2

the latter book. We find here a tendency to generalize the notion of angle, curvilinear angles being illustrated.' According to Günther, Dürer was the first to define parallel lines as lines that are everywhere the same distance apart. As we shall see, this definition became quite common from this time on. The geometry of the single opening of the compasses, which was first seriously considered by Abul Wafa (cir. 988), was employed by Dürer in an approximate construction of the regular pentagon.

The book of Orontius Fineus and those of the Italian writers of the sixteenth century mentioned above show a strong tendency to hold to the special. The path from the particular to the general is a long and tedious one. To illustrate, they rarely used results from a previous exercise, taking frequently a page to re-explain in detail the most simple processes. The treatment of the mensuration of triangles by Fineus illustrates this still better. Right triangles, subdivided into isosceles and scalene, are first considered. Then acute-angled triangles, divided into equilateral, isosceles, and scalene. The isosceles case is subdivided into the cases where the equal sides are either greater or less than the base, then obtuse-angled triangles are considered, subdivided into isosceles and scalene. The value of an altitude is given in the latter case. In all these cases, where the principle is exactly the same, the complete work is given. Heron's formula is then used, but first on right triangles and then scalene. The same sort of repetition is employed in the treatment of the various quadrilaterals. In modern practice we follow Euclid in treating many of the special forms after the general principle is established. These writers treated triangles from the standpoint of mensuration just as the Greeks before Euclid treated them from a logical point of view.1

The illustrations in the books mentioned also have a significance for us. Now that drawings were no longer put in narrow margins, we find frequently a large part of a page given to an illus

1 Mercator used these same notions in his geometry over a hundred years later (1678).

'Cantor holds that this can be traced back to Pappus and the earlier Greeks. See Cantor, I, pp. 421, 700.

1

tration which indicates some phase of human activity. This is invariably in the form of some problem in surveying. Here is a picture showing a man in the act of surveying a stream, there one in which the height of a tower is being found. All this makes the book more attractive and is an aid to the reader. This illustrating of a scientific book is a commendable thing.. Textbook writers to-day are reminding themselves that a picture on the page is of pedagogical value.

GERMANY

Although the Gymnasia and other secondary schools which were founded in the sixteenth century had as one of their functions the preparation of students for the universities, this did not mean that geometry as taught in the universities was carried over into the curricula of the secondary schools. This transference of the Euclidean system was not fully accomplished for over 200 years. Before the foundation of these preparatory institutions, most of the universities had under their direction schools similar to the above, preparatory to the Faculty of Arts in which instruction was given in Latin, logic, rhetoric, and in arithmetic."

2

We have already seen that at the end of the fifteenth century the universities demanded at the most the first six books of Euclid for the master's degree. Cantor says that during the first half of the sixteenth century it was customary to read the first five books of the "Elements" in the universities. In 1521, Melanchthon demanded and received a chair in mathematics at the University of Wittenberg. A little later there was one professor for elementary mathematics and one for the superior. Ratke and Reinhold each occupied the first of these two chairs, and the teacher of "Mathesi inferior" discoursed on the elements of arithmetic and geometry."

That the professors in the universities were becoming more

1 We shall see later that Euclid as such was never generally used in the German schools.

'Suter, p. 48.

3 Cantor, II, p. 394.

4

* Günther, Ens. Math., pp. 237-264; Paulsen, op. cit., pp. 154-155; Hartfelder, Philipp Melanchthon als Præceptor Germaniæ, in Monumenta Germaniæ Pædagogica, 7, p. 310.

interested in Euclid is seen from some of their writings. At the University of Vienna, Johann Vögelin, in 1528, wrote his "Elementale geometricum ex Euclidis geometria. ""1 At Basel, in 1533, Simon Grynæus the elder published his edition of Euclid with the Commentaries of Proclus. This was the first edition of Euclid printed in Greek. Also at Basel (1562), Xylander printed the first German edition of the "Elements," which consisted of the first six books only.3

We may conclude that in the sixteenth century the teaching of Euclid remained practically the same in the Universities. But we observe an increased interest in the translation of the "Elements." By the middle of this century it was translated from the Latin into the Greek, and most important of all into the vernacular, which would indicate an increased interest in its study.

