Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

Planck length. It is given by the algebraic combination Planck length = √ ħG / c3 .

Space and time would no longer make sense below these scales. They’re the end of the line.

ultimately proved that π is greater than 3 + 10/71 and less than 3 + 10/70. Forget about math for a minute. Just savor this result at a visual level: 3 + 10 / 71 < π < 3 + 10 /

now known as the method of exhaustion because of the way it traps the unknown number pi between two known numbers.

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.

Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process.

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,

He proved that each newly created triangle had one-eighth as much area as its parent triangle.

says he hopes that future mathematicians will use it to solve problems that eluded him. Today this secret is known as the Method.

Mathematicians don’t come up with the proofs first. First comes intuition. Rigor comes later.

that the conclusion is true. Whatever its logical status, Archimedes’s Method has an e pluribus unum quality to it. This Latin phrase, the motto of the United States, means “out of many, one.”

This is the beginning of integral calculus.

After all, those same ratios, 3:2 and 4:3, held special significance to the ancient Greeks because of their central role in the Pythagorean theory of musical harmony.

Galileo and Kepler ventured beyond the static world of Archimedes and explored how things moved.

The challenge for Galileo, Kepler, and other like-minded mathematicians of the early seventeenth century was to take their beloved geometry, so well suited to a world at rest, and extend it to a world in flux.

the ancient Greek astronomer Aristarchus to propose a sun-centered universe almost two millennia before Copernicus did.

The only way out of this paradox (as Archimedes himself realized when reacting to Aristarchus’s sun-centered cosmology) would be if all the stars were immensely distant, effectively infinitely far away from the Earth.

Although Galileo did not invent the telescope, he improved it

1611, he observed that the moon had mountains, the sun had spots, and Jupiter had four moons

Galileo was the first practitioner of the scientific method.

One of the simplest and most surprising is this: The odd numbers 1, 3, 5, 7, and so forth are hiding in how things fall.

To time the ball’s descent he used a water clock. It worked like a stopwatch. To start the clock he would open a valve.

“The distances traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity.”

certain distance in the first unit of time. Then, in the next unit of time, it will roll three times as far. And in the next unit of time after that, it will roll five times as far as it did originally.

So Galileo’s odd-number rule seems to be implying that the total distance fallen is proportional to the square of the time elapsed.

Galileo also discovered a law for its speed. As he put it, the speed increases in proportion to the time of falling.

So in this law of falling bodies, Galileo was instinctively thinking about instantaneous speed, a differential calculus concept

He coaxed a beautiful answer out of nature by asking a beautiful question. Like an abstract expressionist painter, he highlighted what he was interested in and cast the rest aside.

In 1962 Brian Josephson, then a twenty-two-year-old graduate student at the University of Cambridge, predicted that at temperatures close to absolute zero, pairs of superconducting electrons could tunnel back and forth through an impenetrable insulating barrier, a nonsensical statement according to classical physics.

Neurosurgeons use arrays of hundreds of Josephson junctions to pinpoint the sites of brain tumors and locate the seizure-causing lesions in patients

the longitude problem was solved by a new kind of clock, developed in the mid-1700s by John Harrison, an Englishman

For GPS, it works like this: When the signals from the four satellites arrive at the receiver, your GPS gadget compares the time they were received to the time they were transmitted.

The numerological pattern that enraptured Kepler was his discovery that the square of the period of revolution of a planet is proportional to the cube of its average

distance from the sun.

The farther a planet is from the sun, the slower it moves and the longer it takes to complete its orbit.

scientific style and disposition. Where Galileo was rational, Kepler was mystical.

Differential calculus cuts complicated problems into infinitely many simpler pieces.

Integral calculus puts the pieces back together

They start with derivatives—the relatively easy techniques for slicing and dicing—and

Its name derives from the Arabic word al-jabr, meaning “restoration” or “the reunion of broken parts.”

Hindu mathematicians invented the concepts of zero and the decimal place-value system for numbers.

That gave calculus the infinitely precise real numbers it needed to describe the continuity of space, time, motion, and change.

Pierre de Fermat and René Descartes, independently linked algebra to geometry.

This connection between linear equations and lines suggested the possibility of a deeper connection, one between nonlinear equations and curves.