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cardioid twice the size of the inner envelope, but this at once, although gradually, changed into a curve that became more and more curtate as the pen receded farther from the center at each revolution, until, in the middle of its compound period, when cos + cos me was equal to zero, it momentarily drew the arc of a circle with the radius 2. After this the lobes of the curve repeated themselves in inverse order, while their axes kept on swinging in the same direction.

A study of Fig. 1 shows that the points of intersection of the lobes are arranged in radial lines at equal angular intervals, and that the points of tangency of the curve with the two envelopes are also spaced equiangularly.

The Outer Envelope a Cardioid.-There is a second way of drawing an envelope that is a cardioid. In the first case we placed the pen at the center of the disk when the phase of each of its two harmonic components was 90°. Now let us make the phase 0° at the center. The first component A, if used alone, will then trace the circle p = sin 0, and the two together will trace p = sin Proceeding as before, we find the envelope

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and p1 + sin @ or (1 + sin 0) 2, which are identical, or rather coincident, the first being traced as usual by the positive extremity of p, say by point P, and the other by a point P' at the constant distance of 2 from P in the negative direction of p. There is then practically only one envelope, which we may in

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We then have a rosette in the inner envelope of Fig. 3 and in the outer one of Fig. 4, using the latter expression in the mechanical sense. In the mathematical sense, however, there are two envelopes in Fig. 4, not coincident, but lying at

right angles to one another. The direction of the mathematical axes has also undergone a change. Their position in these and subsequent figures may readily be deduced from the respective equations, and will for that reason no longer be referred to.

When n = 3 we see that the usual inner envelope is a rosette in Fig. 5, and that the outer one in Fig. 6 is an equal one. Fig. 7 shows a septifolium as an inner envelope. The outer one was not drawn because it would have been almost totally black, as Fig. 6 leads us to suspect, on account of the great number of its close and overlapping lines.

From a mathematical point of view all these seven figures present envelopes that are rosettes. When the common phase of the two components A and B is made 90° at the center of the disk, as in Figs. 1, 3, 5, 7, we have two envelopes, an inner one which is a rosette, and an outer one in which the radius vector of the inner one is increased by 2. When the common phase is 0° at the center, as in Figs. 2, 4, 6, there are also two envelopes, both being equal rosettes. They are coincident when n is odd, but crossed equiangularly when n is even. From a mechanical standpoint, the first class of figures may be said to have rosettes as their inner envelopes and the second to have corresponding equal rosettes as their outer envelopes, the number of lobes being doubled in the latter case when n is even.

The Ratio of the Periods of the Components, or the Value of m.-In all the cases presented m was taken as 15/16 or 16/15, the harmonic component A making 15n revolutions and B 16n, while the disk rotated 15 or 16 times. The number of revolutions of the disk, 15 or 16, must be 1/n that of one of the components. We may select either except when n is a factor of the one used, for then, as soon as the pen has run through one complete compound cycle, it will begin to retrace the curve already drawn, so that the figure will present a disappointing appearance of incompleteness, since it will have only one nth as many lines as it ought to have. For this reason the disk had to make 15 turns for n = 2 and 16 for n = 3. For n = 7, 15 were made, but 16 would have done equally well. For n = 5 (not shown) they had to be 16.

The Starting Phases of the Components.-When the phases of the components A and B were 90° and the pen was set down at the center of the disk, the inner envelopes it traced were the rosettes shown in Figs. 1, 3, 5, 7. The identical figures, only turned at right angles, were drawn when the pen was started in phase 0° on the mechanical Y axis at the distance + 2 from the center, that is, at the upper end of a lobe. The reason is that in a quarter of a turn of the disk one of the components A or B advances exactly one or n quarters of a period also and the other only one-fourth of 1/15 or 1/16 more or less. This difference is insensible in practice when m - 1 is very small. In like manner the identical "outer-envelope" rosettes, turned at right angles, resulted, Figs. 2, 4, 6, when the pen was started on the Y axis at the distance + 2 in phase 90° instead of at the center in phase 0°. From this it follows that the pen may be started in any equal phase a of its components and set down on the mechanical Y axis at the


distance 2 sin a from the center in the last case and 2 cos a in the first, in order to draw the same respective rosette, which will then be turned through the angle a on the disk. For this reason we might call the rosettes in Figs. 1, 3, 5, 7, "cosine" rosettes and those in Figs. 2, 4, 6, "sine" rosettes.

