For a small writer it makes little sense anymore. This is a bit frustrating — for both the reader and myself, but hey! Thus for the audience, continuity is fragmented.
And of course, that workaround is only useful as you are publishing the book. Carry On Book , when the whole book is available in Medium, the sequential releasing is no longer in effect.
The other parts are hyperlinked to the earlier ones. The downside to that is that such unlisted stories are unavailable for generating income through the Members program of Medium. Medium does provide a publication header on each story, that a reader can tap to get to the homepage of the publication, but I found it useful to add a standard footer image to each article as well, that provides the same function, as it is more useful — in my opinion — for the reader, after reading an article that they enjoyed, to be able to jump up to the homepage of the publication, rather than having to scroll up to the header.
I wanted something that was clearly setoff from my text in a different typeface, but not overshadowing it in any way either.
I realized that what I wanted was a font size and style much like that of the attribution found underneath images on Medium. My solution was to do exactly that, only with a non-visible and diminutive image.
I found a 1-pixel transparent gif and I place that where I want the notification to appear. Navigation was another problem. Thus, your menu structure is normally restricted to just a top-level list of sections or groupings, each of which can only have a single story, or a list of stories without any deeper structure — you can only have a collection of stories that share a tag, a single story, or a page of featured stories.
I place it just above the start of the text, underneath the title. I did this because my book has a structural flow, and not just a collection of articles. Being able to move back-and-forth between sections makes sense for the kind of book I am publishing, where the reader may want to refer to another part of the text for needed information.
It seems he has lost interest in life. At times this looks annoying, intentional and brings tears because he was a sharp contrast to everything written above. How do I bring back smile on his face? How does it feel to be deeply connected to god? What happens to our life? What happens to us? A group of politicians and their followers had barged into my office one day.
They wanted funds for elections. I politely explained to them that we do a lot of good to the world directly, and we wish to use our funds for such purposes and not for any political reasons. Some of the followers reacted in extremely disrespectful tones. Every human being has multiple intelligence. For example, your ability to understand, appreciate and perform music is one form of intelligence - musical intelligence. This is completely different from your ability to think in pictures - and if you are capable of thinking in pictures, if you are creative, then that's another form of intelligence - spatial intelligence.
About infinithoughts. We just summed an infinite series and calculated a limit, as we did earlier when we discussed why 0. We could use algebra instead. To do so, we first need to figure out where each runner is on the track at an arbitrary time t seconds af- ter the race begins. Since Achilles runs at a speed of 10 meters per second and since distance equals rate times time, his distance down the track is 10t.
To solve this equation, subtract t from both sides. Then divide both sides by 9. So from the perspective of calculus, there really is no paradox about Achilles and the tortoise. If space and time are continuous, everything works out nicely. Zeno Goes Digital In a third paradox, the Paradox of the Arrow, Zeno argued against an alternative possibility — that space and time are fundamentally discrete, meaning that they are composed of tiny indivisible units, something like pixels of space and time.
The paradox goes like this. If space and time are discrete, an arrow in flight can never move, because at each instant a pixel of time the arrow is at some definite place a specific set of pixels in space.
Hence, at any given instant, the arrow is not moving. It is also not moving between instants be- cause, by assumption, there is no time between instants. Therefore, at no time is the arrow ever moving. Philosophers are still debating its status, but it seems to me that Zeno got it two-thirds right. In a world where space and time are discrete, an arrow in flight would behave as Zeno said. It would strangely materialize at one place after another as time clicks forward in discrete steps.
And he was also right that our senses tell us that the real world is not like that, at least not as we ordinarily perceive it. But Zeno was wrong that motion would be impossible in such a world.
We all know this from our experience of watching mov- ies and videos on our digital devices. But because of our percep- tual limitations, it would look like a smooth trajectory. Sometimes our senses really do deceive us. Of course, if the chopping is too blocky, we can tell the differ- ence between the continuous and the discrete, and we often find it bothersome.
On the analog clock, the second hand sweeps around in a beautifully uniform mo- tion. It depicts time as flowing. Whereas on the digital clock, the second hand jerks forward in discrete steps, thwack, thwack, thwack.
It depicts time as jumping. Infinity can build a bridge between these two very different conceptions of time. Imagine a digital clock that advances through trillions of little clicks per second instead of one loud thwack.
