Geometric Progression Quotes

That's what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you to another book, and another bit there will lead you onto a third book. It's geometrically progressive - all with no end in sight, and for no other reason than sheer enjoyment.

That's what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you onto another book, and another bit there will lead you onto a third book. It's geometrically progressive—all with no end in sight, and for no other reason than sheer enjoyment.

That’s what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you onto another book, and another bit there will lead you onto a third book. It’s geometrically progressive—all with no end in sight, and for no other reason than sheer enjoyment.

The pause - that impressive silence, that eloquent silence, that geometrically progressive silence which often achieves a desired effect where no combination of words, howsoever felicitous, could accomplish it.

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.

The true splendor of science is not so much that it names and classifies, records and predicts, but that it observes and desires to know the facts, whatever they may turn out to be. However much it may confuse facts with conventions, and reality with arbitrary divisions, in this openness and sincerity of mind it bears some resemblance to religion, understood in its other and deeper sense. The greater the scientist, the more he is impressed with his ignorance of reality, and the more he realizes that his laws and labels, descriptions and definitions, are the products of his own thought. They help him to use the world for purposes of his own devising rather than to understand and explain it. The more he analyzes the universe into infinitesimals, the more things he finds to classify, and the more he perceives the relativity of all classification. What he does not know seems to increase in geometric progression to what he knows. Steadily he approaches the point where what is unknown is not a mere blank space in a web of words but a window in the mind, a window whose name is not ignorance but wonder.

The months of the year, from January up to June, are a geometric progression in the abundance of distractions.

I quickly learned that reading is cumulative and proceeds by geometrical progression: each new reading builds upon whatever the reader has read before.

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"?

With scientific discovery and invention proceeding, we are told, at the rate of geometric progression, a generally passive and culture-bound people cannot cope with the multiplying issues and problems. Unless individuals, groups, and nations can imagine, construct, and creatively revise new ways of relating to these complex changes, the lights will go out. Unless man can make new and original adaptations to his environment as rapidly as his science can change the environment, our culture will perish.

...chemists did it in their tubes and doctors did it with patience, but only a techie would do it in geometric progression.

I am a Geometric Progression of all the Books I have ever read.

Another means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions or property in geometrical progression as they rise. Whenever there are in any country uncultivated lands and unemployed poor, it is clear that the laws of property have been so far extended as to violate natural right. The earth is given as a common stock for man to labor and live on. Letter to James Madison, October 28, 1785

love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you onto another book, and another bit there will lead you onto a third book. It’s geometrically progressive

That's what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you onto another bool, and another bit there will lead you onto a third book. It's geometrically progressive-all with no end in sight, and for no other reason than sheer enjoyment.

That's what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you onto another book, and another bit there will lead you onto a third book. It's geometrically progressive - all with no end in sight, and for no other reason than sheer enjoyment.

I am conscious that an equal division of property is impracticable. But the consequences of this enormous inequality [in Europe] producing so much misery to the bulk of mankind, legislators cannot invent too many devices for subdividing property,...[One] means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions of property in geometrical progression as they rise.

String theory, therefore, is rich enough to explain all the fundamental laws of nature. Starting from a simple theory of a vibrating string, one can extract the theory of Einstein, Kaluza-Klein theory, supergravity, the Standard Model, and even GUT theory. It seems nothing less than a miracle that, starting from some purely geometric arguments from a string, one is able to rederive the entire progress of physics for the past 2 milleninia. All the theories so far discussed in this book are automatically included in string theory.

Normal persons deprived of sensation progress from having mild to severe hallucinations, starting out with what looks very much like form constants (geometric patterns, mosaics, lines, rows of dots) and building to more developed, dream-like juxtapositions of perceptions the longer they remain in isolation.

A strong back is a big help but not even the strongest back was built for that treatment, and there combine not just at the kidneys, ad rill down the thighs and up the spine and athwart the shoulders, the ticklish weakness of gruel or water, and an aching that increases in geometric progression, and at length, in the small of the spine, a literal sensation of yielding, buckling, splintering, and breakage: and all of this, even though the mercy of nature has strengthened and hardened your flesh and anesthetized your nerves and your powers of reflection and imagination, reaches in time the brain and the more mirrorlike nerves, and thereby makes itself much worse than before.

There are numerous brain rhythms, from approximately 0.02 to 600 cycles per second (Hz), covering more than four order of temporal magnitude. Many of these discrete brain rhythms have been known for decades, but it was only recently recognized that these oscillation bands form a geometric progression on a linear frequency scale or a linear progression on a natural logarithmic scale. leading to a natural separation of at least ten frequency bands. The neighbouring bands have a roughly constant ratio of e = 2,718 - the base for the natural logarithm. Because of this non-integer relationship among the various brain rhythms, the different frequencies can never perfectly entrain each other. Instead, the interference they produce gives rise to metastability, a perpetual fluctuation between unstable and transiently stable states, like waves in the ocean. The constantly interfering network rhythms can never settle to a stable attractor, using the parlance of nonlinear dynamics. This explains the ever-changing landscape of the EEG.

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.

But it happens at least equally often in the history of science that the understanding of the component parts of a composite system is impossible without an understanding of the behavior of the system as a whole. And it often happens that the understanding of the mathematical nature of an equation is impossible without a detailed understanding of its solutions. The black hole is a case in point. One could say without exaggeration that Einstein's equations of general relativity were understood only at a very superficial level before the discover of the black hole. During the fifty years since the black hole was invented, a deep mathematical understanding of the geometrical structure of space-time has slowly emerged, with the black hole solution playing a fundamental role in the structure. The progress of science requires the growth of understanding in both directions, downward from the whole to the parts and upward from the parts to the whole. A reductionist philosophy, arbitrarily proclaiming that the growth of understanding must go only in one direction, makes no scientific sense. Indeed, dogmatic philosophical beliefs of any kind have no place in science.

