work have an appearance of reality. The subject-matter of the book is concerned only with the construction of plane figures, no proofs being given.

Ozanam produced a work' in 1699 which is an exact reproduction of Le Clerc's book with the exception of some slight alterations of a few of the figures. Ozanam's book is in both French and German. No acknowledgment is made of the text of Le Clerc, but there can be no doubt but that it was simply an edition of the former work.2

Up to the time of the expulsion of the Jesuits (1762), the secondary education of France was largely under their control, and mathematics was generally neglected by them. The opinion of the Abbé Fleury in his "Traité du choix et de la méthode des études" (1686)3 is indicative of the value set upon the teaching of mathematics. He chose as indispensable studies for all, religion and morals, civility, and logic and metaphysics. Grammar, arithmetic, economics, and jurisprudence were considered as studies of second degree; while among those of third degree were Latin, rhetoric, history, natural history, and geometry. The Jesuit College of Clairmont (now the Lycée Louis le Grand) certainly recognized in the early part of the seventeenth century the value of algebra and geometry, if one is to judge from a collection of texts or pamphlets printed under the name of the college and written by some of the professors. The author of one of these pamphlets states that the order of Euclid should not be followed in using no theorem unless it has been proved. "The elements of geometry can be taught far more easily and briefly, and not less thoroughly, if some theorems are assumed, to be proved later when there is need." Here is a suggestion of good pedagogy that anticipates the "Perry Movement" of to

1 Neue Übung der Feldmess-kunst. So wohl auff dem Papier als auff dem

Feld.

2 For a more extended discussion concerning these related works, see Graf, Die Geometrie von Le Clerc und Ozanam, ein interessantes mathematisches Plagiat aus dem Ende des XVII. Jahrhunderts. In Abhandlungen zur Geschichte der mathematischen Wissenschaften, 1899, pp. 115-122.

3 Lantoine, Histoire de l'enseignement secondaire en France au XVIIme et au début du XVIIIme siècle, pp. 188-192.

* Bound as one volume under the heading, Bussey, Encyclopædia mathematica collegii Claromontani Parisiensis, societatis Jesu, 1638.

day. The practice of printing texts or outlines of the mathematical subjects was not confined alone to the College of Clairmont. In 1689 a pamphlet of thirty pages was printed by Riviere in the Collège Mazarin under the title, "Theses mathematicæ de geometria elementari tam speculativa quam practica." Theorems are merely stated. The practical features of the work are shown in the problems, constructions, and the mapping of figures from the field.

The influence of the Port-Royalists in perfecting the methods of mathematics and science should not be passed unnoticed.2 Arnauld, whose epoch-making geometry we have considered above, was a prominent member of this society, as was also the great Pascal.

After the first quarter of the eighteenth century, the teaching of mathematics grew more in favor from year to year in the colleges. It was not until 1789, however, that mathematics began to flourish there. By 1730 text-books in geometry began to be common in the colleges. Previous to this time, dictation by the professors was the common practice. That this method was in use at the beginning of the eighteenth century is shown in the preface of Rivard's text,5 where the author remarks that his book is compiled from notes on the lectures delivered by his professor of philosophy. "They began to suppress dictation made by professors of philosophy in the class, and put in the hands of pupils elementary books, written in French, which different authors began to prepare on all sides.""" Sicard mentions the books of Rivard, Clairaut, and La Caille as being used in the class work. While the colleges by 1730 were using books printed in French, the university did not pursue a similar course until 1789.8 As the book of Rivard was

The "Perry Movement" will be considered in Chapter VII.

2 Cadet, Port-Royal Education, trans. Jones.

3 Sicard, op. cit., p. 352.

5

Ibid., pp. 204, 351.

Rivard, Élémens de mathématiques, 1744 (4th ed.).

6 Sicard, op. cit., p. 204.

'The Abbé Leroy in his "Lettre sur l'education publique," Brussels, 1777, p. 241, also mentions the use of Rivard's book in the colleges in 1730.

9 The students petitioned for French texts in the different branches of philosophy, so the university asked its professors to write elementary books for the different courses. The first written was one on physics. Sicard, p. 358

used in the colleges (1730) before the publication of the other works above mentioned, it is stated with some certainty that this book was the first elementary geometry in the French language to have any general use in the schools of France.

We learn one other important fact from the preface in Rivard's book. The author states that his book is compiled from notes taken in the class of philosophy at the university. This shows us the nature of the teaching at the University of Paris at the beginning of the eighteenth century. In geometry it corresponded to what is given now in the average American high school. The nature of Rivard's book gives us the right to make this assertion. The "Élémens de mathématiques" comprises in a single volume arithmetic, algebra, geometry, and trigonometry. The geometry is divided into three parts, the geomery of lines, of surfaces, and of solids. The author does not treat proportion in his geometry, but refers to the chapter devoted to that subject in his algebra. The subject of incommensurables receives little attention. Parallel lines are defined in terms of their equidistance, and the fundamental theorems on parallels are not rigorously proved. Thus Rivard says1 in one of his proofs, "the corresponding angles are equal, as one can see." On the whole, the book is written in a scholarly fashion, notwithstanding some logical lacunes as just shown. The aim of the book is evidently to improve the student in logical thinking. The practical aim is not in evidence, for there are no applications to surveying, and mensuration is not prominent.

