It will of course be understood that the fusion of algebra and geometry as here understood has nothing to do with the question of teaching the two subjects simultaneously, say two days in the week for one and three days for the other. This plan has many advocates, although on the whole it has not been well received in this country. But what is meant here is the actual fusing of algebra and geometry day after day,--a plan that has as yet met with only a sporadic success, but which may be developed for beginning classes in due time. FOOTNOTES: [37] _Al-jabr wa'l-muq[=a]balah_: "restoration and equation" is a fairly good translation of the Arabic. [38] Or be carried along at the same time as a distinct topic. [39] With a single year for required geometry it would be better from every point of view to cut the plane geometry enough to admit a fair course in solid geometry. CHAPTER IX THE INTRODUCTION TO GEOMETRY There are two difficult crises in the geometry course, both for the pupil and for the teacher. These crises are met at the beginning of the subject and at the beginning of solid geometry. Once a class has fairly got into Book I, if the interest in the subject can be maintained, there are only the incidental difficulties of logical advance throughout the plane geometry. When the pupil who has been seeing figures in one plane for a year attempts to visualize solids from a flat drawing, the second difficult place is reached. Teachers going over solid geometry from year to year often forget this difficulty, but most of them can easily place themselves in the pupil's position by looking at the working drawings of any artisan,--usually simple cases in the so-called descriptive geometry. They will then realize how difficult it is to visualize a solid from an unfamiliar kind of picture. The trouble is usually avoided by the help of a couple of pieces of heavy cardboard or box board, and a few knitting needles with which to represent lines in space. If these are judiciously used in class for a few days, until the figures are understood, the second crisis is easily passed. The continued use of such material, however, or the daily use of either models or photographs, weakens the pupil, even as a child is weakened by being kept too long in a perambulator. Such devices have their place; they are useful when needed, but they are pernicious when unnecessary. Just as the mechanic must be able to make and to visualize his working drawings, so the student of solid geometry must be able to get on with pencil and paper, representing his solid figures in the flat.