294 Professor Hamitton on Conjugate Functions, 
It is not here asserted that every or any Algebraist belongs exclusively to any one of these three 
schools, so as to be only Practical, or only Philological, or only Theoretical. Language and Thought 
react, and Theory and Practice help eachother. No man can be so merely practical as to use frequently 
the rules of Algebra, and never to admire the beauty of the language which expresses those rules, nor 
care to know the reasoning which deduces them. No man can be so merely philological an Algebraist 
but that things or thoughts will at some times intrude upon signs; and occupied as he may habitually be 
with the logical building up of his expressions, he will feel sometimes a desire to know what they mean, 
or to apply them. And no man can be so merely theoretical or so exclusively devoted to thoughts, and 
to the contemplation of theorems in Algebra, as not to feel an interest in its notation and language, its 
symmetrical system of signs, and the logical forms of their combinations ; or not to prize those practical 
aids, and especially those methods of research, which the discoveries and contemplations of Algebra have 
given to other sciences. But, distinguishing without dividing, it is perhaps correct to say that every Alge- 
braical Student and every Algebraical Composition may be referred upon the whole to one or other of 
these three schools, according as one or other of these three views habitually actuates the man, and emi- 
nently marks the work. 
These remarks have been premised, that the reader may more easily and distinctly perceive what the 
design of the following communication is, and what the Author hopes or at least desires to accomplish. 
That design is Theoretical, in the sense already explained, as distinguished from what is Practical on the 
one hand, and from what is Philological upon the other. The thing aimed at, is to improve the Science, 
not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of 
thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor 
failures of symmetry in expression. And that confusions of thought, and errors of reasoning, still darken 
the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit 
of love and honour have studied Algebraic Science, admiring, extending, and applying what has been al- 
ready brought to light, and feeling all the beauty and consistence of many a remote deduction, from 
principles which yet remain obscure, and doubtful. 
For it has not fared with the principles of Algebra as with the principles of Geometry. No candid 
and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by 
Eucuip in his Elements, two thousand years ago; though he may well desire to see them treated in a 
clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in 
the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the 
plan of the argument. But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine 
of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a 
greater magnitude may be subtracted from a less, and that the remainder is less than nothing ; that two 
negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by 
the other, and that the product will be a positive number, or a number denoting a magnitude greater than 
nothing ; and that although the square of a number, or the product obtained by multiplying that number 
by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called 
imaginary, can be found or conceived or determined, and operated on by all the rules of positive and 
negative numbers, as if they were subject to those rules, although they have negative squares, and must 
therefore be supposed to be themselves neither positive nor negative, uor yet null numbers, so that the 
magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, 
nor even equal to nothing. It must be hard to found a Screncz on such grounds as these, though the 
