involved. We are doing for the whole of algebra to-day exactly what one of the most recent texts does in its attempt to introduce problems in physics. As an example of putting the cart before the horse pedagogically, the following is hard to beat. A set of problems on moments is introduced in this way: "When a straight bar is supported at some point, O, (Fig. 1) and masses m, m2, etc.,

are hung from the bar as indicated in the figure, it is found that when the bar is in equilibrium, the following equation always holds:

Then follow simple problems on teeter boards, etc. The most general equation possible first, and then simple numerical cases! If we are to introduce the equation of physics in algebra, let us give the experimental basis first; let us follow that with numerous numerical problems, and gradually lead up to this general equation. Or, if it seems better, we may give all the algebra needed, and leave the physics to the physics teacher. Let me give an example of what can be done, one so simple that it can be given to pupils before they reach the high school.

In this way we get the general equation np=c, in which n, p. and c represent numbers. Then there are other series of simple examples leading up to the equations

n =/; and p = n

and there is an algebraic relation between these three equations. The area of a rectangle leads to a similar equation -a, in which, as before, I, w, and a are always numbers.

Now let us turn to physics and notice the equations of this particular form.

(In every case the letter stands for a number.) M=m, [Momentum mass X velocity.]

pvc [Boyle's Law.]

W=Fd [work force X distance.]

I contend that if a pupil has the generalized arithmetic indicated above, he has all the algebra he needs for his physics; but abstract additions, subtractions, multiplications, and divisions, factoring, G. C. F.'s, and L. C. M.'s, the theory of exponents, radical equations, not to mention imaginaries, give no inkling that algebra is primarily a generalized arithmetic.

I have yet to find a pupil studying electricity, to whom the expression "difference of potential" means anything until a number of experimental cases have come under his observation. No amount of explanation will do the business; but in some way a little explanation coupled with considerable experience gradually leads to a grasp of the idea.

In algebra, pupils do not realize that the letters represent numbers. We may insist that pupils write "Let the number of units," instead of "Let r distance;" but this will not answer; the pupil looks upon this as mere quibbling. I believe, on the other hand, that, if we have series of problems, beginning with numerical examples and ending with the most genera! expression possible in each case, the pupil will soon see that the letters in algebraic expressions represent numbers, because he has followed the real derivation of the formulae.

What would a first year algebra, centered about the problem in the way suggested above, look like? I shall leave you to answer this question; I simply hope that I shall make enough of an impression for you to notice, as problems come up in your algebra classes, how little and simple manipulation is required. Find a complicated case of addition, subtraction, multiplication, or division in the solution of a problem, if you can. Notice the

extent to which you need factoring. Keep a lookout for complicated cases of L. C. M's and G. C. F's. If you don't find problems giving sufficiently complicated manipulations to suit you, make up such a problem; but, if you have any regard for your pupils, don't try it on a first year high school class.

I hope before long to see an algebra based on this principle. I hope to see also a great variety of really practical problems, problems which the pupils will regard as practical; but, while this body of practical problems is gradually growing, we might as well use those we have now. I have heard teachers sneer at those old clock problems. Were it not for the fact that I should prefer a really practical problem, I can see no objection to the clock problems. A clock is certainly within the experience of the ordinary first year high school pupil, while the quantitative side of physical phenomena is not, unless the school itself gives a sufficient amount of laboratory experience. A great number of our practical problems will probably be derived from physics; but it is the height of folly to use them unless they are preceded by ample laboratory work. The most important part of the work is the derivation of the general formula from the laboratory experience. The substitution in this general formula is a matter of secondary importance. I grant that it is a necessary part of our work, of course; but a pupil, who has derived his own formula from his laboratory measurements, will have no difficulty in substituting for special cases. The point I wish to emphasize is this: In our desire for practical problems, let us be careful to keep within the real experience of the pupils. A paragraph on Boyle's Law in the algebra is not a sufficient introduction to justify algebraic problems which involve this law. A clock problem would be better.

Mathematics has been taught for centuries. We might think that the method of teaching would be nearly perfect by this time. The fact that only recently have instructors varied to any extent from the old Greek geometer, Euclid, is evidence that improvement has not been very rapid. All our mathematics, as we have it to-day, was put in shape for us by philosophers, by men who think in general terms. The history of Greek philosophy and the history of Greek mathematics deal with the same persons. And in modern times D'Alembert, Descartes, and Leibnitz are equally great in philosophy and mathematics. After these philosophers got their mathematics, they stated their conclusions in a logical order, not by any means the order in which they

reached these conclusions. We foolishly try to teach their generalization to first year high school pupils, on the theory that if the pupils once get these generalizations, they will be able to use them. The fact is that ninety-nine out of every hundred never think of applying the generalization when the time comes, and don't know how to do it when it is suggested to them. The contention of this paper is that the pupil should make his own generalizations; that he should make the generalizations before he applies them; and that only after he has had long experience in reaching general statements from a number of specific examples, should he think of taking any other person's generalization for granted.

If it is claimed that this is really a yielding to the demands of a practical age clamoring for a brand of mathematics that will "do things," I confess that it is; and I trust that the demands will be strong enough to force the exclusion from our elementary course in arithmetic, algebra, and geometry, of those sections of the subjects which do not succeed in "doing something" at the time they are introduced. And I express this desire. not simply because I want the mechanic and artisan to receive what they need, but because I believe this will be the best possible training for the abstract mathematician. The material demands of a practical age are actually forcing better pedagogy in abstract mathematics.

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CONSTRUCTION OF CONIC SECTIONS BY PAPER-FOLDING. BY ALFRED J. LOTKA,

Research Laboratory, Laurel Hill Works, L. I.

A method has been described for constructing a parabola as the envelope of the creases formed on folding a piece of paper in such manner that a fixed point always falls upon a fixed straight line.

The other conic sections also can be similarly obtained, if for the straight line a circle is substituted**, as is shown by the ac

FIG.I.

companying examples (Figs. 1 and 2), and by the following analytical demonstration.

Referring to Fig. 3, let C be the center of the fixed circle, and F the fixed point.

*S. Row (W. W. Beman and D. E. Smith) Geometric Exercises in Paper-Folding, 1901, D. 116.

**It is, of course, necessary to use translucent paper (tracing paper), or, if using opaque paper, to mark the fixed point on the back of the sheet, and on the edge of a perforation made in the same.