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.

Absolute continuity of motion is not comprehensible to the human mind. Laws of motion of any kind become comprehensible to man only when he examines arbitrarily selected elements of that motion; but at the same time, a large proportion of human error comes from the arbitrary division of continuous motion into discontinuous elements. There is a well known, so-called sophism of the ancients consisting in this, that Achilles could never catch up with a tortoise he was following, in spite of the fact that he traveled ten times as fast as the tortoise. By the time Achilles has covered the distance that separated him from the tortoise, the tortoise has covered one tenth of that distance ahead of him: when Achilles has covered that tenth, the tortoise has covered another one hundredth, and so on forever. This problem seemed to the ancients insoluble. The absurd answer (that Achilles could never overtake the tortoise) resulted from this: that motion was arbitrarily divided into discontinuous elements, whereas the motion both of Achilles and of the tortoise was continuous. By adopting smaller and smaller elements of motion we only approach a solution of the problem, but never reach it. Only when we have admitted the conception of the infinitely small, and the resulting geometrical progression with a common ratio of one tenth, and have found the sum of this progression to infinity, do we reach a solution of the problem. 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 it deals 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 arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history. But to arrive at these laws, resulting from the sum of all those human wills, man's mind postulates arbitrary and disconnected units. The first method of history is to take an arbitrarily selected series of continuous events and examine it apart from others, though there is and can be no beginning to any event, for one event always flows uninterruptedly from another. The second method is to consider the actions of some one man—a king or a commander—as equivalent to the sum of many individual wills; whereas the sum of individual wills is never expressed by the activity of a single historic personage. Historical science in its endeavor to draw nearer to truth continually takes smaller and smaller units for examination. But however small the units it takes, we feel that to take any unit disconnected from others, or to assume a beginning of any phenomenon, or to say that the will of many men is expressed by the actions of any one historic personage, is in itself false. It needs no critical exertion to reduce utterly to dust any deductions drawn from history. It is merely necessary to select some larger or smaller unit as the subject of observation—as criticism has every right to do, seeing that whatever unit history observes must always be arbitrarily selected. Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) 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.

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

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.

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.

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.

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.

Some technological invention is made, like that of a steam engine or a printing press, for example; or some discovery of scientific method, like that of analytical geometry or the infinitesimal calculus; or [pg 020] some discovery of natural law, like that of falling bodies or the Newtonian law of gravitation. What happens? What is the effect upon the progress of knowledge and invention? The effect is stimulation. Each invention leads to new inventions and each discovery to new discoveries; invention breeds invention, science begets science, the children of knowledge produce their kind in larger and larger families; the process goes on from decade to decade, from generation to generation, and the spectacle we behold is that of advancement in scientific knowledge and technological power according to the law and rate of a rapidly increasing geometric progression or logarithmic function.

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.

However, all man's desired geometric progressions, if a high rate of growth is chosen, at last come to grief on a finite earth.

Geometric progression means a small advantage up front can turn into an insurmountable lead, eventually.” I

A Solution Waiting for a Problem Engineers tend to develop tools for the pleasure of developing tools, not to induce nature to yield its secrets. It so happens that some of these tools bring us more knowledge; because of the silent evidence effect, we forget to consider tools that accomplished nothing but keeping engineers off the streets. Tools lead to unexpected discoveries, which themselves lead to other unexpected discoveries. But rarely do our tools seem to work as intended; it is only the engineer’s gusto and love for the building of toys and machines that contribute to the augmentation of our knowledge. Knowledge does not progress from tools designed to verify or help theories, but rather the opposite. The computer was not built to allow us to develop new, visual, geometric mathematics, but for some other purpose. It happened to allow us to discover mathematical objects that few cared to look for. Nor was the computer invented to let you chat with your friends in Siberia, but it has caused some long-distance relationships to bloom. As an essayist, I can attest that the Internet has helped me to spread my ideas by bypassing journalists. But this was not the stated purpose of its military designer. The laser is a prime illustration of a tool made for a given purpose (actually no real purpose) that then found applications that were not even dreamed of at the time. It was a typical “solution looking for a problem.” Among the early applications was the surgical stitching of detached retinas. Half a century later, The Economist asked Charles Townes, the alleged inventor of the laser, if he had had retinas on his mind. He had not. He was satisfying his desire to split light beams, and that was that. In fact, Townes’s colleagues teased him quite a bit about the irrelevance of his discovery. Yet just consider the effects of the laser in the world around you: compact disks, eyesight corrections, microsurgery, data storage and retrieval—all unforeseen applications of the technology.* We build toys. Some of those toys change the world. Keep

Furthermore, we suggest a fractal self is capable of growth and a kind of metamorphosis. Daoists refer to a seasoned human cooperator and facilitator, working adroitly, with a natural ease in the smooth, orderly, adaptive spirit of wuwei, as a sage. Such an individual is typically embedded in a particular affinitive complex system. An affinitive system is virtually anything in nature or human endeavor that is avidly sought by an individual in pursuit of vocation or avocation-a business, social, educational, artistic, scientific, or governmental enterprise, and so forth. Such systems typically develop chaotic structures and behaviors; envisioned as geometrical forms, they often constitute complicated attractors; around the edges of their coherent existence they would tend to be fractally organized, transcending classic dimensionality (see introduction). The sage tends to develop into a leader or catalyst within his or her affinitive system as he or she progressively "evolves" over time into increasing levels of intimacy and coherence with the system.

There is a limit to the number of people who can be assigned to this kind of work because a geometric progression soon takes place with watchers watching watchers until no one is doing anything else.