885 



NOTHING. 



NOTHING, DIFFERENCES OF. 



9SO 



One Som breve 

 2 Minims 



or 

 4 Crotchets f 



or 

 8 Quavers f 



or 

 16 Semiquavers 



32 Deuiiscmiquavera 



r 



. is equal to . 

 c* 



And Lence it will also appear, that one minim is equal to two 

 crotchets, &c. ; one crotchet to two quavers, &e., &c. 



The word note is frequently used as a synonym of sound ; thus we 

 tay a high, low, loud, or soft note, or the note A ; a flat note, &c. Some 

 confusion occasionally arises out of thia double meaning ; but the 

 musical nomenclature is very imperfect, and any attempt to improve 

 thia or any other branch of the art meets with little support from its 

 professors. 



NOTHING. In the article INFINITE will be found as much upon 

 this word a* 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 

 heads : their meanings are somewhat different. The first word, 

 not/liny, 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 201. 

 generally as twenty pounds, no shillings, no pence, and no farthings. 

 Hut if thi.s 20. had been the amount of a number of sums, the symbol 

 A'20 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 

 si^niitication, because such signification is useless. This point is of 

 some importance to the mathematician, as justifying a use of the 

 symbol where it might seem redundant. The 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 nn sheep," though it implies no more of positive conception 

 than the more simple phrase " an ox," may be a proper description 

 where the second would be no such thing. 



The cijififr is 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 numbers. 



The term KI-O considers rather as a starting point of magnitude 

 than as the symbol for the recognition of absence of all magnitude, and 

 really ilenutes, 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 both are less than zero. 

 Whenever magnitude is considered in connection with modifications, 

 the zero ami even the nothing "f 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 

 nee 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 the one extremity actually reaches the other, 

 length a destroyed, but one indication of position still remains, the 

 fixed extremity, or what was the fixed extremity so long as the line 

 had length. Different points (nothing! of length) still tell something 

 about the positions of the different lines which left them ; and there 

 are as many uotlan'/i 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 tcrui) with those of the restricted or arithmetical sense. [ALGEBRA.] 

 All direction however has disappeared when a line is reduced to a 

 point ; and considerations arising out of this, the principles of which 

 api-ar in FRACTIONS, VANISHING, will be applied in the article 

 K!T. 



It might seem as if, in the consideration of the term zero, we had 

 commenced an explanation of negative quantities, and had obtained a 



justification of the phrase less than zero, if not of less than nothing. 

 This may be true to a certain extent, too limited however for the pur- 

 poses of algebra, and not sufficiently expressive of the actual meaning 

 of the words. When the distinction of positive and negative quantities 

 is explained and adopted, the terms greater and less are no longer used 

 in their simple arithmetical meaning, but take a wider signification, 

 such as will allow old theorems of arithmetic to remain true under the 

 same phraseology as before. After an express extension of signification 

 has been accorded to these terms, it is not wonderful that uses of them 

 should be perfectly allowable which could not be made if we retained 

 the old significations. Those who use the extended meanings, without 

 fully understanding and admitting them, will make a mystery of 

 algebra : those who refuse to make the extensions, and yet charge others 

 who do not refuse with falling into all the absurdities which extended 

 uses without extended meanings present to themselves, are precisely in 

 the condition of the honest tar who asserted that the French were such 

 fools as not to know the difference between a cabbage and a shoe (chou). 

 But those again who, professing to use extended meanings, do not take 

 care to make their logic conformable to them, but neglect to distinguish 

 between premises which are true of one set of meanings and not of the 

 other, will fall into such mistakes as would be made by him who should 

 conclude that blood is salt water, because both circulate in arms (of the 

 human body and of the sea). 



Admitting the scale of positive and negative numbers, 



. . . -3, -2, -1, 0, +1, + 2, +3,... 



it is obvious that on the right of 0, on which we have quantities 

 common to pure arithmetic and algebra, we pass from the greater to 

 the less by moving our eyes from right to left ; while on the left we 

 have no meaning at all of greater and less yet established. Let us 

 agree then that we are to pass from what we will call the greater to 

 what we will call the less by passing from right to left in all cases ; and 

 there is no mystery in our meaning when we say that all negative 

 quantities are less than 0, and that 10 is less than 5. 



But is this convention a purely arbitrary one ? We answer that it 

 rather bears the character of interpretation [INTERPRETATION] than of 

 convention. Having new modes of quantity, with corresponding 

 extensions of addition and subtraction, we are rather to ask what greater 

 and less might to mean than what, with liberty of choice, we shall make 

 them mean. The great characteristics of greater and less (or more and 

 less) in arithmetic are, that the more you add the more you get, and 

 that the more you take away the less you leave, and rice versd. The 

 preceding extensions of greater and less are the only ones which will 

 allow of these theorems remaining universally true. Thus 



3 + ( 5) = 2, add more, and 3 + (-4) = - 1, 

 and 1 is greater than 2. Again, 



5-( 12) = 7, subtract less, -5-(-13) = 8, 



and 8 is greater than 7. 



It is however to be remembered, though no rule has been laid down 

 upon the subject, that it may be gathered from the practice of writers 

 that the terms smaller and diminution do not accompany less in its 

 extended meaning. The former term is particularly used in the Diffe- 

 rential Calculus to denote an approach to 0, which in a negative 

 quautity is algebraic increase, and in a positive quantity diminution. 

 And many, perhaps all, writers on the Differential Calculus, are lax iu 

 their use of all the comparative terms, sometimes employing them in 

 the algebraical and sometimes only in the arithmetical sense. The 

 inconvenience is not very great, as a student must have learned to 

 contend with greater difficulties than those of an unexplained use of 

 dubious terms, before he is able to make his way to the higher mathe- 

 matics. But it may be useful to give him a hint that, in reading works 

 of analysis, he would do well at first always to stop for a moment when 

 the word greater or less occurs, and a&k himself whether the problem 

 requires and allows the extended signification or not, and to make some 

 mark of distinction iu every place. This will at once ensure the sound- 

 ness of the first reading, and facilitate the second. 



NOTHING, DIFFERENCES OF. This name is given to certain 

 numbers which are used in so many different theorems that it is worth 

 while to tabulate them, and to consider them as fundamental numbers 

 of reference. They were first specifically noted in this point of view 

 by the late Bishop Briukley. We shall here confine ourselves to a 

 description of their derivation, an expeditious mode of calculating them, 

 a table of some of their values, and one instance of their use. 



If we take a series of terms a, b, c, e, &c., and form the successive 

 differences of a [DIFFERENCES, CALCULUS or], the symbols A a, A-'a, 

 &c., have a meaning which refers to the excess of b above a, &c. If 

 then a should happen to be=0, the symbols AO, A ; '0, &c., may stand 

 for finite quantities : for instance in 



2 

 21 



5 ? 1 2 &c. A0 = 2, A-0 = l. 

 9*1 

 14 5 



But as the preceding series is a set of values of J.r(j! + l) 1, in 

 which the first term is (-c = l gives 41.2 1), it would be necessary 