Transition Envelopes.-The principle just stated may be applied to show the transition from the (mechanical) outer- to the inner-envelope cardioid. Thus Figs. 8-12 are intermediate between Figs. 2 and 1. In all of these seven figures the starting point was at the center of the disk, but the phases were taken at 15° intervals from 0° to 90°. In Fig. 2 the phases of the pen at the start were 0°. In Fig. 8 the phases were 15°, in Fig. 9, 30°, in Fig. 10, 45°, in Fig. 11, 60°, in Fig. 12 75°, and finally in Fig. 1, 90°. The transition may thus be readily followed, and the axis of the envelope seen to swing round with uniform speed.

This identical series of seven transition envelopes might have been obtained by keeping the starting phases of the components at 90° and setting down the pen on the mechanical Y axis at the distance of twice the sines of 0°, 15°, ... 90° from the center. In this case the axes of the envelopes would have remained stationary on the mechanical X axis. When the pen is set beyond the distance 2 sin 90°, the envelopes become curtate. Their inner faces will be the outer ones in Figs. 2, 8-12, 1, while their outer ones will tend to become more circular.

Unequal Starting Phases of the Components. Instead of starting the pen with its components in equal phases, phase differences of any magnitude may be used. By studying the usual generation of Fig. 1 as given before, the initial position of the pen on the mechanical Y axis may so readily be deduced from the position it has there corresponding to the given phase difference, that numerical exemplifications are not necessary. The application to rosettes in general is also sufficiently obvious.

Finally Fig. 13 shows a transition envelope for an even value of n intermediate between Figs. 3 and 4.


By ERNEST B. LYTLE, University of Illinois.

This audience is no doubt familiar with the wonderful growth of high schools in the United States. In 1906 there were 52,394 pupils enrolled in the high schools of Illinois; in 1916 there were 102,870 enrolled, an increase of 96 per cent in ten years in Illinois alone. (Illinois High Schools, L. W. Smith, p. 9.) In the light of the recent marvellous growth in the number and community importance of our high schools, it is not necessary to argue here the great significance

1 Address read before the Ann Arbor meeting of the Mathematical Association of America, Sept. 5, 1919; also before the Illinois Section of the Association, Nov. 22, 1919.

of training high school teachers. Colleges must see the problem and attack it vigorously. In training able teachers our colleges and universities not only render the high schools and their communities a great service but they help themselves by improving the preparation of the students coming to them for further education.

Our particular present interest is to discuss the part mathematicians are now taking in teacher preparation and to suggest ways and means of increasing their service in this direction. Good teaching requires (1) knowledge of subject matter and (2) a sympathetic understanding of students. There is no upper limit to the amount of scholarship desirable, the only question is how much scholarship is possible under given conditions. The few people who believe one can know too much to teach well are looking in the wrong place for the trouble; it is not too much scholarship but rather too little sympathetic understanding of students which makes many failures in teaching. However true it may be that some great scholars are poor teachers because they have permitted themselves to grow unsympathetic and impatient with immature minds, yet sympathetic understanding of the difficulties of less mature persons is by no means incompatible with high scholarship; we all know many master teachers who possess both scholarship and sympathy for the learner.

While recognizing no upper bound to knowledge of subject-matter desired in a teacher, it is still pertinent to inquire, "How much mathematics is it reasonable to require of prospective teachers of mathematics under present conditions?" Today our colleges are quite generally requiring from 20 hours (one hour a day for two school years) to 30 hours (one hour a day for three years) in mathematics to obtain their official recommendation to teach in our high schools. In many cases prospective teachers elect more than the minimum requirement from the regular advanced courses in mathematics. The usual sequence of courses through the calculus is quite uniformly required; but beyond this there seems to be little absolute requirement and considerable variation in practice. Most colleges making any pretense at high school teacher training offer a special Teachers' Course which considers the values, methods, subject matter, texts, reform movements, and best literature of secondary mathematics. The University of Michigan, in 1893, was the first college to offer a course on the teaching of algebra and geometry and by 1912 there were 34 American colleges offering such courses (Bibliography 2, p. 6). In 1916 there were 40 out of 100 selected colleges and universities offering some such teachers' course (Bibliography 8, p. 395). When offered such a course is required, or very strongly advised, for recommendation to high school mathematics positions. Most students elect a course in the theory of equations and determinants as an advanced course in algebra; not quite so uniformly they elect a course in projective or modern geometry. You are no doubt familiar with the discussions in the MONTHLY on the value, methods and content of such an advanced course in geometry. As brought out in these discussions the value and importance of such a course in modern geometry is not 1AM. MATH. MONTHLY, Jan., 1914, p. 30, 31, 32; Feb., 1914, p. 63; Apr., 1918, p. 159.

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