Tally

“ 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 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. ”

“ Thus, in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions, not of physical things or of mental occurrences.
Instead of speaking of a general term, such as “man,” as the subject of which a number can be asserted, we may, without making any serious change, take the subject as the class or collection of objects—i.e. “mankind” in the above instance—to which the general term in question is applicable. ”

“ I count, but I count, in ordinary life, what I take to be real objects, existent quite apart from my counting. Suppose that I count ships seen from the shore. There, says common sense, are the ships, sailing by themselves, and quite indifferent to whether anybody counts them or not. In advance of the counting, the ships, in so far as they are a real collection, have their number. ”

“ Suppose that you are going to count some large number, for example, to measure a number of yards of cloth. Opposite to yourself suppose a man to be placed, who keeps his eye upon you, and holds up a finger for every yard which he sees you measure. When ten yards have been measured he will have held up ten fingers, and will not be able to count any further unless he begin again, holding up one finger at the eleventh yard, two at the twelfth, and so on. ”

“ I noticed some of the company prepare their mouths and draw themselves up at the prospect of some agreeable forfeit. "Let us play at counting," said Charlotte. "Now, pay attention: I shall go round the circle from right to left; and each person is to count, one after the other, the number that comes to him, and must count fast; whoever stops or mistakes is to have a box on the ear, and so on, till we have counted a thousand." ”

“ It is said in the Bible that God has ordered all things according to weight, number and measure. Omnia in pondere et numero et mensura disposuit Deus. Weight is equilibrium, number is quantity, measure is proportion—these three, and these are the eternal or divine basis of the science of Nature. Here now is the formula of equilibrium: Harmony results from the analogy of contraries. Number is the scale of analogies, the proportion of which is measure. The entire occult philosophy of the Zohar might be termed the science of equilibrium. ”

“ No! We are mere prosaic mortals: it is beyond us!
In p. 49 I see some strange remarks about ratios. First look at Def. 44. 'When a magnitude is contained an exact number of times in two magnitudes of its kind, it is said to be their common measure.' (The wording is awkward, and suggests the idea of their having only one 'common measure'; but let that pass.) 'The ratio of two magnitudes of the same kind is the number which would express the measure of the first, if the second were taken as unity.'
'The measure of the first'! Do you understand that? ”

“ The first part of the work deals with psychological questions connected with the concepts, plurality (Vielheit) , unity (Einheit) , and sum (Anzahl) , while the second treats of the symbolical ideas of plurality and sum, and shows how the fact that we are restricted to symbolical number-concepts in arithmetic, determines its character. The author rejects the logical method, which is so strongly advocated by many writers. For him number is the result of psychical processes (p. 130) . Notions like unity and plurality cannot be logically defined, but rest upon ultimate psychical data. ”

“ If every term of a collection can be hooked on to a number, and all the finite numbers are used once, and only once, in the process, then our collection must have just as many terms as there are finite numbers. ”

“ 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." ”

“ Any fact which we know or have been told about our problem is called a datum. The number of pounds of sugar we are to buy is one datum; the price per pound is another.
The plural of datum is data. It is a good plan to write all one's data on one column or page of the paper and work one's sum on the other. This leaves the first column clear for adding to one's data if one finds out any fresh one. ”

“ If we divide the sum of the years of the life of all the individuals inscribed in a table of mortality by the number of these individuals we shall have the mean duration of life which corresponds to this table. ”

“ To those meanwhile who had already drawn numbers ranging from about a hundred and fifty or so to some two hundred and twenty or thirty, the remainder of the drawing was still a matter of anxious interest. For of course their own chances very materially depended on the sort of men who drew the numbers below them. And every time a low number was drawn by some man who it was pretty clear would be rejected by the medical examiners, a murmur of disappointment might be heard among the crowd. ”

“ 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. ”

“ 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. ”

“ Let us imagine a lottery composed of n numbers, of which r are drawn at each draw. The probability is demanded of the drawing of s given numbers in one draw. To arrive at this let us form a fraction whose denominator will be the number of all the cases possible or of the combinations of n letters taken r by r times, and whose numerator will be the number of all the combinations which contain the given s numbers. ”

“ Knowledge being scientific, as it can apply number and measure, an attempt is in order to see if a scientific order of values cannot be built "on a number and measure scale of force," ascent in the scale signifying increase of value, descent diminution of value—all other estimations being prejudices, naïvetés, misunderstandings. ”