We would no longer be able to tell the difference between that kind of digital clock and a true analog clock. Likewise with movies and vid- eos; as long as the frames flash by fast enough, say at thirty frames a second, they give the impression of seamless flow. And if there were infinitely many frames per second, the flow truly would be seamless. Consider how music is recorded and played back.
This is a quintessential analog experience. Whereas when you listen to her on digital, every aspect of her music is minced into tiny, discrete steps and converted into strings of 0s and 1s.
So in everyday life, the gulf between the discrete and the con- tinuous can often be bridged, at least to a good approximation. For many practical purposes, the discrete can stand in for the continu- ous, as long as we slice things thinly enough. In the ideal world of calculus, we can go one better. With limits and infinity, the discrete and the continuous become one.
Zeno Meets the Quantum The Infinity Principle asks us to pretend that everything can be sliced and diced endlessly. Imagining pizzas that can be cut into arbitrarily thin pieces enabled us to find the area of a circle exactly. The question naturally arises: Do such infinitesimally small things exist in the real world? Quantum mechanics has something to say about that.
Its terminology, with its zoo of leptons, quarks, and neutrinos, sounds like something out of Lewis Carroll. The behavior it describes is often weird as well. At the atomic scale, things can happen that would never occur in the macroscopic world. For instance, consider the Riddle of the Wall from a quantum perspective. This effect is known as quantum tun- neling. It actually occurs. This means there is some small but nonzero probability that the electron will be detected on the far side of the barrier, as if it had tunneled through the wall.
With the help of calculus, we can calcu- late the rate at which such tunneling events occur, and experiments have confirmed the predictions.
Tunneling is real. Alpha particles tunnel out of uranium nuclei at the predicted rate to produce the effect known as radioactivity.
Tunneling also plays an important role in the nuclear-fusion processes that make the sun shine, so life on Earth depends partially on tunneling. And it has many technologi- cal uses; scanning tunneling microscopy, which allows scientists to image and manipulate individual atoms, is based on the concept. We have no intuition for such events at the atomic scale, be- ing the gargantuan creatures composed of trillions upon trillions of atoms that we are.
Fortunately, calculus can take the place of intu- ition. By applying calculus and quantum mechanics, physicists have opened a theoretical window on the microworld. The fruits of their insights include lasers and transistors, the chips in our computers, and the LEDs in our flat-screen TVs.
Maxwell made the same assumption in his theory of electricity and magnetism; so did Newton in his theory of gravity and Einstein in his theory of relativ- ity. All of calculus, and hence all of theoretical physics, hinges on this assumption of continuous space and time.
That assumption of continuity has been resoundingly successful so far. But there is reason to believe that at much, much smaller scales of the universe, far below the atomic scale, space and time may ultimately lose their continuous character. At such small scales, space and time might seethe and roil at random. They might fluctuate like bubbling foam. Although there is no consensus about how to visualize space and time at these ultimate scales, there is universal agreement about how small those scales are likely to be.
They are forced upon us by three fundamental constants of nature. One of them is the gravitational constant, G. It measures the strength of gravity in the universe. It is bound to occur in any future theory that supersedes them. The third constant is the speed of light, c. It is the speed limit for the universe. No signal of any kind can travel faster than c. This speed must necessarily enter any theory of space and time because it ties the two of them together via the prin- ciple that distance equals rate times time, where c plays the role of the rate or speed.
In , the father of quantum theory, a German physicist named Max Planck, realized that there was one and only one way to combine these fundamental constants to produce a scale of length.
That unique length, he concluded, was a natural yardstick for the universe. In his honor, it is now called the Planck length. It is given by the algebraic combination!
Space and time would no longer make sense below these scales. These numbers put a bound on how fine we could ever slice space or time. Take the largest pos- sible distance, the estimated diameter of the known universe, and divide it by the smallest possible distance, the Planck length.
That unfathomably extreme ratio of distances is a number with only sixty digits in it. I want to stress that — only sixty digits.
Us- ing more digits than that — say a hundred digits, let alone infinitely many — would be colossal overkill, way more than we would ever need to describe any real distances out there in the material world. And yet in calculus, we use infinitely many digits all the time. As early as middle school, students are asked to think about num- bers like 0.
We call these real numbers, but there is nothing real about them. The requirement to specify a real number by an infinite number of digits after the decimal point is exactly what it means to be not real, at least as far as we understand reality through physics today. If real numbers are not real, why do mathematicians love them so much? And why are schoolchildren forced to learn about them? Because calculus needs them. From the beginning, calculus has stubbornly insisted that everything — space and time, matter and energy, all objects that ever have been or will be — should be re- garded as continuous.
