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EXPLAINED AND ILLUSTRATED;
COMPLETE KEY TO THE SCHOOL EUCLID,
100 ADDITIONAL EXAMPLES.
CHARLES MANSFORD, B.A.,
Mathematical Tutor in the Westminster Training College.
SOLUTION OF GEOMETRICAL EXERCISES.
On page 102 of the School Euclid
a short summary
is given of the methods to be adopted in solving Geometrical Exercises. The subject demands, however, a more comprehensive treatment than the limits of the text-book would allow, and a more complete account of these methods, with fuller illustrations of them, will now be given.
1. GENERAL OUTLINE.-The discussion of a proposition in Euclid consists, as is well known, of three distinct parts, which occur in the following order, viz. : (1.) The Particular Enunciation. (2.) The Construction. (3.) The Demonstration. The same order of thought should be carefully maintained in the discussion and solution of every geometrical exercise. The particular enunciation should be thoroughly mastered and distinctly set forth in words before the construction or demonstration is attempted. By strictly adhering to this direction, the points and lines which are sought in the problem will be clearly distinguished from those which are known, and the truths to be proved in the theorem will be in like manner distinguished from the truths admitted. Until this is done, and done effectually, no progress is possible. He who would succeed in this work must be content to take one step at a time, and not be in a hurry to anticipate the difficulties involved in the construction and demonstration of an exercise before he has well considered, and carefully written out, the particular enunciation. Attention to
this direction will save the student from the subsequent perplexity and failure which are certain to overtake him if he neglects it.
But at the next step the discussions in Euclid cease to be of any real service to us. For although the constructions and demonstrations given, lead us surely to the desired end, they afford us no clue to the method by which they were first discovered. It is precisely at this point, therefore, that the real difficulty in the solution of an exercise occurs. In the demonstration' to which the construction' is auxiliary, we see the intermediate steps by which the hypothesis is connected with the conclusion, and what we need is a method which will enable us to discover for ourselves what these steps are in any given case. A particular account will presently be given of such methods as are available for this purpose. Meantime, if we agree to call this. process of inquiry, coming between the particular enunciation and construction, the Analysis, an exercise when fully drawn out will consist of four parts, viz.: (1.) The particular enunciation. [(2.) The Analysis.] (3.) The construction. (4.) The demonstration. In the propositions of Euclid the Analysis is omitted, but it is evident that some such process was conducted at that stage, although it is not written down. In like manner the line of inquiry by which the solution of an exercise is effected need not always be stated, and in many cases it will not be necessary to commit it to paper at all. What is insisted upon here is, that this is the proper place for the consideration of the analysis, and we now proceed to explain how it is to be conducted. The subsequent construction and demonstration will then follow in their proper order, precisely as in Euclid's propositions.
2. SIMPLE EXERCISES.—It sometimes happens that exercises are given as 'riders' to a particular proposition. In such cases the proposition itself, or some truth involved in it, or established in the course of the
demonstration, will afford the key to the solution of the exercise. More frequently it is left to us to discover the proposition implicated in the solution, our choice being limited to a particular book in Euclid, or to that part of a book which precedes the point at which the exercise is introduced. In these cases our choice is determined by the particular enunciation, which brings into comparison the truths to be proved and the truths admitted. Suppose, for example, the exercise to be one in the first book of Euclid; then it will require us to prove the equality or inequality of certain lines, angles, triangles, or parallelograms.
When two lines are to be proved equal, we shall consider whether they belong to the same triangle; if 80, we shall be thrown back upon proposition 5, and shall endeavour, by aid of the hypothesis, to show that the angles opposed to these sides are equal.
But if the lines to be proved equal form parts of different triangles, we shall then endeavour to prove these triangles equal, choosing according to the particulars given in the hypothesis, props. 4, 8, or 26.
If on the other hand the lines do not form parts of triangles, we shall consider whether, by the introduction of any constructive lines, they may be made to become so, and then proceed as before.
But in all these cases the hypothesis will give the relations between certain known lines and angles, for only in this way could we, by aid of the above propositions, prove the equality of the lines in question. Should the hypothesis however assert the equality of certain areas, we should then be driven to props. 35 to 38. In all cases, therefore, we have to consider what propositions in Euclid contain the same result as that required in the exercise. If on examination the hypothesis of the exercise agrees with or is included in that of the proposition, the solution is immediately obtained. Where this is not the case, a comparison of the one hypothesis with the other will usually suggest the step by which