244 T HEORT of the 



though as well acquainted as the other with the proper words to 

 denote every portion or fragment of the great congeries of thought. 



There is reafon to think, that there are much greater dif- 

 ferences among mankind, with refped\ to that capacity or com- 

 prehenfivenefs of mind, by which they take in, or attend to, at 

 once, a variety of objedls and relations, than there are with 

 refpedl to the conception or fimple apprehenfion of any one of 

 them by itfelf. And that comprehenfivenefs of mind, which 

 is in truth a mod valuable talent, both with a view to fpecula- 

 tion and adion, may be improved by various means, efpecially 

 by frequent exercife, and may be allifted by many expedients. 



A PERSON who, when he firft begins the ftudy of mathe- 

 matics, can apprehend only the axioms and the fimpleft propo- 

 fitions, after a few months or years employed in that ftudy, 

 will eafily apprehend, not only the propofition, but the demon- 

 ftration of complex theorems, which are mafles of co-exiftent 

 thoughts, that could not be exprefled by the fucceffion of words 

 in lefs than feveral minutes, nor by the arrangement of words 

 in lefs than feveral pages. 



The fucceffion, and even the beft arrangement of words are 

 found fo unfuitable for the expreffion of fuch combinations of 

 thoughts as occur in many mathematical propofitions, that other 

 expedients are very generally and properly employed to aflift us in 

 making or in communicating thefe complex operations of thought. 



Diagrams and algebraical formulse anfwer thefe purpofes ad- 

 mirably .well. Neither of them, ftridly fpeaking, is effential 

 to mathematical demonftration ; but both of them are highly 

 ufeful in it, and many good mathematicians would be at a 

 fland if they were deprived of them. A good conJlruBion or 

 diagram will fuggeft inftantaneoufly the whole congeries of 

 thought which conftitutes both the propofition and the demon- 

 ftration of a theorem. A good exprejfwn in algebra anfwers 

 nearly the fame purpofe ; and fuggefts, almoft inftantaneoufly, 

 fuch a mafs of thought, without confufion, as never could 



have 



