Enumeration

“ 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 neglect them, and merely collect examples of the sequence A-p, this is 'simple enumeration'; and although simple enumeration, when the instances of agreement are numerous enough, may give rise to a strong belief in the connection of phenomena, yet it can never be a methodical or logical proof of causation, since it does not indicate the unconditionalness of the sequence. For simple enumeration of the sequence A-p leaves open the possibility that, besides A, there is always some other antecedent of p, say X; and then X may be the cause of p. ”

“ When the enumeration, moreover, is thus bent from the direct order of progression, and is made to form, if we may so say, a kind of corner for the purpose of taking a different course, what meets us at that corner is mentioned twice over,—namely, at the close of the preceding series, and at the head of the deflection specified. ”

“ Thus we get the ordinary system of enumeration, and there can be no manner of doubt that man's ten fingers are the root of it. We are told in treatises on arithmetic that it would have been much more convenient if we had agreed to count by twelves instead of by tens; and possibly this may be true. But if it be, we have so much the more evidence, if evidence be needed, that the basis of the system of counting was not determined by theoretical considerations, but by the simple elementary fact of the number of human digits being ten and not twelve. ”

“ I, A. B. do solemnly swear (or affirm) that I will make a just and perfect enumeration and description of all persons resident within the division assigned to me by the marshal of the district of ⁠ (or the secretary of the territory of ⁠ as the case may be) and make due return thereof to the said marshal (or secretary) agreeably to the directions of an act of Congress, intituled “An act providing for the second census or enumeration of the inhabitants of the United States,” according to the best of my ability. ”

“ So the census grew, from the simple enumeration of the people in 1790, under the very brief schedule which has been given, not only through various additions to the population schedule, but by the introduction of inquiries relating to production, until nearly all the material conditions surrounding the people, their industries, their wealth, their taxation, their carrying trade, fisheries, schools, health—nearly everything, in fact, that the social scientist wishes to know concerning the people and their conditions—are embodied in the census. ”

“ Once more, the child is made to count, and in the process his conceptions of number are cast into a decimal shape, one in which each higher factor is made up of ten of the next lower, till he comes to feel that such tenfoldness is a natural characteristic of enumeration. ”

“ Under that principle, "where general words follow specific words in a statutory enumeration, the general words are construed to embrace only objects similar in nature to those objects enumerated by the preceding specific words." ”

“ Some such principle as this is required if the method of simple enumeration is to be valid.
But this brings us to our other question, namely, how is our principle known to be true? Obviously, since it is required to justify induction, it cannot be proved by induction; since it goes beyond the empirical data, it cannot be proved by them alone; since it is required to justify all inferences from empirical data to what goes beyond them, it cannot itself be even rendered in any degree probable by such data. ”

“ If we could only observe in others, or could in our own case recall to mind how long it is before a child can actually count three, or clearly separate from itself the external objects it perceives, or learns to distinguish between any two objects, or between them and itself, we shall be forced to admit that the first basis of enumeration has an empirical origin, and that it is on such, consequently, that all mathematical science is built up and founded. ”

“ 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 not easy to find a definition for the Science of Numbers that shall be both brief and comprehensive. We may say, however, that it is a method of enumeration by means of which the life of men and things may be fixed and rounded out to a state of approximate completion by being brought into harmony with the laws of Omnipotence. ”

“ When the genus is separated at once into individual objects the process is not logical division, but simple enumeration. Logical division implies a separating into smaller class terms, each term being a genus of still smaller subdivisions. This process may be continued till the last division gives individuals as species. Enumeration takes place when the first subdivision results in a list of individuals. To illustrate ”

“ To attempt a minute enumeration would, indeed, be an impracticable task; and considering the delicate nature of others, which are still the subject of heated controversy, where the ashes are scarcely yet cold, which cover the concealed fires of former political excitements, it is sufficiently difficult to detach some of the more important from the mass of accidental matter, in which they are involved. ”

“ The more instances we examine, and the more care we employ, the more assurance shall we acquire, that the enumeration, which we form from the whole, is complete and entire. ”

“ The induction which proceeds by simple enumeration is puerile, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too small a number of facts, and those only the most obvious. But a really useful induction for the discovery and demonstration of the arts and sciences should separate nature by proper rejections and exclusions, and then conclude for the affirmative, after collecting a sufficient number of negatives. ”

“ For even the greater generalizations, which embrace comprehensive Kinds containing under them a great number and variety of infimæ species, are only empirical laws, resting on induction by simple enumeration merely, and not on any process of elimination—a process wholly inapplicable to this sort of case. ”

