SYMBOLS and NOTATION. The word symbol (from the Greek sgmbolon, at'aLt3oitov) means " that which is taken with," and a symbol is a mark which is always attached to some one particular meaning.
Notation (nota, a known mark) is the method of selecting and assigning meaning to symbols, and the theory of notation (if it yet deserve the name) includes the consideration and choice of symbols, with the formation of rules of selection, so as to take the symbols which are best adapted for the purpose.
This subject *gilt be treated in a very wide manner ; fur all marks with understood meanings are symbols, from written words to direction-posts. A picture is a symbol, the force of which lies in the resemblance to its object, and many of the earliest symbols must have been pictorial. It is obvious that a general treatment of the subject would hardly be within the power of any one person, and that its extent would be enormous, though it would be desirable to have it discussed in a more general form than has yet been attained, in order that its different parts might receive aid from the rest. Symbols are to the progress of civilisation precisely what mechanism ie to that of the arts,—not a moving force, perfectly dead in themselves, but capable of being made the medium by which the power is conveyed to its desti nation, and adapted to its object. They are the instruments of our first thoughts and the originators of new ones. The process by which the earliest symbols called out a yet higher intelligence than that which produced them, which last was again employed in perfecting the symbols themselves, and so on alternately, exactly resembles what has taken place in the mechanical arts. Tho earliest and rudest tools were first employed to make better ones; and every improvement in the use ---- r wee has found one of its best applications in the construction of itself.
We propose in this article to treat particularly of mathematical notation, which, like language, has grown up without much looking to, at the dictates of convenience and with the sanction of the majority. Resemblance, real or fancied, has been the first guide, and analogy has succeed ed.
Signs are of two kinds : 1st, those which spring up and are found in existence, but cannot be traced to their origin • 2ndly, those of which we know either the origin, or the epoch introduction, or both. Those of the first kind pass into the second as inquiry advances.
[ALrilAtirr.] In our present subject we have mostly to deal with the second class.
Mathematical marks or signs differ from those of written language in being almost entirely of the purely abbreviative character, since it is possible that any formula might be expressed in words at length. We say possible, because it is barely so, not meaning thereby to imply that the mathematical sciences could over have flourished under a system of expressions in words. A well-understood collection of notions, how ever extensive, becomes simple as a matter of conception by use and habit, and thus becomes a convenient resting-point for the mind and a suitable basis for new combinations of ideas. Now it is the charac teristic of the advance of human knowledge that the mind never grapples at once with all that is contained in the notions under use for the time being, but only with some abstraction derived from a previous result, or some particular quality of that result. Hence no symbol which should contain the representative of every idea which occurred in the previous operations would ever be necessary ; and more than this, it would even be pernicious from its complexity, as also from its suggesting details which are not required. The generalisation, or rather abstraction, which is the distinctive character of the civilised tanguage as compared with the savage (though the latter is not wholly without it), must be the ruling process of mathematical notation, as it is of the advance of spoken language; and iu this point of view tho connection of our subject with speech presents more analogies and gives more instruction than its comparison with the written signs of speech. The latter is a bounded subject. When once it is agreed how the different modifications of sound shall be represented, written language follows immediately ; nor do the infinite modes of using words require any modification of the method of writing them. In our modem works, for instance, it would be difficult to find many artifices of notation with which to compare the never-ceasing varieties of mathematical signs. In mentioning the marks of punctuation and reference, the italics for emphatic words, and the varieties of print by which notes are distinguished from text, &c., we have almost exhausted the list.