The geometry of La Caille2 (1741) has the same disregard for logical rigor in the treatment of parallels. In the introduction it is stated that this book was prepared for use in the Collège Mazarin and that four hours per week would be given to elementary mathematics. Clairaut also in 1741 published his "Élémens de geometrie." It is even less rigorous in its logic than the geometry of Rivard. The subject of parallels is treated in the same loose fashion. Constructions are frequently given

1 Op. cit., p. 24.

3

2 Leçons élémentaires de mathématiques.

3 Ces leçons seront expliquées au Collège Mazarin, tous les ans depuis la Saint Martin jusqu'au Carême, les Lundis, Mardis, Jeudis, & Vendredis, à une heure & un quart après-midy."

without proofs in any strict sense.1 Clairaut published anotherTM edition in 1765. It has the features just mentioned, but herethe author boldly attacks the rigor of Euclid. In the preface he says, "It was necessary that Euclid should prove that intersecting circles have not the same center, because he had to contend with the obstinacy of the Sophists. It was necessary to have this geometry as a logic, but to-day times have changed." The author says he neglects many propositions because they are of no use in themselves. Our reformers to-day could not be more radical in this respect. In other ways less striking, Clairaut shows pedagogic insight. He does not begin his geometry with the customary definitions and principles, but places them where needed in the body of the text. In the preface the author calls attention to this feature. From the standpoint of aim,. Clairaut states that his book has the double object, the measuring of land, and the discovery of geometric principles. On the whole the book is noteworthy in that the author recognized that a book logically perfect is not necessarily the best book for use in the schoolroom. He was ready to sacrifice logic for the sake of interest and practical necessity.

2

In the last half of the eighteenth century, as the military schools increased the efficiency of their work, mathematics received more attention, and texts written for use in these schools began to appear. One of the first of these was the "Cours de mathématiques" of Étienne Bézout, which includes mechanics, arithmetic, geometry, algebra, and astronomy. The geometry includes plane and spherical trigonometry as separate subjects. Like those of Rivard and Clairaut, the geometry is divided into three parts, the geometry of lines, of surfaces, and of solids. It is. interesting to note that the geometry of planes in space is treated under the heading of surfaces. This is not the usual arrangement, for solid geometry generally includes the treatment of planes in space. Bossut in 1789 wrote a text on mathematics with a particular object in view. In the preface the author states that the mission of his book is to help make uniform the teaching of mathematics in the different royal military schools. It is based on the course given at the Ècole Royale

1 Thus from a point without a line it is required to draw a perpendicular to that line. The author solves this by basing it on the case where the point is on the given line. But the proof is lacking in the latter case.

2 p. xiij.

• Cours de mathématiques.

Militaire at Paris.1 A review of the book will give us an idea of the nature of the geometry taught there at that time. As was customary in France and Germany, the various branches of elementary mathematics were embraced in one book. This contained arithmetic, algebra, geometry, and trigonometry. Chapters I-VII are on plane geometry. Chapter VIII considers the "properties of planes," and Chapters IX-XII are on the geometry of solids. Chapter XIII contains the elements of plane and spherical trigonometry. This is followed by a chapter on the "applications of algebra to geometry," in which the conic sections are treated. The book has the general characteristics of the books of Clairaut and Bézout, that is, it emphasizes the practical phase of the subject; but not so much as one would expect since it was especially prepared for the military schools.3 Its logic is faulty in places, in particular in the treatment of parallel lines.

Legendre's "Éléments de géométrie," which appeared in 1794, obtained a wide-spread popularity both in Europe and in the United States. Its general recognition was due to two causes. It abandoned the sequence of Euclid and so simplified the subject-matter. Secondly, it was logically sound, and hence was recognized by the mathematical world. Legendre departed from Euclid in two important respects. First, his sequence differs from that in Euclid. The sequence in the first six books of Euclid is as follows: Book I is on the geometry of lines; Book II, on areas; Book III, on circles; Book IV, on regular figures; Book V treats the theory of proportion; and Book VI applies this to plane figures. Legendre's sequence in plane geometry is: Book I, on lines; Book II, on circles; Book III, on proportion applied to plane figures; and Book IV, on regular polygons and the mensuration of the circle. Book III in Legendre includes parts of Book I and Book VI of Euclid. Legendre does not treat the theory of proportion, but refers the reader to the treatment of that subject in arithmetic and algebra.

1 In the preface, the author also says that his book may be a little "extreme" for those pupils not intending to be military engineers; e. g. for those preparing for the infantry or cavalry. He says the "learned professors know how to abridge the work in such cases."

'One finds here the use of aa for a2, which is a relic of the old algebraic symbolism.

'It was also used in the Collège of Sorize, as was also the geometry of Bézout. Leroy, op. cit., p. 132.