This article deals with the plight of logic in Geometry in our schools. Almost all branches of science take recourse in logic for their axiomatic treatments. Even social sciences are no exceptions. But mathematics is logic personified. William Whitehead and Bertrand Russel have shown that mathematics is symbolic logic.
It is more so with geometry because it was in geometry that logic was first introduced. But the irony is that it is in geometry that logic is in its lowest ebb in our schools today.
The idea of logical proof was first brought in by Thales of Millitus in Geometry. This idea was experimented and extended by great geometers like Plato.
Soon Geometry came of age. In 300 B.C. Euclid wrote his monumental work “The Elements” in thirteen volumes. Though he drew upon his predecessors, the whole design was his own. He built his system completely on logic. Taking five axioms, five postulates and twenty-three definitions, he systematically deduced 465 theorems in a logical chain.
It is a profound contribution to human thought in general and to geometry in particular. It was thought to be the final word on geometry. For more than two thousand years intellectuals, statesmen, rulers, politicians, lawyers, judges, teachers and theologians had been studying it to develop their power of reasoning, to sharpen their wits and intelligence, arguing capacity and decision making, (Abraham Lincoln studied the whole work twice.
But unfortunately even this great masterpiece began to reveal some discordant notes after two thousand years. Mathematicians had been challenging Euclid’s fifth postulate. They thought that it could be proved as a theorem using the other postulates as though his axioms were not ‘independent’. For an axiomatic system to be reliable, its set of axioms must be ‘consistent’, and independent. Consistency means that none of the axioms can be logically derived from some other axioms.
Mathematicians wanted to clear Euclid of all blemishes. In a bold experiment they tried a different form of the fifth postulate by keeping the other postulates intact. To their surprise they arrived at a new geometry which was absolutely consistent within itself. Thus, arose non-Euclidean geometries such as Labotchevskian and Rienmanian which had their applications in Physics. So much so, with regard to the fifth postulate he was saved.
But many other flaws were discovered. Euclid had tried to define everything. But in an axiomatic approach, it is impossible to define everything and we have to start with a few undefined concepts. Moreover, his axioms were insufficient. To remove these flaws Hilbert introduced a lengthy set of axioms. This made the structure so abstract that it could not be used in secondary schools. But the general feeling was that Euclid must be saved. So Birkhoff gave some modifications to Gilbert’s axioms by assuming all properties of real numbers and creating the real ‘ruler’ and ‘protractor’ axioms. Thus Euclid was again brought to school level. While such attempts were taking place to save Euclid, a number of other experiments were carried out in geometry in general. This gave rise to several breakthroughs. Descartes invented Analytical Geometry. This was a new approach which had tremendous impact on many branches of mathematics.
Then came projective geometry, n-dimensional geometry, differential geometry, vector geometry, transformation geometry of Felix Klein, and so on. This poured in an infinite wealth of geometrical facts. This renaissance necessitated some inevitable changes in the content and methods of geometry in schools.
The traditional deductive geometry of Euclid required a whole life time to learn comparatively a very small portion of what has been discovered so far. It taxed the brain of the learner too much. So came the slogan “Down with Euclid”.
In the Second International Congress on Mathematical Education held in England in 1972, the major objectives for geometry in our schools were observed as follows: (a) An understanding of the basic facts about geometric figures in the plane and solids in space. (b) An understanding of the basic facts about geometric transformations such as reflections, rotations and translations. (c) An application of the deductive method. (d) Integration of the geometric ideas with other parts of mathematics. (e) Euclidean geometric course does not achieve these objectives. Dr. J.N. Kapur in his paper on “What shall we teach in school geometry” observes, “Once we are prepared to give up axiomatic and Euclidean approach, we can give the students the wonderful vistas of three and four dimensional spaces and of the fascinating symmetries in painting, sculpture, molecules and other objects of nature.” Elsewhere he has expressed his views against both traditional and modern geometries. On the contrary, he is for a unified course on transformation geometry, matrices and vectors.” In the background of all that has been said above, let me have a critical appraisal of the geometry syllabus in our schools. Tranquility prevailed in our schools before 1976 because Euclidean geometry was taught by traditional methods (not Hilbertian). The Government of Tamilnadu introduced New Geometry in our schools in 1976. It was based on the book written by Edwin and Moise which had followed Birkhoff’s approach. It was a beautiful axiomatic treatment. New concepts such as between-ness, inside and outside of an angle and many other new symbols were precisely defined. The present author wrote a reference book for teachers. But this syllabus was totally withdrawn from the next academic year. This was the death knell of axiomatic geometry in our schools. Since then various experiments have been carried out one at every revision of our syllabus, all mix-ups of axiomatic and practical geometry. More importance was given to knowing facts rather than the method. The old generation of teachers who knew Euclidean geometry are almost out of the scene (by retirement). The new set of teachers are ignorant of those methods because they have not studied it at school. So they follow what is found