Arithmetic Progression Quotes

Certainly, the terror of a deserted house swells in geometrical rather than arithmetical progression as houses multiply to form a city of stark desolation. The sight of such endless avenues of fishy-eyed vacancy and death, and the thought of such linked infinities of black, brooding compartments given over to cob-webs and memories and the conqueror worm, start up vestigial fears and aversions that not even the stoutest philosophy can disperse.

In One Dimensions, did not a moving Point produce a Line with two terminal points? In two Dimensions, did not a moving Line produce a Square wit four terminal points? In Three Dimensions, did not a moving Square produce - did not the eyes of mine behold it - that blessed being, a Cube, with eight terminal points? And in Four Dimensions, shall not a moving Cube - alas, for Analogy, and alas for the Progress of Truth if it be not so - shall not, I say the motion of a divine Cube result in a still more divine organization with sixteen terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this - if I might qupte my Lord's own words - "Strictly according to Analogy"? Again, was I not taught by my Lord that as in a Line there are two bonding points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series: 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have eight bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to analogy"?

That second man has his own way of looking at things; asks himself which debt must I pay first, the debt to the rich, or the debt to the poor? the debt of money, or the debt of thought to mankind, of genius to nature? For you, O broker! there is no other principle but arithmetic. For me, commerce is of trivial import; love, faith, truth of character, the aspiration of man, these are sacred; nor can I detach one duty, like you, from all other duties, and concentrate my forces mechanically on the payment of moneys. Let me live onward; you shall find that, though slower, the progress of my character will liquidate all these debts without injustice to higher claims. If a man should dedicate himself to the payment of notes, would not this be injustice? Does he owe no debt but money? And are all claims on him to be postponed to a landlord's or a banker's?

His play had become wilder and wilder at each hole in arithmetical progression. If he had been a plow, he could hardly have turned up more soil. The imagination recoiled from the thought of what he would be doing in another half hour if he deteriorated at his present speed.

Schools normally schedule one subject, for example, Japanese, the first period, when you just do Japanese; then, say, arithmetic the second period, when you just do arithmetic. But here it was quite different. At the beginning of the first period, the teacher made a list of all the problems and questions in the subjects to be studied that day. Then she would say, “Now, start with any of these you like.” […] This method of teaching enabled the teachers to observe - as the children progressed to higher grades - what they were interested in as well as their way of thinking and their character. It was an ideal way of teachers to really get to know their pupils.

Your Lordship tempts his servant to see whether he remembers the revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, for more knowledge. Doubtless we cannot see that other higher Spaceland now, because we have no eye in our stomachs. But, just as there was the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there was close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant? In One Dimension, did not a moving Point produce a Line with two terminal points? In Two Dimensions, did not a moving Line produce a Square with four terminal points? In Three Dimensions, did not a moving Square produce—did not this eye of mine behold it—that blessed Being, a Cube, with eight terminal points? And in Four Dimensions shall not a moving Cube—alas, for Analogy, and alas for the Progress of Truth, if it be not so—shall not, I say, the motion of a divine Cube result in a still more divine Organization with sixteen terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this—if I might quote my Lord’s own words—“strictly according to Analogy”? Again, was I not taught by my Lord that as in a Line there are two bounding Points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, “strictly according to Analogy”? O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a fourth Dimension; but, if I am right, my Lord will listen to reason. I ask therefore, is it, or is it not, the fact, that ere now your countrymen also have witnessed the descent of Beings of a higher order than their own, entering closed rooms, even as your Lordship entered mine, without the opening of doors or windows, and appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am henceforth silent. Only vouchsafe an answer.

When we're all so terribly alone. The least we can do in this life is love one another... just a hug and a kiss...' He was right, of course; but how could you love everyone? If only enough of us loved enough--perhaps by some arithmetical progression, everyone would be given this gift. But that was useless. Speculation for those of us left behind, who were not going to hurl ourselves off a building in the pressure of a summer heat wave, a lover's quarrel, a drug, I thought as we sat there now. There was no such end for the rest of us, or glorious legacy of love

Pythagoras, as everyone knows, said that 'all things are numbers'. This statement, interpreted in a modern way, is logically nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music, and the connection which he established between music and arithmetic survives in the mathematical terms 'harmonic mean' and 'harmonic progression'. He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares and cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or, as we should more naturally say, shot) required to make the shapes in question. He presumably thought of the world as atomic, and of bodies as built up of molecules composed of atoms arranged in various shapes. In this way he hoped to make arithmetic the fundamental study in physics as in aesthetics.

