founded in 1636, at which time arithmetic and geometry were taught in the last year of the three years' course. One day a week was given to these studies together for three-fourths of the year. In 1655, when the four years' course was adopted, mathematics was still taught in the senior year. In 1726 printed texts began to appear, the first printed geometry used being that of John H. Alsted (1558-1638).1 About this time Euclid was used at Harvard for the first time. By 1726 Yale University (founded in 1701), as well as Harvard, taught arithmetic and geometry in the senior year. Euclid was used as a text at Yale in 1733. In 1744 geometry was taught there in the second year instead of the fourth, and had the same position in 1777. This change did not occur at Harvard before 1787. It was not until 1818 that geometry began to be taught there in the first year. At the University of Pennsylvania (founded in 1755) the descent of geometry through the classes was more rapid, it being taught in the first year in 1758 together with arithmetic and algebra. The geometry included the first six books of Euclid in the first year, and in the second year Euclid XI and XII, together with plane and spherical trigonometry and applied mathematics.

It was not until 18442 that Harvard required geometry for entrance, and even then merely the elementary notions were demanded. In 1865 the demands were no more rigid, as can be seen from the title of the text in which the student was to be prepared. Yale in 1855 followed Harvard's example by requiring two books from Playfair's Euclid, a higher standard than that set at Harvard. In 1887 all of plane geometry was required, nothing being said about Euclid.

The other colleges and universities that came into existence had varying courses and requirements for admission, but they approximated on the whole the standards set at Harvard and Yale. A particular exception was found in some of the institu

Ward's " Mathematics" was used between 1726 and 1738. The work was not strict in its logic.

2 Broome, A Historical and Critical Discussion of College Admission Requirements, p. 45. Cajori gives the date 1843.

3 The book was, Hill's Second Book in Geometry, Parts I and II, or “An Introduction to Geometry as the Science of Form." The student was supposed to study as far as p. 130.

tions in the South. The work in these universities was seriously hampered during the Civil War and their standards were necessarily lowered.

Although it was not until 1844 that the universities began to insist on geometry as a requirement for admission, it does not follow that previous to that time geometry was not taught in schools of lower grade. The academies which came into existence soon after the Revolutionary War were not at first preparatory schools, their courses of study even including some university subjects. At Phillips Exeter Academy, in 1818, geometry was taught in the fourth or highest class of the classical course. In the three years' English course geometry was taught in the second year, together with plane trigonometry and its applications to the mensuration of heights and distances. A separate course was also given during this year in the mensuration of surfaces and solids. The third year included the applications of mathematics.1

2

Geometry found a small place in the Colonial grammar schools. These schools before the middle of the eighteenth century were not preparatory schools for the universities, and the mathematics taught there "smacked of trade." Geometry was then taught with navigation and surveying. One of the oldest grammar schools with a different aim was the Boston Latin School, whose function was to prepare students for college. During the period from about 1815 to 1828, geometry was taught in the fourth and fifth years of the five years' course. The work seems to have been of a high standard.

3

The development of the American high school began at the end of the first quarter of the last century. The first of these was the English High School at Boston, founded in 1821. The movement scon spread and, according to the estimate of Dr. W. T. Harris, by 1860 there were forty high schools in the United States. The first course of the Boston English High School comprised three years of work. In the second year, the program included geometry, trigonometry with applications to 1 Brown, The Making of Our Middle Schools, pp. 230-238.

2 Ibid., pp. 128-134.

3

Catalogue Boston Latin School, 1635-1885, with historical sketch, pp.

the mensuration of heights and distances, and the mensuration of surfaces, and solids.1 The character of this course indicates a preparation for immediate practical service. The courses at the university during this same period had much in common with the above.

By 1844, when the universities began to place geometry on the list of entrance requirements, the high schools naturally took up more seriously the teaching of the subject. Further stimulus was given to the work when the University of Michigan in 1871 inaugurated the system of accrediting schools. This system permits students to enter the university without examination after the university has inspected the work in the high schools. The system was adopted by other state universities in the west and is now an established institution. The east has been slow in adopting this plan, the older institutions in particular still admitting by examination only.

