Sum

“ For not the hundredth part of the sum, but the half of the sum they press for and exact; and this when he of whom it is exacted has a wife, is bringing up children, is a human being, and is filling their threshing floor, and their wine-press by his own toils. ”

“ Just as we go out with the sum total of our present actions upon us, so we see that we come into it with the sum total of our past actions upon us; that which takes us out is the very same thing that brings us in. ”

“ We might say that, according to the above, the final end in its totality must be divided among the sum of individuals, and that hence, if the final end is a determined and complete Whole, the sum of individuals must also be a complete Whole; and this remark furnishes us opportunity for an additional statement, which opens a wide prospect. For, in so far as the final end is to become visible, it must be apportioned among a determined and fixed sum of individuals, since it is visible only in the form of individuals. ”

“ There is another manner of proceeding to guess the figure left out, by letting the person chuse the sum he pleases to be multiplied, but then you must ask him to shew you the sum he means to have multiplied, and to let you add one figure at your option. ”

“ To raise the number to ä higher power means in point of form to go on multiplying a number with itself an indefinite amount of times.—Since this third type of calculation exhibits the complete equality of the sole existing distinction in number, viz. the distinction between Sum or amount and Unity, there can be no more than these three modes of calculation. ”

“ You have no standard to estimate by, no measure to make comparisons with. Given a certain small quantity of higher pleasure in collision with a certain large quantity of lower, how can you decide between them? To work the sum you must reduce the data to the same denomination. You must go to quantity or nothing; you decline to go to quantity, and hence you can not get any result. But if you refuse to work the sum, you abandon the greatest amount of pleasure principle. ”

“ I won’t name the sum, even to as old a friend as yourself, ’Squire; for the man who lets the world peep into his purse, will soon find it footing him up, like a sum in arithmetic. You’ve gentlemen in town, however, who sometimes get more for a single case, than I can ’arn in a twelvemonth.”
“Still, considering your beginning, and late appearance at the bar, Timms, you are doing pretty well. Do you lead in many trials at the circuit?” ”

“ Properly speaking, this sum cannot be done.
Multiplication is merely a contracted form of addition: it means taking a number or quantity a certain number of times. Every multiplication can be proved by addition. All numbers are abstract or concrete—3 is abstract, £3 is concrete. ”

“ After thousandths, come ten thousandths, hundred thousandths, mil- Konths, &c. but for ordinary uses we seldom come down to so small a fraction. Multiplication is performed as if there were no comma ; but in the sum total, as many figures must be cut off at the right hand of the comma, as are cut off in both the multiplier and sum multiplied. ”

“ The sum, or the All of pleasures is a self-contradiction, and therefore the search for it is futile. A series which has no beginning, or, if a beginning, yet no end, can not be summed; there is no All, and yet the All is postulated, and the series is to be summed. ”

“ He is guided by a sum of impressions, not one of which is emphatic or distinguished from the rest, not one of which is essential, not one of which is conceived, but all of which together drive him to a conclusion to which nothing but that sum-total leads. ”

“ If he works a sum on the principle that two and two make five, his answer will be hopelessly wrong. But if the wall stands, if the sum comes right, the laws, the principles, cannot be wrong, though they may be different from others. ”

“ One sum of money is distinguishable from another only by its amount. The character and tendency of the process M—C—M, is therefore not due to any qualitative difference between its extremes, both being money, but solely to their quantitative difference. ”

“ A sum of money is divided among a number of persons; the second gets 8d. more than the first, the third gets 1s. 4d. more than the second, the fourth 2s. more than the third, and so on. If the first gets 6d. ”

“ Then, how much ought the value of any to be? Clearly, at any point of the curve, the value of will be the sum of all the little ’s from up to that level, that is to say, . And as each is equal to , it follows that the whole will be equal to the sum of all such bits as , or, as we should write it, . ”

“ If there is nothing which is at once an x and a z to the same y, the sum of the two given numbers would give the required number. Suppose, for example, that we had given the average number of friends that men have, and the average number of enemies, the sum of these two is the average number of persons interested in a man. ”

“ The sum total of what is experienced by mankind is a selection from the sum total of what exists, and any general character exhibited by this selection may be due to the manner of selecting rather than to the general character of that from which experience selects. ”

“ 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. ”

“ When a man reasoneth, he does nothing else but conceive a sum total, from addition of parcels; or conceive a remainder, from subtraction of one sum from another; which, if it be done by words, is conceiving of the consequence of the names of all the parts, to the name of the whole ”

“ Value being a ratio, it is impossible for all values to rise or fall simultaneously. The sum of subjective values may increase or decrease,—indeed it is one of the great objects of human endeavour to increase the sum of want-satisfying power,—but the sum of the ratios between these subjective values is constant. ”

“ In no other way can values be arranged one above another in a scale: in no other can a person conveniently calculate the sum of his possessions; and it is easier to ascertain and remember the relations of many things to one thing, than their innumerable cross relations with one another. ”

“ This view is supported by the statutory requirement that the forbidden thing must be 'intended to circulate as money, or to be received or used in lieu of the lawful money of the United States.' One of the lexical definitions of the word 'sum,' and the sense in which it is most commonly used, is 'money.' 'Sum. (2.) A quantity of money or currency; any amount indefinitely, as a sum of money, a small sum, or a large sum.' Webster's Dic. 'For a less sum than one dollar' means exactly the same thing as for a less sum of money than one dollar. ”