First, as to putting the clock back. Would you think I was joking if I said that you can put a clock back, and that if the clock is wrong it is often a very sensible thing to do? But I would rather get away from that whole idea of clocks. We all want progress. But progress means getting nearer to the place where you want to be. And if you have taken a wrong turning then to go forward does not get you any nearer. If you are on the wrong road, progress means doing an about-turn and walking back to the right road and in that case the man who turns back soonest is the most progressive man. We have all seen this when doing arithmetic. When I have started a sum the wrong way, the sooner I admit this and go back and start again, the faster I shall get on. There is nothing progressive about being pig-headed and refusing to admit a mistake. And I think if you look at the present state of the world it's pretty plain that humanity has been making some big mistake. We're on the wrong road. And if that is so we must go back. Going back is the quickest way on.

Now to picture the mechanism of this process of construction and not merely its progressive extension, we must note that each level is characterized by a new co-ordination of the elements provided—already existing in the form of wholes, though of a lower order—by the processes of the previous level. The sensori-motor schema, the characteristic unit of the system of pre-symbolic intelligence, thus assimilates perceptual schemata and the schemata relating to learned action (these schemata of perception and habit being of the same lower order, since the first concerns the present state of the object and the second only elementary changes of state). The symbolic schema assimilates sensori-motor schemata with differentiation of function; imitative accommodation is extended into imaginal significants and assimilation determines the significates. The intuitive schema is both a co-ordination and a differentiation of imaginal schemata. The concrete operational schema is a grouping of intuitive schemata, which are promoted, by the very fact of their being grouped, to the rank of reversible operations. Finally, the formal schema is simply a system of second-degree operations, and therefore a grouping operating on concrete groupings. Each of the transitions from one of these levels to the next is therefore characterized both by a new co-ordination and by a differentiation of the systems constituting the unit of the preceding level. Now these successive differentiations, in their turn, throw light on the undifferentiated nature of the initial mechanisms, and thus we can conceive both of a genealogy of operational groupings as progressive differentiations, and of an explanation of the pre-operational levels as a failure to differentiate the processes involved. Thus, as we have seen (Chap. 4), sensori-motor intelligence arrives at a kind of empirical grouping of bodily movements, characterized psychologically by actions capable of reversals and detours, and geometrically by what Poincaré called the (experimental) group of displacement. But it goes without saying that, at this elementary level, which precedes all thought, we cannot regard this grouping as an operational system, since it is a system of responses actually effected; the fact is therefore that it is undifferentiated, the displacements in question being at the same time and in every case responses directed towards a goal serving some practical purpose. We might therefore say that at this level spatio-temporal, logico-arithmetical and practical (means and ends) groupings form a global whole and that, in the absence of differentiation, this complex system is incapable of constituting an operational mechanism. At the end of this period and at the beginning of representative thought, on the other hand, the appearance of the symbol makes possible the first form of differentiation: practical groupings (means and ends) on the one hand, and representation on the other. But this latter is still undifferentiated, logico-arithmetical operations not being distinguished from spatio-temporal operations. In fact, at the intuitive level there are no genuine classes or relations because both are still spatial collections as well as spatio-temporal relationships: hence their intuitive and pre-operational character. At 7–8 years, however, the appearance of operational groupings is characterized precisely by a clear differentiation between logico-arithmetical operations that have become independent (classes, relations and despatialized numbers) and spatio-temporal or infra-logical operations. Lastly, the level of formal operations marks a final differentiation between operations tied to real action and hypothetico-deductive operations concerning pure implications from propositions stated as postulates.

The many years that we all spend in schools learning skills like reading, writing, and arithmetic—as well as the additional learning that happens on the job and on our own—makes us more productive and, in some cases, is intrinsically rewarding. It is also a contribution to the nation’s capital stock.

By May 2013, Khan Academy included more than 4,100 videos, most no more than a few minutes long, on subjects ranging from arithmetic to calculus to physics to art history. These videos had been viewed more than 250 million times, and the Academy’s students had tackled more than one billion automatically generated problems.15

On a stationary military frontier between a civilization and a barbarism, time always works in the barbarians' favour; and, besides this, the barbarians' advantages increases in geometrical progression at each arithmetical addition to the length of the line which the defenders of the civilization have to hold

large-frame pattern recognition, and the most complex forms of communication are cognitive areas where people still seem to have the advantage, and also seem likely to hold on to it for some time to come. Unfortunately, though, these skills are not emphasized in most educational environments today. Instead, primary education often focuses on rote memorization of facts, and on the skills of reading, writing, and arithmetic—the

