1.1. Preface.

The following Lessons have been prepared for use in a brief course in Algebra offered in the first part of the second semester, freshman year, at Wellesley College. An effort has been made to give unity to the work by proposing one general problem, - the discovery of the roots of Higher Numerical Equations. Everything has been omitted which does not bear directly on the solution of this problem. It hardly need be added that in so brief a presentation of the subject only the more important theorems could be included. The essentials of the Theory of Logarithms have been presented, preliminary to the study of Trigonometry.

The works chiefly consulted in writing the Lessons, and to which special indebtedness should be acknowledged, are the treatises of Todhunter, Hall and Knight, Chrystal, Burnside and Panton, Serret, and Bertrand.

ELLEN HAYES.

Wellesley, Mass,
1891.

2.1. Preface.

In revising the Lessons on Higher Algebra various minor changes have been made in the articles of the first edition, and a number of articles added. A section on Determinants has also been introduced. In explanation of the brevity of these Lessons, it ought to be said, that the mathematical course for which they were written includes further work for more advanced students, in which the text-book used is Burnside and Panton's Theory of Equations.

The presence, in an algebra book, of so unusual a feature as a section on the nature of mathematical reasoning, calls for more than a passing word.

It cannot have escaped the notice of those who are acquainted with the prevailing methods of instruction in mathematics and logic that, on the one hand, the average teacher of mathematics gives little or no attention to the nature of the processes according to which mathematical reasoning is conducted; while, on the other hand, the teacher of logic almost ignores mathematics as a source of illustrations of the principles with which he is dealing. A preface is not the place to discuss the causes or effects of this divorce of two subjects which might go hand in hand; but I must express my conviction that when we discover those methods of instruction which are essential to sound education, they will be found to include logic for the student of mathematics, and mathematics for the student of logic. Since writing the Appendix I have received the Report of the Committee on Secondary School Studies, appointed at the meeting of the National Educational Association, July 9, 1892, including the Conference Report of the Committee on Mathematics, of which Professor Simon Newcomb, of the Johns Hopkins University, is chairman. This Conference Report confirms me in the belief that any successful effort to improve the teaching of mathematics, both in colleges and secondary schools, must recognise the claims of logic. "The very fact that demonstrative geometry is the most elaborate illustration of formal logic in the entire curriculum of the student, makes the consideration of these elementary principles of logic more interesting and profitable in this connection than in any other." (Report: Mathematics, p. 115.) The analytical branch of mathematics, however, scarcely falls behind the geometrical in affording 'elaborate illustration of formal logic'; and the identity of the logic of algebra and the logic of geometry is probably best shown by putting algebraic and geometrical illustrations side by side.

The closing section of this book does not undertake to give any formal or complete presentation of the logic of mathematics; much less does it furnish those outlines of ratiocination which are essential to a proper understanding of mathematical reasoning. As far as possible, technical terms have been avoided. There is no mention of 'contraposition,' or an 'undistributed middle term,' or an 'illicit process.' These and like matters the individual teacher will introduce or omit according to his own judgment. I have only insisted on the necessity of distinguishing between the conditional proposition mathematics and the causal proposition of inductive science; and I have emphasised the fact of the constant recurrence of the distributed predicate - or equal quantification of the terms P and Q - in the mathematical proposition: All P is Q.

As regards methods of presenting this proposed combination of logic and mathematics, much might be said; but I may here only suggest that good reasons can be advanced for viewing the "Science of Proof, or Evidence," itself as an observational inductive science, and teaching it as such. It would obviously be absurd to expect students to re-evolve the Systems of Aristotle and Mill, - as absurd as to expect them to re-discover the laws of physics or biology; but, on the other hand, it is not necessary to teach logic after the all too common dogmatic fashion. For instance, one might first allow a student to chance upon the fallacy of the illicit process of the major term, and then encourage him to set to work to determine wherein lies the error. After he has discovered that cases following a certain type are invariably incorrect, it will be time enough to give him the technical term for what he has observed.

In conclusion, whatever may be its faults, the object of the Appendix will be secured if those who read it are stimulated to study formal logic for the purposes of mathematics, and are thence led to explore that larger and more important realm known as Inductive Logic, the principles of which are of daily concern in the pursuit of science and in the conduct of life.

ELLEN HAYES.

Wellesley, Mass,
February 1, 1894.

3.1. Preface.

Part II of this work is a revision of the author's Lessons on Higher Algebra prepared several years ago in view of the need of a suitable text-book for a brief course in algebra required in the freshman year at Wellesley College.

In developing a short course in higher algebra, it has seemed desirable and has been found possible to give unity to the work by proposing one general problem, - the determination of the roots of higher numerical equations. Most of the matter of Part II will be found to bear on this problem.

