the schools preferred the dogmatic books, but from 1765, when Anitschkof1 published his series of mathematics, the demonstrative method passed definitely into the Russian works on elementary mathematics.

About this time (1759), the Gymnasia first sent pupils to the university. Before this the students had come exclusively from the seminaries and ecclesiastical academies.

By 1786 the public schools had been organized. The parish schools prepared pupils for the district schools, those of the district for the Gymnasia, and the Gymnasia for the university. In the so-called principal public schools, which from 1781 to 1786 had not yet been consolidated with the Gymnasia, geometry was taught in the highest class. The text-book3 used represented the transitory period between the dogmatic and the demonstrative method.

At this time (1786), a special school, the Institute of Pedagogy, was created for the preparation of teachers for the public schools. The Gymnasia later took over this function, the Institute preparing only for the Gymnasia. Each school, including the university, had as its duty the preparation of teachers for the inferior schools which preceded it in the school system.

The program of the Gymnasia at St. Petersburg from 1811 to 1816 embraced in the first class, arithmetic and algebra; in the second class, geometry and plane trigonometry; in the third class, applications of algebra to geometry, and the conic sections. In the fourth or highest class was included the applications of mathematics to physics.

To recapitulate, as late as the middle of the seventeenth century geometry was taught in Russia after the method of medieval instruction. It was embraced in the quadrivium and was entirely practical in its nature. By the beginning of the eighteenth

1 One of these was on geometry, theoretical and practical, on the lines of Wolf in Germany.

'Beer und Hochegger, op. cit., p. 105. Also see Bobynin.

3 Bobynin gives its title, Le petit manuel de géométrie. It is divided into three sections: The measure of lengths, the measure of areas, and the measure of solids. Sixteen theorems are demonstrated. The rest is composed of definitions and problems, for the most part practical and dogmatically explained.

century the practical needs of warfare stimulated an interest in geometry and it was taught then in special schools with other branches of mathematics. It found a place in the newly founded Gymnasia in 1725, where it was valued more for its scientific character. Text-books in geometry began now to be used, the first appearing in 1732. The ecclesiastical schools took up the study in 1743, but as an elective. After 1786, the Gymnasia increased their functions by preparing teachers for the lower schools, and the teaching, which had formerly been characterized by memory work and learning by rule, began to lose its dogmatic character and appealed more to reason. It seems plausible that this change was finally accomplished by means of the new aim given to the teaching in the Gymnasia, that of the preparation of teachers.

HOLLAND

The seventeenth century, which was the most brilliant in the history of Holland,' was also the epoch which produced its great geometers. During the seventeenth and eighteenth centuries about thirty books on geometry, forty on arithmetic, and twenty on applied mathematics were published.

The teaching of geometry in Holland began with the practical. The relation of geometry to drawing and perspective, and its applications to engineering, surveying, and the marine, were seen and appreciated. The Academy for Engineers was founded in 1660, and practical geometry was taught there from the beginning, the theoretical part being considered of minor importance.

Some of the treatises of the seventeenth and eighteenth centuries show the same practical trend. Ozanam in 1698 published at Amsterdam a complete course of mathematics. It was divided into five parts as follows: I, Introduction to mathematics, and the "Elements" of Euclid; II, Arithmetic, trigonometry, and sine tables; III, Geometry and fortifications; IV, Mechanics and perspective; V, Geography and gnomonics (dialling). Here we see attention given to Euclid, but the aim of the series is toward the practical. The work of De Graaf 1 Cardinaal, L'enseignement mathématique en Hollande. In Ens. Math., 1900, pp. 317-339.

2 Such men as Jean de Witt, C. Huygens, F. van Schooten, and J. Hudde.

(1694) treats essentially the same subjects, but the "Elements" as such is not included. These treatises are on the same lines as those followed later by Wolf in Germany. During this period works on geometry alone appeared, and usually under the name of the 'Elements" of Euclid.

There is evidence that there was some independence of Euclid in the last half of the eighteenth century, for one geometry published announced a departure from the Euclidean method. This was the text of J. H. van Swinden (1746-1823) of Amsterdam. The texts of J. de Gelder (1765-1840) show a rigidity of method that indicates interest in the logical side. He wrote two geometries, the "Principles of Geometry" and the "First Principles of Geometry" (1827). The more advanced of his geometries was intended, as the preface indicates, for use in the university and for those interested in mathematics. The second was for use in the Latin schools.1 In the preface of the latter work it is stated that the decree of 1815 required from future students of the university some knowledge of mathematics. We thus see a demand for preparatory schools in the modern sense. But the preparation in geometry was not thorough. The university did not continue the work from where the schools left it, but reviewed and enlarged upon it. We conclude therefore that in the first half of the nineteenth century the mathematics of the Latin schools was restricted to first principles.

