this be stopped and that the learning be rational. It is not to be judged that this evil was entirely corrected.

ENGLAND

About 150 years elapsed after Adelard of Bath translated Euclid from the Arabic into Latin before the "Elements" began to be taught at Oxford University. Roger Bacon (1212-1294), who lectured there and at Paris, recognized the importance of geometry and stimulated mathematical interest at both of these universities. Writing at the end of the thirteenth century, he says that at Oxford, few, if any, residents read more than the definitions and the enunciations of the first five propositions of Euclid. By the middle of the fifteenth century a little greater interest was taken in this study, for we learn that at the same university, from 1449 to 1463, the first two books were read."

Added interest was given to the study of Euclid over a hundred years later (1570), when Sir Henry Billingsley translated the "Elements" from the Greek into English.3 Previous to this time there had been no professorship of mathematics at Oxford or Cambridge. About 1570 Sir Henry Savile began to give unpaid lectures on the Greek geometers at Oxford, and in 1619 the Savilian professor was Briggs, a Cambridge man, who began lecturing on Euclid I. 9, where Savile had left off. Cambridge followed the example of Oxford, and in 1663 the Lowndean professorship of mathematics was founded in that university. About the same time Isaac Barrow, the teacher of Newton, made a complete edition of Euclid, having published in 1660 an English translation for two of his pupils at Trinity College, Cambridge. This remained a standard for about fifty years. In 1702, William Whiston edited Tacquet's' Euclid. This remained a standard until the appearance of Simson's book,

1 Ball, A History of the Study of Mathematics at Cambridge, p. 3.

2 Ibid., p. 9; Gow, p. 207.

3 Ball, op. cit., pp. 22-23.

1 Gow, p. 208. The problem is "To bisect a given angle."

5 In 1665.

Ball, op. cit., p. 46.

'The Euclid of Tacquet was printed in Antwerp. It was popular on the continent. See p. 67 above

About 1730 the usual texts of Euclid were the editions of Barrow, Gregory, or Whiston.1 The next edition of Euclid to have widespread popularity was that of Robert Simson, which appeared in 1756. The editions of Euclid, following Simson, were more or less based on this book. The texts of Playfair (1795) and Todhunter (1862) obtained great popularity in England and also in America. We thus see a wide-spread interest in Euclid from the standpoint of the text-books. There was no writing on the practical side of geometry such as prevailed on the continent up to about the middle of the seventeenth century. Nor do we find texts combining the practical with the logical. Euclid reigned supreme.

5

4

The period from 1660 to 1730 marked the time when the study of Greek geometry was at its height in England. The universities were now giving their attention to the new turn mathematics had taken after the invention of the differential calculus by Newton. With this new material to work with and to give their time to, we should expect that Euclid would receive some attention in the secondary schools. Just when this transition began it is hard to say. During the eighteenth century the average age of freshmen at the universities was gradually increasing, and when boys stay longer at school, they necessarily begin to learn higher subjects. Hence there is strong probability that during this century Euclid was gradually being studied in the schools, and it may be safely guessed that its place among the school books dates only from the middle of the last century at the earliest." If geometry was studied in the schools by the middle of the eighteenth century, the attention given to it must have been very slight. The great "Public Schools" were certainly very tardy in admitting Euclid into their course of study. We learn that Dr. George Butler, head master at

1 Ball, op. cit., pp. 92-93.

1 Gow, p. 208.

One MS. of the fourteenth century treated on the surveying of heights and distances. See Halliwell, Rara mathematica, pp. 56-71, where the work appears.

4 Gow, p. 208.

5 Ibid., (note 2).

Gow remarks (p. 208, note 2) that he can find no useful information on the curriculum of a public school before 1750.

4

Harrow from 1805 to 1829, introduced a little Euclid "lightly glanced at by the Sixth Form once a week." In the program of 1829, the Sixth Form (the highest) studied Euclid and vulgar fractions one period a week,' but geometry was not required at Harrow before 1837.2 At the Edinburgh Academy (1835-36) the fifth class studied Euclid I; the sixth class, Books I-IV, when algebra was begun; the seventh class, the six books of Euclid completed. In the highest class, the seventh, trigonometry and algebra were both studied. By 1839 at Rugby," we find the Fourth Form studying Euclid I, propositions 1-15, and algebra being begun. By the close of the Sixth Form, Euclid VI was studied. At the Edinburgh Institution" six books of Euclid and the appendix of Playfair were studied. At this same time we learn something of the work and aims of the St. DomingoHouse School. This school prepared students in three lines: for the universities, for the army, navy, and engineering, and for trade. From the years of eight to ten the pupil gained ideas of the geometric figures; from ten to twelve he studied plane figures; from twelve to fourteen the geometry of solids together with trigonometry and algebra. From the years of fourteen to sixteen (the fifth class), spherical trigonometry, land surveying, navigation, and mechanics were studied. Those who intended to enter the universities studied in the sixth class Euclid I-VI in Latin.' Eton College admitted geometry into the course in 1836, but it was not required until 1851.8

In brief, the teaching of geometry in England has been of the extreme Euclidean type. The books in use have been the editions of Euclid, which excluded even the practical work in mensuration,

1 Williams, Harrow, p. 85 and appendix c.

2 Staunton, The Great Schools of England, p. 27. Also see Quarterly Journal of Education, Vol. III, p. 4, and Bache, Report on Education in Europe, p. 396ff.

