Euler Quotes

Read Euler, read Euler, he is the master of us all.

Nothing takes place in the world whose meaning is not that of some maximum or minimum.

Logic is the foundation of the certainty of all the knowledge we acquire.

You look like Euler's equation," he murmured as he looked me up and down. Nerd translation: Euler's equation is said to be the most perfect formula ever written. Simple but elegant. Beautiful.

When Euler died, he simply said I am finished and collapsed, to which someone in the audience muttered darkly" Another conjecture of Euler is proved

I could solve all the Diophantine equations, extend Newton’s work on infinite series, complete Euler's analysis of prime numbers, and it wouldn’t matter.” She looked at Isabelle. “Ella is the beauty. You and I are the ugly stepsisters. And so the world reduces us, all three of us, to our lowest common denominator.

Madam, I have just come from a country where people are hanged if they talk.

e^(iπ)+1 = 0

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Shut up about Leibniz for a moment, Rudy, because look here: You—Rudy—and I are on a train, as it were, sitting in the dining car, having a nice conversation, and that train is being pulled along at a terrific clip by certain locomotives named The Bertrand Russell and Riemann and Euler and others. And our friend Lawrence is running alongside the train, trying to keep up with us—it’s not that we’re smarter than he is, necessarily, but that he’s a farmer who didn’t get a ticket. And I, Rudy, am simply reaching out through the open window here, trying to pull him onto the fucking train with us so that the three of us can have a nice little chat about mathematics without having to listen to him panting and gasping for breath the whole way.

But that would mean it was originally a sideways number eight. That makes no sense at all. Unless..." She paused as understanding dawned. "You think it was the symbol for infinity?" "Yes, but not the usual one. A special variant. Do you see how one line doesn't fully connect in the middle? That's Euler's infinity symbol. Absolutus infinitus." "How is it different from the usual one?" "Back in the eighteenth century, there were certain mathematical calculations no one could perform because they involved series of infinite numbers. The problem with infinity, of course, is that you can't come up with a final answer when the numbers keep increasing forever. But a mathematician named Leonhard Euler found a way to treat infinity as if it were a finite number- and that allowed him to do things in mathematical analysis that had never been done before." Tom inclined his head toward the date stone. "My guess is whoever chiseled that symbol was a mathematician or scientist." "If it were my date stone," Cassandra said dryly, "I'd prefer the entwined hearts. At least I would understand what it means." "No, this is much better than hearts," Tom exclaimed, his expression more earnest than any she'd seen from him before. "Linking their names with Euler's infinity symbol means..." He paused, considering how best to explain it. "The two of them formed a complete unit... a togetherness... that contained infinity. Their marriage had a beginning and end, but every day of it was filled with forever. It's a beautiful concept." He paused before adding awkwardly, "Mathematically speaking.

Certainly not! I didn't build a machine to solve ridiculous crossword puzzles! That's hack work, not Great Art! Just give it a topic, any topic, as difficult as you like..." Klapaucius thought, and thought some more. Finally he nodded and said: "Very well. Let's have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit." "Love and tensor algebra?" Have you taken leave of your senses?" Trurl began, but stopped, for his electronic bard was already declaiming: Come, let us hasten to a higher plane, Where dyads tread the fairy fields of Venn, Their indices bedecked from one to n, Commingled in an endless Markov chain! Come, every frustum longs to be a cone, And every vector dreams of matrices. Hark to the gentle gradient of the breeze: It whispers of a more ergodic zone. In Reimann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our asymptotes no longer out of phase, We shall encounter, counting, face to face. I'll grant thee random access to my heart, Thou'lt tell me all the constants of thy love; And so we two shall all love's lemmas prove, And in bound partition never part. For what did Cauchy know, or Christoffel, Or Fourier, or any Boole or Euler, Wielding their compasses, their pens and rulers, Of thy supernal sinusoidal spell? Cancel me not--for what then shall remain? Abscissas, some mantissas, modules, modes, A root or two, a torus and a node: The inverse of my verse, a null domain. Ellipse of bliss, converge, O lips divine! The product of our scalars is defined! Cyberiad draws nigh, and the skew mind Cuts capers like a happy haversine. I see the eigenvalue in thine eye, I hear the tender tensor in thy sigh. Bernoulli would have been content to die, Had he but known such a^2 cos 2 phi!

I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain ... But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further—then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty.

Just before they boarded the yacht, Cassandra glanced at Tom and reached up to a delicate necklace she'd worn constantly since the day he'd given it to her. She touched the little charm, made in the shape of Euler's infinity symbol, that hung at the hollow of her throat. And as always, the private signal made him smile.

The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal.

Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsot has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.

The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error.

That this blind and aging man forged ahead with such gusto is a remarkable lesson, a tale for the ages. Euler's courage, determination, and utter unwillingness to be beaten serves, in the truest sense of the word, as an inspiration for mathematician and non-mathematician alike. The long history of mathematics provides no finer example of the triumph of the human spirit.

Only one number can stake any claim to any special status, and that is zero – the origin – upon which all other numbers depend. It is the perfect balance point of all the other numbers, which is why the monad is the “container” of all other numbers, their source. There it is, slap bang in the middle of the Euler unit circle, controlling all. It’s the SOUL of the circle.

