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extant, how much or rather how little geometry did he really know! And of what he did know how little was he in a position to turn to practical account ! He could describe in full, and with chapter and verse for every statement, how according to Euclid it was possible to draw a straight line through a given point parallel to a given straight line ; but he had never worked such a problem in practice, he was probably unacquainted with the use of the set-square, and unaware that Euclid's laborious method had been modified by modern draughtsmen so as to give an accurate result in a quarter of the time.
He might be thoroughly convinced, and able to convince the world, that in obtuse-angled triangles if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted, without the triangle, between the perpendicular and the obtuse angle ; but he did not know this fact in the portable form a2 = 12 + (2 – 2 bc cos A, and failing this, he was not in a position to use it to calculate the third side of a triangle, given two sides and an angle, or to calculate the angles, given the sides.
It is hardly to be expected that a text-book written in 300 B.C. would be up to date. However little human reasoning processes may have altered, account must be taken not only of the numberless applications of geometry in modern times, but also of the great improvement and simplification in the expression of mathematical relationships brought about by the use of algebra and the arabic system of numera
tion. The idea of directed number and the systematic use of this idea in the geometry of Descartes must find a place in any adequate treatment of geometry in the modern school. At first, however, the forces of conservatism were sufficient to relegate the systematic numerical work to the domain of algebra and arithmetic, but it is noteworthy that among the first reforms was the reduction of the longwindedness of Euclid by the use of directed quantities, so as to include in one enunciation kindred propositions for which separate enunciations had formerly been required. The best instance of this process is to be found in the treatment of the “algebraic” propositions of Book II. А similar innovation was the extension of the idea of an angle to include reflex angles, whereby a more intelligible treatment (though none the less sound) of the angle properties of the circle was obtained. These innovations are now so well established that it will probably surprise some of the younger generation of teachers to learn that they were innovations at all, and that much controversy was excited by the first attempts to tamper with the sacred text.
The next innovation was more daring—no less than a deliberate interference with the holy sequence
propositions. It was one of Euclid's self-imposed laws not to employ any construction until the possibility of the construction had been proved. Accordingly, he dared not bisect an angle to prove the equality of the angles at the base of an isosceles triangle, for he required this very proposition for the proof of his construction for bisecting an angle. His overcoming of the difficulty by the construction of the Pons Asinorum is a masterpiece of human ingenuity. As such, however, it has not always been recognized by the successive generations of
schoolboys who have been dragged across the bridge, or if so recognized the object of the construction has been apt to be misinterpreted. Tired of the wearisome obstacle, the sympathetic conductor decided to show his charges how to jump the ditch. “Let it be granted,” he said, “that an angle can be bisected ;” and, with a side glance at the now purely monumental structure, the convoy passed on its way. Thus was opened up a whole field of interesting innovation. For long it had been felt that Euclid's classification of subject-matter, however admirable, was in many places disturbed by the need for preserving a sound sequence of propositions. If one short cut could be taken, why not others ? The benefits derived from greater simplicity and more satisfactory arrangement would more than compensate for any loss of rigidity by departure from the strict deductive line. Where the argument failed and a fresh postulate would be too violent a remedy_why, the engineers were calling out for measurement—“Let us measure.” So it came about that for a time every teacher of mathematics did that which was right in his own eyes, and the pupil progressed somewhat haltingly -partly by reasoning, partly by measurement, and, where these failed, by guesswork—still somewhat bored, but in truth not half so bored as he used to be. There was life in the new movement that had been lacking in the old system, and the pupil could feel, what he had not felt before, that all this geometry meant something and led somewhere ; it was connected with real things. It is true that results based purely on measurement are often unsatisfying, and the measurement programme as carried out by mathematical teachers devoid of any acquaintance with the methods of Science bore traces of its amateur conception. Many
fell victims to the heuristic craze, and would be found pretending to their classes that they were discovering the Theorem of Pythagoras by measuring the sides of a number of right-angled triangles or laboriously counting squares. Still, to the average boy or girl a line three inches long is more substantial and more satisfactory than the line AB. The pupil is no longer expected to be merely a passive recipient of the colourless doctrine with occasional interludes of retailing the said doctrine or working out some puzzle related to it. He is now given something to do as well as to learn, and he feels that he will be judged not so much by what he can say as by what he can do. That is why he is less bored than formerly.
In the last ten years something like order has gradually evolved out of the chaos of systems that followed the fall of Euclid. It has now become customary to set apart the first year of a course in geometry for the assimilation of geometrical ideas and the acquirement of a knowledge of certain fundamental geometrical facts. This process is carried on by practical workdrawing with ruler, compass, protractor, and set-square, paper-folding, cardboard-modelling and such-like exercises. The facts referred to consist of the elementary propositions concerning angles at a point, parallel straight lines, and congruent triangles. These propositions are not proved in the sense of being referred to dependence upon something simpler and more fundamental than themselves. They are assumed, and worked with, until they become realized and familiar, and are intended to provide a basis for the purely theoretical work that is to follow. This last is taken up in the second year, and the system is virtually that of Euclid with an increased number of axioms and
numerous omissions of the less important propositions. Throughout the theoretical work it is customary to continue the practical exercises for purposes of illustration, and in order to keep the work in touch with reality.
Although this system is in general use there are many teachers to whom it does not commend itself. The teacher is no longer a law unto himself, but provides himself with a law in the shape of a text-book. As text-book writers differ widely in the order of their “ facts ” and propositions, as well as in the selection of propositions to be treated as unproved “facts,” it follows that there is great diversity of practice between school and school, and, where the shoe rather pinches the teacher, between schools and external examining bodies. In such anomalies is to be found the excuse for the demand one sometimes hears for the re-establishment of a universally recognized sequence of propositions. There is really little strength behind this demand. To be sure, we have lost Euclid's best feature, his well-knit web of reasoning secure in every loop. The fabric has been somewhat roughly torn. Can we now unite the broken threads so as to have it whole and strong as it was before ? Is it any use putting a new patch on the old garment ? Not a bit. The garment was well-nigh worn out before the rent was made at all. Mathematicians working on the frontiers of the subject have in the last fifty years cast much light on the foundations of Euclid. The close examination of fundamental assumptions has revealed that Euclid made implicitly many more assumptions than his explicit axioms or postulates, and that even of the axioms themselves one at least, that dealing with parallels, embodies no necessary truth. Even