Two writings of al-Hassār have survived. The first, entitled Kitāb al-bayān wa t-tadhkār [Book of proof and recall] is a handbook of calculation treating numeration, arithmetical operations on whole numbers and on fractions, extraction of the exact or approximate square root of a whole of fractionary number and summation of progressions of whole numbers (natural, even or odd), and of their squares and cubes. Despite its classical content in relation to the Arab mathematical tradition, this book occupies a certain important place in the history of mathematics in North Africa for three reasons: in the first place, and notwithstanding the development of research, this manual remains the most ancient work of calculation representing simultaneously the tradition of the Maghrib and that of Muslim Spain. In the second place, this book is the first wherein one has found a symbolic writing of fractions, which utilises the horizontal bar and the dust ciphers i.e. the ancestors of the digits that we use today (and which are, for certain among them, almost identical to ours) [Woepcke 1858-59: 264-75; Zoubeidi 1996]. It seems as a matter of fact that the utilisation of the fraction bar was very quickly generalised in the mathematical teaching in the Maghrib, which could explain that Fibonacci (d. after 1240) had used in his Liber Abbaci, without making any particular remark about it [Djebbar 1980 : 97-99; Vogel 1970-80]. Thirdly, this handbook is the only Maghribian work of calculation known to have circulated in the scientific foyers of south Europe, as Moses Ibn Tibbon realised, in 1271, a Hebrew translation. [Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa]

Do not fail to observe in this connection the following two facts. One of them is that the magnitude of the terms of any geometric progression whose ratio (no matter how small) is 2 or more will overtake and surpass the magnitude of the corresponding terms of any arithmetical progression, no matter how large the common difference of the latter may be. The other fact to be noted is that the greater the ratio of a geometric progression, the more rapidly do its successive terms increase; so that the [pg 018] terms of one geometric progression may increase a thousand or a million or a billion times faster than the corresponding terms of another geometric progression. As any geometric progression (of ratio equal to 2 or more), no matter how slow, outruns every arithmetic progression, no matter how fast, so one geometric progression may be far swifter than another one of the same type.

My contention is that while progress in some of the great matters of human concern has been long proceeding in accordance with the law of a rapidly increasing geometric progression, progress in the other matters of no less importance has advanced only at the rate of an arithmetical progression or at best at the rate of some geometric progression of relatively slow growth. To see it and to understand it we have to pay the small price of a little observation and a little meditation.

The world to-day is full of controversy about wealth, capital, and money, and because humanity, through its peculiar time-binding power, binds this element “time” in an ever larger and larger degree, the controversy becomes more and more acute. Civilization as a process is the process of binding time; progress is made by the fact that each generation adds to the material and spiritual wealth which it inherits. Past achievements—the fruit of bygone time—thus live in the present, are augmented in the present, and transmitted to the future; the process goes on; time, the essential element, is so involved that, though it increases arithmetically, its fruit, civilization, advances geometrically.

The question is of utmost importance both theoretically and practically, for the law—whatever it be—is a natural law—a law of human nature—a law of the time-binding energy of man. What is the law? We have already noted the law of arithmetical progression and the law of geometric progression; we have seen the immense difference between them; and we have seen that the natural law of human progress in each and every cardinal matter is a law like that of a rapidly increasing geometric progression. In other words, the natural law of human progress—the natural law of amelioration in human affairs—the fundamental law of human nature—the basic law of the time-binding energy [pg 090] peculiar to man—is a Logarithmic law—a law of logarithmic increase. I beg the reader not to let the term bewilder him but to make it his own. It is easy to understand; and its significance is mighty and everlasting. Even its mathematical formulation can be understood by boys and girls. Let us see how the formulation looks.

He has translated Virgil’s Aeneid . . . the whole of Sallust and Tacitus’ Agricola . . . a great part of Horace, some of Ovid, and some of Caesar’s Commentaries . . . besides Tully’s [Cicero’s] Orations. . . . In Greek his progress has not been equal; yet he has studied morsels of Aristotle’s Politics, in Plutarch’s Lives, and Lucian’s Dialogues, The Choice of Hercules in Xenophon, and lately he has gone through several books in Homer’s Iliad. In mathematics I hope he will pass muster. In the course of the last year . . . I have spent my evenings with him. We went with some accuracy through the geometry in the Preceptor, the eight books of Simpson’s Euclid in Latin. . . . We went through plane geometry . . . algebra, and the decimal fractions, arithmetical and geometrical proportions. . . . I then attempted a sublime flight and endeavored to give him some idea of the differential method of calculations . . . [and] Sir Isaac Newton; but alas, it is thirty years since I thought of mathematics.

Who refuses to do arithmetic is doomed to talk nonsense - from "Progress and it's Sustainability

Human teams are not bound by arithmetic - you cannot add, subtract, multiply or divide people!