Mathematics was also taught in the Gymnasia, the programs of which were regulated by the royal decrees of 1815, 1816, and 1826. The mathematical programs of this class of schools were more extended than those of the Latin schools, although both of these types of schools prepared for the university. Parvé2 says that in the programs of the Gymnasia the limits of algebra and arithmetic were quite well defined, but in geometry there was nothing definite. As a consequence solid geometry disappeared from the programs. This lack of uniformity may explain why the university had to give its attention to the elementary work in geometry. Parvé also says that the methods in these schools were often defective. There was much learning by

'Similar plans of grading text-books are employed by the text-book writers of to-day.

2 Cardinaal (op. cit., p. 328) refers to a report of Parvé, inspector of secondary schools, in 1850.

heart, and the pupils were forced to demonstrate mechanically. Parvé sought to unify the work in the preparatory schools and so he set definite limits to the elementary mathematics. For the subject-matter of geometry he recommended the following: Fundamental notions, congruence and similarity of plane figures, circles, areas of plane figures, regular polygons, and the geometry of solids. Thus it was not until the middle of the last century that the subject-matter of geometry in the secondary schools of Holland assumed its present proportions.

Thus the teaching of geometry in Holland up to the beginning of the eighteenth century was essentially of a practical nature. During this time we find little reference to the teaching of Euclid, although the "Elements" went through several editions. During the eighteenth century there appeared some texts showing an independent treatment of logical geometry, and by the end of that century the teaching of geometry was well established in the secondary schools. Up to the middle of the nineteenth century the method of teaching was that of learning by heart, which, it is safe to suppose, continued even later.

OTHER EUROPEAN COUNTRIES

Geometry did not gain a foothold in secondary instruction in Austria2 until the middle of the nineteenth century. During the sixteenth century the Jesuits dominated the field of superior education and mathematics was sacrificed to the benefit of the classics. After the expulsion of the Jesuits in 1773, petty reforms were introduced, but the teaching of mathematics was not yet systematized in the Gymnasia. Some of these, however, in 1775 offered courses in geometry that treated the subject in a practical yet scientific manner.3 In 1805 there was a plan to teach mathematics in the two higher classes of the Gymnasia, but it was not carried fully into effect, for in 1841 geometry was not yet in the program of studies actually in use. But the idea was

1 Cardinaal, op. cit., p. 328.

2 Simon, L'enseignement des mathématiques au gymnase autrichien. In Ens. Math., 1902, pp. 157-166; Baran, Geschichte der alten lateinischen Staatschule und des Gymnasiums in Krems, pp. 115-135; Beer und Hochegger, op. cit., Vol. I, pp. 266-532; Paulsen, op. cit., p. 695. 'Monumenta Germaniæ Pædagogica, 30, pp. 106-128.

being favored and the belief was getting stronger that the teaching of mathematics should be in the hands of specialists. In 1848 the Gymnasia were entirely reorganized. The course of study of eight years was divided into two courses of four years each. The first besides being preparatory was also a course in general culture. The last treated the subjects in a more scientific manner and prepared for the university. Before this time mathematics received one-half an hour a week. Now mathematics received three hours, and natural science, which had not been taught before, received the same number. The first teaching was characterized by emphasizing quantity rather than quality. By 1856 the teaching became more profound and systematic, and in 1884 the teaching of mathematics ranked equally with the languages and history.

Bulgaria1 also had a late mathematical development. This began after the Turkish yoke was thrown off. It was not until 1839 that a work of any kind was printed in the Bulgarian language, the first geometry appearing in 1867.2 It was not until 1850 that the Gymnasium programs gave geometry its place as a separate subject.3

In Switzerland, the study of geometry in the secondary schools was slighted until the beginning of the eighteenth century, when it began to take rank with the other subjects. In the Gymnasium at Basel, in 1717, under the influence of John Bernoulli, geometry, together with geography and history, occupied a place on the school program. As Sturm's "Mathesis juvenilis" was used, one can judge that the work was of a character adapted to young minds.

THE UNITED STATES

As was the case in Europe, geometry was first taught in the United States in the universities, and so continued until after the middle of the nineteenth century. Harvard College was 1 Sourek, L'enseignement mathématique en Bulgaria, Ens. Math., 1905, pp. 257-270.

2 By V. Gruev, printed in Vienna.

3 Rein, Encyklopädisches Handbuch der Pädagogik, I, p. 804.

4 Burckhardt, Geschichte des Gymnasiums zu Basel, pp. 87-319.

5 Cajori, The Teaching and History of Mathematics in the United States. Unless otherwise stated, Cajori is the authority here referred to.