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), Brieg1 (1581), and Lower Saxony" (1585) make no mention of any of the mathematical branches. 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 Jahr hunderts, 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. 2 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. 3 Günther, Ens. Math., pp. 252-253. Ibid., p. 27. The "sphæra" was mathematical astronomy. The astronomy mentioned above must have. been general astronomy. 5 Ibid, pp. 251-254. used and the work was practical. At the Pädagogium in the same place arithmetic was the only mathematical subject taught,* and at Kurfalz2 (1615) the "sphæra" and geometry were taught only as electives. At the Gymnasium at Erfurt, arithmetic was the only mathematical subject taught up to 1615.3 In the program extending from 1619 to 1624, mention is made of one period being given to mathematical instruction in the first class, but no mention is made of geometry.* In 1643, and also in 1647, the first and second classes combined studied the "sphæra" one period per week. Arithmetic was studied one period per week in all three classes, and geometry was taught although not especially mentioned. At this time Schröter's geometry was used at Erfurt. It dealt with both plane and solid geometry, but was entirely practical, and included the rudiments of surveying." Euclid was not taught at this time, for in 1666 the rector at Erfurt recommended the teaching of the "Elements." The teachers answered that it was too difficult." The same advice was given in 1671, with the recommendation “that in mathematics the ‘Elements' of Euclid together with 'ocular demonstration' be given one period weekly." The result was that the mathematics teacher, Gruvius, believing Euclid too hard, wrote in 1671 a geometry based on the "Elements." This was used in the school at Erfurt. The books of both Gruvius and Schröter contained many exercises of a practical nature. At Gotha (1605, 1626) arithmetic and geometry were taught in the fifth of the six classes. In the Gymnasium at Jochimsthal10 (1607), geometry was taught in the highest of the three classes. At the Giessen" Pädagogium (1605-1623), considerable 1 Vormbaum, II, p. 24. Cf. Hellman, Mathematischer Unterricht an den Erfurter evangelischen Schulen im 16. und 17. Jahrhundert, II, pp. 4-5. Hereafter referred to as Hellman. 2 Günther, Ens. Math., pp. 251-254. 3 Hellman, p. 4. 4 Ibid., p. 5. 5 Ibid. Ibid., p. 7. 8 Ibid, p. 10. Simon Stevin in his Tomus secundus mathematicorum hypomnematum de geometria praxi emphasized the practical side of geometry. Ed. 1605. 'Vormbaum, II, p. 30. 10 Ibid., pp. 72, 75. 11 Monumenta Germaniæ Pædagogica, 28, p. 29. attention seems to have been given to the study of mathematics, the work including the "sphæra," geometry, geodesy, and cosmography. Mathematics and geography were studied in the Gymnasia at Beyreuth1 (1664) and at Liegnitz2 (1673). The school "Ordnung" of the state of Brunswick3 (1688) provided for the teaching of practical geometry in the Ritterakademie at Wolfenbüttel, the work, however, being done in private hours together with the arithmetic. In the Gymnasia at Gorlitz' (1609), Beuthen (1614), Soest (1618), Moers' (1635), Stralsund (1643), Kronstadt (1644, 1657), and Halle1o (1661), the study of geometry was still neglected. The same was true at the Lyceum at Sorau11 (1650) and in the Latin Schools at Emden12 (1621) and at Frankfort-on-the-Main13 (1654). The church schools of the state of Brunswick gave little attention to the study of mathematics. We do not find that geometry was on their programs before 1741.14 Thus the seventeenth century saw geometry not yet generally taught in the secondary schools. The influence of Euclid was just beginning to be felt in some of the schools. On the whole, practical geometry was more commonly taught than in the previous century. There was the usual practical work in connection with surveying. As for method, little can be said. In logical geometry we recall that the rector at Erfurt in 1671, recommended the teaching of Euclid with "ocular demonstrations." This would seem to indicate some sort of explanation from a diagram. In the eighteenth century, we find mathematics and science becoming more prominent in the secondary work. This would be expected after the great development in these branches in the previous century, in which the names of Galileo and Kepler 1 Vormbaum, II, p. 629. 2 Ibid., p. 649. Monumenta Germaniæ Pædagogica, 8, pp. 244, 266. Vormbaum, II, p. 98. 'Ibid., p. 117. • Ibid., pp. 206-207. 1 Ibid., p. 273. 8 Ibid., pp. 379-382. Ibid., p. 384. 10 Ibid., pp. 522-529. 11 Ibid., p. 396. 12 Ibid., p. 260. 13 Ibid., p. 438. 14 Monumenta Germaniæ Pædagogica, 7, pp. 49-196. were associated with physics and astronomy, Descartes with the invention of analytic geometry, and Newton and Leibnitz with that of the calculus. Some of the schools were quite late in beginning the teaching of geometry. At the Gymnasium at Zittau, no geometry was taught before 1707. It seems that it became elective about this time. Under the influence of Christian Pescheck, the instruction became more common, but it was still elective and was taught in private classes. It was not until 1726 that regular instruction was given in mathematics in the upper classes, but no mention is made that this included instruction in geometry.2 As for the program of 1740 and 1760, the work in mathematics was enlarged, but no special mention is made of geometry. From 1783 to 1798, the work in geography was widened, but the mathematics was reduced. In the upper classes it was again made elective for private work. In 1803 the standards were raised, and students had regular hours in which some practical geometry was taught (two periods), but in their regular leisure hours (7-10 or 6-9, and 1-3) mathematical science was studied. In the leisure hours on Saturdays and Sundays (from 11 to 12), scientific geometry was studied by the combined Prima and Secunda. Thus we see that, at Zittau, it was not until the beginning of the nineteenth century that practical geometry was required, and at that time the more scientific study of the subject was made elective in private classes. In the Gymnasium at Erfurt from 1713 to 1743, geometry was still largely taught in connection with geography. The geometry of Gruvius was taught in the second class, but the study was not likely beyond the elementary constructions.' In 1762 each of the three classes had from one to one and one-half hours of mathematics per week. In the first class, "geometry" and "stereometry" are mentioned. In the Gymnasium at Görlitz (1770), the study of mathematics was optional in the 1 Friedrich (p. 29) states that the rector in 1707 believed that geometry and astronomy should be taught only in private classes. |