Superlative

“ But why talk in superlatives, as if something that is beautiful could be surpassed? ”

“ On the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. The three, for instance, are regarded as a whole when we name them three. If, however, we count these three as one, two, three, then the number of times we count is an abstract number. Thus number in the abstract is the number of times that the act of counting is performed in any particular case. ”

“ Division teaches how to divide a number into two or more equal parts, as in the division of prize-money.
[39] Division is of great importance, whether political, ecclesiastical, commercial, civil, or social. Nothing is more likely to destroy your opponents than a split. Divide et impera is the true Machiavelian policy of all governments.
Numbers, that is the multitude, are to be divided, in a variety of ways,—by mob orators, or by mob-sneaks, or by parliamentary flounderers, or by mystifying pulpit demagogues. ”

“ SAPIENTIA. When numbers of any magnitude are set down in order. the first set down is called the "minor term" and the last the "major." When, in making a division, we say by how many the number is to be divided, we give the "divisor," but when we enumerate how many there are in each of the parts we set forth the"quotient." ”

“ The number of operations in a group may be either finite or infinite. When it is finite, the number is called the order of the group, and the group is spoken of as a group of finite order. If the number of operations is infinite, there are three possible cases. When the group is represented by a set of geometrical operations, for the specification of an individual operation a number of measurements will be necessary. In more analytical language, each operation will be specified by the values of a set of parameters. ”

“ Numerical quantities, to be added or subtracted, must be in the same denomination; we cannot, for instance, add 55 shillings and 100 pence, any more than we can add 3 yards and 2 metres.
31. Relative Position in the Series.—The above method of dealing with addition and subtraction is synthetic, and is appropriate to the grouping method of dealing with number. We commence with processes, and see what they lead to; and thus get an idea of sums and differences. If we adopted the counting method, we should proceed in a different way, our method being analytic. ”

“ For the articles of a statistical table, I think the last census of Congress presented what was proper, as far as it went, but did not go far enough. It required detailed accounts of our manufactures, and an enumeration of our people, according to ages, sexes, and colors. But to this should be added an enumeration according to their occupations. We should know what proportion of our people are employed in agriculture, what proportion are carpenters, smiths, shoemakers, tailors, bricklayers, merchants, seamen, &c. ”

“ It is the obvious one that when the cases which exist are shared among several kinds, it is impossible that each of those kinds should be a majority of the whole: on the contrary, there must be a majority against each kind, except one at most; and if any kind has more than its share in proportion to the total number, the others collectively must have less. ”

“ Thus, by taking a little trouble, the historian can apply one constant measure to his judgments of increase and decrease, as he reviews social changes. He must subdivide his community into enumeration units, and against each unit, at each convenient date, he must enter a record of his judgment that the trait, activity, interest, or relation under investigation can be predicated of a large or of only a small majority, of a large or of only a small minority, of the individuals composing the enumeration unit. ”

“ If we formed our groups, not by equal numbers, but by equal amount of variation, we should probably find, in accordance with the law of averages, that the central group in which the mean characteristics prevailed was much more numerous than the extremes, perhaps twice, or even three times, as great as either of them, and forming such a series as the following—10 maximum, 30 mean, 10 minimum development. ”

“ An aggregate whose elements are real numbers is said to “extend to infinite values” if, after any number , however great, is specified, it is possible to find in the aggregate numbers which exceed in absolute value. Such an aggregate is always infinite. The “neighbourhood of a number (or point) for a positive number ” is the aggregate of all numbers (or points) for which the absolute value of denoted by , does not exceed .
5. General Notion of Functionality.—A function of one variable was for a long time commonly regarded as the ordinate of a curve ”

“ Yet wait until the end and you will see that the division has been made with mathematical accuracy—that the measure, the quality of the soil, the slope of the field, the distance from the village—everything in short has been taken into account, that the reckoning has been correctly done and, what is most important, that every one of those present who were interested in the division is certain of the correctness of the division or settlement. The cry, the noise, the racket do not subside until every one is satisfied and no doubter is left. ”

“ We commence our present division—and shall follow the like course with the others—with its higher forms, and, proceeding in descending order, shall in each conclude with those humble rudimentary forms in which the rigid divisions of classification are obliterated, and the only differences which can be assigned are, at best, but little more than arbitrary. ”

“ He runs up a few rounds of the multiplication table just to see if they are all there. He ceases to think at all, looks steadily out of the window and sees nothing. He ceases to count anything in the census. ”

“ Propor′tionless, without proportion or symmetry of parts.—Proportional representation, representation in a parliament, &c., according to the number of electors in an electoral district.—Compound proportion, a proportion between two quantities, each of which is the product of two or more, those of the first set being in their order proportional to those of the other; In proportion, in the degree or measure, according; Inverse, or Reciprocal, proportion, an equality of ratio between two quantities and the reciprocals of other two, as 6 : 3 = 1⁄2 : 1⁄4. ”

“ In a House of 670 Members, or any other even number, if divided into two parties, the majority (in the sense he uses the word—viz., the difference) must always be an even number. It is true that the division lists sometimes show a majority which is an odd number, but in such a case an odd number of Members must have been absent from the division. ”

