INTRODUCTION.

"The scientific part of Arithmetic and Geometry would be of more use for regulating the thoughts and opinions of men than all the great advantage which Society receives from the practical application of them: and this use cannot be spread through the Society by the practice; for the Practitioners, however dextrous, have no more knowledge of the Science than the very instruments with which they work. They have taken up the Rules as they found them delivered down to them by scientific men, without the least inquiry after the Principles from which they are derived: and the more accurate the Rules, the less occasion there is for inquiring after the Principles, and consequently, the more difficult it is to make them turn their attention to the First Principles; and, therefore, a Nation ought to have both Scientific and Practical Mathematicians." -JAMES WILLIAMSON, Elements of Euclid with Dissertations, Oxford, 1781.

THE preceding arraignment is nearly as pertinent to-day in this country as it was in England more than a century ago. But so far as Geometry is concerned blame no longer rests with the scientific mathematicians. Their investigations of First Principles have not only furnished us with Euclid in his purity, but have developed entirely new and equally consistent geometries, under postulates alternate to Euclid's petition of the angle-sum of a rectilineal triangle. Thus has been fulfilled what must at least have opened up as dim vistas to Euclid's mind when he discerned the necessity for assuming, or petitioning as the old geometers called it, his indemonstrable postulate.*

* Called variously the 5th postulate, or the 11th or 12th axiom.

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Still further, scientific mathematicians, besides offering the true Euclid in available text-books with desirable additions and extensions, have corrected several errors in definitions and demonstrations which constituted the sole blemishes in the most perfect work ever performed by a single man. There is no longer good excuse for teachers choosing texts which present the postulate as a common notion or axiom; to say nothing of such as baldly omit the whole doctrine of ratios and proportionality. There is a momentous difference between ratios and fractions, and text-books which present a proportion simply as an equality of fractions have set up a miserable cause of stumbling. They consider "merely a special case of no importance, whose only excuse for existence lies in the general case omitted."* Incommensurability is the rule, commensura

bility the exception.

On the other hand, when we consider Arithmetic and Algebra the cap of censure fits the other head. If our scientific mathematicians have furnished satisfactory textbooks in these subjects, I am not acquainted with them. All of us who are teaching mathematics must agree with good old Williamson when he complains, in the dissertation already quoted, that he found it more difficult "to make a rational arithmetician than an enlightened geometer."

Let me hasten to say that the apparently controversial tone of this preface springs from no polemical spirit. I approach the task I have set myself with utmost modesty; nay, oppressed by a sense almost of presumption in attempting to clarify what so many have left confused. But so sorely needed is a successful accomplishment of what I

* Catalogue Univ. of Texas, 1891-1892.

attempt, that an honest effort needs no apology. I wish also to explain that the present treatment takes its form from the immediate practical aim in view; viz., that of a syllabus for a rapid review of such ground of arithmetic and algebra as will best prepare for the study of what goes in our curricula by the name of "higher algebra," with special adaptation to the needs of that large portion of my classes who are taking the course in order to qualify as teachers in the public schools.

I write this Introduction, and dedicate the little work to the teachers in the common schools, however, in the hope of attaining a wider usefulness, in the way of awakening in some Practical Mathematician a desire to make rational arithmeticians" of the youths whose studies he is directing. It is proper to explain still further that, working away from any great library, I have been compelled to prepare this matter for printing without having time to procure a few published works which I would like to see before committing myself to publication.

I must not be understood as advancing anything new to mathematicians, though I know of no English text-book which consistently expounds and maintains the theories of number and algebra here presented. The work is addressed, not to mathematicians, but to inquiring students and teachers. A sound doctrine of number and its algebra seems to be left by our text-books to chance inference, or deferred to stages seldom reached in undergraduate courses of study. A straightforward development, comprehensible by beginners, of the number concept would be of immense service in mathematical instruction.

For six years I have given my classes the substance of this syllabus as the best explanation I could offer of diffi

culties which could not honestly be avoided. In July, 1894, I read in the current issue of the Monist an article by Hermann Schubert, writing in Hamburg, on Monism in Arithmetic, enunciating a unifying principle which he called the Principle of No Exception, referring it originally to Hankel. Of course some such principle is more or less clearly in the mind of every student of mathematics, but having never read Hankel's own statement, I cannot say whether his Principle of Permanence is substantially identical with the developing principle I set forth, or rather in line with the notion of algebra as "the science which treats of the combinations of arbitrary signs and symbols, by means of defined, though arbitrary, laws," *- the view of the famous Dean of Ely, and the long line of algebraists of whom he is the prototype. The bare statements of such a principle from radically different standpoints might be confusingly similar to one not fully alive to the fundamental variance. For example, in Schubert's statement of his Principle of No Exception, I recognized what I conceive to be a somewhat inadequate expression of the postulate I had called the Principle of Continuity (I still prefer this name as pointing with direct emphasis to its cardinal outgrowth- the conception of number as continuous), whereas in the next preceding issue of the same journal he is at utter variance with me in declaring that, "all numbers, excepting the results of counting, are and remain mere symbols, nothing but artificial inventions of mathematicians."

* Peacock's Report on the Recent Progress and Present State of certain branches of Analysis, in the British Association Report for 1833, p. 195. Cf. also Peacock's Treatise on Algebra, 1830, republished and enlarged in 1842.