I propose in these papers to demonstrate, or rather to indicate the nature of the demonstration of, the proposition that—All Geometry, whether projective or metrical, may logically be regarded as merely a particular concrete application of the general theory of Order, as defined by Boundaries.

It is true that Geometry, as ordinarily understood, is something more than this; it is concerned with the actual subjective conceptions we entertain of Space, and so far provides matter of discussion to the Psychologist; and it is concerned with the actual objective measurements we make upon physical bodies in Space, and so far concerns the Physicist; while in both these respects it extends beyond the region of pure Logic into those of Epistemology and Philosophy generally. But so far as the pure Mathematician is concerned, every geometrical theorem is embraced in the Theory about to be discussed.

In its most simple and general form the proof by recurrence of a formula containing constants and one variable consists in giving a number of successive integral values to the variable, and showing by trial that the formula is true for those values. The next step is to show that the form of the result remains the same when n+1 is substituted for n, or that each result depends upon the preceding one. If this second step is neglected, as it often used to be, the proof seems to be invalid.

This proof by mathematical or successive induction has been very generally accepted by mathematicians and neglected by logicians.

While it is clearly impossible for the average high school teacher of mathematics to become familiar with all the modern branches of this subject, it is desirable that he should not be totally ignorant of any extensive branch. The views of a number of eminent mathematicians often furnish one of the simplest as well as one of the most reliable introductions to the nature and the scope of a difficult subject. The following list of quotations has been prepared for the purpose of providing such an introduction for one subject. The list could easily be extended, but the variety and the standing of the mathematicians quoted are perhaps sufficient to inspire confidence. The qnotations are arranged chronologica1ly, beginning with 1892. several of them were published in the American Mathematical Monthly, volume x. 1903, p. 87