TO THE TEACHER ¶before an exercise implies that the exercise should be discussed in class. before a theorem indicates that the proof of this theorem is not required by certain examining bodies; their schedules should be consulted. In many theorems an indication of the gist of the proof is given in small type in square brackets. This may be reproduced, or not reproduced, by the pupil when writing out the theorem, according to the wish of the teacher. Notes and Answers, mainly concerned with Practical Geometry, contains the answers to the few numerical exercises and a few notes on teaching. Note the very full index at the end of the book. CHAPTER I INTRODUCTORY In Junior Geometry and Practical Geometry much freedom has been allowed in arriving at theorems; what seemed obvious has often been assumed to be true without definite proof. In Theoretical Geometry the ideal is that each theorem shall be deduced from previous theorems or explicit assumptions; and a greater degree of caution is required before accepting as true that which seems at first sight correct. The method, in fact, is that of the Greek geometricians; and the theorems selected for proof are based upon the text-book written by Euclid, a professor in the Greek University of Alexandria (about 300 B.C.). Euclid was perhaps not the greatest of Greek mathematicians; men like Archimedes, Apollonius and Hipparchus made more remarkable discoveries; but his book was an extraordinary achievement and the foundation of most of the work done in geometry till quite recent times. Formerly it was thought that Euclid had reduced the science of geometry to its simplest assumptions, but it is now known that certain other fundamental assumptions must be made which are not stated in Euclid's work. For this reason we do not attempt to enumerate the root assumptions of the subject, some of which indeed are very difficult to understand. To this extent we fall short of the ideal described above. Definitions which are merely explanations of words used are not given in the text, but will be found in a list at the end of the book. But if the argument depends on the definition chosen, this is given in the text. The pupil must not be discouraged at having sometimes to prove the apparently obvious in Theoretical Geometry; the proving of a fact (and not the fact itself) is the primary object of the work. It should be borne in mind that we are forging links in a logical chain which hangs from certain hooks, which are the assumptions on which we base our work; one unsound link lets the rest of the chain fall to S. H. T. G, 1 the ground. On the other hand, it is not logically unsound to assume the truth of a theorem without proof, provided we realise that we are adding to our list of assumptions, adding one more to the hooks (assumed to be firm) from which our chain hangs. No doubt the ideal is to have as few hooks as possible; but it is better to have a larger number of hooks than to have unsound links in our chain. In due course the pupils will find a certain intellectual pleasure in logical geometry, even in proving the obvious; some even may, at a later stage, work at the foundations of the subject and test the soundness of the fundamental assumptions. HINTS ON SOLVING RIDERS Always draw a good figure; and a large figure if it is at all complicated. The figure should not, in general, be drawn with instruments. It is important to acquire the ability to draw figures freehand; such figures should be neat, lines that are to be equal should be approximately equal, parallel lines should look parallel and right angles should look right angles. Always state the data and mark them in the figure. A large percentage of the failures to solve a rider are due to not realising all that is given; hence the importance of writing out the data. The data in a rider need not, in general, be put out so fully as in a formal theorem: record the essential points that are not obvious from the figure, and record them as far as possible in symbols and in their simplest form. E.g. Instead of saying "ABCD is a parallelogram," it may be helpful to write AB is || to DC, AD is || to BC. Again, instead of saying "M is the mid-point of the base BC of a AABC,” Say "BM = MC" (it is obvious from the figure that ABC is a triangle). Always state concisely what has to be proved. Method of attack. First look at the figure and see what given facts have been indicated there, think what other facts follow from them; if possible, indicate them in the figure. Then read the data again and see whether there is anything more that you can use. If you still do not see the proof, start from the other end and think thus: "The result will be true if I can prove this equal to that; which will be true if I can prove this other fact," and so on. When in difficulties, it is always a sound plan to see whether you have used all the facts in the data and what theorems bear on those facts or what theorems are suggested by particular words in the data. In the earlier work we very often prove lines equal by proving two triangles congruent; so look out for triangles that appear to be congruent and then try to prove them congruent. The equality of angles often depends on lines being parallel or on the angle-sum of a triangle. A little later, triangles are often proved equivalent (i.e. equal in area) by means of parallels. Still later, equal ratios, or a result such as PQ. PR = PN. PD, at once suggest similar triangles. In writing out a complicated rider it is a good plan to give in brackets a brief outline of your method of proof. This is frequently done in the theorems. See Th. 8, 20, 29. In constructions, for example when asked to construct a circle to satisfy given conditions, it is often well to sketch in the required circle and to consider the properties of the resulting figure. CHAPTER II ANGLES AT A POINT, PARALLELS, ANGLE-SUM ANGLES AT A POINT. DEF. When one straight line stands on another straight line and makes the adjacent angles equal, each of the angles is called a right angle: and the two straight lines are said to be at right angles or perpendicular to one another. THEOREM 1. If a straight line stands on another straight line, the sum of the two angles so formed is equal to two right angles. Data The st. line AO meets the st. line BC at O. To prove that L BOA + L AOC = 2 rt. 4 S. Construction through O perpendicular to BC. Proof and Draw OD to represent the line L BOA = L BOD + 4 DOA, LAOC = L DOC - DOA, .. L BOA + L AOC = L BOD + 4 DOC = 2 rt. ▲ s. B Fig. 1. Constr. Q. E. D. COR.* If any number of straight lines meet at a point, the sum of all the angles made by consecutive lines is equal to four right angles. THEOREM 2 [CONVERSE OF TH. 1]. If the sum of two adjacent angles is equal to two right angles, the exterior arms of the angles are in the same straight line. Data The sum of the adjacent 8 BOA, AOC = 2 rt. ▲ s. To prove that Construction BOC is a straight line. Produce BO to D. * Abbreviation for "corollary," which means a subsidiary theorem arising from the main theorem (Latin corolla, a little crown). |