were reduced from forty-seven to forty-four, and in 1892 this was reduced to forty-two.1 We thus see that the impetus given to the study of mathematics in 1816 did not last and this was because influences were brought to bear that made the classics predominant, and hence the time given to mathematics was correspondingly diminished."

In 1788 the Prussian government passed a regulation requiring all candidates to pass an examination before entering the universities. This was revised in 1812, but it appears, however, that the universities still admitted certain persons by special permission who had not passed these examinations. In 1844 an ordinance was passed that the Gymnasia exclusively were to prepare students for these examinations.3

During the first three decades of the nineteenth century the teaching of mathematics in the Gymnasia of Germany was not very intensive. Thus trigonometry was not taught as a distinct subject in Bavaria before 1860.* It has already been mentioned that during the eighteenth century the practical side of geometry was still emphasized and that Euclid as such was not taught in the secondary schools. To-day, in Germany, geometry is taught with reference to the other mathematical branches; the texts are not Euclidean in the strict sense of the term. As there was no sudden change in the method of teaching geometry during the last century, it is safe to judge that this unifying process in the teaching of mathematics was gradually evolved from the methods laid down by Wolf and his followers in the eighteenth century.

To summarize, the first logical geometry to be taught in Germany was in the universities. Previous to this time, the church schools taught only a few definitions and some practical exercises. In the sixteenth century, the secondary schools gave some little attention to geometry, but the work on the whole seems to have been optional. In the seventeenth century it was more widely taught in the secondary schools, where it seems to have been largely practical. During this century texts began to appear 1 Russell, op. cit., p. 312.

2 Pietzker, L'enseignement mathématique en Allemagne pendant le XIXme siècle, Ens. Math., 1901, pp. 77-78.

3 Ibid, pp. 78-79.

4 Günther, Ens. Math., p. 263.

which were used in the secondary schools. In the eighteenth century, better books were written, combining the logical and practical features of the work. Such were the texts of Wolf, Sturm, and Kästner. The latter book was quite academic in character, and, as it was much used, we can see that the schools were paying more attention to the pedagogic character of the work. During this century, the Realgymnasia were established, and mathematics obtained in these schools a ranking along with the classics. At this time geometry began to influence the work of the elementary schools, as simple constructions and other practical work began to be taught in the Realschulen. More attention was now paid to demonstrative work and pupils were more and more cautioned not to learn their work by heart.

Two features in particular stand out in this development. Geometry, which began in the universities, gradually worked its way down into the lower schools. Instead of being regarded as a finishing study at the close of the eighteenth century, it was regarded as a necessary preliminary study for the higher work. The other feature to be mentioned is the practical character of geometry when it found a place in the secondary schools. The universities in the Middle Ages taught the applications of geometry, but separate from the logical work. In the early secondary schools, the geometry taught was in connection with its applications in astronomy and geography. In the eighteenth century, attention was still given to applications in surveying in connection with the regular class work. The tendency grew, however, to confine the teaching to the logic and the ordinary applications in mensuration. In the nineteenth century, as will be shown later, the tendency has been to intensify the study of the several mathematical branches, the practical in geometry being limited almost entirely to its own field.

FRANCE

It was not until the eighteenth century that science found any recognition in the secondary schools of France. Previous to this time literary education occupied almost exclusively the field, and what science did exist was taught in the higher classes as a branch of philosophy, just as history was associated with the humanities.1 Mathematics was not included as a branch of 1 Sicard, Les études classiques avant la révolution, pp. 188-189.

philosophy but occupied a separate field, although Rollin and others looked with disfavor on this isolation of the subject.1

In the sixteenth century there was practically no science or mathematics taught in the colleges, although Erasmus advocated a place for natural history, geography, physics, mathematics, and even history. We recall that in 1536 the first six books of Euclid were required for the master's degree at the University of Paris. The statutes of 1598 (art. 40) show that in the Faculty of Arts the sciences and mathematics were passed over almost without mention. It was recommended that students study the sphere3 at 6 A.M., "with the help of some books of Euclid." The effect of this program, a heritage of the Middle Ages, was almost nil. We can thus see that if mathematics was so little studied at the university at the end of the sixteenth century, the colleges could have paid but little attention to that subject. The newly discovered sciences of the seventeenth and eighteenth centuries made a change in the attitude of the institutions, and by the middle of the latter century science and mathematics were generally recognized.

The practical side of geometry was studied fully as much as in Germany, if one is to judge from the texts printed. The interest of Orontius Fineus in the practical side of geometry has already been mentioned. Charles Bouvelles was another writer of the sixteenth century. His geometry was not practical in the same sense as that of Fineus, for it included neither mensuration nor surveying. Neither was it on the lines of Euclid, although it showed the Euclidean influence in the nature of the subject-matter. This work appeared first in Latin in 1503,5 and was followed in 1542 by an edition in French. As was customary in later texts 1 Thus at the Collège Mazarin mathematics was a separate course. 'Sicard, op. cit., p. 190; Douarche, L'université de Paris et les Jesuites,

P. 154.

This had reference to "sphæra," or mathematical astronomy.

