104 



the more extensive use of this palpable reasoning wherever it 

 was practicable, — notably the late Dean Cowie, formerly 

 Mathematical Professor at Woolwich. And it is a remarkable 

 fact that we owe one of the most striking and beautiful 

 concrete demonstrations — that of determining the volume of 

 the pyramid — to a blind man, Sir Isaac Newton's friend, Dr. 

 Sanderson, who can only have arrived at this concrete demon- 

 stration by abstract reasoning. 



The method I am about to bring under your notice was 

 chiefly elaborated by my friend Monsieur Edouard Lagout, a 

 distinguished engineer in the service of the French Govern- 

 ment, and has been very extensively adopted in the technical 

 schools connected with the War Department and the Ministries 

 of Agriculture and Public Works, as well as in the primary 

 and normal schools. To explain it I have had certain models 

 prepared. And I must ask you to excuse my beginning by 

 showing you some very simple self-evident demonstrations, as 

 in describing the system it is necessary to begin with its 

 elements — with the simplest problems. 



You will see that the models would themselves explain most 

 of the definitions necessary to be understood, so I need not 

 occupy time therewith. And furthermore, we will to-night, if 

 you please, take for granted the usual axioms. 



TLAN1' GEOMETRY. 



If we take twelve separate square inches as shown by these 

 models and arrange them together in three rows as in Figure 1, 

 it is evident that they form a rectangular oblong or parallelo- 

 gram with an area of twelve square inches, for we can eoinri 

 the square inches that exactly cover the parallelogram. 



Again, if we arrange nine of them in three rows, as in 

 Figure 2, it is evident that they form a rectangle of an area of 

 nine square inches, for we can count the square inches that 

 exactly cover it. 



It is equally evident that we can tell the area of any other 

 rectangle we make by arranging any number of these square 

 inches by counting them. But if we made the rectangle of a 

 great number it would take a long time to count them, so that 

 we should find it easier to count the number of rows there are 

 in the rectangle and the number of square inches in each row 

 and to multiply the two numbers together ; for we find that this 

 will give the same number as the counting of all the square 

 inches will. Thus, in the first figure there are three rows with 

 four in each row, and three times four are twelve, the same 

 number as we counted ; and in the second figure there are three 

 rows with three in each row, and three times three are nine, 

 which is also the same number as we counted. In like manner, 



