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NATURAL GEOMETRY. 



By Alfred Mault, Engineering Inspector to the 

 Central Board of Health. 



At a time when the desirability and importance of imparting 

 technical education to all classes are generally admitted, an 

 effort to render such education easier to both teacher and pupil 

 is worth consideration. Intelligent reasoning is the basis of 

 all such education. And of such reasoning mathematical is 

 the most important, and perhaps the most difficult, to the young 

 and uneducated. 



There are two ordinary methods of learning mathematics : 

 one, the Euclidian, which follows a road to a goal that the 

 traveller does not see until he arrives at it ; and the other, the 

 method of most books on arithmetic and mensuration, which 

 shows the goal without pointing out the road that leads to it. 

 The Euclidian mode is wearisome to the young pupil, as h/e 

 cannot see the use of proving in the abstract certain mathe- 

 matical truths, even if his reasoning powers are sufficiently 

 trained to do more than learn the propositions by rote. The 

 idea that this is all that his reasoning powers are capable of 

 doing has led the authors of arithmetical manuals to content 

 themselves with giving him dry rules without troubling to 

 prove to him the correctness of these rules. I propose to 

 bring under your consideration to-night a third method, in 

 which the pupil can see the end of the reasoning from the 

 beginning, and can recognise the utility and necessity of each 

 step that has to be taken to logically arrive at that end. I do 

 not wish to decry the Euclidian method, but to explain and 

 vindicate it to untrained minds — to show by a train of concrete 

 reasoning, visible and tangible, that, geometry is not a mere 

 course of dry abstract induction, but a logical series of steps, 

 each one of which it is necessary to climb in order to reach 

 stages of truth yet in advance. 



There is no novelty in applying concrete reasoning to the 

 solution of mathematical propositions. It is not only allowable, 

 but necessary. Euclid himself used it perforce, though as little 

 as possible. " But seeing that, lie was driven to use it so early 

 as in the fourth proposition of the first book of his Elements, 

 it may be said that all his work is partly founded upon concrete 

 reasoning. Many distinguished mathematicians have advocated 