Accordingly, everything can and should be quantified by real numbers. In this idealized, imaginary world, we pretend that everything can be split finer and finer without end. The whole theory of calculus is built on that assumption. If all we ever used were decimals with only sixty digits of precision, the number line would be pockmarked and cratered.
There would be holes where pi, the square root of two, and any other numbers that need infinitely many digits after the decimal point should exist. If we want to think of the totality of all numbers as forming a continuous line, those numbers have to be real numbers. They may be an approximation of reality, but they work amazingly well. Reality is too hard to model any other way. With infinite decimals, as with the rest of calculus, infinity makes everything simpler.
His name was Archimedes. For one thing, there are a lot of funny stories about him. Several portray him as the original math geek. Both of these ideas have countless practical applications. It also underlies all of naval architecture, the theory of ship stability, and the design of oil-drilling platforms at sea. And you rely on his law of the lever, even if unknowingly, every time you use a nail clip- per or a crowbar.
Archimedes might have been a formidable maker of war ma- chines, and he undoubtedly was a brilliant scientist and engineer, but what really puts him in the pantheon is what he did for math- ematics. He paved the way for integral calculus. To say he was ahead of his time would be putting it mildly.
Has anyone ever been more ahead of his time? Two strategies appear again and again in his work. The first was his ardent use of the Infinity Principle. To probe the mysteries of circles, spheres, and other curved shapes, he always approximated them with rectilinear shapes made of lots of straight, flat pieces, fac- eted like jewels. By imagining more and more pieces and making them smaller and smaller, he pushed his approximations ever closer to the truth, approaching exactitude in the limit of infinitely many pieces.
This strategy demanded that he be a wizard with sums and puzzles, since he ended up having to add many numbers or pieces back together to arrive at his conclusions.
Specifically, he mingled ge- ometry, the study of shapes, with mechanics, the study of motion and force. Sometimes he used geometry to illuminate mechanics; sometimes the flow went in the other direction, with mechanical arguments providing insight into pure form. It was by using both strategies with consummate skill that Archimedes was able to pen- etrate so deeply into the mystery of curves.
Squeezing Pi When I walk to my office or go out with my dog for an evening stroll, the pedometer on my iPhone keeps track of how far I walk. The distance traveled equals stride length times the number of steps taken. Archimedes used a similar idea to calculate the circumference of a circle and to estimate pi.
Think of the circle as a track. It takes a lot of steps to walk all the way around. The path would look something like this. Each step is represented by a tiny straight line.
By multiplying the number of steps by the length of each one, we can estimate the length of the track. And so the approximation is sure to underestimate the true length of the circular track. Archimedes did a series of calculations like this, starting with a path made up of six straight steps. He began with a hexagon because it was a convenient base camp from which to embark on the more arduous calculations ahead.
The advantage of the hexagon was that he could easily calculate its pe- rimeter, the total length around the hexagon. Why six? Of course, six is a ridiculously small number of steps, and the resulting hexagon is obviously a very crude caricature of a circle, but Archimedes was just getting started. Once he figured out what the hexagon was telling him, he shortened the steps and took twice as many of them.
He did that by detouring to the midpoint of each arc, taking two baby steps instead of striding across the arc in one big step. A man obsessed, he went from six steps to twelve, then twenty-four, forty-eight, and, ultimately, ninety-six steps, working out their ever-shrinking lengths to migraine-inducing precision. That required him to calculate square roots, a nasty chore to do by hand. Forget about math for a minute.
The squeeze technique that Archimedes used building on ear- lier work by the Greek mathematician Eudoxus is now known as the method of exhaustion because of the way it traps the unknown number pi between two known numbers. The bounds tighten with each doubling, thus exhausting the wiggle room for pi.
Circles are the simplest curves in geometry. Yet, surprisingly, measuring them — quantifying their properties with numbers — transcends geometry. How to measure the length of a curved line or the area of a curved surface or the volume of a curved solid — these were the cutting-edge questions that consumed Archimedes and led him to take the first steps toward what we now call integral calculus.
Pi was its first triumph. He avoided doing all that because pi was not a number to him. It was a magnitude, not a number. We no longer make this distinction between magnitude and number, but it was important in ancient Greek mathematics. It seems to have arisen from the tension between the discrete as represented by whole numbers and the continuous as represented by shapes.
