the demands of education. The student needs to make abstractions, to get along with a figure drawn on a plane, and to be able to work independent of the sphere or its photograph.

The teacher will do well to add to the treatment usually given some little discussion of recent features for which we are indebted to the Germans. A considerable saving is effected in "producing" lines, planes, and curved surfaces, in treating prisms, pyramids, cylinders, and cones, by the introduction of the notion of prismatic, pyramidal, cylindrical, and conical surfaces and spaces. The concepts are simple, and by their use a number of proofs are considerably shortened. The prismatoid formula, introduced by a German, E. F. August, in 1849, should also have place on account of its great value in mensuration. Euler's theorem, which states that in the case of a convex polyhedron with e edges, v vertices, and fƒ faces, e +2=ƒ+v, also deserves place, both for the reasoning involved and its interesting application to crystallography. These additions are easily made, whatever text-book is in use, and teachers will find them of great value. The objection on the score of difficulty is groundless.

The one-to-one correspondence between the polyhedral angle and the spherical polygon should also be noted, a correspondence not always sufficiently prominent in our text-books. This relation may be set forth as follows:

"Since the dihedral angles of the polyhedral angles have the same numerical measures as the angles of the spherical polygons, and the face angles of the former have the same numerical measure as the sides of the latter, it is evident that to each property of a polyhedral angle corresponds a reciprocal property of a spherical polygon, and vice versa. This relation appears by making the following substitutions:

Polyhedral Angles.

a. Vertex.

b. Edges.

c. Dihedral Angles.
d. Face Angles.

Spherical Polygons.

a. Centre of Sphere.
b. Vertices of Polygon.
c. Angles of Polygon.
d. Sides.

"In addition to the correspondence between polyhedral angles and spherical polygons, it will be observed that a relation exists between a straight line in a plane and a great-circle arc on a sphere. Thus, to a plane triangle corresponds a spherical triangle, to a straight line perpendicular to a straight line corresponds a greatcircle arc perpendicular to a great-circle arc, etc." It may also be mentioned, in passing, that the word "arc" is always understood to mean "great-circle arc," in the geometry of the sphere, unless the contrary is stated.

A further relationship of interest in the study of solid geometry is that existing between the circle and the sphere, and illustrated in the following statements:

Hence may be anticipated a line of theorems on the sphere, derived from those on the circle, by making the following substitutions:

1. Circle, 2. circumference, 1. Sphere, 2. spherical surface, 3. line, 4. chord, 5. diameter. 3. plane, 4. circle, 5. great circle."

The advantage in noticing this one-to-one correspondence is evident if we consider some of the theorems. In the following, for example, a single proof suffices for two propositions :

If a trihedral angle has two dihedral angles equal to each other, the opposite face angles are equal.

If a spherical triangle has two angles equal to each other, the opposite sides are equal.

The generalization of figures already mentioned in speaking of plane geometry here admits of even more extended use. It is entirely safe to take up the mensuration of the volume or the lateral area of the frustum of a right pyramid, and then let the upper base shrink to zero, thus getting the case of the pyramid

as a corollary, or let it increase until it equals the lower base, thus getting the case of the prism; the prism would, however, naturally precede the frustum. So for the frustum of the right circular cone, and the cone and cylinder, a method not only valuable from the consideration of time, but also for the idea which it gives of the transformation of figures.

Most of these suggestions can be used to advantage with any text-book. text-book. Some are doubtless used already by many teachers, and it is hoped all may be of value.

CHAPTER XIII

THE TEACHER'S BOOK-SHELF

Although in this work considerable attention has already been paid to the bibliography of the subject, a few suggestions as to forming the nucleus of a library upon the teaching of mathematics may be of value. It has been the author's privilege, after lecturing before various educational gatherings, to reply to many letters asking for advice in this matter, and so he feels that there are many among the younger generation of teachers who will welcome a few suggestions in this line.

In the first place, the accumulation of a large number of elementary text-books is of little value. The inspiration which the teacher desires is not to be found in such a library; such inspiration comes rather from a few masterpieces. Twenty good books are worth far more than ten times that number of ordi. nary text-books. Hence, in general, a teacher will do well never to buy a book of the grade which he is using with his class; let the book be one which shall urge him forward, not one which shall make him satisfied with the mediocre.