Sum Quotes (41 quotes)

“The Universe repeats itself, with the possible exception of history.” Of all earthly studies history is the only one that does not repeat itself. ... Astronomy repeats itself; botany repeats itself; trigonometry repeats itself; mechanics repeats itself; compound long division repeats itself. Every sum if worked out in the same way at any time will bring out the same answer. ... A great many moderns say that history is a science; if so it occupies a solitary and splendid elevation among the sciences; it is the only science the conclusions of which are always wrong.

*Question:*Why do the inhabitants of cold climates eat fat? How would you find experimentally the relative quantities of heat given off when equal weights of sulphur, phosphorus, and carbon are thoroughly burned?

*Answer:*An inhabitant of cold climates (called Frigid Zoans) eats fat principally because he can't get no lean, also because he wants to rise is temperature. But if equal weights of sulphur phosphorus and carbon are burned in his neighbourhood he will give off eating quite so much. The relative quantities of eat given off will depend upon how much sulphur etc. is burnt and how near it is burned to him. If I knew these facts it would be an easy sum to find the answer.

*Replying to G. H. Hardy’s suggestion that the number of a taxi (1729) was “dull”*: No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 1³ + 12³ and 9³ + 10³.

A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history. … Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of man) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.

Calculating machines do sums better than even the cleverest people… As arithmetic has grown easier, it has come to be less respected.

Complex organisms cannot be construed as the sum of their genes, nor do genes alone build particular items of anatomy or behavior by them selves. Most genes influence several aspects of anatomy and behavior–as they operate through complex interactions with other genes and their products, and with environmental factors both within and outside the developing organism. We fall into a deep error, not just a harmful oversimplification, when we speak of genes ‘for’ particular items of anatomy or behavior.

Each part of the project had a specific task. These tasks were carefully allocated and supervised so that the sum of their parts would result in the accomplishment of our over-all mission.

Entropy theory is indeed a first attempt to deal with global form; but it has not been dealing with structure. All it says is that a large sum of elements may have properties not found in a smaller sample of them.

Four circles to the kissing come,

The smaller are the benter.

The bend is just the inverse of

The distance from the centre.

Though their intrigue left Euclid dumb

There’s now no need for rule of thumb.

Since zero bend’s a dead straight line

And concave bends have minus sign,

The sum of squares of all four bends

Is half the square of their sum.

The smaller are the benter.

The bend is just the inverse of

The distance from the centre.

Though their intrigue left Euclid dumb

There’s now no need for rule of thumb.

Since zero bend’s a dead straight line

And concave bends have minus sign,

The sum of squares of all four bends

Is half the square of their sum.

I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are permutations and aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then from the top down, the result is always different. Again if you multiply a number by another number before you have had your tea, and then again after, the product will be different. It is also remarkable that the Post-tea product is more likely to agree with other people’s calculations than the Pre-tea result.

If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.

If you go far enough out you can see the Universe itself, all the billion light years summed up time only as a flash, just as lonely, as distant as a star on a June night if you go far enough out. And still, my friend, if you go far enough out you are only at the beginning of yourself.

If you would make a man happy, do not add to his possessions but subtract from the sum of his desires.

Just as a tree constitutes a mass arranged in a definite manner, in which, in every single part, in the leaves as in the root, in the trunk as in the blossom, cells are discovered to be the ultimate elements, so is it also with the forms of animal life. Every animal presents itself as a sum of vital unities, every one of which manifests all the characteristics of life. The characteristics and unity of life cannot be limited to anyone particular spot in a highly developed organism (for example, to the brain of man), but are to be found only in the definite, constantly recurring structure, which every individual element displays. Hence it follows that the structural composition of a body of considerable size, a so-called individual, always represents a kind of social arrangement of parts, an arrangement of a social kind, in which a number of individual existences are mutually dependent, but in such a way, that every element has its own special action, and, even though it derive its stimulus to activity from other parts, yet alone effects the actual performance of its duties.

Living is like working out a long addition sum, and if you make a mistake in the first two totals you will never find the right answer. It means involving oneself in a complicated chain of circumstances.

Lucy, dear child, mind your arithmetic. You know in the first sum of yours I ever saw there was a mistake. You had carried two (as a cab is licensed to do), and you ought, dear Lucy, to have carried but one. Is this a trifle? What would life be without arithmetic, but a scene of horrors.

Newton has shown us that a law is only a necessary relation between the present state of the world and its immediately subsequent state. All the other laws since discovered are nothing else; they are in sum, differential equations.

Not only do the various components of the cells form a living system, in which the capacity to live, react, and reproduce is dependent on the interactions of all the members of the system; but this living system is identical with the genetic system. The form of life is determined not only by the specific nature of the hereditary units but also by the structure and arrangement of the system. The whole system is more than the sum of its parts, and the effect of each of the components depends on and is influenced by all previous reactions, whose sequence is in turn determined by the whole idiotype.

