2

of Euclid were read.1 Cologne (founded in 1388) also did not require any mathematics for the baccalaureate, but three books of Euclid were required for the licentiate. At the University of Bologna, geometry was taught as early as 1383 by Antonio Bilotti of Florence, but it was associated with astrology. The University of Leipzig (founded in 1409) required the "Sphæra materialis" for the baccalaureate, and for the licentiate, Euclid, besides other requirements. The universities of Italy seem to have required less geometry than did the other universities mentioned. The interest there was primarily in astrology. Reference has already been made to this in the teaching at Bologna during the fourteenth century. As late as 1589, in the same university, Galileo taught during two years, the "Elements" of Euclid, the "Sphæræ" of Sacrobosco and of Theodosius, and the "Quadripartium" of Ptolemy, the latter being a work on astrology.1 We learn also that in 1598 mathematical lectures were given at Pisa on this latter work.5

Regarding then the mathematical instruction in the universities of the Middle Ages, it appears that more advance was made in the German institutions. In those of France, England, and Italy, the work was less extensive. The study of Euclid was prescribed at Oxford in the thirteenth century, but it was not until the latter half of the fourteenth century that the study was taken up in any serious way by the various universities. By that time candidates for the master's degree were studying, at the most, the first six books of the "Elements." For the bachelor's degree, during this same period, little or no Euclid was required.

The methods of instruction were essentially the same in the different universities of the Middle Ages. As texts were

1

Günther, p. 211.

2

Suter, p. 79

3 Gherardi, Einige Materialien zur Geschichte der mathematischen Facultät der alten Universität Bologna, ins Deutsch übersetzt von Maximilian Curtze, p. 20.

4

5

Gherardi, op. cit., pp. 14-15.

Hankel, p. 357; Rashdale, The Universities of Europe in the Middle Ages, Vol. I, pp. 249-250.

Kaufmann, Die Geschichte der deutschen Universitäten, Zweiter Band: Entstehung und Entwickelung der deutschen Universitäten bis zum Ausgang des Mittelalters, pp. 355-356.

very rare owing to the great labor involved in copying by hand, it was the custom of the professors to read from the text while the students took notes. When the reading was interspersed by the commentaries of the professor, these were generally dictated to the students.1 Sometimes the students read instead the professor's manuscript and copied from it. The method of reading and copying was varied by discussions on some parts of the text. These took the form of disputations, which were usually held once a week. We are to consider that Euclid was taught in conformity with these methods.2

1 At Paris there was a statute against such dictation. Rashdale, op. cit., Vol. I, p. 438.

2 For accounts of the methods of class instruction in the Universities of

the Middle Ages see: Paulsen, Geschichte des gelehrten Unterrichts auf den deutschen Schulen und Universitäten vom Ausgang des Mittelalters bis zur Gegenwart, pp. 18-19 et al.; Günther, pp. 192-197; Rashdale, op. cit., Vol. I, pp. 220-250, 436-438, Vol. II, pp. 452-457; Kaufmann, op. cit., pp. 342-370; Compayrè, Abelard and the Origin and Early History of Universities, pp. 167-184.

CHAPTER V

THE TEACHING OF GEOMETRY FROM THE YEAR 1525 TO THE PRESENT TIME

The end of the fifteenth and the beginning of the sixteenth century marked an epoch in the history of Europe. In 1453, Constantinople was taken by the Turks and Greek learning found its way into Italy. The scholasticism of the Middle Ages had worn itself out and educated Europe was ready for the intellectual feast awaiting it. The printing-press, which was invented about this time, was a powerful agent in bringing this about. Thus was the Renaissance inaugurated. In 1492, America was discovered and a great stimulus was thereby given to trade. It was at this time that changes were wrought in the aim and work of the schools and the universities that have a bearing on our topic. By the beginning of the sixteenth century, the universities. were regularly teaching Euclid. It was natural under the influence of the new education that the universities should extend their work. This meant that the more elementary work should be taken up by other schools. And so by the time of the Reformation we find arising the different institutions under the names of Gymnasien, Pädagogien, Lyceen, etc.1 About the same time (1525) mathematical instruction reached down into the German Volksschulen. We thus see the beginning of the pressure of the university on the secondary schools, and secondly, that of the secondary on the elementary. This presents a serious. problem with us to-day. To understand better the development of the teaching of geometry, we shall first examine some of the early texts that were printed in various countries, especially in Italy, Germany, and France.