It is the author's experience that students of the mathematics usually offered in college need to have an elementary algebra at hand for reference. Part I has been prepared to meet this need. It is also believed that for the purposes of high schools and preparatory schools, Part I will prove a helpful supplement to the ordinary algebra text-books, since it emphasises certain important aspects of the subject which at present are being more or less neglected. In writing this part, two classes of secondary school students have been kept in mind: those whose mathematical studies will end with the high school course, and who are entitled on that account to know something of the spirit and methods of the mathematical fields which they are not going to enter; and secondly, those who expect to advance to mathematical work of a more difficult grade, and who ought not to find an abrupt break between the elementary course of the preparatory school and the higher work of the college or university.

ELLEN HAYES.

Wellesley, Mass,
January 1, 1897.

4.1. Preface.

This little book has been written for two classes of persons: those who wish, for purposes of culture, to know, in as simple and direct a way as possible, what the calculus is and what it is for; and students primarily engaged in work in chemistry, astronomy, economics, etc., who have not time or inclination to take long courses in mathematics, yet who would like "to know how to use a tool as fine as the calculus."

The 'pure' mathematician will note the omission of various subjects that are important from his point of view; but for him there are admirable and lengthy treatises on pure calculus. Also the student whose experience has led him to conceive of mathematical study as the doing of interminable lists of exercises, will be surprised and, possibly, disappointed. This book is a reading lesson in applied mathematics. Fancy exercises have been avoided. The examples are, for the most part, real problems from mechanics and astronomy. This plan has been pursued in the conviction that such problems are just as good as make-believe ones for purposes of discipline, and a good deal better for purposes of knowledge. The time-honoured method of presenting calculus is much as if travellers should be stopped and made to pound stone on the high way, so that they never get anywhere or even know what the road is for. The following pages are a protest against the conventional method; for I am wholly in sympathy with a remark made by Professor Laster F. Ward, in his Outlines of Sociology: "There is no more vicious educational practice, and scarcely any more common one, than that of keeping the student in the dark as to the end and purpose of his work. It breeds indifference, discouragement, and despair."

A chapter on analytic geometry has been introduced, in the hope that teachers will try the plan of presenting the elements of the calculus and of analytic geometry together. There is no good reason either for keeping them distinct or for presenting analytic geometry first.

To three works I have to express my deep obligation. The spirit manifest in them has been my chief encouragement in preparing this book. I refer to Greenhill's Differential and Integral Calculus, Perry's Calculus for Engineers, and Nernst and Schönflies' Einführung in die mathematische Behandlung der Naturwissenschaften.

We have in these works, let us hope, an indication of the role which the calculus is to play in schemes for liberal and scientific education in the not far distant future.

ELLEN HAYES.

Wellesley College,
September, 1900.

5.1. On Science and Mathematics.

... the majority of students decline to study astronomy. It might prove difficult, and - disturbing thought - it might have some mathematics in it. These same students cannot give an intelligent account of the cause of the change of seasons; they cannot give any account at all of their own watches as the astronomical instruments that they are. They cannot tell Jupiter from Sirius; and as for the stars as suns with attendant worlds, their sizes, distances, motions, and constitutions; the galaxy, nebulae, - what claim have these things on persons absorbed in the little play of their brief hour?

In what has gone before I have spoken of science only as a body of knowledge, the achievement of the human intellect in discerning, discriminating, and classifying facts, and in discovering their relations of sequence. But, in so far as science is presented to you for study in college, quite half its value resides in the method of science. What is this method? In outline it is this: A fact, or group of facts discriminatingly classified, claims the observer's attention. Viewed as an effect, it is required to find antecedent facts which have operated as cause. A hypothesis - that is, a provisional solution of this causal problem - is framed: perhaps several are framed.

Then more observations must be made or experiments performed, to test these hypotheses. Material must be impartially collected and justly dealt with. Conclusions must not be drawn until warranted by the evidence. No hypothesis may pass to the status of explanation or of law until it stands all the tests that can be devised. The truth, without regard to the labour involved in attaining it and without regard to the consequences, is the dominant consideration with every real scientist. The obvious mental advantage of scientific training is, therefore, one in behalf of the powers of observation and judgment: its moral advantage consists in impressing the lesson that truth is to take precedence of all else, - our schemes, our tastes, our desires, our prejudices. If to seek truth and to know when it is found is a mark that pre-eminently distinguishes a being as a human being, the scientific method permits no comparison with other methods. There are others. You have only to watch the politician, the theologian, the metaphysical philosopher, the literator, to perceive what the other methods are. Do not misunderstand me. Science has no infallible recipe for making superior persons out of hopelessly inferior material. It is not difficult to find students and teachers of science who are to the last degree unscientific. They are called botanists, physicists, astronomers, and so on; but their lives, professionally and socially, exhibit bias, prejudice, and partial judgments. The power of calm comparison and estimation of evidence seems to be largely, if not wholly, lacking in them. Yet I must affirm that, if physics, rightly studied, does not make a person accurate, nothing will; if botany, rightly studied, does not lead him to observe, nothing will; if geology, rightly studied, does not train his reasoning powers, nothing will. If all of these together do not lead him to set truth above everything else, it will probably be in vain to invoke other agencies.