In the sixteenth century very little attention was given to he study of geometry in the secondary schools. Even the practical was not universally taught, this being particularly noticeable in the evangelical schools. Melanchthon assigned a small place to mathematics in the programs of the middle schools.1

5

The Gymnasium at Nuremberg was founded in 1526 on a base half academic. The teacher in mathematics was the celebrated Schoener, the manufacturer of globes. Mathematics was assigned an advantageous place and even when the first success of the school diminished, the classes in mathematics were attended in a satisfactory manner. "This," says Günther, "is comprehensible in a center of traffic and industry." Geometry does not seem to have been taught in the Nuremberg Gymnasium in the sixteenth century. In 1556 the program of the Cathedral School at Wurtemberg provided for "Rechnen" and "Lectio sphærica." No mention is made of geometry. On the other

1 Cantor II, p. 394.

2 De Morgan, article, Eucleides, pp. 71-72.

3 According to Cantor (II, pp. 550-551) Scheubel printed some of the arithmetical books of the "Elements" (VII, VIII, and IX). De Morgan (p. 73) gives more than this, adding also books IV to VI inclusive.

4 Günther, Ens. Math., pp. 237-264.

5 Ibid, pp. 249-250.

• Friedrich, Über die erste Einführung und allmähliche Erweiterung des mathematischen und naturwissenschaftlichen Unterrichts am Gymnasium zu Zittau, p. 27. Hereafter referred to as Friedrich.

hand, at Strassburg, the program of 1578 included some geometry.1 Arithmetic was taught in the Secunda, and in the Prima (highest class) the elements of astronomy and a few theorems from Book I of Euclid. No mention is made of geometry being studied at Zittau, and at St. Afra in Saxony geometry was not on the program in 1602. In the Gymnasium at Zwickau, Saxony, opportunity was given as early as 1521 for the study of geometry, but it was optional. Students who wished could listen to lectures on arithmetic, geometry, and astronomy, the classes meeting on Saturdays. The regular course assigned "Rechnen" to the Quinta and astronomy to the Tertia.

The study of the "sphæra" and arithmetic generally constituted the work in mathematics in the schools of the sixteenth century. This is shown in the school "Ordnungen" of Goldberg (1546), Würtemberg (higher cloister schools, 1582), Brandenburg (1564), and Cologne (1543). At the Augsburg Gymnasium (1576) arithmetic was taught in the fifth class but not in the higher classes. Geometry was not taught in the regular work but some mathematics was given in public lectures. The "Ordnungen" of Kursaxony1o (1528), and of the church schools of Brunswick" (1543), Wittenberg12 (1533), Hanover13 (1536), Schleswig-Holstein1 (1542), Pomerania, 15 (1563), Brieg (1581), and Lower Saxony" (1585) make no mention of any of the mathematical branches.

16

By the end of the sixteenth century the teaching of geometry was the exception in the secondary schools of Germany. There

1 For thirty years after its foundation (1538) not even arithmetic was taught. Russell, German Higher Schools, p. 42.

In Germany, the classes are numbered the reverse of the practice in the United States.

5 Vormbaum, Die evangelischen Schulordnungen des achtzehnten Jahrhunderts, Vol. I, p. 54. Hereafter referred to as Vormbaum.

"seems to have been two causes for this. The universities were still teaching it, and with increasing success, if we are to judge by the interest taken in the editing of the "Elements." Secondly, there was no demand for it in these schools from the practical side, for, as Günther1 points out, the Gymnasia were interested in furnishing functionaries for the state and pastors for the churches.

In the seventeenth century, as a result of the Thirty Years War, the educational institutions of all classes were nearly destroyed, and hence little progress could be expected in the teaching of geometry. The reforms of Ratke touched only a little on mathematics, but those of Comenius tended to unite the study of mathematics and natural science of his time. According to Günther, it is difficult to prove that any school was influenced in its mathematical program by this great teacher, but one can admit an indirect influence, in view of the fact that his "Orbis pictus'' was admitted into the schools. Regarding the character of the mathematical work of the schools in the latter part of this century, "the conception of academic study formed in the last quarter of the seventeenth century remained about 150 years as a model. The method and contents of the mathematical program in general remained the In those days the young mathematician had a notion of heterogeneous matters-he needed general knowledge allied to mathematics, hence there was little intensity. This arrangement persisted into the first decade of the nineteenth century."

'same.

At St. Afra in Saxony the mathematical program of 1602 was arithmetic for the first two classes, while the highest class studied the "sphæra" and the first rudiments of astronomy. No geometry was taught, and, according to Friedrich, these same conditions existed up to the beginning of the eighteenth century. In 1605 the rudiments of geometry were taught at the Gymnasium of Coburg, and the subject was obligatory.5 No text was

1 Op. cit., Ens. Math., p. 250.

'This book brought the pupil in touch with life's activities by means of pictures. Some of these touched upon applied geometry. See Compayré, The History of Pedagogy, tr. by W. H. Payne, p. 126.

'Günther, Ens. Math., pp. 252-253.

4 Ibid., p. 27. The "sphæra" was mathematical astronomy. The astronomy mentioned above must have been general astronomy. 5 Ibid, pp. 251-254.