1 Suter, p. 48

2 Günther, p. II

SOME EARLY PRINTED BOOKS ON GEOMETRY

Up to the time of the invention of printing, two types of geometries had appeared. There was only one kind of logical elementary geometry, Euclid. The practical geometries were of one general standard inasmuch as they were almost entirely independent of Euclid, but they varied somewhat in the sequence and content of their subject-matter. These two types continued after the fifteenth century. The practical geometries began to decrease in number about the middle of the seventeenth century, by which time a third type of geometry began to be common. This type might be called a practical Euclid, a geometry built on the logical lines of Euclid, but recognizing the practical value of the subject.1

The first important printed work on mathematics was the "Summa de arithmetica geometria proportioni et proportionalita" (1494)2 of the Italian Paciuolo, commonly known as Lucas di Borgo. The part devoted to geometry is much like the “Practica geometria" of Leonardo of Pisa, which was written nearly 300 years earlier. The figures are drawn in the margin as in the latter work. In both books the drawings are very poor, often inaccurate. In the figures representing solids there is an entire lack of perspective, this being shown particularly in the pyramid, the cone, and the cylinder. Many of the problems in mensuration are identical in both, and the treatment of solids is very much the same.

The following are some of the differences: By the time of Paciuolo the influence of Euclid in southern Europe was being more widely felt, so we are not surprised to find in his geometry some parts of Euclid. Part I of Paciuolo begins with definitions, followed by two chapters corresponding to Books I and II of Euclid. The first book gives no proofs, merely stating the propositions. In this respect we are reminded of the treatment of Euclid in the geometry of Boethius. The second book of Euclid is more fully explained. Thus the construction of the problem on the Golden Section is carefully given. The next chapter, IV, corresponds to Book VI of Euclid, which treats of

1 This type will be mentioned below in connection with the teaching of geometry in the various countries.

2 Edition of 1523 referred to here.

proportion applied to plane figures. Paciuolo gives more attention than does Leonardo to the use of instruments for simple surveying. Here we find the quadrans (square form), the plumb quadrans, the use of the staff and shadow, and the mirror for finding heights. When compared with the book of Leonardo, Paciuolo's has two rather prominent features: its recognition of Euclid, and the attention given to field problems.

The next important practical geometry appeared in France in 1556, written by Orontius Fineus.1 He, like Paciuolo, ranked as a prominent writer on mathematics during the first half of the sixteenth century. It is significant that such writers wrote practical geometries. Euclid of course was considered unchangeable, but the fact that these men and others wrote on practical geometry shows where their real interest lay. As we shall see, these writings had an influence on mathematical teaching.

2

Fineus abandoned the placing of figures in the margin, putting them in the body of the text. The drawings are generally good. Euclid is not recognized directly, the subject-matter being devoted to mensuration, applied both to the ordinary geometric figures and to field problems. In this latter work, the various instruments in common use in surveying are exemplified.

In our teaching to-day we are beginning to realize that the geometry should be more experimental. The transit and planetable are being employed by our best teachers in some forms of simple surveying. It is of value to consider the instruments used for such work in the Middle Ages and up to the seventeenth century. The various practical geometries during this period generally described the use of these instruments. The most common instruments were the astrolabe, quadrans, plumb-quadrans, staff (and shadow), "La Croce," the ordinary square, the baculus or Jacob's staff, and the mirror. The astrolabe was used in Gerbert's geometry in finding heights and distances. The Arabs were familiar with it in Gerbert's time, for we learn from Cantor (I, pp. 705-706), that a certain Arab, As-Sâgânî, who died in 990, was a maker of astrolabes. The astrolabe was a circular instrument generally a foot or less in diameter. In using it, say for finding heights, the observer sighted across its center, the instrument being held 1 De re & praxi geometrica, libri tres.

In the practical geometry of Leonardo of Pisa they are in the margin