"Is not mathematics as important as science?" I note this question in your reply to my last letter. Let us consider.

In the curriculum of a well-known university the prerequisite for one course in mathematics is stated as "a certain facility in abstract reasoning." The framer of the prerequisite had a right to use any word he chose, but he had no right to employ one in an unusual sense without explaining. Mathematics does not require "facility in abstract reasoning," as the term reasoning is generally understood; nor does the study of mathematics cultivate the power of reasoning. In its realm there are no evidences to be gathered and weighed, no hypotheses to be framed, no causal relations to be searched for, no laws of nature to be disclosed, and no deductions to be made from such laws. That is, the opportunity for such training as science affords is not afforded at all by mathematics. "We train your reasoning power 'while you wait' in the mathematical class-room" is one of those standard announcements passed on from one generation of teachers to another: it is zealously recited by persons who, whatever they may know in mathematics, have plainly never given any thought to comparative logic.

Nevertheless, you should, by all means, learn some mathematics. Fortunately, it is the easiest of subjects to do alone. No libraries, no laboratories, and - I might almost add - no teachers are required. Neither need you be afraid of missing the way in its so-called reasoning. If you run against any serious obstacle, the fault is probably in the text-book. Lay it aside and try another on the same topic. Among the elementary indispensable branches I should put the calculus. You ought to know enough of this language to be able to read some of the "rhymes of the universe" that are written in it. Take a couple of hours each morning next summer, you and your brother, and you can do enough calculus to serve all ordinary purposes. But beware lest you use up your time in merely acquiring a fatal facility in working exercises. Find some real examples relating to falling bodies, to the path of a baseball, to the energy of a fly-wheel, and the value of the calculus, as an aid in exact science, will need no further emphasis. The pathetic feature of study in pure mathematics is that the student not only has no idea what it is all for, but he is rather proud of the notion that it is not "for" anything. To be sure, so far as any one knows yet, much of it is not; but some of it is. Under the time-honoured method the young mathematician adds page to page and chapter to chapter, and he is as one breaking stones on a road that for him leads nowhere. Gamma functions and Fourier's series, he is impressively assured by his teacher, are used in science. Where and how he never finds out, because in his eagerness to specialise in mathematics he has failed to learn even the rudiments of the sciences that invoke the aid of an integral or a series. The undergraduate mathematical "specialist" is looked upon with peculiar awe by his companions. The admiration commanded by his supposed performances exceeds the admiration that would be caused if, for instance, he read Arabic or could stand on his head in the gymnasium. Meanwhile, under the workings of the elective system, he is probably ignorant of the Laws of Motion - not to mention several other laws that it might be to his advantage to learn.

You tell me that your freshman teacher advises you to continue mathematics. That is, it is suggested that you may become one of the mathematical elect. Why this advice? The Latin argument over again: you can succeed in mathematics, and you may want to teach it some time. You will recall my letter concerning Latin. However, I must add a word further. Suppose you do some time teach mathematics, let us say the algebra and geometry and trigonometry of a high school. In the sequel you will understand, if you find it difficult to believe now, that pure mathematics is of small use to you in preparation for that work. It is all-important that you know algebra for what it really is, - a language, unrivalled for compactness and freedom from ambiguity, for expressing quantitative propositions and conducting quantitative discourse. There can be little doubt that the difficulties and disgusts, the hours wasted, the failures incurred, in its study, are largely owing to the grievous lack on the part of the teacher, and the text-book, of any insight into the true character of algebra. Geometry has its own peculiar ill-treatment to cry out against. It is rendered formal and unreal, a matter of blackboard diagrams, because sight is lost of its natural office, - the expression of linear relations in our actual environment. These difficulties will never be reached and overcome by more and higher pure mathematics, but by a practical preparation on the teacher's part through study of those sciences in which algebra and geometry are used. A sound first-year course in astronomy will incidentally invest solid geometry with an interest and obvious value that it cannot possibly otherwise acquire. Devise a diagram of your own to prove that proposition in Appendix J of Davis' Physical Geography, "If a body revolves without rotation, every part of it is subject to equal and parallel centrifugal forces." Begin now to make a collection of all the problems of this sort that you can find. Practical questions in physics will go far toward rescuing algebra from its present detested status.

I had slowly counted 12 when we heard a low thud, as if the rock had struck the edge of the cliff, then a fainter and fainter echo till the last rumbling seemed to die away in the depths of the nether world.

What was the probable depth of the barranca where the basalt boulder was rolled in? When you come to teach, give your boys and girls this from a real book of real travels, and observe its effect on their opinion of quadratic equations. The algebra book is yet to be written in which the author's first endeavour shall be not to make the book merely "interesting," or easy, or exhaustive, but to reveal to its readers the common-sense of algebra.