268 On Definitions. 



plained for that purpose. The number of these terms, in different 

 branches of knowledge, is widely different. And in many branches, 

 numerous and complicated inquiries are to be made before we pro- 

 ceed to the use of definitions. 



Of all the sciences, Arithmetic is that in which definitions are the 

 most precise, and in which the least preparatory instruction is re- 

 quired. 



Before proceeding to lay down the necessary definitions in this 

 science, it is only requisite to settle by agreement the meaning of 

 the words, one, sum, and difference. When the force of these ex- 

 pressions is thoroughly understood, we can proceed with advantage 

 to the definitions of all the common numbers. Thus, two is a num- 

 ber equivalent to the sum of one and one ; three, to the sum of two 

 and one, and so throughout the scale. When we have, for example, 

 defined five to be the sura of four and one, our reason or our recol- 

 lection informs us, that it is also equal to the sum of three and two. 

 And operations analogous to this are, to a boundless extent, mas- 

 tered with ease, by every human being. Another set of arithmeti- 

 cal definitions, are the names of the different modes of reasoning 

 upon numbers. Thus, when the same number is several times 

 added to itself, this operation is termed multiplication ; and of this 

 kind are all the various rules or operations in this science, and in this 

 manner they are defined, or receive their name. 



It is proper to add, that the various branches of Algebra, or ana- 

 lytical science, is only a continuation of Arithmetic ; and that the 

 whole doctrine of Fluxions, Infinitesimals, and Functions, are merely 

 an extension of the same great subject. They are all founded in 

 accurate definitions, and require nothing further to be taken for 

 granted, than the meaning of the three terms mentioned above. 



Arithmetic, in this extensive sense, possesses also the singular 

 property, that it is, in the strictest view, independent of all the other sci- 

 ences, while its rules and results are applicable to every one of them. 



Geometry requires the previous admission of more principles than 

 numbers, and though independent of material objects, has yet a 

 closer affinity with them. It requires that we be agreed as to what 

 is meant by a point, a line, a surface, a body, and, in the opinion of 

 most geometricians, by a straight line. The term Angle ought, per- 

 haps, to be also added to the list. When the meaning of these terms 

 has been clearly determined, and understood by those who are to 

 use them, the whole remaining superstructure of geometrical reason- 

 ing is established upon accurate definitions. Upon these are sup- 