4 Sicard, op. cit., p. 190. Cf. Hahn, Das Unterrichts-Wesen in Frankreich mit einer Geschichte der Pariser Universität, pp. 99-100.

Bouvelles, Livre singulier et utile, touchant l'art et pratique de géométrie, Paris, 1542. His Le livre de l'art et science de géométrie, 1511, was the first geometry printed in French (see "Nouvelle Biog. Générale," Vol. 6, Paris, 1862).

2

the preface contains some remarks on the origin and history of geometry. Mention is made of Pythagoras, Archimedes, and Euclid. Geometry and arithmetic being compared, the author says arithmetic is the soul and geometry the body, "both have attributes in common, but geometry is dependent on arithmetic.""1 This shows how far Bouvelles was from the influence of the "Elements" of Euclid. Four chapters are given to plane geometry and two to solid. The book closes with two chapters on the relation of geometry to symmetry as seen in the animate and inanimate world. The author proves few theorems, some being mere statements of facts. He sometimes generalizes from special cases without giving the necessary steps. Thus he states that in an isosceles right triangle, the acute angles are each one-half of a right angle, and hence the angle sum of the triangle is two right angles. He then concludes that the angle sum in any triangle is two right angles. In solving the problem, "To find a mean proportional between two given lines," Bouvelles says, by way of introduction, that the Germans are accustomed to drink and eat on square tables, and the French on tables longer on one side than the other. "It is proposed," he says, "to reduce a French table to a German table." There is a suggestion here, at least, of good pedagogy. The book on the whole may be classed as one in which the logical aim is in evidence, but it is also clear that the author sought to touch the lives of the people.

4

The geometry of Peter Remus (1569), while essentially logical, shows a marked departure from Euclid in the sequence of subject-matter and the treatment thereof. Parallels are treated as in Euclid but the theory of proportion is omitted.

In the beginning of the seventeenth century, Jean Errard3 wrote his edition of Euclid, and in 1678, Nicholas Mercator

1 Bouvelles, op. cit., p. 4.

2 Thus when showing how to draw two lines equidistant, he says that two lines having a common perpendicular will not meet, by virtue of this perpendicularity, fol. 8 (verso).

3 Bouvelles' book was very popular, for it went through many editions. The Latin was printed in 1511. The first French edition was printed in 1542. This was followed with some alterations by editions in 1547, 1551, 1555, 1566, 1605, and 1608.

4 Edition of 1627 referred to here.

Les neuf premiers livres des élémens d'Euclide, 1605.

produced a work' showing the influence of Euclid, but its independence of the "Elements" is quite marked. The author emphasizes the idea of motion, using this especially in his treatment of proportion. Parallel lines are defined as lines everywhere the same distance apart. Mercator attempts to prove the fundamental theorems on parallels without assuming the axiom of parallels or its equivalent, and hence the reasoning is fallacious. The logic is also faulty in other places.

The geometry of Arnauld, which appeared in 1667, also shows an independence of Euclid. The book is primarily a logical geometry and has for its aim the presentation of plane geometry in a logical manner, but "in a new order and with new demonstrations of the most common propositions. Arnauld breathed a new spirit into elementary geometry and had great influence on the succeeding text-books. Thus through these centuries we see independence of Euclid in France as well as in Germany.

The geometry of Le Clerc (1669) is an interesting book from a pedagogical standpoint. As the author states, it represents a new and singular method. The left page of the open book is given up to the written work and the right page to pictures which are supposed to illustrate the theorem or problem under discussion. Le Clerc goes beyond the "Orbis pictus" of Comenius, which appeared eleven years earlier. Some of the representations are ridiculously fanciful. Thus, he is considering the problem, "Through a given point to draw a line parallel to a given line." The picture which corresponds to this is that of four men with swords, two of whom are fighting a duel. Two of the swords are parallel in position while the other two represent auxiliary lines. Le Clerc certainly carried his idea to the extreme, but his fundamental idea was correct, that of making the

1 In geometriam.

2 See preface, Arnauld, Les nouveaux élémens de géométrie. Ed. 1683. The text is fully described by Bopp in his Antoine Arnauld, der Grosse Arnauld als Mathematiker, in Abhandlungen zur Geschichte der Mathematischen Wissenschaften, 1902, pp. 189-336.

3 Thus, see Lamy, Les élémens de géométrie ou de la mesure de l'entendue, 1710. In the preface, the author remarks that he does not follow Euclid's sequence, but, like M. Arnauld, follows the natural order.

Le Clerc, Pratique de la géométrie sur le papier et sur le terrain.