in Mathematical Reasoning. 375 
to justify: the reasons which have induced me to do so, are to be 
found in the incomplete manner, in which these parts of the sub- 
ject are treated by the generality of our elementary writers, a cir- 
cumstance which impedes the subsequent progress of the student 
more than is perhaps usually allowed. To those who may here- 
after employ themselves in supplying a considerable desideratum, 
in English Mathematics, by composing an introductory treatise on 
the application of Algebra to Geometry, I may be permitted to 
recommend a copious developement of its very first elements, not 
merely a detailed account of the changes of the situation or direc- 
tion of lines, by changes in the signs, or magnitudes of quantity, 
or by the circumstances of the radicals, in which they may be in- 
volved : but by way of giving practical illustrations of the precepts 
so delivered, they should be accompanied by a series of examples, 
in which every circumstance, attending the section of two or more 
right lines, should be explained with scrupulous minuteness: the 
influence of such a course of reading, will be sensibly felt by the 
student as he proceeds; and the uninviting dryness of the results 
will be relieved by a judicious selection, which may render them 
valuable points of reference, in his future reading.* How much 
such a system is calculated to assist all our enquiries in theoretical 
mechanics will be allowed by those, who have had, in some measure, 
to form it for themselves, at the moment when the natural difficul- 
ties of the subject were sufficient to require their undivided atten- 
tion. Iam indeed inclined to refer a considerable portion of those 
difticulties, which are so much complained of by English students, 
well versed in the mathematical science of their own country, 
when they first open the works of the continental geometers, to 
* These observations were written prior to the publication of the Analytical Geometry 
of Mr. Lardner, and to the still more recent work of Mr. Hamilton. I am happy that 
one considerable impediment to the progress of the English student is at length removed. 
