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

54

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) 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

'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.

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

vertically. Knowing the distance to the object and the readings on the astrolabe, the observer was enabled by proportion to find the required height. The quadrans in its various forms was used in essentially the same way. The mirror was a hemispherical surface used generally for finding heights. It was placed on the ground, and the observer placed himself so as to see the reflection of the object in the mirror. Knowing his own height and two distances, he found by proportion the required height. The use of the square is interesting and could well be employed in school practice to-day. It is required to find the distance to a given object from a given point. A staff is placed on the given point. An ordinary square is placed in a vertical plane with the right angle uppermost on the top or by the side of the staff so that the observer can sight along the longer arm at the distant object. Holding the square fixed, he next sights along the shorter arm, marking on the ground the point determined by this line of sight. The observer knows the height of the staff and the distance from the point just determined to the foot of the staff. By proportion, he finds, as a third proportional, the required distance.

Practical geometries on the lines laid down by Fineus were quite common in Italy in the sixteenth century. A book by Cosimo Bartoli1 is a close reproduction of the one by Fineus. In many cases the figures and wording are identical. In 1567 appeared the work of Pietro Cataneo,2 which is concerned with mensuration only. The work of Silvio Belli3 (1569) deals primarily with the surveying of heights and distances. The book of Gargiolli1 (1655), which followed the plan of Fineus, shows that as late as the middle of the seventeenth century there was still an interest in books that dealt with the mensuration of plane and solid figures and with the surveying of heights and distances.

Another type of geometry, which illustrates the correlation between algebra with geometry, is represented by the work of Gloriosus (1627). The author works many problems by the

aid of algebra, in which he refers to the work of Tartaglia and Vieta of the previous century. In one problem simultaneous quadratics are involved.1 A book by Gloriosus, which appeared twelve years later, shows that there was current an interest in proving theorems not found in Euclid."

The sixteenth century saw a change in the aim of practical geometry in Germany. Under the influence of Peurbach and Regiomontanus in the fifteenth century, and before them Jordanus Nemorarius in the thirteenth, geometry saw its application largely in the fields of astronomy and surveying. Under the influence of Albrecht Dürer the sixteenth century applications were directed towards architecture and the building arts.* Dürer's "Underweysung der Messung mit dem Zirckel" (1525) indicates these lines of application. That the book had an influence outside of Germany is shown in the Italian treatise of Bartoli (1589) mentioned above, where some plans of sections of a cone are given that are almost identical with those in Dürer's work. Various spirals and fanciful designs show the aim in

1 Some symbolism employed is worth noting. A, is used for our a3. The former symbol was first used by Vieta (1600). The symbols for plus and minus are given as and. These are variations of the modern symbols first given in print by Widmann in 1489. The symbol for square root is given as B, which can be traced back in print to Paciuolo (1494). No sign is used here for equality.

The symbolism contained in some manuscript notes written in the margin of an edition of Euclid by Sebastian Curtius may be mentioned in this connection. The symbol of equality is written ∞. As Descartes (1637) first employed this sign, and since the book of Curtius was printed in 1618, it is evident that the marginal notes were inserted at least after 1637. The notes further give □ and as symbols for the words "square" and "rectangle," respectively, used in written explanations just as we find them in our texts to-day. See Curtius, Die sechs ersten Bücher Euclidis, 1818. The copy in which the above-mentioned notes are inserted is owned by Professor David Eugene Smith.

'Exercitationum mathematicarum. Decas tertia. In this is given a controversy over the proof of the theorem that the three altitudes of a triangle are concurrent. The proof finally given by the author is fallacious.

3 A work on elementary geometric constructions appeared anonymously in Germany in the fifteenth century under the title Geometria deutsch. It was the first printed book on geometry in the German language. See Günther, pp. 347-354; Cantor, II, pp. 450-452.