side. The three kinds of triangles are employed, however, the acute, right, and obtuse angled.1 From the pedagogical point of view, the order in which Gerbert places a series of definitions is of interest. He defines solid, surface, line, and point in the order named. Euclid used the reverse order. It is recognized to-day that Euclid's order is not pedagogical, and that the proper method is to proceed from that which is the most familiar in experience, the solid, and arrive lastly at the conception of the point.

As has been said, before Gerbert the geometry in the church schools did not include any practical applications. That the geometry of Gerbert may have had some influence on later teaching is shown from the fact that in some of the church schools there were exercises in finding heights and distances. At the monastery of St. Gall, geometry was studied in the "open." "Die Geometrie wurde nicht nur in der Schulstube, sondern am liebsten im Freien studiert. . es wurde die Höhe vom Erdboden bis zum Kirchturmhahn gemessen, oder ein jüngst dem Kloster vermachtes Gut wurde abgesteckt."2

To summarize briefly, the teaching of geometry in the church schools of the middle ages was limited to the learning of definitions and the performing of a few simple constructions with rule and compasses. The subject was generally taught but there was no real instruction as we understand it to-day. The influence of Euclid was practically nil. After Gerbert there was a little work also in the finding of heights and distances, the astrolabe and the mirror being the field instruments employed. The method of the recitation was such that an ability to memorize was the main desideratum.3

OTHER BOOKS ON PRACTICAL GEOMETRY BEFORE THE RISE OF THE UNIVERSITIES. BOOKS THAT INFLUENCED THE UNIVERSITY TEACHING

Since we are concerned with the practical as well as the logical in the teaching of geometry, it is necessary that we discuss, somewhat briefly, certain mathematical works chiefly of the thir

2 Ibid., pp. 113-114 (ref. Dümmler, Ekkehart, iv, etc., s. 23ff).

3 Ibid., pp. 79-80

teenth century that influenced the practical mathematics of the early universities and affected, although indirectly, the character of the geometry taught in many secondary schools of the sixteenth and succeeding centuries.

The first of these writers was Leonardo of Pisa, who was the first in Europe to write on algebra, and who is famous also as having spread the knowledge of the Arabic (Hindu) system of numerals in Europe. But it is his "Practica geometriæ," which appeared in 1220, that has claim for attention here. By this time Euclid had been translated twice into the Latin, but Leonardo's work in its general treatment was not at all Euclidean. It must be classed as a practical geometry as its title indicates, although, as Professor Cantor2 points out, the stereometric propositions are drawn from Books XI, XII, XIII, and XIV of the "Elements.”3 The book is divided into eight parts preceded by an introduction. In the latter are the usual definitions, and also statements of some important geometric facts given in Euclid as theorems. Thus in considering the propositions involved when two parallel lines are cut by a third line, Leonardo states merely which angles are equal and which are supplementary. The definitions are followed by a discussion of the different units of measurement and their uses, arithmetical and geometric. Part I deals with number operations illustrated geometrically, and also treats proportion. The author shows his acquaintance with Euclid where he draws a circle containing two intersecting chords, forms the equation of the proper products, and adds that it is the same as shown in Euclid. Part II treats square root. Part III contains the mensuration of the various plane figures. Part IV, the various ways of dividing plane figures into equal areas by auxiliary lines. Part V, cube root. Part VI, the mensuration of solids, and Part VII, the mensuration of heights and distances with reference to the use of the geometric quadrans. Part VIII gives further work on the mensuration of circles and their inscribed and circumscribed polygons. We thus see that Leonardo was not concerned with geometry from the Euclidean standpoint.

'The "Practica geometria" has been edited by Boncompagni (1862). Unless otherwise stated our authority is this edition.

2 Cantor, II, p. 39.

Cantor states that Curtze holds that Leonardo may have had access to the translation of the "Elements" by Gherardo of Cremona.

But a certain logical consistency is observed in the treatment of his subject-matter. He begins with definitions. As the work is to be concerned with practical geometry, there is next considered units of measure and the operations of arithmetic illustrated geometrically. Proportion and square root precede the work in the mensuration of plane figures. Before the mensuration of solids is taken up, cube root is explained. After the mensuration in three-dimensional space, the author proceeds to the mensuration of heights and distances, the quadrans being the field instrument employed. This work with the quadrans really involves the rudimentary principles of trigonometry,' and thus we see that Leonardo, like Heron of Alexandria, placed what was essentially trigonometry after the treatment of solid geometry. In brief, Leonardo of Pisa systematized the subject-matter of practical geometry. His book stood as a type for the later Italian practical geometries.