in the textbook verbatim, without a critical approach. In the Second International Congress there was a consensus among the geometers. The Transformation Geometry should be taught in schools, because, (i) children find it easy. (ii) children like it (ii) teachers, even those who were previously antagonistic to mathematics rapidly adopted it (iii) it is also axiomatic, but in our schools it was introduced in Std.XII once and was withdrawn in the next revision of the syllabus. In the new syllabus introduced in Std. XI the next year (1966) Euclidean geometry is given the appearance of axiomatics in the sense that mentions are made about definitions, axioms, etc. But in fact it is only a make-belief logical treatment, because undefined terms are not specified, axioms are not complete and theorems are not derived (barring the theorems 6 and 17). This may be because too much of stress should not be given to deductive reasoning. Postulates on congruence of triangles (SAS, ASA, SSS and HL) are not used to prove theorems except in theorems 6 and 17). Even there no mention is made as to which postulate has been made use of. While introducing SAS and ASA postulates, some kind of explanations using superposition method is given in Tamil medium book. But we should not forget that Euclid himself was rapped by mathematicians for his method of proof by “superposition” on many counts. The author is of the opinion that with what is available, we can make the course much more interesting by injecting some more deductive logic. Deductive reasoning is not to be shunned. In fact it in geometry that deductive logic is easy because of its visual appeal. It need not be highly rigorous of course. Why don’t we treat it as a game? A game has certain rules to follow. Following the rule one has to move the balls. Here the ball is what is given (data). The goal is what is to be proved (conclusion). The rules are the rules of logic (deductive reasoning). If the child moves the ball adhering to the rules with one eye on the ball and the other eye on the goal, he is sure to win. If this kind of approach is followed, (of course with simple rules only), then children will really enjoy geometry. In the light of the above observations, let me give my own views of development of the subject, without changing any of the postulates or definitions given and keeping the changes to the minimum. Better give some name to each postulate. This will facilitate easy reference. For example, say, (1) A1 One line through two-points postulate (2) A2 One point on two-lines postulate (3) A3 Parallel Postulate (4)A5 through A8-SAS, ASA, SSS, HL Postulates In the book axiom 6 is given as, “If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.” This is really the definition of a parallelogram. There is no necessity to change it into a postulate. A still more serious defect is that a kind of explanation given to SAS, ASA, etc. in terms of ‘superposition’ for which Euclid himself was rapped by mathematicians. Now I do not suggest anything in theorems 1 to 5 though some of them are axioms in Edwin and Moise. The syllabus writers have the freedom to effect these changes, so long as there is no contradiction in the development. But the absence of protractor postulate is really felt there. What is now needed is sufficient practice in the skill of using SAS, ASA, SSS and HL postulates. But the book proceeds with some properties of triangles given as theorems without proofs. (7 to 13) It seems that importance is given for knowing geometrical facts rather than knowing the reasons. Children are not even asked to verify them by drawing and measuring. Incorrect and misleading figures add to the confusion. Some consolidated lists of geometrical facts have been given. Against each of them the fact whether it is a postulate or a theorem could have been mentioned in brackets with greater advantage. Here some unwarranted confusions have been brought in. Circum-centre is internationally accepted as S, but it has been changed into O in theorem 11, where O is the accepted ortho-centre. Again ortho-centre has been symbolized as H in theorem 13. These will lead to a lot of confusion in higher classes. Coming to the quadrilateral section, some age-old definitions have been changed. Usually a definition contains the minimum necessary conditions. Anything more is superfluous. In the book a quadrilateral has been defined as “a figure enclosed by four sides and having four angles and four sides and two diagonals.”Mention about angles and diagonals is superfluous here. At the same time the necessary condition that it is a plane figure is not mentioned in the definition. In its place we can have, “A quadrilateral is a closed figure on a plane formed by four lime segments meeting only at their end points.” This will exclude the classical complete quadrilateral also. Another suggestion is, as already mentioned, the properties of a parallelogram (opposite sides and angles equal, diagonals bisect each other, etc. and the four converses), properties of rhombus, square, rectangle, etc. could have been easily proved as theorems using triangle congruence postulates or even as exercises giving detailed hints in brackets. Let us conclude with one more thing. The concept of angle has not been defined properly. Angles as union of rays and measure of angle has not been differentiated. Angle is said to be that which is contained that which is contained between two rays. But between-ness postulate is a much higher concept which Euclid himself did not give, the absence of which led to geometrical fallacies such as, all obtuse angles are right angles, etc., proved by Gauss. Here ‘right angle’ has been defined in terms of degree measure, but there is no protractor postulate. ‘Straight angle has been defined here, but it does not auger well with the definition of ‘angle’ given earlier in which there is a hint about ‘between-ness’ which contradicts the definition of a ‘straight angle’. So it is better to discard the definition of a ‘straight’ angle. Some loud thinking among teachers is the need of the hour.