No. 149.] 207 



NEW ELEMENTS OF GEOMETRY, AND NEW METHOD OF DE- 

 MONSTRATING THE LAWS OF GEOMETRICAL OPERATION. 



Sijjiopsis of a lecture delivered by Seba Smith, author of " JYew Ele- 

 ments of Geometry y''^ before the menibers of the American Institute^ 

 Jan. 21, 1851. 



After adverting to the great importance and value ol geometry, 

 as one ot the principal foundations of human knowledge, and 

 the high estimation in which it has been held by sages and phi- 

 losophers of all ages, the lecturer presented a brief comparison 

 between the geometry of the ancients, particularly the Greeks, 

 and the geometry of the moderns. The Greek method of invest- 

 igating geometrical subjects, was by rule and compasses, and 

 arithmetical calculations. They worked upon real magnitudes, 

 things which they could see, and feel, and measure, and com- 

 pute with rule and compasses. The moderns, on the contrary, 

 during two centuries past, following the example of Descartes, 

 have confined their geometrical investigations almost entirely to 

 the algebraical process ; working upon ideal quantities, mere ab- 

 stractions, which they could neither see nor feel to be true repre- 

 sentatives of magnitude. Hence the reason why the Greeks 

 made great and rapid advances in geometry, while in modern 

 times the science has made no advancement at all, but remains 

 upon the same level where the Greeks left it. Hence the reason 

 too why with the Greeks geometry was a popular study, and its 

 beauties and benefits diffused among the many ; while with the 

 moderns it is considered dry, tedious, and forbidding ; and its 

 knowledge is limited to but very few. So common and necessary 

 was a knowledge of this science considered among the Greeks, 

 that Plato had inscribed over the entrance to his academy the \ 

 words " Let no one who is ignorant of geometry enter here." 

 But in our modern colleges, instead of all the pupils being re- 

 quired to understand geometry when they go in, not even one in a 

 hundred of them have a respectable knowledge of it when they 

 come out. Mr. Smith contends that when the true principles of 

 geometry are understood, as laid down in his volume of " New 

 Elements of Geometry," lately published, and the proper method 