“ A similar proceeding will apply wherever each individual relate belongs to one individual correlate exclusively. If we want the number of x's per y, and no x belongs to more than one y, we have only to divide the whole number of x's of y's by the number of y's. ”

“ A specific ten thousand persons present is much more effective and, being a round number, is a sufficient indication that no actual count has been made. In all cases where there is a choice between a specific and a general term, the specific one should be used. ”

“ The aggregate of all individual things occurring in a definite concept, or the common characteristics of which make up this concept, is called a group. Such a group may consist of a limited or finite number of members, or may be unlimited, according to the nature of the concepts that characterize it. Thus, all the integers form an unlimited or infinite group, while the integers between ten and one hundred (or the two-digit numbers) form a limited or finite group. ”

“ The endeavours of the individuals engaged count cumulatively, to such effect that doubling their forces will more than double the aggregate efficiency; and conversely, reducing the number will reduce the effectiveness of their work by something more than the simple numerical proportion. ”

“ The totality of entities which lend themselves to a given type of order, form a collection, or an aggregate; and if the order of a given aggregate is to be unambiguous, the members of this aggregate must be uniquely determined, and must be mutually exclusive: they must be what we shall later denote by the term elements of the aggregate. Externality and ‘elementarity’ are only different consequences of the same relation, which we denote by the term extension, and which thus together with the relation of order, will constitute the basis of our definition of space and time. ”

“ In the system of counting every place, or column, counting from the right, has a value ten times greater than the one in the place or column nearest on the right. Thus in the number 36,542 the first figure on [71] the right represents “ones,” the next ten times as much or “tens,” the next ten times as much again or “hundreds,” and so on. We really read this number backward when we name it, for in handling it in any way we have to start with the last figure, representing the “ones.” The number really means two ones, four tens, five hundreds, six thousands and three ten thousands. ”

“ It was then judged that the number of people, as therein provided, was the best criterion for fixing the proportion of each state, and that proportion in each state to be raised in the most easy manner for the people. ”

“ This fact is more accurately summarized in the coefficients of correlation, table XV, p. 37.
As seen from its heading, table XXII gives the systems in order of achievements. These serial standings are derived from tables III and IV. Reading from the top, system XXIII has an average serial standing of one, being lowest in both reasoning and fundamentals; system XXV ranks three in average serial standing, being fourth from lowest in reasoning and second from lowest in fundamentals; the readings for the other systems are similar. ”

“ Any group falls into subdivisions, the members of each of which are approximately equal, when measured by any standard, because the classification is imperfect. If we make it more refined, the subdivisions must be subdivided again. ”

“ We count when we co-ordinate in turn one member of a group after another with the members of the number series that succeed one another until [86] the group to be counted is exhausted. The last number required for the co-ordination is called the sum of the members of the counted group. Since the number series continues indefinitely, every given group can be counted. ”

“ If a quantity being given, first one quantity be both added to it and subtracted from it, and then another greater or less, the proportion of the remainder to the aggregate, is greater where the less quantity is added and substracted, than where the greater quantity is added and substracted. ”

“ We have, then, considered variation of units in a repeated series, where they may vary and where they must, and we find the real value of repetition to appear in inverse proportion to the individual significance of the separate units; the more interesting or expressive the unit is in itself with individual significance, the less do we want it repeated ”

“ Composite pictures, are, however, much more than averages; they are rather the equivalents of those large statistical tables whose totals, divided by the number of cases, and entered in the bottom line, are the averages. ”

“ The four groups were even over even ratios, which we decided could be reduced in various manners to the other groups by dividing until one number of the ratio was no longer even; then we eliminated the two other groups which had odd numbers divided by either odd or even numbers, because the first or top number had to be twice the second or bottom number, and therefore could not be odd; this left the last group we are now to greet, even divided by odd.
Boy: Wonderfully put, Socrates. It is amazing how neatly you put an hour of thinking into a minute. ”

“ The theoretic proportion is, that numbers should be exactly balanced,—that one sovereign should equal six hundred lords, that six hundred lords should equal six hundred and fifty-eight commoners, and that these should represent twenty-nine millions of people. ”

“ The process of counting is nothing but a setting up of a one-to-one correspondence between the objects to be counted and certain words, 'one,' 'two,' 'three,' etc., in the mind. Many savage peoples have discovered no better method of counting than by setting up a one-to-one correspondence between the objects to be counted and their fingers. ”

“ In division, what is called the Italian method of arranging the divisor and quotient, appears to be preferable to the common one, as it places them in such a manner as to be easily multiplied by each other, and as it agrees with algebraic notation. ”

“ Constituencies returning several members are formed. A representative is given to every group of electors which attains to a definite proportion of the whole, the proportion depending upon the number of members to be returned. ”

“ The mass of data that has not been seen, or very imperfectly seen, that has been misunderstood, or incorrectly recorded, the many pitfalls of omission and commission cause one at times to wonder at the temerity of those who feel confidence in what is termed the comparative method. ”