with the power of leaving a track in the sky, the track would be an ellipse only, unencumbered with axes. What are the axes, then? Indications of invisible forces? Not so. No line of attraction at any point of the orbit corresponds to the minor axis. The axes are human devices to enable us to measure and express the various elements of the orbit." "Well, but," exclaims perhaps some clever pupil, "if the straight lines are unreal, if they mean nothing, why were they invented, and why were we made to study them?" Such reasonable criticism is a great help to the teacher. He proceeds to picture the state of Astronomy in the days when nothing was known of the movement of the planets. He describes the first bewildered effort of the human mind to represent to itself the path of these wanderers. He shows how some man may have at last conceived the brilliant idea of projecting imaginary straight lines across the sky from one fixed constellation to another, thus forming a sort of background of measuring-rods; how the constellations, with these imaginary connecting lines, might be copied on a tablet, and the path of the planet registered thereon from day to day; and how Science might grow up by man inventing modes of measuring and registering curves which the living forces of Nature were describing in Space.
After such a lesson, the class goes on with its work with renewed interest and quickened intelligence. But how would the case be if a group of other teachers were present, who should comment on the lesson in this wise? One says "All the good books present curves with axes; you think yourself cleverer than our best writers." A
