June 1, 1883.] 



♦ KNOWLEDGE ♦ 



333 



general mctliods which lie must apply to problems in order to obtain 

 Eolations for himself. The mathematical teacher who simply solves 

 the problems brought to him by his pupils docs little to show how 

 snch problems are to be treated. He should exhibit to his pupils 

 the train of thought which leads him to apply such and sucli pro- 

 cesses to the solution of a problem. And more than tliis — a good 

 tutor will show his pupils where they might be led astray by im- 

 perfect methods; he will try the effects of steps which he himself 

 knows to be bad, and thus show his 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 scries, will learn more than he 

 would from seeing any number of problems merely solved. 



Geometsicai. Deduciioxs. 



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 ii'iJcr 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 

 tliis 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 

 tliat the proposition to which the deduction is appended shall be 

 made use of in the solution. It will occasionally happen, with 

 carelessly-constructed riders, that a simpler solution, not involving 

 this proposition, is available; but generally there can bo no diffi- 

 culty in so arranging the proof as to introduce the proposition on 

 which the deduction is supposed lo be founded. 



A deduction may be given as an exorcise 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 bo 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, deduciblo as cornllaries) 

 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 propo- 

 sitions 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. 



CONSTBCCTIOX. 



The first step in the solution of a geometrical problem is the con- 

 struction of a figure which shall afford a clear conception of what 

 we have to do or prove. There arc some who insist that no one 

 deserves to be called a geometrician who makes use of well-drawn 

 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 con- 

 sideration, there is nothing better calculated to lead the student to 

 observe new properties-— or properties new to him — than the con- 

 struction 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 con- 

 cerned, is the most valuable fruit of geometrical studies. 



The beginner should oven use mathematical instruments, and 

 should spare nn 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 rela- 

 tions described in the proposition to be investigated. Simple as this 



seems to be, there are a few 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 



A C D B 



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 every one can bisect a 

 lino, 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 lino into five, seven, or nine equal 

 parts. 



{To be continued.) 



(£^iir Cbess Column. 



By Mephisto. 



THE TOURXAMENT. 

 SCORE UP TO TUESDAY NIGHT. 



There was no fresh play last week, as it was deemed desirable to 

 allow the players to decide their drawn battles. The beginning of 

 the second round was, therefore, postponed to Monday, the 28th 

 inst. Of the games which came to a decisive conclusion, the con- 

 test between English and Mason proved doubly interesting, as in 

 the first instance it was a very well-fought game, as may be seen 

 from its perusal ; secondly, it was a success gained by a likely can- 

 didate for high honours. Mason by this victory passed Steinitz and 

 Tschigorin, coming out second. In spite of the time granted, there 

 yet remains one draw to be played off between Rosenthal and 

 English, either of whom, may yet slightly improve their position by 

 winning. 



The score of the players before the beginning of the second 

 round stood as follows : — 



Zukertort 12 



Mason 9i 



Steinitz 9 



Tschigorin 9 



Blackburne 8} 



Bird 7 



Winawcr 7 



English 6 J 



Rosenthal 6 i 



Mackenzie 5 



Xoa 3i 



Sellman 3i 



Skipworth 3 



Mortimer 



• Some of the matters here treated of may seem trivial, but in 

 examinations they will be found not unimportant. It must, indeed, 

 be remembered that these papers have special reference to the 

 requirements of young mathematicians preparing for examination. 



