I make a few remarks on what may seem to some trivial matters, yet which are of importance to beginners.
In giving the proof at the board, there is no need to use three letters, and drag children by their help round every angle; we can write a Greek letter or a number, as we constantly do in trigonometry, or we could colour the angles; say the red is equal to the blue, and let the children write out the propositions in an abbreviated form first; or we might adopt the convenient and concise plan of Professor Henrici: let capitals stand for points, small letters for lines, and let angles be represented by the small letters with ∠ prefixed. Thus we have line PQ or a; PR or b; and ∠ QPR or ∠ ab; anything to avoid tediousness is good; children are so bored by verbosity.
Riders need not be always mere lines without any human or scientific interest. Suppose instead of saying—From two points to draw lines to a given line, which shall make equal angles with the given line, we say—Let CD be a mirror or a wall, a ray or a ball strikes it at P, draw the direction it will take after—or, There is a big house A, and a little house B, near a river—the man in B has to fetch water for A daily, where should he draw the water so as to go the shortest possible distance?
The method of determining the distance of the moon can be made clear long before a child is able to conceive the trigonometrical ratios, and if we are able to arouse an interest in astronomy, we may excite ardour in some which will make hard thought and work delightful. The distant prospect of the mountain top has a wonderful power of leading us on. The writer can never forget the joyful enthusiasm with which she threw herself into the study of mathematics in consequence of hearing courses of lectures on astronomy from Mr. Pullen of Cambridge, Gresham Professor of Astronomy, and the late Vice-Chancellor of Cambridge has described to her the power which the first realisation of the wonders of the boundless universe had over him when a boy of fourteen.
Mr. Glazebrook has suggested that some insight may be given to those who have no high mathematical ability into what seems so marvellous to the uninitiated, the development of curves from equations.
Algebra.The close relation between algebra and geometry becomes apparent in Euclid, Book II., but this might be shown somewhat earlier by methods such as those recommended by Mr. Wormell in the first pages of Plotting or Graphic Mathematics. We can see by a figure that 1 + 2 + 3 + 2 + 1 = 32, and lead the pupil on to the general proposition which is in constant use, when treating of falling bodies.
Or we can show similarly that the sum of an arithmetical series equals a + l2.