The historical details are murky, but it appears that sometime be- tween Pythagoras and Eudoxus, between the sixth and the fourth centuries bce, somebody proved that the diagonal of a square was incommensurable with its side, meaning that the ratio of those two lengths could not be expressed as the ratio of two whole numbers. In modern language, someone discovered the existence of irrational numbers.
The suspicion is that this discovery shocked and disap- pointed the Greeks, since it belied the Pythagorean credo. This de- flating letdown may explain why later Greek mathematicians always elevated geometry over arithmetic. They were inadequate as a foundation for mathematics. To describe continuous quantities and reason about them, the ancient Greek mathematicians realized they needed to invent some- thing more powerful than whole numbers.
So they developed a sys- tem based on shapes and their proportions. It relied on measures of geometrical objects: lengths of lines, areas of squares, volumes of cubes. All of these they called magnitudes. They thought of them as distinct from numbers and superior to them.
It was a strange, transcendent crea- ture, more exotic than any number. My children certainly were intrigued by it.
They used to stare at a pie plate hanging in our kitchen that had the digits of pi running around the rim and spiral- ing in toward the center, shrinking in size as they swirled into the abyss. For them, the fascination had to do with the random-looking sequence of digits, never repeating, never showing any pattern at all, going on forever, infinity on a platter. We will never know all the digits of pi.
Nevertheless, those digits are out there, waiting to be discovered. Yet twenty-two trillion is nothing compared to the infinitude of digits that define the actual pi. Think of how philosophically disturbing this is.
I said that the digits of pi are out there, but where are they exactly? They exist in some Platonic realm, along with abstract concepts like truth and justice.
On the one hand, it represents order, as embodied by the shape of a circle, long held to be a symbol of perfection and eternity. On the other hand, pi is unruly, disheveled in appearance, its digits obeying no obvious rule, or at least none that we can perceive. Pi is elusive and mysterious, forever beyond reach. Its mix of order and disorder is what makes it so bewitching. Pi is fundamentally a child of calculus. It is defined as the unat- tainable limit of a never-ending process.
But unlike a sequence of polygons steadfastly approaching a circle or a hapless walker step- ping halfway to a wall, there is no end in sight for pi, no limit we can ever know. And yet pi exists. There it is, defined so crisply as the ratio of two lengths we can see right before us, the circumference of a circle and its diameter.
That ratio defines pi, pinpoints it as clearly as can be, and yet the number itself slips through our fingers. Pi is a portal between the round and the straight, a single number yet infinitely complex, a balance of order and chaos. Calcu- lus, for its part, uses the infinite to study the finite, the unlimited to study the limited, and the straight to study the curved. The Infinity Principle is the key to unlocking the mystery of curves, and it arose here first, in the mystery of pi.
Cubism Meets Calculus Archimedes went deeper into the mystery of curves, again guided by the Infinity Principle, in his treatise The Quadrature of the Pa- rabola.
A parabola describes the familiar arc of a three-point shot in basketball or water coming out of a drinking fountain. Actually, those arcs in the real world are only approximately parabolic. A true parabola, to Archimedes, would have meant a curve obtained by slicing through a cone with a plane. Imagine a meat cleaver slicing through a dunce cap or a conical paper cup; the cleaver can make different kinds of curves depending on how steeply it cuts through the cone.
A slice parallel to the base of a cone makes a circle. In more modern lan- guage, a segment of a parabola means the curved region lying be- tween the parabola and a line that cuts across it obliquely. The strategy used by Archimedes was astonishing. He reimag- ined the parabolic segment as infinitely many triangular shards glued together like pieces of broken pottery. His plan was to find all their areas and then add them back together to calculate the curved area he was wondering about.
It took a kaleidoscopic leap of artistic imagination to see a smooth, gently curving parabolic segment as a mosaic of jagged shapes. If he had been a painter, Ar- chimedes would have been the first cubist. To carry out his strategy, Archimedes first had to find the areas of all the shards. But how, precisely, were those shards to be defined? After all, there are countless ways to piece triangles together to form a parabolic segment, just as there are countless ways to smash a plate into jagged bits.
The biggest triangle could look like this, or this, or this: He came up with a brilliant idea — brilliant because it es- tablished a rule, a consistent pattern that held from one level of the hierarchy to the next. He imagined sliding the oblique line at the base of the segment upward while keeping it parallel to it- self until it just barely touched the parabola at a single point near the top.