Notable enough, however, are the controversies over the series 1 – 1 + 1 – 1 + 1 – … whose sum was given by Leibniz as 1/2, although others disagree. … Understanding of this question is to be sought in the word “sum”; this idea, if thus conceived—namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken—has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.

Nothing in the entire universe ever perishes, believe me, but things vary, and adopt a new form. The phrase “being born” is used for beginning to be something different from what one was before, while “dying” means ceasing to be the same. Though this thing may pass into that, and that into this, yet the sums of things remains unchanged.

On Sept 15th [1852] Mr Goulburn, Chancellor of the Exchequer, asked my opinion on the utility of Mr Babbage's calculating machine, and the propriety of spending further sums of money on it. I replied, entering fully into the matter, and giving my opinion that it was worthless.

One summer day, while I was walking along the country road on the farm where I was born, a section of the stone wall opposite me, and not more than three or four yards distant, suddenly fell down. Amid the general stillness and immobility about me the effect was quite startling. ... It was the sudden summing up of half a century or more of atomic changes in the material of the wall. A grain or two of sand yielded to the pressure of long years, and gravity did the rest.

Rulers and generals muster their troops. Magnates muster the sums of money which give them power. The fascist dictators muster the irrational human reactions which make it possible for them to attain and maintain their power over the masses. The scientists muster knowledge and means of research. But, thus far, no organization fighting for freedom has ever mustered the biological arsenal where the weapons are to be found for the establishment and the maintenance of human freedom. All precision of our social existence notwithstanding, there is as yet no definition of the word freedom which would be in keeping with natural science. No word is more misused and misunderstood. To define freedom is the same as to define sexual health. But nobody will openly admit this. The advocacy of personal and social freedom is connected with anxiety and guilt feelings. As if to be free were a sin or at least not quite as it should be. Sex-economy makes this guilt feeling comprehensible: freedom without sexual self-determination is in itself a contradiction. But to be sexual means—according to the prevailing human structure—to be sinful or guilty. There are very few people who experience sexual love without guilt feeling. “Free love” has acquired a degrading meaning: it lost the meaning given it by the old fighters for freedom. In films and in books, to be genital and to be criminal are presented as the same thing.

Science is a capital or fund perpetually reinvested; it accumulates, rolls up, is carried forward by every new man. Every man of science has all the science before him to go upon, to set himself up in business with. What an enormous sum Darwin availed himself of and reinvested! Not so in literature; to every poet, to every artist, it is still the first day of creation, so far as the essentials of his task are concerned. Literature is not so much a fund to be reinvested as it is a crop to be ever new-grown.

The ancients devoted a lifetime to the study of arithmetic; it required days to extract a square root or to multiply two numbers together. Is there any harm in skipping all that, in letting the school boy learn multiplication sums, and in starting his more abstract reasoning at a more advanced point? Where would be the harm in letting the boy assume the truth of many propositions of the first four books of Euclid, letting him assume their truth partly by faith, partly by trial? Giving him the whole fifth book of Euclid by simple algebra? Letting him assume the sixth as axiomatic? Letting him, in fact, begin his severer studies where he is now in the habit of leaving off? We do much less orthodox things. Every here and there in one’s mathematical studies one makes exceedingly large assumptions, because the methodical study would be ridiculous even in the eyes of the most pedantic of teachers. I can imagine a whole year devoted to the philosophical study of many things that a student now takes in his stride without trouble. The present method of training the mind of a mathematical teacher causes it to strain at gnats and to swallow camels. Such gnats are most of the propositions of the sixth book of Euclid; propositions generally about incommensurables; the use of arithmetic in geometry; the parallelogram of forces, etc., decimals.

The following theorem can be found in the work of Mr. Cauchy: If the various terms of the series

sin

is discontinuous at each value (2

*u*_{0}+*u*_{1}+*u*_{2}+... are continuous functions,… then the sum*s*of the series is also a continuous function of*x*. But it seems to me that this theorem admits exceptions. For example the seriessin

*x*- (1/2)sin 2*x*+ (1/3)sin 3*x*- …is discontinuous at each value (2

*m*+ 1)π of*x*,…
The Hypotenuse has a square on,

which is equal Pythagoras instructed,

to the sum of the squares on the other two sides

If a triangle is cleverly constructed.

which is equal Pythagoras instructed,

to the sum of the squares on the other two sides

If a triangle is cleverly constructed.

The law of the conservation of energy is already known, viz. that the sum of the actual and potential energies in the universe is unchangeable.

The mathematics of cooperation of men and tools is interesting. Separated men trying their individual experiments contribute in proportion to their numbers and their work may be called mathematically additive. The effect of a single piece of apparatus given to one man is also additive only, but when a group of men are cooperating, as distinct from merely operating, their work raises with some higher power of the number than the first power. It approaches the square for two men and the cube for three. Two men cooperating with two different pieces of apparatus, say a special furnace and a pyrometer or a hydraulic press and new chemical substances, are more powerful than their arithmetical sum. These facts doubtless assist as assets of a research laboratory.

