was

the feeling grew that all students should not be subjected to the same program. We learn of "special courses" being given in some of the lycées. In 1890 a so-called "modern course' added to the school program,' but the schools were nevertheless still dominated by the classical course. study of 19022 marks an epoch in giving the student great freeThe new plan of dom of choice in his selection of a program. vision has been made for the work in mathematics and science. Liberal proSince the time of Legendre, the French schools have in the main either followed his "Éléments" or used texts based on it. A text that rivalled Legendre's about the beginning of the last century was the geometry of Lacroix. These, says Delambre,' were the two important texts in use between the years 1789 and 1810. Referring to the features of these books, Delambre comments on the efforts of Legendre "to treat some parts wholly analytically" that the book of Lacroix "can serve as an introduction to modern analysis."

and says

That Lacroix was a worthy rival of Legendre in the field of elementary geometry can be seen in his essays on teaching. These show us what was engaging the attention of a teacher and mathematician at that time (1805). Referring to the correspondence between plane and solid geometry, Lacroix says,5 "Many lines and figures traced on a plane are only particular cases of lines, planes, and solids considered in space; and it is indispensable, whenever possible, to teach in close relation those parts of plane and solid geometry that admit of analogous treatment.' Thus we see a suggestion to teach parts of plane and solid geometry simultaneously. The author lays emphasis

6

1 Compayré, The Reform in Secondary Education in France (trans. by Finnegan), Educational Review, 1903, p. 134.

'Plan d'études et programmes d'enseignement dans les lycées et collèges de garçons (Arrêtés du Mai, 1902).

3 Élémens de géométrie (á l'usage de l'ecole centrale des quatres-nations). 7th ed. (1808).

4 Rapport historique sur les progrès des sciences mathématiques depuis 1789, Paris, 1810, p. 3.

5

Lacroix, Essais sur l'enseignement en général et sur celui des mathématiques en particulier, 1838 (1st ed., 1805), p. 297.

The author cites the trouble of finding a correspondence between the congruent triangles of a parallelogram and the non-congruent pyramids of a triangular prism (p. 297).

It is remarked that pupils at the age of fifteen or sixteen were studying solids (p. 299).

on analogy. He gives a suggested sequence of subject-matter as follows:

1. Straight lines with regard to length but not to situation. 2. Combinations of lines, congruent1 and similar triangles. 3. Polygons, congruent or similar.

4. Combinations of lines and circles.

Lacroix then goes on to say, that from the above order, in the first part of the geometry, he builds up by analogy the rest of his work, relative to the measure of areas, to planes, and to solids. The author is keeping in mind the analogies between plane and solid geometry, for, he says, when Euclid I, 47, is placed in the early part of geometry, as in the "Elements," there is no analogous theorem in solid geometry, but when proved under similar triangles (not pure geometry, he says) there is an analogous theorem, which is, "The square of the area of the largest face of a tetrahedron, which has three contiguous faces mutually perpendicular, is equal to the sum of the squares of the areas of the other three faces."

Lacroix says experience teaches him not to separate the teaching of theorems and problems, and that in the construction work great care should be used to have the drawings exact. He also believes that simple surveying should be taught in this connection. The author raises the question whether algebra should precede or follow geometry. Lacroix does not give an absolute answer, but says it depends upon the minds of the pupils. He says geometry, above all other mathematics, should be learned first, provided it is presented with respect to its applications, "either on paper or in the field."

These reflections of Lacroix have a significance to us from the standpoint of present teaching. The practice of pointing out the analogies between plane and solid geometry is commendable. Lacroix even suggests that in some cases the proofs in the two geometries could be given in connection with each other."

3

1 The author uses the word "égal."

2 Lacroix, op. cit., p. 303.

3 As regards style of demonstration, Lacroix says that the "reductio ad absurdum" should be used as little as possible for the reason that the best proof is that which follows in a chain of proofs (p. 307).

This is being agitated in Italy and France at the present time. below, pp. 109-112; 115-116.

See

We get some idea of the teaching in the colleges in 1843, as well as some suggestions as to method, from a report of M. Busset.1 He suggests for a first reform that arithmetic and algebra be applied from the first in the teaching of geometry. As for the aim in teaching of geometry, Busset says it should not be that of developing the science of men, but that of arriving at a knowledge of nature. The rest of the report is concerned with recommendations as to the treatment of specific portions of the subject-matter of geometry, arithmetic, and algebra. It is recommended in the teaching of geometry to employ the principle of duality and to make use of the method of analysis.3

To summarize, we find that geometry was beginning to be seriously studied in the secondary schools of France by the early part of the eighteenth century, and by the last quarter of that century mathematics as a whole was flourishing in the colleges. During the period from 1794 to 1808, when the university and the colleges were suppressed, the interest in mathematics was kept alive in the military schools founded by the government. During the last century these schools and the other government technical schools, by their rigid entrance requirements, have stimulated mathematical study in the secondary schools, and to-day the work in the lycées and colleges is shaped in a large measure by these demands.

The teaching of geometry in the secondary schools of the eighteenth century was not according to the strict logic of Euclid, if we are to judge from the subject-matter of the books which were used at that time. In these the logic was sometimes insecure. Legendre secured a proper balance in his "Éléments." His work was logical but at the same time was usable in schoolroom practice. The influence of Legendre has endured to this day, but the later French geometers did not confine themselves to the plan of Legendre. We have seen that early in the last century recommendations were given by Lacroix that showed an independence of method.

Concerning the method of study, it can only be said that up to the early part of the eighteenth century, learning by heart was the practice. Recommendations were made at that time that

1 De l'enseignement des mathématiques dans les collèges, p. II, ff. 2 Busset, op. cit., p. xi.

3 Ibid.,

P. 28.

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.2

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.

* 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.

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.

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