SECTION I.

GEOMETRICAL PROBLEMS.

INTRODUCTION.

THE object of these hints to the solution of geometrical problems is to show the student how he should deal with deductions which are proposed to him in examination. of problems are given, with several fully solved, and hints supplied for the solution of others; but these are often of little use to the student. The average mathematical student requires to learn-not how to solve this or that problem, nor what construction will help him in any particular case: but what are the general methods which he must apply to problems in order to obtain solutions for himself. mathematical teacher who simply solves the problems brought to him by his pupils does little to show how such problems are to be treated. He should exhibit to his pupils the train of thought which leads him to apply such and such processes to the solution of a problem. And more than this: a good tutor will show his pupils where they might be led astray by imperfect methods; he will try the effects of steps which he himself knows to be bad, and thus show his

There are many books in which sets

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pupils what methods to avoid as well as what methods to apply. One problem thus dealt with is worth a dozen which are merely solved; and I believe the student who will carefully go through the examples which I shall take to pieces (so to speak) in the following series, will learn more than he would from seeing any number of problems merely solved.

I. GEOMETRICAL DEDUCTIONS.

Geometrical deductions are problems which are intended to be solved by the application of recognised geometrical methods and propositions. They are divided into several classes.

A geometrical deduction is termed a rider when it is given as an exercise on a particular proposition. It generally happens that the difficulty of a deduction is greatly diminished when it is given in this way, for we know in what direction to seek for a solution. When a deduction is presented as a rider, it is, of course, expected that the proposition to which the deduction is appended shall be made use of in the solution. It will occasionally happen, with carelesslyconstructed riders, that a simpler solution, not involving this proposition, is available; but generally there can be no difficulty in so arranging the proof as to introduce the proposition on which the deduction is supposed to be founded.

A deduction may be given as an exercise on a particular book of Euclid, or on a given set of propositions. In such a case, it is, of course, expected

that no later books or propositions (as the case may be) shall be made use of.

Or, a deduction may be given as an exercise on Euclid, generally-in which case it is expected that no methods which are not used by Euclid shall be applied to the solution of the problem; and, further, that no proposition not contained in Euclid, or not readily deducible from Euclid's propositions, shall be made use of.

Lastly, there are deductions of a more advanced character, and propositions which present themselves in the solution of problems in other subjects, such as trigonometry, optics, mechanics, and so on. In treating deductions of this sort, it is allowable to make use of several well-known geometrical problems not established by Euclid, nor obviously deducible (that is, deducible as corollaries) from his propositions. Hence these properties may themselves be presented as exercises on Euclid-and in fact most of them will be found in collections of deductions. It seems better, however, to direct the student's attention specially to propositions of this sort, since their importance is apt to be lost sight of when they are included in a long list of deductions. It is possible that I may on some future occasion attempt to gather together all those propositions which may fairly be looked on as subsidiary. Some of them are very simple, others less so; but the student should have all of them at his fingers' ends, since they are of continual service in geometrical processes.

The first step in the solution of a geometrical problem is the construction of a figure which shall afford a clear conception of what we have to do or prove. There are some who insist that no one deserves to be called a geometrician who makes use of welldrawn figures. To solve a difficult problem when the illustrative figure is unlettered, or when ovals are drawn for circles, waved lines for straight ones, and so on, may be all very well for the advanced mathematician. Indeed, a good geometrician should be able to take up a list of problems and solve the major part without pen or paper. But it seems to me a great mistake to insist that the learner should increase the difficulties he naturally has to encounter by making difficulties for himself. And independently of this consideration, there is nothing better calculated to lead the student to observe new properties

or properties new to him-than the construction of a well-drawn figure. He is led to notice relations which would otherwise escape him. Thence he learns to seek for the proof of such relations, to satisfy himself that they are real-not apparent. And it is this habit of being always on the watch for new properties which serves as the most efficient aid in the solution of geometrical problems, and which, also, so far as mathematical progress is concerned, is the most valuable fruit of geometrical studies.

The beginner should even use mathematical

instruments, and should spare no pains in the exact construction of his figures. But after awhile, all that will be necessary is that the figures should be drawn, free-hand, so as to represent as closely as possible the relations described in the proposition to be investigated. Simple as this seems to be, there are some points which deserve to be attended to. A few illustrations will serve better than formal rules :—

Suppose a problem spoke of a trisected line: the student would probably draw a line, as A B (Fig. 1), and then divide it as nearly as possible into three equal parts, in C and D. This is not the best plan: he should draw a line, as A D, bisect it as nearly as possible in C, and then produce it to B, so that D B

may be as nearly equal to D C as possible. He will thus have a line much more exactly trisected than by the former method, since everyone can bisect a line, or produce it till the part produced is equal to the adjacent part, whereas many fail in the attempt to trisect a line. Similar remarks apply to the division of a line into five, seven, or nine equal parts.

Suppose we had to solve such a problem as the following:-From a given point outside the acute angle contained by two given straight lines, to draw a straight line so that the part intercepted between the two given straight lines may be equal to the part between the given point and the nearest line.