upper classes.1 As at Zittau, those taking this work met for instruction in their leisure hours on Saturday and Sunday. In the Fürstenschulen about the same time, the first year and a half was given to "Rechnen," and the remaining year and a half to the beginnings of geometry and applied mathematics." Geometry was prescribed in 1702 at the Pädagogium3 at Halle where the "Elementa geometria" of Andrew Tacquet was "explained" in connection with field practice.

The school "Ordnungen" of some of the German states and cities not mentioned above show an increased interest in the study of geometry. We learn that the study was prescribed in the Gymnasia of Baden (1705), at the Dormstadt Pädagogium in Hesse (1752), and in Baden-Durlach' (1767) where great efforts were made to train the teachers for the work. The "Ordnung" of the city of Brunswick shows that about 1745 geometry and arithmetic were first taught in private classes in the Gymnasia Katharineum and Martineum. It was not until 1801, however, that geometry was a prescribed study in the latter institution. Even then the Prima class met but once a week for this work.

Since the mathematical books of Christian Wolf exerted a wide influence on teaching in the secondary schools, it would be well to get some idea of their nature. Wolf wrote two important series of mathematical works, the "Elementa matheseos universæ" and the "Auszug aus den Anfangsgründen aller mathematischen Wissenschaften." The latter has many features like the former, the larger work, but is far simpler and more practical. The author evidently intended it primarily for use 1 Friedrich, p. 32.

2 Heym, Zur Geschichte des mathematischen und naturwissenschaftlichen Unterrichts an Gymnasien, insbesondere an der Thomasschule in Leipzig, pp. 17-18. Hereafter referred to as Heym.

'Vormbaum, III, p. 91. According to Hellman (p. 12) geometry was optional in 1721

Tacquet, Elementa geometria planæ ac solidæ, quibus accedunt selecta ex Archimedes theoremata, 9th ed., 1694. Books I-VI, XI-XII of Euclid are generally followed, but the author shows independence of Euclid especially in his definition of parallel lines and in his "parallel axiom."

5 Monumenta Germaniæ Pædagogica, 24, p. 355.

Ibid., 27, p. 273.

'Ibid., 24, pp. LXII-LXIII, 121.

8 Ibid., 7, p. 196ff.

in secondary schools.1 Both series were used, however, in the universities of Germany. We learn two things in this connection: geometry was still taught in the universities, and secondly, as we shall see, it was not necessarily according to Euclid. Wolf's "Elementa matheseos universe," which first appeared in 1714, consists of five volumes. The first includes arithmetic, geometry (plane and solid), plane trigonometry, and analysis (including algebra, analytics, and calculus). The remaining volumes treat principally of mechanics, optics, perspective, geography, astronomy, navigation, fortifications, and architecture. The geometry is divided into two parts, plane and solid. The plane geometry consists of six chapters. Chapter I is devoted to definitions and axioms. Chapter II, to properties of lines, employing many constructions. Chapter III, to parallels and triangles. Chapter IV, to circles. Chapter V, to regular figures; and Chapter VI, to mensuration of plane figures. The solid geometry comprises five chapters. The first is devoted to definitions; Chapter II treats of planes; Chapter III, of the construction of solids; Chapter IV, of the mensuration of solids; and Chapter V is on gauging. The general sequence of subjectmatter in the plane geometry resembles Euclid, but it is far from being copied after the "Elements." The theory of proportion is omitted, all facts on proportion being referred to the corresponding chapter in the author's arithmetic, which explains proportion algebraically. The treatment of parallel lines is not Euclidean. The author defines parallels in terms of their equidistance. On this as a basis, Euclid's parallel axiom is avoided, and Wolf therefore commits himself to illogical proofs. The area of a rectangle is found without considering the incommensurable case. Arithmetical computations are frequently inserted and frequent reference is given to the applications of geometry to surveying. On the whole, the geometry of Wolf, although based on logic, shows an independence of Euclid. The sequence of propositions in many

1 The Anfangsgründe was used in the institutions at Halle and in other Gymnasia and universities of Germany. Vormbaum, III, p. 246; Heym, pp. 8-9; Hellman, p. 13.

2 It was printed in Latin and went through many editions. Reference is here made to the 9th, which appeared in 1732. This shows the great popularity of the work.

instances is totally unlike that in the "Elements," and the methods of proof are many times different. The practical nature of the text and the simplified treatment of the logic (in some cases fallacious) marks a further departure from the geometry of the Greeks. The fact that Wolf's books were used in both the universities and the secondary schools shows that the teaching of geometry in Germany was by no means dominated by Euclid.1

Besides the books of Wolf, the geometries of Sturm (16351703) and Kästner (1719-1800) show progress in aim and method. The "Mathesis juvenilis "2 of J. C. Sturm was written with special reference to academic teaching. The part devoted to geometry is, like the texts of Wolf, largely of a practical nature. The author grades the work for the different Gymnasium classes and also gives directions to the teachers who may use the books. Kästner3 first treats the logical and then applies the same in mensuration and simple surveying. Euclid's axiom of parallels is used and the fundamental theorems on parallels are proved rigidly, but so briefly that it is necessary to supply some of the steps. Incommensurables are also treated in a scientific manner. On the whole the author seeks to make his work logically sound and at the same time to make it practical. Like the "Mathesis juvenilis" of Sturm, the text was specially prepared for academic

use.