“ While section 2 expressly provides for an enumeration of persons, Congress has repeatedly directed an enumeration not only of the free persons in the States, but also of those in the territories, and has required all persons over eighteen years of age to answer an ever-lengthening list of inquiries concerning their personal and economic affairs. ”

“ The actual enumeration shall be made within three years
after the first meeting of the Congress of the United States, and within
every subsequent term of ten years, in such manner as they shall by law
direct. The number of representatives shall not exceed one for every
thirty thousand, but each State shall have at least one representative ”

“ The enumeration is at best provisional, but in its totality cannot go far wrong in establishing the measure of a man. It includes the qualities which can hardly be determined otherwise than by an impressionistic judgment, as well as those appraised by actual achievement under test. ”

“ Sometimes people will speak of mere counting as one of the simplest operations in the world; but on occasions, as I shall show, it is far from easy. Sometimes the labour can be diminished by the use of little artifices; sometimes it is practically impossible to make the required enumeration without having a very clear head indeed. An ordinary child, buying twelve postage stamps, will almost instinctively say, when he sees there are four along one side and three along the other, "Four times three are twelve;" while his tiny brother will count them all in rows, "1, 2, 3, 4," etc. ”

“ Better methods of enumeration have become known, through our own experience at three censuses taken under the act of 1850, and through the experiences of other nations in conducting similar services. Even the conditions of the country have greatly changed. ”

“ In this latter respect as furnishing the key to the outer world in so far as the objects of the latter are a matter of direct enumeration,—capable of being counted,—it is the first great step in the conquest of nature. ”

“ Bookish men, taken from speculative pursuits and set to work on something they know nothing about, have generally been found or thought to do it ill; therefore philosophers are unfit for business, etc., etc. All these are inductions by simple enumeration. Reasons having some reference to the canons of scientific investigation have been attempted to be given, however unsuccessfully, for some of these propositions; but to the multitude of those who parrot them, the enumeratio simplex, ex his tantummodo quæ præsto sunt pronuncians, is the sole evidence. ”

“ It may give a better conception of the amount of medusæ in this extent, if we calculate the length of time that would be requisite, with a certain number of persons, for counting this number. Allowing that one person could count a million in seven days, which is barely possible, it would have required that eighty thousand persons should have started at the creation of the world to complete the enumeration at the present time! ”

“ In the present stage of political and social science, no intelligent person would, were the question a purely original one, propose a longer interval between the periodical enumerations of the country than five years. ”

“ The enumeration of the examples of any higher order is a completely unsolved problem.
As to classification, it is largely a matter of individual taste—perhaps an æsthetic question, for there is beauty in the law and order of numbers. A man once said that he divided the human race into two great classes: those who take snuff and those who do not. I am not Pg 120sure that some of our classifications of magic squares are not almost as valueless. ”

“ The enumerating letters (A, B, etc.) of the first, or main, divisions are indented the space of the usual paragraph; the subdivisions of the first divisions are so far indented that their enumerating letters or figures are in alignment with the first letter of the first word under the division above. This mode of indention is continued with the next subdivisions, thus putting the enumerating letters or figures of the respective divisions or subdivisions in perpendicular alignment. This mode of enumeration and indention can be illustrated diagrammatically:
89. ”

“ And be it further enacted, That immediately after the valuations and enumerations shall have been completed, as aforesaid, the principal assessor in each assessment district, shall, by written notifications, to be publicly posted up, in at least four of the most public places in such district, advertise all persons concerned, of the place where the said lists, valuations and enumerations may be seen and examined ”

“ Somewhat to my surprise I discovered a comparatively simple equation which represented the census enumerations very closely and which, notwithstanding the fluctuations in the various factors which affect the growth of population, followed the general course of this growth with remarkable fidelity, as will be seen by the following table, which shows the population as given by the Census Bureau and as determined by the empirical formula. ”

“ A colon is used before a quotation, enumeration, or observation, that is
introduced by as follows, the following, or any similar expression; as,
Send me the following: 10 doz. "Armstrong's Treasury," 25 Schulte's
Manual, etc.
A semicolon (;) between parts that are subdivided by commas.
The semicolon is used also between clauses or members that are
disconnected in sense; as, Man grows old; he passes away; all is
uncertain. When as, namely, that is, is used to introduce an example or
enumeration, a semicolon is put before it and a comma after it ”

“ 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. ”

“ In counting the all of the kind is the all of our purpose; in classification the all is the all of nature. Then we must remember that in mathematics, number is taken as the representative of the other concomitant properties of quantity and that they are reduced to number by measurement, while in classification kinds are used to represent the other properties and they are reduced to kinds by logical convention. While in conventional counting we consider kinds in series, so in classifying the bodies and properties of nature we are compelled to consider kinds in series. ”