REPORT ON CERTAIN BRANCHES OF ANALYSIS. XOO 



furnish. Great simplicity in the exposition and exemplification 

 of first principles, a perfect knowledge of the consequences to 

 which they lead, and great forbearance in not making them an 

 occasion for the display of the peculiar opinions or original re- 

 searches of their authors. 



There is, in fact, only one elementary work which is entitled 

 to be considered as having made a very near approach to per- 

 fection. The Elements of Euclid have been the text-book of 

 geometers for two thousand years ; and though they labour 

 under some defects, which may or may not admit of remedy, 

 without injury to the body of the work, yet they have not re- 

 ceived any fundamental change, either in the propositions them- 

 selves, or in their order of succession, or in the principles of 

 their demonstrations, in the propriety of which geometers of 

 any age or country have been found to acquiesce. It is true that 

 both the objects and limits of the science of geometry are per- 

 fectly defined and understood, and that systems of geometry 

 must, more or less, necessarily approach to a common arrange- 

 ment, in the order of their propositions, and to common prin- 

 ciples as the bases of their demonstrations. But even if we 

 should make every allowance for the superior simplicity of the 

 truths to be demonstrated, and for the superior definiteness of 

 the objects of the science to be taught, and also for the superior 

 sanction and authority which time and the respect and accept- 

 ance of all ages have assigned to this remarkable work, we may 

 well despair of ever seeing any elementary exposition of the prin- 

 ciples of algebra, or of any other science, which will be entitled to 

 claim an equal authority, or which will equally become a model to 

 which all other systems must, more or less, nearly approximate. 



There are great difficulties in the elementary exposition of 

 the principles of algebra. As long as we confine our attention 

 to the principles of arithmetical algebra, we have to deal with 

 a science all whose objects are distinctly defined and clearly un- 

 derstood, and all whose processes may be justified by demon- 

 strative evidence. If we pass, however, beyond the limits which 

 the principles of arithmetical algebra impose, both upon the re- 

 presentation of the symbols, and upon the extent of the opera- 

 tions to which they are subject, we are obliged to abandon the 

 aid which is afforded by an immediate reference to the sensible 

 objects of our reasoning. In the preceding parts of this Report 

 we have endeavoured to explain the true connexion between 

 arithmetical and symbolical algebra, and also the course which 

 must be followed in order to give to the principles of the latter 

 in their most general form such a character as may be adequate 

 to justify all its conclusions. But the necessity which is thus 



