shown in the working out of the curriculum and in the limits assigned to the selection of subject-matter. Parallel with this the study of mathematics has been carried on with greater intensity, which has resulted in a certain isolation in the several mathematical branches. In the United States and England this is especially noticeable, while in Germany and France the relation between the several branches has received some recognition. We have shown that for the most part the teaching of geometry in the early secondary schools of the various countries was associated with its applications in surveying. The tendency in the nineteenth century has been away from any such applications. It is only within the last few years that we hear of demands being made for a closer relationship between the teaching of pure and applied mathematics in the secondary schools.

The secondary schools of the United States began to prepare more generally for the university by the middle of the nineteenth century. In England, also, the "Public Schools" were beginning to require the teaching of Euclid, which means that those schools were now preparing in geometry for the university. In Germany, before this time, the Gymnasia were given the sole privilege of thus preparing students. We can therefore see a special reason why the study of mathematics has become more intensive.

As regards changes in method, it is hard to say just when demonstrative work, as we have it to-day, began. The early Greeks employed the Socratic method in their teaching. Euclid showed the world a system of demonstrative geometry. In the early universities, the pupils learned from dictation or lectures. In the secondary schools in Germany in the sixteenth and seventeenth centuries similar methods were employed. The students usually learned the work by heart and recited it. During the seventeenth century in Germany, demonstrative work, as opposed to working by rule, began to be more emphasized, and in the next century the custom of explaining propositions was common. In Russia the eighteenth century showed this same transition. The work previously had been taught dogmatically, the students learning by heart and working by rule. Now they were taught to demonstrate their propositions and apply them rationally. In England the students practiced the demonstrative method in their study of Euclid. Though the nineteenth

century saw the dogmatic method formally discredited, yet traces of it remained in those countries where the text-book has been of prime importance and hence where the pupils have had the tendency to learn by heart. England and the United States especially come under this heading.

In this development three educational aims have been apparent, the practical, the logical, and that of preparing for advanced work in mathematics. These have been associated in greater or less degree in the various countries and institutions, considered.

CHAPTER VI

PRESENT-DAY TEACHING OF GEOMETRY

This chapter will deal primarily with the teaching of geometry in secondary schools. In the United States the term secondary school is commonly applied to the high school. It represents the last four years' work in a twelve years' course. All students take the same work during the first eight years whether they intend going into the high school or not. In England, France, and Germany the secondary school is not the continuation of the elementary school as with us, but a separate institution which prepares usually for general culture or the "higher callings." The boy generally enters these institutions in those countries at the age of nine and remains nine years.

GERMANY

The three classes of secondary schools in Germany1 are the Gymnasia, the Realgymnasia, and the Oberrealschulen. These all have courses of nine years. In some of the smaller cities the last three years are omitted and the institutions are called the Progymnasia, the Realprogymnasia, and the Realschulen. The Gymnasium is the classical institution with both Latin and Greek. The Realgymnasium does not offer Greek, and the Oberrealschule offers neither Latin nor Greek. The last two institutions lay more emphasis on mathematics. In the Prussian Lehrplan2 for 1901, it is prescribed that the Gymnasia give thirty-four periods per week to "Rechnen" and mathematics, the Realgymnasia forty-two periods, and the Oberrealschulen forty-seven periods.

1 For an interesting account of the present teaching of mathematics in Prussia, see Young, The Teaching of Mathematics in the Higher Schools of Prussia.

' Centralblatt für die gesammte Unterrichtsverwaltung in Preussen, 1901 p. 473ff

In the Gymnasia, geometry is begun in the third year (Quarta), when the pupil is eleven or twelve, the work being of a propadeutic character. Two periods a week are given to this work. Here the student is made familiar with the fundamental conceptions of straight line, angles, and triangles, and does construction work with the compasses and rule. By the end of the seventh year (Obersecunda), the student has completed plane geometry, and during the next two years has completed solid geometry. Algebra is begun in the fourth year and continued throughout the course. Trigonometry is begun in the sixth year and also continues parallel with the other mathematical subjects. The mathematical course is practically a unit, geometry being begun first. Then algebra enters and finds application in the geometry. Trigonometry is begun when similar triangles are being studied in geometry, and the two subjects are made mutually dependent on each other.

In the Realgymnasium, geometry is begun also in the third year, but more time is given in this school to mathematics, so we find solid geometry entering into the work in the sixth year, when the boy is about fourteen years of age. In the Oberrealschule the first work in geometry is given in the second year (Quinta). The work on the whole is like that in the Realgymnasium, but in the higher classes it is more extensive. Plane geometry is studied even after solid geometry is begun in both of these latter schools, but the work is not the standard Euclidean geometry, for it includes the theory of harmonic points and rays, and symmetry.

The schools in Frankfort-on-the-Main do not follow strictly the above programs. In the Goethe Gymnasium, for instance, we find these differences: Geometric drawing and mensuration are begun in Quinta. Solid geometry is begun in Untersecunda and taught for four years. Plane geometry is also continued and becomes a study of the conic sections, analytically considered. The work more nearly coincides with that of the Realgymnasium described above.

The first work in geometry in the German secondary schools is entirely propedeutic, beginning in Quarta1 or Quinta according to the school. The pupil learns here how to use the proper in1 In the Goethe and Kaiser Friedrich Gymnasia in Frankfort, the writer observed very rigid logical work in Quarta.

struments and gains conceptions of the different geometric figures. The work of the following year links very closely with this preparatory work, but the logical now assumes prominence. The course includes the subject-matter of Euclidean geometry, but Euclid as a text has no place in the German schools. In fact, very little dependence is placed upon any text. The value of modern geometry is being recognized, for in the programs mention is made of harmonic points and rays, and the theory of transversals, and algebraic and trigonometric methods are freely applied to the work in geometry. In fact, there is such an interlacing of the work that one sometimes finds it difficult to name the subject that is being taught.1

In the German schools, the classroom is a place for instruction, not a place where the pupils "say" their lessons. The lesson in geometry begins by the teacher calling a boy to the single small blackboard behind the teacher's desk. The figure is drawn, the teacher meanwhile questioning either the boy at the board or the other members of the class. A single student rarely gives an entire proof except in resumé. The teacher is careful that all members of the class participate, the method being that of class instruction as opposed to individual instruction. The work is very thorough, if necessary the whole hour being given to a theorem with its applications and corollaries. Time is not lost, for the students understand the work, and when the foundations are once laid they are secure. Very little homework is assigned, the exception generally being in connection with some construction problem. With teachers who know their subjects, who know how to economize time, and who know how to instruct, the mathematical education of the German boy is well provided for.

FRANCE

The lycées2 are the great institutions for secondary education in France. They correspond to the German institutions just described, but with this difference: In Germany, the courses vary with the different institutions, while in France, as in the United States, one institution gives a choice of several courses.

1 This is particularly true of the upper classes.

2 In the smaller cities are found "collèges," which are also secondary institutions, but the lycées are of higher standard.