4 Ibid., p. 396ff (ref. Journal of Education, London, Vol. VII, p. 235ff).

7 The early age at which pupils studied geometry at this school argues

that it must have been taught there for some years.

8 Sharpless, English Education, p. 14; Staunton, op. cit., p. 27

texts like those of Wolf in Germany, or Clairaut in France, not being in favor. As we shall see later, the teaching to-day closely follows these traditional lines. The universities were the first to teach Euclid, beginning as early as the thirteenth century. Toward the middle of the eighteenth century perhaps a little geometry was taught in the schools, but it was not until about the middle of the nineteenth century that the study of Euclid became common in the secondary schools of England.

RUSSIA

Mathematical works under the form of practical arithmetic and "the primitive art of measuring lines, areas, and volumes" have existed in Russia for a long time.1 In the latter half of the sixteenth century, the government undertook the task of surveying the empire. Those who carried out this practical work possessed special knowledge drawn from various manuscripts on surveying. When this survey was undertaken it showed the necessity for mathematical instruction in the schools, but the government refused to further any plan to bring this about. Medieval instruction must have characterized the work in the schools as late as 1660, judging from a book used at that time. This was a sort of encyclopædia which embraced the seven liberal arts of the Middle Ages. The geometry included was essentially surveying.

The teaching of mathematics was neglected until Peter the Great saw the need of re-organizing the army and navy on the model of that of western Europe. So at the beginning of the eighteenth century there were created many special schools where the teaching of mathematics took a prominent part. In these special schools the teachers either dictated the lessons

1 Bobynin, L'enseignement mathématique en Russie. Aperçu historique. In Ens. Math., 1899, pp. 77-100; Hippeau, L'instruction publique en Russie, pp. xx-92; Beer und Hochegger, Die Fortschritte des Unterrichtswesens in den Culturstaaten Europas, vol. 2, pp. 5-105; Also see the article on Russia in Baumeister, Die Einrichtung und Verwaltung des höhern Schulwesens in den Kulturländern von Europa und in Nord Amerika, I2, pp. 561-576. Where authority is not cited below, Bobynin is the scource consulted.

2 Hippeau, op. cit., p. xx; Beer und Hochegger, op. cit., p. 5. Also see Bobynin

to the pupils or read to them from their own note books, or from the books accepted as manuals. In the main the training was that of working by rule and applying what had been learned by heart.

In 1725 the Gymnasia1 of Russia were created. Before this time geometry was taught (as were the other branches of mathematics) with reference to its utility, largely in connection with the engineering of warfare. The aim of the Gymnasium was not only to prepare for the practical professions, but also to develop pure science.2 Geometry was taught in the two higher classes.

For thirteen years after the founding of these Gymnasia, mathematical teaching was done without text-books. The first geometry to be used as a text was written by Krufft3 (inspector of Gymnasia) in 1732. Its title was "Kurtze Einleitung zur theoretischer Geometrie zum Gebrauch der studirenden Jugend in dem Gymnasia bey der Academie der Wissenschaften in Saint Petersburg," showing that the book was intended for academic The attempt of the Gymnasia to develop mathematics from a scientific point of view did not prove popular at first, for the attendance steadily declined for several years.

use.

Another development in mathematical teaching in Russia was occasioned by the ecclesiastical schools taking up this study in 1743. The study was optional, however, even arithmetic being elective.

During this period most of the books on mathematics held to the dogmatic method. Such was the "Géométrie pratique" (1760) of Étienne Nasarof, which contained only definitions and problems with rules and solutions. A notable exception was the first algebra in the Russian language by Nicolas Mouravief (1752), in which book an attempt was made to replace the ancient dogmatic method by the demonstrative. Bobynin says

1 The first teachers came from Germany. In the special schools just mentioned, some of the teachers came from England.

Beer und Hochegger (op. cit., p. 50) give 1747 as the date of the founding of the first Gymnasium in St. Petersburg. Baumeister (op. cit., p. 561) gives 1725, the same as given by Bobynin.

'The ecclesiastical schools emphasized the classical side of education. Krufft also wrote a text on mathematical geography. Both of these were in German, but were later translated into Russian