While it is not a subject that has any place in a school, save perhaps for incidental conversation with some group of enthusiastic students, it may interest the teacher to consider this proposition in connection with the fourth dimension just mentioned. Consider these triangles, where [L]_A_ = [L]_A'_, _AB_ = _A'B'_, _AC_ = _A'C'_. We prove them congruent by superposition, turning one over and placing it upon the other. But suppose we were beings in Flatland, beings with only two dimensions and without the power to point in any direction except in the plane we lived in. We should then be unable to turn [triangle]_A'B'C'_ over so that it could coincide with [triangle]_ABC_, and we should have to prove these triangles equivalent in some other way, probably by dividing them into isosceles triangles that could be superposed. [Illustration] [Illustration] Now it is the same thing with symmetric spherical triangles; we cannot superpose them. But might it not be possible to do so if we could turn them through the fourth dimension exactly as we turn the Flatlander's triangle through our third dimension? It is interesting to think about this possibility even though we carry it no further, and in these side lights on mathematics lies much of the fascination of the subject. THEOREM. _The shortest line that can be drawn on the surface of a sphere between two points is the minor arc of a great circle joining the two points._ It is always interesting to a class to apply this practically. By taking a terrestrial globe and drawing a great circle between the southern point of Ireland and New York City, we represent the shortest route for ships crossing to England. Now if we notice where this great-circle arc cuts the various meridians and mark this on an ordinary Mercator's projection map, such as is found in any schoolroom, we shall find that the path of the ship does not make a straight line. Passengers at sea often do not understand why the ship's course on the map is not a straight line; but the chief reason is that the ship is taking a great-circle arc, and this is not, in general, a straight line on a Mercator projection. The small circles of latitude are straight lines, and so are the meridians and the equator, but other great circles are represented by curved lines. THEOREM. _The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle._