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,' which is concerned with mensuration only. The work of Silvio Belli3 (1569) deals primarily with the surveying of heights and distances. The book of Gargiolli (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.3 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

A, is used for our a3.
The symbols for plus

1 Some symbolism employed is worth noting. The former symbol was first used by Vieta (1600). and minus are given as Hand. 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.

the latter book. We find here a tendency to generalize the notion of angle, curvilinear angles being illustrated.1 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 Wafa3 (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.

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).

2 Op. cit., p. 162.

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

See above, p. 21.

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

1

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.

Günther, Ens. Math., pp. 237-264; Paulsen, op. cit., pp. 154-155; Hartfelder, Philipp Melanchthon als Præceptor Germania, 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." 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

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.5 "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.

* Günther, Ens. Math., pp. 237-264.

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.