“ All the numbers which the pupil learns to understand and manipulate are in fact abstractions; all the operations are abstractions; percent, discount, interest, height, length, area, volume, are abstractions; sum, difference, product, quotient, remainder, average, are facts that concern elements or aspects which may appear with countless different concrete surroundings or concomitants. ”

“ How can number 45 be divided into four such parts that, if to the first part you add 2, from the second part you subtract 2, the third part you multiply by 2, and the fourth part you divide by 2, the sum of the addition, the remainder of the subtraction, the product of the multiplication, and the quotient of the division, be all equal? ”

“ The simplest numerical truths are seen by Intuition; when we endeavour to deduce the more complex from these simplest, we employ such maxims as these:—If equals be added to equals the wholes are equal:—If equals be subtracted from equals the remainders are equal:—The whole is equal to the sum of all its parts. ”

“ To us, each action stands and is seen in relation to the sum of our own individual actions, and the proportion when it bears to that sum is very different from the proportion it bears to the whole sum of action in general. It is easy, therefore, to conceive how different the subjective view of action must be from the objective, and how far a history of action such as Mr. ”

“ Macheath. But if I could raise a small Sum—Would not twenty Guineas, think you, move him?—Of all the Arguments in the way of Business, the Perquisite is the most prevailing—Your Father’s Perquisites for the Escape of Prisoners must amount to a considerable Sum in the Year. Money well tim’d, and properly apply’d, will do any thing. ”

“ The principal multiplied by its time and divided by the interest, and the quotient being multiplied by the factor less one is the time. The sum of principal and interest divided by unity added to the interest on unity is the principal. ”

“ In the first number are 4 units, and in the second 9: these will, when the numbers are added together, contribute 13 units to the sum. Again, the 3 tens in the first and the 9 tens in the second will contribute 12 tens to the sum. The 8 hundreds in the first and the 7 hundreds in the second will add 15 hundreds to the sum; and the [Pg 15] thousand in the first with the 2 thousands in the second will contribute 3 thousands to the sum ”

“ If there be three continual proportionals, the sum of the extremes, together with the mean twice taken, the sum of the mean and either of the extremes, and the same extreme, are continual proportionals. ”

“ The value of the first, must always be estimated according to its usefulness after the modification it has received is entirely 362destroyed, and when by the nature of the thing both must be consumed together, then the total value is the sum of both. The value of the second must be estimated according to the labour it has cost to produce it. ”

“ In fact, the first attempts which you made to set sums, you got the figures so many, and of so high value, that you couldn’t add them without carrying; so you reduced them by little and little, until you just got the figures barely small enough to make the amount less than ten; and thus you made the sums as difficult as they could be made, without carrying; and this you gave her for her first lesson. The thing which you were to come to in the end, you took as the beginning. ”

“ Interest at any per cent, can also be performed in the same manner, without any extra figures. Fractions of every denomination, can be summed up instantly, without reducing them to a common denominator. ”

“ The principal might yield a greater sum,
Yet handled ill, amounts but to this crumb; ”

“ Example: A person was born in 1860, and is now 38 years of age.
122. There are 3 numbers in continued proportion—the middle number is 60, and the sum of the others is 125. Find the numbers.
123. A lends B a certain sum at the same time he insures B’s life for £737 12s. 6d., paying annual premiums of £20; at the end of three years and just before the fourth premium is to be paid, B dies, having never repaid anything. What sum must A have lent B in order that he may have just enough to recoup himself, together with 5 per cent. compound interest on the sum lent and on the premiums? ”

“ Among these general laws must be included the same two which we have noticed as ultimate principles of the Science of Number also, and which are applicable to every description of quantity; viz., The sums of equals are equal, and Things which are equal to the same thing are equal to one another; the latter of which may be expressed in a manner more suggestive of the inexhaustible multitude of its consequences, by the following terms: Whatever is equal to any one of a number of equal magnitudes, is equal to any other of them. ”

“ 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. ”

“ Propose to a person of the company to multiply, by whatever number he pleases, one of the three sums which you will give him on a piece of paper; desire him to strike out whatever figure he pleases of the product of his multiplication, let him change and invert the order of the remaining figures after the defalcation he has chosen. ”

“ While teaching arithmetical sums, the teacher cannot give the example that if A has ten rupees in his pocket and B steals them without consent then calculate how many will remain in the pocket of A cannot be an example of subtraction. ”

“ We all know how the size of sums of money appears to vary in a remarkable way according as they are being paid in or paid out. ”

“ When Mrs Evergreen said: “My husband’s age is represented by the figures of mine reversed; he is older than I am, and the difference between our ages is one-eleventh of their sum,” he was 54, and she was 45.
Return to description
[380]
17 This is the completed multiplication sum—
Image
Return to description
18
Add 3 to 10, and then divide
Till 8 the sum has satisfied—
is solved by writing 13 in Roman numerals, XIII. ”

“ Number supplies the means of measuring other quantities, by the assumption of a unit of measure of the appropriate kind: but where nature supplies the unit, number is applicable directly and immediately. ”

“ The oft-times repeated assumption that x plus y equals a leads ultimately to the fixed belief that a is an attainable result, whatever values may be assigned to the other factors. If we assign concrete dollars to the abstract x and y, a theoretically becomes concrete dollars as well. ”

“ So much for the theses to be maintained. The detailed proof of these contentions will best be given in connection with a critical account of various versions of quantity theory doctrine. Attention will be given in this summary to the expositions of Nicholson, Mill, Taussig, and Kemmerer, and very special attention to I. Fisher, though some other writers will also be taken into account. ”