thought must be regarded as a disjoined succession of dead results, but as living insights in one line, each piercing deeper and deeper. 112. In this light, note that the revelation in antiquity of fractional and surd numbers, and the recognition of number as positive and negative which has prevailed for two centuries (these may be regarded as the "first" (§ 78) and second extensions of the number-concept), are both merely special cases of the universal principle here advanced. 113. To generalize is to see in a multiplicity of objects similar relations to one form of mental activity that knows those objects. But until one sees the need of a deeper principle than that which he has hitherto employed, he does not seek a way leading from what is known to him to knowledge beyond. Any idea is at first bare of manifold essential relations, external and internal. By reflection such relations are slowly revealed. During the process the idea may seem derivative from the relations (Cf. geometric definitions of number, § 25); but finally this looseness must be reduced to order, and then all its belongings are seen to unfold from the idea itself, "first the blade, then the ear, after that the full corn in the ear.” 114. What has just been said would do for a description. of the famous dialectic which Hegel describes as "the self-movement of the notion (Begriff)." Indeed, it is not much more than a paraphrase of its description by Dr. Harris, "Seize an imperfect idea and it will show up its imperfection by leading to and implying another idea as a more perfect or complete form of it. Its imperfection will show itself as dependence on another." (Italics mine.) 115. I know no other method by which the teacher can lead a student to attain for himself a concept of number adequate to any comprehension of modern mathematical analysis. Each tentative idea of number must pass over into the next deeper as the result of further and further insight into the subject. It remains to apply in characteristic cases the Principle of Continuity, discovering from the formula of definition of any operation the nature of the resulting number, as well as the efficacy of any such new phase of number in any combination in the defined operations. XII. SIGNIFICANCE AND EFFICACY OF NUMERICAL OPERATIONS UNDER THE ULTIMATE CONCEPT. 116. The very first application of the Principle of Continuity to the generalization of the operation Subtraction, displays a number sui generis, which is of immense importance in analysis. The formula of definition of subtraction is (vide § 42) b a + a = - b. Then a a = what number? The formula declares that it is a number which, added to a, makes a; that is, it is a number which has no efficacy in addition, and therefore none in subtraction. The best and only unprejudicial name for this number is zero. Its symbol in arithmetic and in the algebra of number is 0. I trust that at least it has been made clear to the student that it is only the very primary and crudest concept of number which would consider zero "nothing;" for although of no efficacy in addition or subtraction, it will presently be seen to exert extraordinary effect in every other operation. I entreat the student not to slip at this point; for the human mind, once made sensible of its powers, will never afterwards suffer its conception to be clogged by the tyranny of material categories. Moreover, it may quite commonly be found necessary to translate. into correct terms much discourse in mathematical treatises, even when written by men eminent for skill and learning, to say nothing of inadequate or erroneous presentations in works on physics and applied mathematics in general. For example, you may read a Trigonometry which defines the trigonometric ratios not as numbers, but as sects (pieces of straight lines); yet you can often catch the author adding one of his bits of straight lines to 2 or 32, and in a context where he really means the number 2 or 32, etc. Occasionally you will meet denial or even ridicule of all that I endeavor to lead you to see, and perhaps by a man of world-wide fame. For example, in a didactic treatise on Mathematics by De Morgan, published in a serial Library of Useful Knowledge, London, 1836, zero is conceived to be "nothing"; for on page 23 one reads, "Above all, he must reject the definition, still sometimes given of the quantity — a, that it is less than nothing. It is astonishing that the human intellect should ever have tolerated such an absurdity as the idea of a quantity less than nothing; above all, that the notion should have outlived the belief in judicial astrology and the existence of witches, either of which is ten thousand times more possible." The truly astonishing thing concerning the human intellect is that such a man as De Morgan could have written this sentence, familiar as he must have been with Newton's distinction, "Quantitates vel Affirmativæ sunt seu majores nihilo, vel Negativæ seu nihilo minores." But, although deficiency is quite as quantitative as excess, the whole remark is impertinent; for zero is not "nothing." Negative numbers are unquestionably less than zero. Yet, taking him at his own word, De Morgan should have hesitated before ridiculing as crazy the careful dictum of as powerful and piercing an intellect as has ever served man's will. 117. Before investigating the efficacy of zero in other operations, let us look into further results of the generalization of subtraction. What are the properties of the resulting number in the operation ba, if ba? Consider the results of the following series of operations, 1+2; 1+1; 1; 1−1; 1-2; 1-3; 14, etc. Here we have a series of numbers which at first decrease by 1, viz., 3; 2; 1; 0. The subsequent numbers respectively answer the questions, what number added to 2 makes 1, added to 3 makes 1, added to 4 makes 1? Now, in these operations the sums remain the same, and the given summands in each case increase by 1; it is clear, therefore, that the required summands must decrease by 1. Moreover, these numbers in additive combination nullify 1, 2, 3, etc.; that is, make the sum in each case zero. Thus, 1+ (12) = (1 + 1) − 2 = 0; 2 + (1 − 3) = (2 + 1)-3=0; 3+ (1 − 4) = (3 + 1)-4=0. Such reflections reveal an unending series of discrete numbers decreasing from zero, each less than the preceding by 1. Their effect in nullifying 1, 2, 3, etc., in addition, renders appropriate the appellations positive and negative to primary numbers and these now discerned. These terms are established terms in logic, and are expressive of just such a relation of clean-contradictory as has been discovered in these modes of number. On this score, either might be called positive and the other negative; but every propriety commends the course adopted - primary numbers are positive, and such results as we have just considered, negative. That negative number finds unlimited corroboration in adaptation to the facts of other sciences, has been amply illustrated (§ 109); but its existence for pure mathematics is nowise dependent upon such circumstances. Negative number should never be defined or explained by such oppositions as right and left, up and down, forward and backward, north and south, past and future, capital and debt; but always in its essential character as number. 118. Writing pos. for positive, and neg. for negative, it is evident that pos. a + neg. b = pos. a pos. 1 pos. b; for = -pos. 1, etc. -pos. 2 neg, 1, therefore, by definition of subtraction, 1 pos. 2 neg. 1 pos. 1; but = 2-pos. pos. Also, since subtraction is the inverse of addition, 119. Hereby Section 42 is completely generalized, and the common rule about "signs" and parentheses for additions and subtractions established without restriction. 120. We are arrived now at a matter of extreme importance, viz., the dual significance of the signs + and One of the most salient imperfections of ordinary textbooks is their failure to make a clear-cut distinction between the essentially double meaning of +, and of -. Too often the operational significance alone is defined, although on the next page you may find a complacent statement a+ (− a) = 0; whereas, if + means add, and means subtract, a + (− a) means, "starting with a, add and then subtract a," of course, with the result a. And under a purely operational definition such an expression as a / − b is like a "sentence" made by writing words on |