By 1762 some practical geometry' began to be taught in the Realschulen, which had recently come into existence. Hellman3 tells us that "pure practical geometry without proofs" was

'The names of Gesner and Ernesti are associated with the teaching of mathematics at the Thomasschule at Leipzig in the eighteenth century. The latter wrote a book, Initia doctrina solidioris, in classical Latin, 1736 (Paulsen says in 1755), containing arithmetic, geometry, physics, and astronomy. It went through five editions up to 1796. The geometry was based on Euclid and laid emphasis on formal discipline. See Starke, Die Geschichte des mathematischen Unterrichts in den höhern Lehranstalten Sachsens, p. 21.

'In two volumes. Edition of 1711, 1716 referred to here.

3 Anfangsgründe der Arithmetik, Geometrie, ebenen und sphärischen Trigonometrie, und Perspectiv (4th ed., 1786).

This included constructions with the rule and compasses.

5 Op. cit., p. 15.

taught in the schools at Erfurt. We thus see the working down of elementary geometry into the intermediate grades.1

By the end of the eighteenth century, then, geometry was generally taught in the secondary schools of Germany. It was not the pure logic of Euclid, but was based on logic. Freyer at Halle and Sturm at Altdorf were far from using Euclidean proofs.2 The texts used show that the practical element was not ignored. The universities still taught the subject, but they did not adhere to the "Elements," judging from some of the texts used. Also, the Realschulen began to teach a little geometry of a practical nature. Some general tendencies in the teaching of mathematics during this century are mentioned by Günther. Francke in the Pädagogium at Halle gave mathematics for the first time an equal footing with the other subjects. Semler at Halle, Hecker at Berlin, and others dared to abandon somewhat important points in the old plan of study in placing the classical languages in the "rear" and bringing mathematics and natural science into positions of importance.

3:

Concerning methods of teaching geometry, we observe in the eighteenth century some hints on schoolroom practice that are not at variance with methods employed in Germany to-day. Demonstrative work on the part of the pupil seems to have been more insisted on at Halle. From the "Schulordnung" for the Pädagogium at Halle in 1721, the recommendation is given that students be prepared in geometry so that they can demonstrate more easily. We learn that it is directed that figures be drawn on the board, and that pupils copy them in their books. This shows that the blackboard was then used and that the practice of keeping note-books in geometry was in vogue. In the geometry taught in the Pädagogium at Halle in 1721, the teach

1 The Realschulen occupy a place intermediate between the common schools (Volksschulen) and the Gymnasia. They prepare students primarily for the life of the middle classes.

2 Hellman, p. 12.

3 Ens. Math., pp. 262-263.

* Günther refers to Beier, Die Mathematik im Unterrichte der höhern Schulen von der Reformation bis zur Mitte des 18. Jahrhunderts, im Bericht über die Realschule II. Ordnung zu Crimmitzschau auf das Schuljahr 1878-79, p. 21.

5 Vormbaum, III, p. 246. Cf. Heym, pp. 8, 9.

ing aim was two-fold, that of sharpening the wit and of making the work practical.1 The form of questioning suggested in the above mentioned "Ordnung" shows efforts to stimulate exact thinking. A line is drawn on the board and the following questions and answers given:

ist.

1. Was ist das? A. eine Linie. 2. Warum ist es eine Linie? A. 3. Was ist denn nun eine Linie?

weil es in die Länge gezogen ist. A. was in die Länge weg gezogen

(Dies ist das erste Merckmahl, woran man eine Linie von andern Sachen unterscheidet: aber noch undeutlich.)

4. So ist ja dieser lange Tisch auch eine Linie? A. nein.

5. Warum nicht? A. weil er breit und dick ist, dass ich viel Linien drauf und dran ziehen könte.

6. Was muss den bey einer Linie nicht seyn? A. keine Breite noch Dicke.

7. Was muss man aber da seyn? A. die Länge.

8. Was ist nun eine Linie? A. eine Länge ohne Breite und Dicke. (Das ist nun nichts anders, als die ordentliche Definition einer Linie: und zugleich auch der Weg, wodurch die mathematici zu solcher Definition kommen.)

2

We learn, also, that in the Fürstenschulen at Halle the authorities recommended that the pupils be encouraged to learn their geometry understandingly and not by heart. These statements as to methods employed refer to the schools at Halle only, but as the work there was of a high order, it tells us perhaps the most that can be said regarding method.

The beginning of the nineteenth century marked the period of the reorganization of the Gymnasia of Prussia. Under the influence of Frederick Wolf and Humboldt, the first general course of study took effect in 1816. Two or three recitations per week for each class in mathematics had sufficed before this time, As six periods per week were now given to mathematics, it can be judged that this study began to rank on a more nearly equal footing with the classics. But in 1827 a reaction set in and the number of periods per week was reduced to four, and in 1837 to three periods for the Quinta, Quarta, and Tertia. In 1882 the total number of periods devoted to all the classes in mathematics in the Gymnasia was thirty-four. In the same year the total number of periods per week in the Realgymnasią

1 Vormbaum, III, p. 247.

2 Ibid., pp. 631-632.