This reaction, it may be said, started as early as 1832, the time when Benjamin Peirce, the first American worthy to be ranked with Legendre, Wallis, Abel and the Bernouillis, became professor of mathematics and natural philosophy at Harvard University. Since that time the mathematical courses in our leading Universities have been enlarged and strengthened until now the opportunity for research work in mathematics as offered, for example, at the University of Chicago, Harvard, Yale, Cornell, Johns Hopkins, Princeton, Columbia and others, is as good as is to be found anywhere in the world. For example, the following are the subjects offered at Harvard for the Academic year 1899-1900: Logarithms, Plane and Spherical Trigonometry; Plane Analytical Geometry; Plane and Solid Analytical Geometry; Algebra; Theory of Equations. - Invariants; Differential and Integral Calculus; Modern Methods in Geometry. - Determinants; Elements of Mechanics; Quaternions with application to Geometry and Mechanics; Theory of Curves and Surfaces; Dynamics of a Rigid Body; Trigonometric Series. Introduction to Spherical Harmonics. - Potential Function; Hydrostatics. - Hydrokinematics. - Force Functions and Velocity-Potential Functions and their uses. - Hydrokinetics; Infinite Series and Products; The Theory of Functions; Algebra. - Galois's Theory of Equations; Lie's Theory as applied to Differential Equations; Riemann's Theory of Functions; The Calculus of Variations; Functions Defined by Linear Differential Equations; The Theory of Numbers; The Theory of Planetary Motions; Theory of Surfaces; Linear Associative Algebra; the Algebra of Logic; the Plasticity of the Earth; Elasticity; and the Elliptic and the Abelian Transcendants.

While the great activity and real progress in mathematics is going on in our higher institutions of learning, a like degree of activity is not yet being manifested in many of our colleges and academies and the Public Schools in general. It is not desirable that the quantity of mathematics studied in our Public Schools be increased, but it is desirable that the quality of the teaching should be greatly improved. To bring about this result is the aim of this book.

It does not follow, as is too often supposed, that any one familiar with the multiplication table, and able, perhaps, to solve a few problems, is quite competent to teach Arithmetic, or "Mathematics," as arithmetic is popularly called. The very first principles of the subject are of the utmost importance, and unless the correct and refined! notions of these principles are presented at the first, quite as much time is lost by the student in unlearning and freeing himself from erroneous conceptions as was required in acquiring them. Moreover, no advance in those higher modern developments in Mathematics is possible by any one having false notions of its first principles.

As a branch for mental discipline, mathematics, when properly taught, has no superior. Other subjects there are that are equally beneficial, but none superior. The idea entertained by many teachers, - generally those who have prepared themselves to teach other subjects, but teach mathematics until an opportunity to teach in their special line presents itself to them, - that mathematics has only commercial value and only so much of it should be studied as is needed by the student in his business in after life, is pedagogically and psychologically wrong. Mathematics has not only commercial value, but educational and ethical value as well, and that to a degree not excelled by any other science. No other science offers such rich opportunity for original investigation and discovery. So far from being a perfected and complete body of doctrine "handed down from heaven" and incapable of growth, as many suppose, it is a subject which is being developed at such a marvelous rate that it is impossible for any but the best to keep in sight of its ever-increasing and receding boundary. Because, therefore, of the great importance of mathematics as an agent in disciplining and developing the mind, in advancing the material comforts of man by its application in every department of art and invention, in improving ethical ideas, and in cultivating a love for the good; the beautiful, and the true, the teachers of mathematics should have the best training possible. If this book contributes to the end, that a more comprehensive view be taken of mathematics, better services rendered in presenting its first principles, and greater interest taken in its study, I shall be amply rewarded for my labor in its preparation.

In this edition I have added a chapter on Longitude and Time, the biographies of a few more mathematicians, several hundred more problems for solution, an introduction to the study of Geometry, and an introduction to the study of Algebra.

The list of biographies could have been extended indefinitely, but the student who becomes interested in the lives of a class of men who have contributed much to the advancement of civilization, will find a short sketch of the mathematicians from the earliest times down to the present day in Cajori's 'History of Mathematics' or Ball's 'A Short History of Mathematics'.

The biographies which have been added were taken from the 'American Mathematical Monthly'. I have received much aid in my remarks on Geometry from 'Study and Difficulties of Mathematics', by Augustus De Morgan.

It yet remains for me to express my thanks to my colleague and friend, Prof F A Hall, of the Department of Greek, for making corrections in the Greek terms used in this edition,

THE AUTHOR.

Drury College, July, 1899.

JOC/EFR May 2013

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