Theory of Conjugate Functions, or Algebraic Couples ; with a Preliminary and 
Elementary Essay on Algebra as the Science of Pure Time. 
By WILLIAM ROWAN HAMILTON, 
MRI. A., F.R.A.S., Hon. M. R. S. Ed. and Dub., Fellow of the American Academy of Arts and 
Sciences, and of the Royal Northern Antiquarian Society at Copenhagen, Andrews’ Professor of 
Astronomy in the University of Dublin, and Royal Astronomer of Ireland. 
Read November 4th, 1833, and June Ist, 1835. 
General Introductory Remarks. 
Tue Study of Algebra may be pursued in three very different schools, the Practical, the Philological, or 
the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contempla- 
tion ; according as ease of operation, or symmetry of expression, or clearness of thought, (the agere, the 
fari, or the sapere,) is eminently prized and sought for. The Practical person seeks a Rule which he 
may apply, the Philological person seeks a Formula which he may write, the Theoretical person seeks a 
Theorem on which he may meditate. The felt imperfections of Algebra are of three answering kinds. 
The Practical Algebraist complains of imperfection when he finds his Instrument limited in power; when 
a rule, which he could happily apply to many cases, can be hardly or not at all applied by him to some 
new case; when it fails to enable him to do or to discover something else, in some other Art, or in some 
other Science, to which Algebra with him was but subordinate, and for the sake of which and not for its 
own sake, he studied Algebra. The Philological Algebraist complains of imperfection, when his Language 
presents him with an Anomaly; when he finds an Exception disturb the simplicity of his Notation, or the 
symmetrical structure of his Syntax; when a Formula must be written with precaution, and a Symbolism 
isnot universal. The Theoretical Algebraist complains of imperfection, when the clearness of his Con- 
templation is obscured; when the Reasonings of his Science seem anywhere to oppose each other, or 
become in any part too complex or too little valid for his belief to rest firmly upon them; or when, 
though trial may have taught him that a rule is useful, or that a formula gives true results, he cannot 
prove that rule, nor understand that formula: when he cannot rise to intuition from induction, or cannot 
look beyond the signs to the things signified. 