The mathematical teaching in the universities was at first influenced by the English, although it is not expressly stated that Euclid was employed as a text until Harvard was nearly 100 years old. The influence of the English increased until the invasion of French mathematics, which began in the first quarter of the nineteenth century. In the programs of the universities, Legendre now began to take the place of Euclid. As we have seen, most of the early texts used in the universities were either the English Euclids or works based on them, the editions of Robert Simson and Playfair being in common use. After the introduction of Legendre, the use of Euclid necessarily decreased until to-day the books of the English type are rarely employed. Some of the most popular geometries that have been based on Legendre are those of Davies (1840) and Chauvenet (1870).

Logical rigor in method was not required before the middle of the eighteenth century, if we are to judge from the nature of a text in common use at that time. Ward's mathematical books

were in use at both Harvard and Yale. His geometry begins with definitions, followed by twenty problems on constructions. Then follow some theorems demonstrated in a loose fashion.

1

Catalog Boston Latin School, pp. 296-301.

2 The influence of the French began when Claude Crozet was appointed professor of mathematics at the West Point Military Academy (1816-1823).

The treatment of parallels in particular is not sound. The way geometry was taught at Dartmouth soon after its foundation in 1769 is seen from a statement of Samuel Gilman Brown: “I remember hearing one of the older graduates say that the first lesson of his class in mathematics was twenty pages in Euclid, the instructor remarking that he should require only the captions of the propositions, but if any doubted the truth of them he might read demonstrations, though for his part his mind was perfectly satisfied." As Ward's book was used at Yale as late as 1777, we may safely judge that Euclid's logic had no very strong hold on the teaching. But during the last quarter of the eighteenth century, when English influence became stronger, Euclid was taught more generally, and hence the teaching of geometry must have been characterized by greater logical rigor.

To summarize briefly, the development of the teaching of geometry in the United States up to the last quarter of the nineteenth century has been as follows: The universities first took up the work and did not generally give it over to the secondary schools until the middle of the last century. Some of the smaller institutions did not make geometry an entrance requirement until about twenty years ago. During the Colonial period, geometry was studied in the grammar schools, and after the Revolution in the newly founded academies. After 1821, when the first high school was established in Boston, geometry found a place in these schools. The teaching in the early universities and in the other schools was at first quite practical. Logic became of more importance when the English Euclids were in greatest favor, during the last quarter of the eighteenth and the first quarter of the nineteenth centuries. Notwithstanding the fact that the French influence, which began about 1817, has tended to make the teaching again more practical, the English influence has been lasting. Excepting for a form of dogmatism that characterized some of the early teaching, the demonstrative method has been the common practice.

The following general conclusions may be drawn in regard to the teaching of geometry since the beginning of the sixteenth century: During the sixteenth and seventeenth cen1 Cajori, op. cit., p. 74.

turies but slight attention was given to the study of geometry in the secondary schools of Germany. In France it received even less attention. In England we find no mention of its being taught outside the universities. In Russia and the United States, secondary schools were not yet created. By the end of the seventeenth century, the teaching of geometry was beginning to be somewhat systematized in Germany, but there was little intensity in the study, general knowledge being the chief desideratum. The geometry taught was largely in connection with geography and surveying. Euclid was thought too hard and was not looked upon with favor.

The eighteenth century showed a change in the teaching of geometry and of mathematics in general. Academic texts now began to be used in the schools. Hitherto geometries were mostly either practical, with little reference to logic, or editions of Euclid, the opposite extreme. This combination of the logical with the practical was especially marked in France and Germany. In England, the books based on Euclid were getting to be more academic in character. In this century geometry was more commonly taught in the secondary schools, this being specially true in Germany, although some of the schools still taught it only as an elective. Simple construction work and exercises in practical problems found their way into the Realschulen in Germany. In France the study of geometry was emphasized in the military schools, but the eighteenth century showed a great development also in the colleges, and by the end of that century geometry and mathematics in general were flourishing in those institutions. In 1794 Legendre, following the example of his less illustrious predecessors in France and in Germany, composed his "Éléments" on lines different from those of Euclid. There is some probability that Euclid was being taught in the secondary schools of England by the middle of this century. In Holland and Switzerland the Gymnasia saw the work being systematized. In Russia, the Gymnasia were being created and were beginning to be interested in the study of geometry as a science. In the United States, the eighteenth century still saw geometry taught only in the universities.

In the nineteenth century the teaching of mathematics in the secondary schools was more completely systematized, this being