NOTHING. In the article INFixTrE will be found as much upon this word as will enable us to dispense with the consideration of the symbol 0, as the limit approached but never attained by the continual diminution of magnitude.
Among the terms used in mathematical language are nothing, cipher, and zero. The etymologies of the two latter terms are explained under those heads : their meanings are somewhat different. The first word, nothing, implies the absence of all magnitude, but its occurrence denotes either that magnitude did exist, or might have existed, or does exist, under similar circumstances in other problems, or in the same problem under different points of view. Were it not for this, the word would be useless : thus we do not consider it necessary to speak of M. generally as twenty pounds, no shillings, no pence, and no farthings. But if this 201. had been the amount of a number of sums, the symbol £20 0 0 would be useful as indicating that the results of an operation (addition) had left no quantities in places where beforehand quantity might have been expected. The term unity would have been useless in the same manner, except as a tacit reference to other units ; anything we please is one of its kind, and accordingly the indefinite article (a or an), which is certainly one in etymology, has lost its definite monadic signification, because such signification is useless. This point is of some importance to the mathematician, as justifying a use of the symbol 0 where it might seem redundant. The 0 and 1 are frequently useful as symbols of distinction where they are not wanted as symbols of operation : in like manner, in common Language, the simple phrase " one ox and no sheep," though it implies no more of positive conception than the more simple phrase " ao ox," may be a proper description where the second would be no such thing.
The cipher is 0 considered in a purely arithmetical point of view, as the mode of denoting a blank column intervening between, or following, or even preceding, columns which contain significant munbers.
The term zero considers 0 rather as a starting point of magnitude than as the symbol for the recognition of absence of all magnitude, and really denotes, not the entire absence of magnitude, but the arbitrary determination to reckon all magnitudes by their excess or defect from a certain zero magnitude. Thus the zero point of the thermometer
does not mean that shown when there is no temperature, but a certain temperature, that of freezing water ; and degrees above and below zero indicate excesses or defects of temperature above or below that standard. It is then perfectly proper to say that ten degrees below zero is a lower temperature than five degrees, and that hotly are less than zero. Whenever magnitude is considered in connection with modifications, the zero and even the nothing of such magnitude may require to be considered with similar modifications, even though all absolute magni tude is lost. Straight lines, for example, admit of consideration with reference not only to their lengths, but also to their positions and directions. Let the straight lines diminish each by an approach of one extremity towards the other, and position and direction still always distinguish each line from the others, though all be of the same linear magnitude (length); when tire one extremity actually reaches the other, length is destroyed, but one indication of position still remaios, the fixed extremity, or what was the fixed extremity so long as the line had length. Different points (nothings of length) still tell something about the positions of the different lines which left them ; and there are as many not/tin/a of length (distinguishable) as there are different points in space. These zeros, as it might be proper to call them, are of most essential consequence, as zeros, in the complete method of con necting the explanations of symbols in algebra (in the widest sense of the term) with those of the restricted or arithmetical sense. [./moznite.] All direction however has disappeared when a line is reduced to a point ; and considerations arising out of this, the principles of which appear in FRACTIONS, VANISHINO, will be applied in the article TANGENT.