plaints of examiners show; the effect on the examinee is a well-known enervation of mind, an almost incurable superficiality, which might be called Problematic Paralysis a disease which unfits a man to follow an argument extending beyond the length of a printed octavo page. Against the occasional working and propounding of problems as an aid to the comprehension of a subject, and to the starting of a new idea, no one objects, and it has always been noted as a praiseworthy feature of English methods, but the abuse to which it has run is most pernicious." 1

The interpretation of solutions-Algebra is generous, says D'Alembert; it often gives more than is asked.2 And it is one of the mysteries which teachers and text-books usually draw about the science, that some of the solutions of the applied problems are not usable, are meaningless.

But there should be no mystery about this. It is a fact, easily explained, that it is not at all difficult to put physical limitations on a problem that shall render the result mathematically correct but practically impossible. For example, if I can look out of the window 9 times in 2 seconds, how many times can I look out in I second, at the same rate? The answer, 4 times, is all right

1 Chrystal, Presidential address of 1885.

2 L'algèbre est généreuse; elle donne souvent plus qu'on ne lui demande.

mathematically, but physically I cannot look out half a time. Similarly, if 5 men are to ride in 2 carriages, how many will go in each, the carriages to contain the same number? Mathematically the solution is simple, but a physical condition has been imposed, "the carriages to contain the same number," which makes the problem practically impossible. A few such absurd cases take away all the mystery attaching to results of this nature, and show how easy it is to impose restrictions that exclude some or all results.

For example, the number of students in a certain class is such as to satisfy the equation 2x2 - 33x - 1400; how many are there? The conditions of the problem are such as to make one root, 20, legitimate, but the other, 1, meaningless. Algebra has been generous; it has given more than was asked.

Consider also the problem, A father is 53 years old and his son 28; after how many years will the father be twice as old as the son? From the equation 53+x=2(28+x) we have x = − 3. We are now under the necessity of either (1) interpreting the apparently meaningless answer, - 3 years after this time, or (2) changing the statement of the problem to avoid such a result. Either plan is feasible. We may interpret" 3 years after" as equivalent to "3 years before," which is entirely in accord with the notion of negative numbers; or we may change the problem

to read, "How many years ago was the father twice as old as the son." Most algebras require this latter plan, one inherited from the days when the negative number was less understood than now.

"Unlike other sciences, algebra has a special and characteristic method of handling impossibilities. If this problem of algebra is impossible, if that equation is insoluble, instead of hesitating and passing on to some other question, algebra seizes these solutions and enriches its province by them. The means which it employs is the symbol." The symbol " - 3," for the number of years after the present time, without sense in itself, is seized and turned into a means for enriching the domain of algebra by the introduction and interpretation of negative numbers.

The further interpretation of negative results, and the discussion of the results of problems involving literal equations, is a field of considerable interest and value; but since most text-books furnish a sufficient treatment of the subject, it need not be considered here.

Conclusion The few topics mentioned in this chapter might easily be extended. It would be suggestive to dwell upon the absurdity of drilling a pupil upon the two artificially distinct chapters on surds and fractional exponents, as our ancestors used to separate the "rule of three" from proportion-matters explainable only

1 De Campou.

by reviewing their history. The theory of fractions, the common fallacy in the proof of the binomial theorem for general exponents, the use of determinants, the complete explanation of division or involution, the questions of zero, of infinity, and of limiting valuesthese and various other topics will suggest themselves as worthy a place in a chapter of this kind. But the limitations of this work are such as to exclude them. The topics already discussed are types, and it is hoped that they may lead some of our younger teachers of algebra to see how meagre is the view offered by many of our elementary text-books, how much interest can easily be aroused by a broader treatment of the simpler chapters, and how necessary it is to guard against the dangers of the slipshod methods and narrow views which are so often seen in the schools. As algebra is often taught, there is force in Lamartine's accusation, that mathematical teaching makes man a machine and degrades thought, and there is point to the French epigram, "One mathematician more, one man less." 2

1 L'enseignement mathématique fait l'homme machine et dégrade la pensée. Rebière's Mathématiques et mathématiciens, p. 217.

2 Un mathématicien de plus, un homme de moins. Dupanloup. Quoted in Rebière, ib., p. 217.

CHAPTER IX

THE GROWTH OF GEOMETRY

Its historical position - Roughly dividing elementary mathematics into the science of number, the science of form, and the science of functions, the subject has developed historically in this order. Hence the chronological sequence would lead to the consideration of geometry before algebra, not only in the curriculum, but in a work of this nature. The somewhat closer relation of arithmetic and algebra, however, explains the order here followed, if explanation is necessary for a matter of so little moment.

Reserving for the following chapter, as was done with algebra, the question of the definition of geometry, we may consider by what steps the science assumed its present form. We shall thus understand more clearly the limitations which the definition will be seen to place upon the subject, we shall see the trend. which the science is taking, and we shall the more plainly comprehend the nature of the work to be undertaken by the next generation of teachers.

The dawn of geometry - The world has always tended to deify the mysterious. The sun, the stars, fire, the sea, life, death, number-these have all