284 THIRD REPORT — 1833. 



imposed upon us of dealing with abstractions of a nature so 

 complete and compreliensive, renders it extremely difficult to 

 give to the principles of this science such a form as may bring 

 them perfectly ^vithin the reach of a student of ordinary powers, 

 and which have not hitherto been invigorated by the sevei'e dis- 

 cipline of a course of mathematical study. 



The range of the science of algebra is so vast, and its appli- 

 cations are so various, both in their objects and in their degrees 

 of difficulty, that it is quite impossible to fix absolutely the 

 proper proportion of space which should be assigned to the 

 developement of its difterent departments. If a system of al- 

 gebra could be confined to the statement of fundamental prin- 

 ciples, and to the establishment of fundamental propositions 

 only, it might be possible to approximate to a fixed standard, 

 which should possess the requisite union of simplicity and of 

 sufficient generality. But it is a science which cannot be taught 

 by an exposition of principles and their general consequences 

 only, but requires a more or less lengthened institution of ex- 

 amples of many of its different applications, in order to produce 

 in the student mechanical habits of dealing with symbols and 

 their combinations. The extent also to which such develope- 

 ments are necessary will vary greatly with the capacities of dif- 

 ferent students, and it would be quite impossible to determine 

 any just mean between diffuseness and compression which shall 

 be best adapted to the wants of the general average of students, 

 or to the systems of instruction followed by the general average 

 of teachers. 



In the early part of the last century the Algebra of Maclaurin 

 was almost exclusively used in the public education of this 

 country. It is unduly compressed in many of its most essential 

 elementary parts, and is also undvdy expanded in others which 

 have reference to his own discoveries. It was written, however, 

 in a simple and pure taste, and derived no small part of its 

 authority as a text-book from the great and well-merited repu- 

 tation of its author. It was subsequently, in a great measure, 

 superseded, in the English Universities at least, by the large 

 work of Sanderson, which was composed by this celebrated 

 teacher to meet the wants of his numerous pupils. It was, in 

 consequence, swelled out to a very unwieldy size by a vast 

 number of examples worked out at great length ; and it laboured 

 under the very serious defect of teaching almost exclusively 

 arithmetical algebra, being far behind the work of Maclaurin 

 in the exposition of general views of the science. At the latter 

 end of the last century Dr. Wood, the present learned and 

 venerable master of St. John's College Cambridge, in conjunc- 



