208 [Assembly 



of teaching the science shall be adopted, it will become one of 

 the most common and most delightful studies of childhood and 

 youth ; its principles and its terms will become familiar as house- 

 hold words, and its admirable discipline and beautiful harmo- 

 nies will be almost universally enjoyed by young and old. 



Mr, Smith proceeded to explain the discoveries he had made 

 in the laws of geometry, which greatly simplify the science, 

 invest it with a new philosophy, and change it in fact, from an 

 abstract to a mechanical science. In this view of the philoso- 

 phy of geometry, he found himself sustained by Sir Isaac New- 

 ton and Dr. Barrow, and he quoted passages from those profound 

 mathematicians directly to the point. Geometry, said Mr. S., 

 has but one single object, and that is the measurement and com- 

 parison of magnitudes. Whether the magnitude be a material 

 substance or a limited portion of space, the object and the oper- 

 ation are precisely the same, viz : the measurement of so much 

 bulk or quantity of extension. Now for the accomplishment of 

 this very simple and direct operation, the early geometers adopt- 

 ed and brought into use three different kinds of units, which they 

 assumed to be unlike in their nature, axii^ incapable of being made 

 measures of each other. The unit of a line, they say, has length 

 and nothing else ; the unit of a surface has length and breadth 

 but no thickness; and the unit of a solid has length, breadth, 

 and tliickness. And for more than two thousand years, the 

 world has been sadly perplexed and bothered, and science greatly 

 retarded, by mixing up these three heterogeneous quantities in 

 all geometrical operations. And during the last two hundred 

 years, these perplexities and difficulties have been doubled by 

 throwing geometry into the hands of algebra, where the three 

 heterogeneous units are subdivided into positive and negative 

 quantities ; thus compelling the operator to make use of six 

 most uncongenial and unlike instruments of measurement, 

 where the nature of the case required but one, and where there 

 never should have been but one. 



These two definitions, "a line is length without breadth," and 

 " a surface is length and breadth without thickness," Mr: Smith 

 wholly repudiates, because he finds them entirely contradicted 

 by the perfect laws of geometrical operation. All solids have 