The oppressive weight of disaster and tragedy in our lives does not arise from a high percentage of evil among the summed total of all acts, but from the extraordinary power of exceedingly rare incidents of depravity to inflict catastrophic damage, especially in our technological age when airplanes can become powerful bombs. (An even more evil man, armed only with a longbow, could not have wreaked such havoc at the Battle of Agincourt in 1415.)

The powers which tend to preserve, and those which tend to change the condition of the earth's surface, are never in equilibrio; the latter are, in all cases, the most powerful, and, in respect of the former, are like living in comparison of dead forces. Hence the law of decay is one which suffers no exception: The elements of all bodies were once loose and unconnected, and to the same state nature has appointed that they should all return... TIME performs the office of integrating the infinitesimal parts of which this progression is made up; it collects them into one sum, and produces from them an amount greater than any that can be assigned.

The same algebraic sum of positive and negative charges in the nucleus, when the arithmetical sum is different, gives what I call “isotopes” or “isotopic elements,” because they occupy the same place in the periodic table. They are chemically identical, and save only as regards the relatively few physical properties which depend upon atomic mass directly, physically identical also. Unit changes of this nuclear charge, so reckoned algebraically, give the successive places in the periodic table. For any one “place” or any one nuclear charge, more than one number of electrons in the outer-ring system may exist, and in such a case the element exhibits variable valency. But such changes of number, or of valency, concern only the ring and its external environment. There is no in- and out-going of electrons between ring and nucleus.

The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.

This formula [for computing Bernoulli’s numbers] was first given by James Bernoulli…. He gave no general demonstration; but was quite aware of the importance of his theorem, for he boasts that by means of it he calculated

91,409,924,241,424,243,424,241,924,242,500.

*intra semi-quadrantem horæ!*the sum of the 10th powers of the first thousand integers, and found it to be
To Nature nothing can be added; from Nature nothing can be taken away; the sum of her energies is constant, and the utmost man can do in the pursuit of physical truth, or in the applications of physical knowledge, is to shift the constituents of the never-varying total. The law of conservation rigidly excludes both creation and annihilation. Waves may change to ripples, and ripples to waves; magnitude may be substituted for number, and number for magnitude; asteroids may aggregate to suns, suns may resolve themselves into florae and faunae, and floras and faunas melt in air: the flux of power is eternally the same. It rolls in music through the ages, and all terrestrial energy—the manifestations of life as well as the display of phenomena—are but the modulations of its rhythm.

Truth is a totality, the sum of many overlapping partial images. History, on the other hand, sacrifices totality in the interest of continuity.

[De Morgan relates that some person had made up 800 anagrams on his name, of which he had seen about 650. Commenting on these he says:]

Two of these I have joined in the title-page:

[Ut agendo surgamus arguendo gustamus.]

A few of the others are personal remarks.

Great gun! do us a sum!

is a sneer at my pursuit; but,

Go! great sum! [integral of

Adsum, nugator, suge!

is addressed to a student who continues talking after the lecture has commenced: …

Graduatus sum! nego

applies to one who declined to subscribe for an M.A. degree.

Two of these I have joined in the title-page:

[Ut agendo surgamus arguendo gustamus.]

A few of the others are personal remarks.

Great gun! do us a sum!

is a sneer at my pursuit; but,

Go! great sum! [integral of

*a*to the power*u*to the power*n*with respect to*u*] is more dignified. …Adsum, nugator, suge!

is addressed to a student who continues talking after the lecture has commenced: …

Graduatus sum! nego

applies to one who declined to subscribe for an M.A. degree.

[Defining Life] the sum of the phenomena proper to organized beings. In consists essentially in this, that organized beings are all, during a certain time, the centres to which foreign substances penetrate and are appropriated, and from which others issue.

…comparing the capacity of computers to the capacity of the human brain, I’ve often wondered, where does our success come from? The answer is synthesis, the ability to combine creativity and calculation, art and science, into whole that is much greater than the sum of its parts.

“She can't do sums a

“Can

The Queen gasped and shut her eyes. “I can do Addition, if you give me time-but I can do Subtraction, under

*bit*!” the Queens said together, with great emphasis.“Can

*you*do sums?” Alice said, turning suddenly on the White Queen, for she didn't like being found fault with so much.The Queen gasped and shut her eyes. “I can do Addition, if you give me time-but I can do Subtraction, under

*any*circumstances!”
“Try another Subtraction sum. Take a bone from a dog: what remains?” [asked the Red Queen]

Alice considered. “The bone wouldn't remain, of course, if I took it—and the dog wouldn’t remain; it would come to bite me—and I’m sure I shouldn’t remain!”

“Then you think nothing would remain?” said the Red Queen.

“I think that’s the answer.”

“Wrong, as usual,” said the Red Queen, “the dog's temper would remain.”

Alice considered. “The bone wouldn't remain, of course, if I took it—and the dog wouldn’t remain; it would come to bite me—and I’m sure I shouldn’t remain!”

“Then you think nothing would remain?” said the Red Queen.

“I think that’s the answer.”

“Wrong, as usual,” said the Red Queen, “the dog's temper would remain.”