and on Algebra as the Science of Pure Time. 297 
intuitions, connected with that notion of time, and fitted to become the sources of a pure Science; and on 
the actual deduction of such a Science from those principles, which the author conceives that he has 
begun. Whether he has at all succeeded in actually effecting this deduction, will be judged after the 
Essay has been read; but that such a deduction is possible, may be concluded in an easier way, by an 
appeal to those intuitions to which allusion has been made. That a moment of time respecting which 
we inquire, as compared with a moment which we know, must either coincide with or precede or follow 
it, is an intuitive truth, as certain, as clear, and as unempirical as this, that no two straight lines can 
comprehend an area. The notion or intuition of OrpER 1n Time is not less but more deep-seated in the 
human mind, than the notion or intuition of OrDER IN SpAcE; and a mathematical Science may be founded 
on the former, as pure and as demonstrative as the science founded on the \atter. There is something 
mysterious and transcendent involved in the idea of Time ; but there is also something definite and clear: 
and while Metaphysicians meditate on the one, Mathematicians may reason from the other. 
Ill. That the Mathematical Science of Time, when sufficiently unfolded, and distinguished on the 
one hand from all actual Outward Chronology (or collections of recorded eyents and phenomenal marks 
and measures), and on the other hand from all Dynamical Science (or reasonings and results from the 
notion of cause and effect), will ultimately be found to be co-extensive and identical with Algebra, so far 
as Algebra itselfis a Science: is a conclusion to which the author has been led by all his attempts, whe- 
ther to analyse what is Scientific in Algebra, or to construct a Science of Pure Time. It is a joint result of the 
inductive and deductive processes, and the grounds on which it rests cou!d not be stated in a few general 
remarks. The author hopes to explain them more fully in a future paper; meanwhile he refers 
to the present one, as removing (in his opinion) the difficulties of the usual theory of Negative and Ima- 
ginary Quantities, or rather substituting a new Theory of Contrapositives and Couples, which he considers 
free from those old difficulties, and which is deduced from the Intuition or Original Mental Form of Time : 
the opposition of the (so-called) Negatives and Positives being referred by him, not to the oppos'tion of 
the operations of increasing and diminishing a magnitude, but to the simpler and more extensive contrast 
between the relations of Before and A fter,* or between the directions of Forward and Backward ; and 
Pairs of Moments being used to suggest a Theory of Conjugate Functions,+ which gives reality and 
meaning to conceptions that were before Imaginary,{ Impossible, or Contradictory, because Mathemati- 
cians had derived them from that bounded notion of Magnitude, instead of the original and comprehensive 
thought of OrpER 1N PRroGreEssion. 
“It is, indeed, very common, in Elementary works upon Algebra, to allude to past and future time, as one among many 
illustrations of the doctrine of negative quantities ; but this avails little for Science, so long as magnitude instead of PROGRES- 
Sion is attempted to be made the basis of the doctrine. 
+ The author was conducted to this Theory many years ago, in reflecting on the important symbolic results of Mr. 
Graves respecting Imaginary Logarithms, and in attempting to explain to himself the theoretical meaning of those remark- 
able symbolisms. The Preliminary and Elementary Essay on Algebra as the Science of Pure Time, is a much more 
recent developement of an Idea against which the author struggled long, and which he still longer forbore to make public, on 
account of its departing so far from views now commonly received. The novelty, however, is in the view and method, not 
inthe results and details: in which the reader is warned to expect little addition, if any, to what is already known. 
¢ The author acknowledges with pleasure that he agrees with M. Caucny, in considering every (so-called) Imaginary 
Equation as a symbolic representation of two separate Real Equations: but he differs from that excellent mathematician in 
_ his method generally, and especially in not introducing the sign »/—1 until he has provided for it, by his Theory of 
Couples, a possible and real meaning, as a symbol of the couple (0.1). 