“ Let me procure a larger sheet, as the measureless sea of infinite and nameless numbers is flowing fast upon me. Next reckon the chances in the case of two persons, and they swell to the vast sum of nine millions, seven hundred and sixty-five thousand, six hundred and twenty-five; while the chances for four men will be the square of that number, and so on for ever. ”

“ But in common practice odds is considered as in the singular number. We always say, "What is the odds;" and I should rank this among the words, which, altho they have the termination of regular plurals, more properly belong to the singular number.
The word gallows is evidently of this class. "Let a gallows be made," say the translators of the Bible, with perfect propriety. Indeed I cannot conceive how any man who has read English authors, can consider this word as in the plural. ”

“ Now when we are referring to averages it must be remembered that the minimum number of different occurrences necessary in order to justify the average is that which enables each of them to present itself once. ”

“ Otherwise the pupil is likely to limit the meaning of, say, four to four sensibly discrete things and to have difficulty in multiplication and division. Measuring, or counting by insensibly marked off repetitions of a unit, binds each number name to its meaning as —— times whatever 1 is, more surely than mere counting of the units in a collection can, and should reënforce the latter. ”

“ It is, however, correct to say, "The signers of the petition were less than a hundred," where the round number, a hundred, is something like a collective noun, and less is thought of as meaning a less quantity or amount. ”

“ And since we are assured that the all-wise Creator has observed the most exact proportions of number, weight, and measure in the make of all things, the most likely way, therefore, to get any insight into the nature of those parts of the creation which come within our [Pg 30] observation must in all reason be to number, weigh, and measure. ”

“ The larger the number we take out of any mixed crowd, the clearer become the symptoms of order, the more nearly will the average length of each selected class be the same. These few cases will serve as simple examples of a property of things which can be traced almost everywhere, to a greater or less extent, throughout the whole field of our experience. ”

“ It is quite true that the fundamental truths of the science of number all rest on the evidence of sense but, having used eyes and fingers upon ten balls or twenty balls, upon ten nuts, or leaves, or sheep, or what not, the child has formed the association of a given number with objects, and is able to conceive of the association of various other numbers with objects. ”

“ To detect the relations of Number considered directly as Number, is not easy: and we might 205 contemplate for a long time a Table of recorded Numbers without perceiving the order of their increase and diminution, even if the law were moderately simple; as any one may satisfy himself by looking at a Tide Table. ”

“ 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. ”

“ 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. ”

“ The difference, therefore, of our conduct in preferring the greater number depends not upon our passions, but upon custom, and general rules. We have found in a multitude of instances, that the augmenting the numbers of any sum augments the passion, where the numbers are precise and the difference sensible. ”

“ If you once commit the addition table thoroughly, you have it fixed for life; whereas if you do not, you have to make the calculation over again every time, and thus lose a vast amount of labor. I have no doubt that there are some in this class who are in the habit of counting, who have ascertained that seven and eight for instance, make fifteen; by counting up from seven to fifteen, hundreds of times. ”

“ When the fingers and toes of the person counting are exhausted, he has recourse to those of another person, if he wishes to count further, although he has then passed the limit of numerical phraseology. ”

“ Counting is fundamental, and counting that is spontaneous, free from sensible observation, and from the strain of reason. A study of these original methods shows that multiplication was developed out of counting, and not from addition as nearly all textbooks treat it. Multiplication is counting. When children count by 4's, etc., they accent the same as counting gymnastics or music. When a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations. ”

“ Sometimes a number above ten, but not over twenty, is indicated with the hands only by counting up to ten in the ordinary way, and then opening all the fingers and counting again, until they reach the requisite amount in excess of ten. ”

“ Here are the count’s notes; they are most of them incomprehensible, owing to the abbreviations he used: probably he thought he would have future opportunities of examining the hieroglyphics, but the most important is clear enough, it runs thus: ‘Let the serpent be thy guide on thy right hand.’ This the count told me referred to the direction to be followed: that is to say, in going and returning through the labyrinth of apartments we must follow the direction in which the figure of the serpent appeared to be travelling. The rest of the notes you can look over at your leisure, if you like. ”

“ As a Number Primer it teaches in a very simple, logical way, the first steps in number. It is easy, and appeals to the natural interest of the child in counting, besides being adapted to his various capacities. ”

“ It is easy for any two people to count the number of men in a room and to agree as to the result; but ask them to say how many tall men, how many with black hair, how many blue-eyed, how many straight-nosed—and you will get a different result each time. ”