Other writers on practical geometry during the thirteenth century were Savasorda, Jordanus Nemorarius, and Robertus Anglicus. According to Curtze, Savasorda wrote his "Liber embadorum" before Leonardo wrote his "Practica geometriæ," and the latter work was based on the book of Savasorda. Jordanus was a German mathematician, who wrote extensively on mathematics. The "De triangulis" represents best his contributions to geometry. It is divided into four books. Some of the propositions in the first book show an acquaintance with Euclid. The other books are largely on mensuration and on arcs and chords of circles. The subject-matter adds nothing not in the book of Leonardo.

Another writer on practical geometry during this pre-university period was Robertus Anglicus (cir. 1271), who lived in France. His "Tractatus quadrantis,' 195 as the name would

1 Leonardo emphasized this work but little, using only three pages for this kind of applied geometry.

2

Curtze, Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance. See Part I, Der "Liber Embadorum" des Savasorda in der Übersetzung des Plato von Tivoli.

3 MS. (Latin 11246) trans. Plato of Tivoli in the Bibliothèque Nationale, Paris.

4 Cantor, II, pp. 73-86.

❝ Der Tractatus Quadrantis in Deutscher Übersetzung aus dem Jahre 1477, herausgegeben von M. Curtze, pp. 43-63. It was recently (1897) published in French by M. Paul Tannery

indicate, was concerned with the measuring of heights and distances. There is some mensuration of plane figures and problems on the contents of casks and vats. The work is more confined to the practical than that of Leonardo or even Jordanus.

It is not hard to understand why this interest in practical geometry was kept alive. When Euclid finally received recognition by the universities the aim was only to learn the "Elements," there being little incentive to add to or improve the subject-matter. But there was a wide field for the applications of geometry. As has been seen, this was taught largely in connection with geography and astronomy. Before considering the teaching of geometry in the universities of the Middle Ages, the works of some other writers up to the sixteenth century will be briefly summarized. Some of these influenced the mathematical teaching in the universities more than any of the practical works already mentioned. The "Sphæra" of Sacrobosco (b. 1244) ranks as one of the most noteworthy of these. It was essentially a mathematical astronomy, and was widely used in the universities.1 The "Geometria speculativa" of Bradwardine. (b. 1390) had a wide influence. Regiomontanus (b. 1436), who received his mathematical instruction from Puerbach at the University of Vienna, added much to practical mathematics in writing his "De triangulis."2 This was on trigonometry, but included much of the work which had formerly been treated in so-called geometries. Here we see a demarcation, and trigonometry as a science assumed more of an independence from this time on. The Germans, Widmann and Adam Riese, who were contemporaries of Regiomontanus, should be mentioned as having written on practical geometry.

The books that have just been considered, beginning with that of Leonardo of Pisa, show no advance in logical geometry, only traces of Euclid's influence being perceptible. But they stimulated the study of practical geometry, and so made an essential contribution to the development of mathematical science. We shall now turn to the influence of Euclid on the mathematical instruction in the universities.

1 Cantor, II, pp. 87-91; Suter, Die Mathematik auf den Universitäten des Mittelalters, p. 67. Hereafter referred to as Suter

'Cantor, II, pp. 254-289.

THE UNIVERSITIES OF THE MIDDLE AGES

The courses of study in the early universities were founded on the seven liberal arts as outlined by Capella and Cassiodorus, and later by Alcuin, and by Hrabanus Maurus at Fulda.1 So the first instruction in geometry could have been little different from that given in the church schools at that time. But a change came when Euclid finally entered the university curriculum.

It was over 200 years after Adelard of Bath translated Euclid from the Arabic before university regulations demanded that a student must attend lectures on mathematics to be entitled to a degree. Previous to this time, however, lectures on mathematics had been given at some of the newly founded universities. Thus, Sacrobosco (John of Holywood) taught astronomy and mathematics the last years of his life (d. 1256) at the University of Paris. It is not improbable that it was his influence that caused Paris to pass a statute (1336), that unless a student had attended lectures on mathematics he was not entitled to a degree. "From a preface to a commentary on the first six books of Euclid, dated 1536, it appears that a candidate for the degree of M.A. was then required to take an oath that he had attended lectures on the said books." An Oxford statute in the thirteenth century required for the licentiate six books of Euclid. But this was a standard beyond the ordinary reading, for as late as 1450, only the first two books were read. In Prague, founded in 1350, the first six books of Euclid were required for the master's degree.5 The statutes of the University of Vienna for the year 1389 required one book of Euclid for the bachelor's degree and five books for the licentiate. At Heidelberg, founded in 1385, no mathematics was required for the bachelor's degree during the remaining years of the fourteenth. century,' and for the licentiate (in 1388) only the first three books 1 Suter, p. 48.

2 Cantor, II, p. 87.

'Gow, p. 207; See Hankel, pp. 354, 355 and Kästner, I, p. 260.

4 Suter, p. 64.

5

• Ibid., p. 77. During the period from 1365 to 1400 lectures were also given on Boethius, which contained some of Euclid. Aschbach, Geschichte der Universität in ersten Jahrhunderte ihres Bestehens, p 93.

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