the easy parts of the subject without any aid whatever from a teacher.

In algebra generally, adoption of the text-book method, in the lessons we heard, was very marked the pupils did all the work, the teacher's duty being to direct their energies and to ask questions, in order to have difficulties explained by the more intelligent members of the class.

Many schools sent excellent work in algebra, to the World's Fair; we admired especially the papers from schools in Massachusetts, California, and Minnesota.

The course of study in geometry differs largely from that current in England, the Euclidian system being generally abandoned. To understand the American method, it is necessary to read one of the books employed, the most popular being Wentworth's, which apparently is thorough and well arranged. It treats of the straight line and the circle first, then of proportional lines and similar polygons, going on to areas of polygons and the properties of regular polygons and circles. The latter part is devoted to solid geometry, the subjects of Euclid XI., and the properties of polyhedrons, the cylinder, cone, and sphere.

This course is generally covered in a year, the less ambitious schools not attempting solid geometry, which, in America, is a College subject. As in algebra, four or five lessons per week are given. Some of the more progressive schools are introducing, into the early years of the High School course or the latter years of the grammar school, a series of

lessons in form, practical geometry, etc., termed in the Boston Latin School, "Objective Geometry," which is intended as a preparation for the more rigid and logical study of later years. The textbook most in favour for this part of the work is Spencer's "Inventional Geometry." We heard an excellent lesson, based on this work, given to a class of backward girls in a good private school in Philadelphia. This book, written by the father of Herbert Spencer, has, unfortunately, long been out of print, but it is now, we believe, reprinted in England, and is well worth the attention of English teachers.

No subject has seemed to us more difficult, or has involved more thought and inquiry than this question of the teaching of geometry. To make any definite report upon it is almost impossible, the differences between American and English education being more marked here than anywhere. One positive statement can be made: the actual properties of figures, the formulæ for areas, etc., are more carefully studied than with us, and seem to be thoroughly well known. This is brought out clearly by the answers to that questioning which is apparently the most important part of the mathematical teacher's duty. If the object of the study of geometry in schools is to fit young people to be engineers, architects, surveyors, designers, etc., then this system is admirable. To estimate the degree of logical training gained by the average boy or girl is a much more delicate matter. The usual method of a lesson is as follows:

Certain pages of the text-book having been set by the teacher, to be studied at home, she calls out about one-third of the class to draw figures on the board, and in turn to demonstrate the proofs of the propositions. The less intelligent pupils have then an opportunity of asking questions, which are generally answered by other members of the class. The teacher merely guides the work. After this some riders may be worked in the same way, the teacher first questioning the clever students, the others listening and understanding as well as they can. The questions from the teachers are as a rule excellent, but there seems to be little of that building up of the new work on the foundation of the old, by the simultaneous activity of teacher and class, that is the ideal of English mathematical teaching in schools.

This ideal was, however, seen in actual practice in the Brooklyn High School for Girls, and the Cambridge High School. In the latter we saw, also, excellent sets of solutions to riders, well arranged and neat. We saw similar sets in the Brearley School, New York. Riders involving numerical calculation, and the drawing of exact and careful figures, occur much more commonly than with us. The effect of this is to make geometry examination papers there much easier than, as a rule, they are in England. The percentage of marks required for a pass is also much higher in America.

The Boston High School for Girls uses no textbook. The girls listen to the demonstrations worked as riders by the more advanced members of

the class, make their own notes, and are tested by the teacher's examination, written at stated periods, on which their promotion depends. In general, the written work is much less than that done in England; "recitation" takes its place. The work sent by schools in California and Minnesota to the Chicago Exhibition appeared to be much above the average.

By the courtesy of the authorities of two schools. we were allowed to give short lessons in geometry to two classes, and were much impressed by the general brightness and intelligence shown by the pupils in solving simple riders. Much of this is, doubtless, due to the recitation method; but we cannot help suspecting there is (as teachers who know both both countries say) some psychological difference between the American and English student of mathematics. To discuss this is, however, beyond the scope of the present inquiry.

Higher subjects are rarely studied in High Schools; some do a little Trigonometry. Advanced schools often take Solid Geometry (properties and mensuration of solids, etc.). We did not find any cases of the teaching of Mechanics as a mathematical subject; no great examination demands it, as does the Matriculation Examination of the University of London. Astronomy is studied much more generally than with us, but from a text-book as an information subject.

There seems to be nothing corresponding to the work done in our best schools by a few girls or boys in the sixth form, for scholarship examinations at the Universities. Indeed the public school

system would hardly allow of such special attention being given to a few. In private schools small classes are the rule, and the work thus seemed more thorough to an English spectator. We heard a good deal of the unpopularity of mathematics among pupils. Some principals considered that girls were conspicuously weaker in that subject than boys. At the same time, the subject is one held in respect by laymen, and considered a part of a liberal education for both sexes, even by persons of comparatively little culture. We were informed, for instance, that a local Board was generally willing to pay for a special mathematical teacher. It is often taught in private boarding schools, and is, we believe, always required for admission to College. Classics does not secure this general recognition, perhaps because of its less practical character.

Where there is so much popular feeling for the study of mathematics, and so much intelligence on the part of the pupils, it is strange that the subject should not raise more enthusiasm among American girls. We cannot but think that, in this subject, study of English methods of teaching might be of advantage to American teachers.

The Teaching of History.

This subject receives much more attention in American schools than in our own. Indeed, this statement holds good for the Universities also, where historical studies claim the devotion of a large number of the ablest undergraduates, and where, as in Johns Hopkins University, the number of