It defined the third corner of the big triangle, the other two being the points where the oblique line cut the parabola. Archimedes used the same rule to define the triangles at every stage in the hierarchy.
At the second stage, for example, the triangles looked like this. Notice that the sides of the big triangle now play the role of the oblique line used earlier.
Next, Archimedes invoked known geometrical facts about pa- rabolas and triangles to relate one level of the hierarchy to the next. He proved that each newly created triangle had one-eighth as much area as its parent triangle. The trick is to cancel all but one of its infinitely many terms by multiplying both sides of the equation for Area by 4 and subtracting the original sum from it.
The magic happens between the next-to-last line and the last line above. A Cheesy Argument Archimedes would not have approved of the legerdemain above. He arrived at the same result by a different route.
He resorted to a subtle style of argumentation often described as double reductio ad absur- dum, a double proof by contradiction.
He never had to summon infinity. So everything about his proof was ironclad. It still meets the highest standards of rigor today. The gist of his argument becomes easy to understand if we put it in everyday terms. Suppose three people want to share four identical slices of cheese. But suppose the three people happen to be mathematicians who are milling around the food table before the seminar, eyeing the last four slices of cheese.
Euclid, cut that leftover slice into quarters, not thirds, and everyone, take a quarter of that leftover slice. Eudoxus, stop whining. One way to look at it is to keep a running tally of how many slices each person gets. After round one, each gets one slice. And so on. In The Quadrature of the Parabola, Archimedes gave an argument very close to this, including a diagram with squares of different sizes, but he never invoked infinity or used the counterpart of the three dots [!
Rather, he phrased his argument in terms of finite sums so that it was unim- peachably rigorous. His key observation was that the tiny square in the upper right corner — the current leftover remaining to be shared — could be made smaller than any given amount by considering a sufficiently large but finite number of rounds.
He shares his private intuition, a vulnerable, soft-bellied thing, and says he hopes that future mathematicians will use it to solve problems that eluded him. Today this secret is known as the Method. I never heard of it in calculus class.
But I found the story of it and the idea behind it enthrall- ing and astounding. He writes about it in a letter to his friend Eratosthenes, the li- brarian at Alexandria and the only mathematician of his era who could understand him. It gives him intuition. And that guides him to a watertight proof. First comes intuition.
Rigor comes later. This essential role of in- tuition and imagination is often left out of high-school geometry courses, but it is essential to all creative mathematics. Our subject is endless. It humbled even Archimedes himself.
What is the Method, and what is so personal, brilliant, and transgressive about it? The Method is mechanical; Archimedes finds the area of the parabolic segment by weighing it in his mind. Or, if you prefer, think of it as being seated at one end of an imaginary seesaw. Next he figures out how to counterbalance it against a shape he already knows how to weigh: a triangle.
From this he deduces the area of the original parabolic segment. Together, they will produce the answer he seeks. The instruction manual reads as follows: Draw the big triangle inside the parabolic segment, as before, and label it ABC. As in the cubist proof, this triangle is again going to serve as an area standard. The parabolic segment will be compared to it and will turn out to have four-thirds its area. Its base is the line AC. And its left side is a vertical line that extends upward from A until it meets the top side at point D.
That fact will become important later. Set it aside for now. The next step is to build the rest of the seesaw — its lever, its two seats, and its fulcrum. The lever is the line that joins the two seats. Last week, around 30, people downloaded books from my site - 9 people donated.
I love offering these books for free, but need some support to continue doing so. You don't need an account and it only takes a minute. You can also support it by buying one of the collections. The great central fact of the universe is that Spirit of Infinite Life and Power that is behind all, that animates all, that manifests itself in and through all; that self-existent principle of life from which all has come, and not only from which all has come, but from which all is continually coming.
If there is an individual life, there must of necessity be an infinite source of life from which it comes. If there is a quality or a force of love, there must of necessity be an infinite source of love whence it comes. If there is wisdom, there must be the all-wise source behind it from which it springs. The same is true in regard to peace, the same in regard to power, the same in regard to what we call material things.
There is, then, this Spirit of Infinite Life and Power behind all which is the source of all. This Infinite Power is creating, working, ruling through the agency of great immutable laws and forces that run through all the universe, that surround us on every side.
Every act of our every-day lives is governed by these same great laws and forces. Every flower that blooms by the wayside, springs up, grows, blooms, fades, according to certain great immutable laws. Every snowflake that plays between earth and heaven, forms, falls, melts, according to certain great unchangeable laws.
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