CONCERNING TERRESTRIAL MAGNETISM. 
77 
sign of the radicals, — so also, as we should expect from the same principles being 
employed in both cases, in the equation of the curve of verticity the branches adapted 
to like poles and those adapted to unlike poles, are expressed in the same equation 
given in my former paper. It required, then, the previous separation of the two 
systems in the magnetic curve as a preliminary step to the separation of those in the 
curve of verticity. Such is the course I have pursued, but I have carefully abstained 
from the insertion of any properties of either curve except those which were essential 
in the determination of the number of points of terrestrial verticity. However, to 
avoid the long and complicated reductions into which I found the algebraical inves- 
tigation was leading me, I have in one instance recurred to a geometrical method of 
investigation, founded on an elegant property of the magnetic curve, (first given by 
Professor Leslie in his Geometrical Analysis, p. 400,) by means of which, and the 
genesis of the curve of verticity which it suggests, enables us to establish the required 
conclusion with ease and simplicity. 
No apology is necessary, I conceive, for the introduction of new methods of inves- 
tigating a problem in pure science, where those already existing are either insufficient 
or inconveniently operose in their application to that problem : nor yet for the em- 
ployment of several different methods which in treatises on pure science are usually, 
and in good taste, kept distinct, when we are investigating a physical problem to 
which any one of them, taken separately, is inadequate. I readily and fully admit 
the desirableness and superior elegance of unity of mathematical method, even in 
physical investigations ; and, doubtless, repeated efforts made by different geometers 
tends to the gradual formation of such united and systematic methods of develop- 
ment : nevertheless it is rarely the case that a unity in the full sense of the word can 
be brought into the mathematics of physical inquiries, — the unity that exists in the 
most perfect of them being more apparent than real. Mere symbolical notation does 
not constitute sameness of method. I have made no attempt of the kind in these 
inquiries, but have employed the language, methods, and notations, that seemed to 
me to be best adapted to obtaining the results after which the conditions and objects 
of the problem led me to search ; or otherwise more apparent symmetry might have 
been easily given to the several investigations into which I have entered. 
I have prefixed a few geometrical lemmas which were necessary to substantiate 
and facilitate the mode of reasoning here employed. Most of them are required in 
the present paper, the earlier one subservient to some of the others, and these for di- 
rect quotation. There is one indeed, (the sixth) which is not essential, but it is 
that all consideration of curves may be advantageously banished from such inquiries. Such a plan may, indeed, 
furnish an elegant abstraction fitted for the higher order of minds and the highest degree of mathematical 
skill : but it would at the same time effectually preclude the possibility of elementary acquirement, and in very 
many cases that of original inquiry in new departments of physical science. The consequence would therefore 
in reality be, to retrograde the science instead of facilitating its progress. Curves, though but so few of them, 
enter into so many branches of philosophical investigation, that science would, save to one in a thousand, be 
rendered unintelligible by their abolition. 
