point. In this way the work may be made as easy or as difficult as desired, and may be made to fit in with any schedule of laboratory time.

6. That each student be provided opportunity to study the transmission and utilization of energy, to trace and measure the loss in all these processes and to study ways of reducing such losses. The facilities for this study will depend greatly upon the equipment and power supply of the school. In any case, it is not difficult to provide various small gear, belt, and sprocket machines which may be operated by means of cords passing around drums attached to the shafts and supporting scale pans. The sprocket wheels of an old bicycle which may be obtained. at small cost from any junk shop provides apparatus for several excellent exercises. These small machines should always be made up from commercial parts and should not be fitted too freely with special ball-bearing or friction reducing devices. They should be large enough for some actual service. Let the pupil deal with the conditions which obtain in practice. The one experiment of this sort most often used, that of the efficiency test of a system of pulleys, done as it usually is with an eightounce balance and special cone-bearing pulleys giving an efficiency that is absurdly high seems to me in these days to be almost if not quite worse than nothing. If the experiment is done let it be with a half-inch rope, a heavy weight and the pulley sheaves of common use.

Where the equipment of the school will permit, this line of study may be extended to a study of speeds and friction of shafting, the general details of the school power plant, efficiency tests of electric and water motors, etc. The teacher should not feel, however, that this larger equipment is absolutely essential to good instruction in the sort of mechanics here suggested. Excellent results may be obtained with the small devices mentioned above which are surely available for every laboratory.

(NOTE: In conclusion, to illustrate the type of instruction suggested as needed, the speaker exhibited lantern slides showing models of a simple truss, roof truss, crane, and arch now in use at Pratt Institute, also of a screw jack arranged for an efficiency test, and of an apparatus for studying rotary motion, and the kinetic energy of a rotating body. These models and suggestions for accompanying laboratory exercises will be shown in later numbers of SCHOOL SCIENCE AND MATHEMATICS.)

CURRENT TENDENCIES IN SECONDARY MATHEMATICS IN

ITALY.*

By J. W. A. YOUNG.

The University of Chicago

In the world wide forward movement in the teaching of mathematics, in the midst of which we stand, each nation must deal with this teaching according to local conditions, yet many of its problems are common to all nations, and each finds encouragement and moral support in the progress made beyond its borders. Consequently it is not strange that I can begin this brief report which I bring you from Italy today with the statement that our own American movement is not unfelt there, but is cited at the beginning of the directions to the Royal Commission, of which I shall speak later.

The following table indicates the names and sequence of the Italian schools:

I discuss the program in mathematics of the classical schools only, in which somewhat less mathematics is taught than in the others.

The subjects taught and their distribution are indicated below:

*Address before the Mathematics Section C. A. S. & M. Teachers, Nov. 30, 1906.

21

1. The time allotted to mathematics is increased. (The additional time is given to algebra and trigonometry.)

2. The time for "fusion" of plane and solid geometry is reduced from two years to one.

(The interweaving of plane and solid geometry has been urged by the Italians for some little time and in the curriculum of 1900, at the instance of the society of teachers, Mathesis, opportunity was given to do so during two years.)*

3. In the curriculum of 1900, the geometric treatment of measures is assigned to the second year of the Liceo and it is forbidden to make applications of arithmetic or algebra to geometry earlier. In the new curriculum the treatment of measures is moved back to first year of the Liceo and the prohibition no longer appears.

4. The new program gives much fuller pedagogic instructions than the old, including the direction that the exercises should be so selected as to show the utility of mathematics in the common affairs of life and in the study of the phenomena of nature. No graphs are included.

This brief sketch may serve to give some idea of the condition of affairs found by the Royal Commission recently appointed. to study the whole subject of reform in secondary schools. On March 27, 1906, this Commission made public the first results of its work in the form of a long list of questions covering the entire field and addressed to all interested. Many of the questions relative to mathematics are of a decidedly "leading" character and no better account of the trend of thought concerning the teaching of mathematics in Italy can be given than to reproduce these questions (in condensed form) herewith:

I. In arithmetic, should not less stress be laid on enunciation of rules and definitions and abstract descriptions of processes?

2. What place should be given to theoretic arithmetic? In so far as it consists of demonstration of processes of arithmetic, should it not be treated as an application of algebra?

3. Would it not be better if geometry were taught by the same instructor throughout the secondary school, instead of one in the ginnasio and another in the liceo?

*For specimens of "fused" treatment see: Paolis, Elementi di Geometria, Turin, 1884. Lazzeri e Bassani, Elementi di Geometria, Livorno 1898. Veronese, Elementi di Geometria, 2nd ed., Padua, 1900. Méray, Nouveaux Éléments de Géométrie, 2nd ed., Dijon, 1903.

4. In teaching geometry, would it not be desirable to lay more stress on graphic exercises, both in solution of problems and in verification of propositions demonstrated?

5. Would it not be well to omit proof of propositions already intuitively evident to pupils?

6. What has been the effect on mathematics of the freedom of election between Greek and mathematics in the last two years of the liceo?

7. In algebra, would it not be better, instead of beginning with generalities concerning negative numbers, followed by operations on polynomials, powers, radicals, etc., to proceed at once to solution of equations, developing the theory as required by them?

8. What is your opinion of the so-called "fusion" of plane and solid geometry?

9. What place should be given to trigonometry? Is it desirable that the little which is given should be distributed, as now, over three years?

IO. Should proportion be treated according to the 5th Book of Euclid? How should the concept of irrational number be introduced?

II. Does it not seem well to you that place should be made in secondary instruction for the fundamental notions of graphic representation of functions and the concept of first and second derivatives?

12. What weight do you give to the wide-spread opinion that only pupils with special aptitude can profitably study mathematics in the secondary school? In how far is this opinion due to defects in the instruction?

13. What means are best to create and augment interest in mathematics? Would it not be efficacious, to teach from the very beginning, the various parts of the subject in close relations with their applications, putting the purpose they serve into the foreground?

14. To what extent are notes on the history of mathematics useful, either in exciting the interest of pupils or contributing to their culture?

15. Should written examinations in mathematics be restored in all classes?

16. What ones of the texts in current use do you regard as didactically the best?

What the character of the replies will be and what their effect on instruction will be, remains to be seen, but that the Commission itself has its face turned in the direction of less prominence for the abstract, more use of the concrete, introduction of graphic methods, and emphasis upon applications, is sufficiently evident.

NOTE ON THE GRIDIRON PENDULUM.
BY C. F. ADAMS,

Central High School, Detroit.

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a

The gridiron pendulum is practically out of date; but for many years it has been a favorite illustration in text books of difference in expansion of different metals. Did it ever occur to many of us that the nine rod pendulum of brass and iron is impracticable, if not impossible? A simple calculation will make this clear. Let L-length of the pendulum from a to b, point of suspension to center of oscillation.

L-as+hc+tb, all iron. Let length of

brass.

The extra length of iron in the pendulum= I also.

If we take the coefficients of iron and brass as 12 and 18 (millionths) we have the equation 12(L+1)=181, the expansion of the iron. and brass being equal.

1=2L

This means that the two brass rods on either side must equal in length twice the length of the pendulum, which is of course impossible. If, however, we use 11 and 19 as representing the coefficients of expansion, which is perhaps more nearly the truth.

1=1.4L.

The construction of a genuine gridiron pendulum on this supposition is virtually impractical. I wonder how many genuine gridiron pendulums the reader has ever seen!