According to my almond-eyed little spy, the great surgeon, may his own liver rot, lied to me when he declared yesterday with a deathhead's grin that the operazione had been perfetta. Well, it had been so in the sense Euler called zero the perfect number. Actually, they ripped me open, cast one horrified look at my decayed fegato, and without touching it sewed me up again.

It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.

Euler’s Formula encapsulates the whole of existence. It contains 0, the number of the monad (ontological zero); the number e that determines exponentiation; the number i that determines the imaginary domain (time); the number 1 that determines the domain of counting numbers (and with 0 creates the binary system of computing), and real numbers (space); -1, the number of the negative domain (antimatter); and the number π that determines the world of the circle and geometry. Euler’s Formula is the unquestionable God Equation.

The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholely ‘useless’ (and this is true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.… The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. It

Never was his remarkable memory more useful than when he could see mathematics only in his mind's eye.

A thought had come to Thozan more than once: the ultimate reason we are set in this world is to break each other's hearts...

As Boy had once heard his father say: Your desire will tell you who you are, you can’t reason with a boner – it’s as simple as that!

You look like Euler's equation,' he murmured as he looked me up and down. Nerd translation: Euler's equation is said to be the most perfect formula ever written. Simple but elegant. Beautiful.

In retrospect, Euler's unintended message is very simple: Graphs or networks have properties, hidden in their construction, that limit or enhance our ability to do things with them. For more than two centuries the layout of Konigsberg's graph limited its citizens' ability to solve their coffeehouse problem. But a change in the layout, the addition of only one extra link, suddenly removed this constraint.

If you were given to thinking of numbers as having human-like qualities, you might picture e^i*pi as a guru into transcendental meditation who'd achieved infinite enlightenment. But there's a problem with that-Euler's formula shows that e^i*pi can never free itslef from worldly concerns. Recall, e^i*pi is really -1 in disguise, and -1 is just a mathism for owing a dollar to your friend, Steve. One hand clapping.

In his eulogy, the Marquis de Condorcet observed that whosoever pursues mathematics in the future will be "guided and sustained by the genius of Euler" and asserted , with much justification, that "all mathematicians...are his disciples.

When Euler discovered that i^i is real, he exclaimed in a letter to a friend that this "seems extraordinary to me"-part of his genius, as well as of his charm, was an inexhaustible capacity to be surprised and delighted by his discoveries.

In his life of seventy-six years, Euler created enough mathematics to fill seventy-four substantial volumes, the most total pages of any mathematician. By the time all of his work had been published (and new material continued to appear for seventy-nine years after his death) it amounted to a staggering 866 items, including articles and books on the most cutting-edge topics, elementary textbooks, books for the nonscientist, and technical manuals. These figures do not account for the projected fifteen volumes of correspondence and notebooks that are still being compiled.

Illuminism is based on Euler’s Formula. Euler’s Formula is the basis of Fourier mathematics. Fourier mathematics is the basis of quantum mechanics. Quantum mechanics is the basis of the scientific world. Therefore Euler’s Formula is the basis of the scientific world, and the basis of everything.

Linking their names with Euler’s infinity symbol means . . .” He paused, considering how best to explain it. “The two of them formed a complete unit . . . a togetherness . . . that contained infinity. Their marriage had a beginning and end, but every day of it was filled with forever. It’s a beautiful concept.” He paused before adding

Euler's general equation stands out because it forged a fundamental link between different areas of math, and because of its versatility in applied mathematics. After Euler's time it came to be regarded as a cornerstone in "complex analysis," a fertile branch of mathematics concerned with functions whose variables stand for complex numbers.

The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered.

Die Mathematik ist es, die uns vor dem Trug der Sinne schützt und uns den Unterschied zwischen Schein und Wahrheit kennen lehrt.

You can save nobody from himself, not even yourself

The imaginaries finally lost their air of impossibility when nineteenth-century mathematicians realized that they're actually perfectly ordinary, law-abiding numerical beings-it's just that they hail from a different dimension. We'll go there later on.

Feynman said, “If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied” Our sentence would be: “The Monadology asserts that the fundamental units of existence are INFINITE, dimensionless, living, thinking points – monads, ZEROS, souls – each of which has INFINITE energy content, all controlled by a single equation – Euler’s Formula – and the collective energy of this universe of mathematical points creates a physical universe of which every objective value is ZERO, but, through a self-solving, self-optimizing, dialectical, evolving process, the universe generates a final, subjective value of INFINITY – divinity, perfection, the ABSOLUTE.” For ours is the religion of zero and infinity, the two numbers that define the soul and the whole of existence. As above, so below.

But just what are imaginary numbers, you may now be asking yourself, and what on earth could it mean to raise e to an imaginary-number power? This chapter concerns mathematicians' long struggle to answer the first of these two questions. Later we'll take up the second one , which inspired Euler to devise the most radical expansion of the concept of exponents in math history. At this point, suffice it to say that affixing an imaginary exponent to a number has a dramatic effect on it-something lime what happens to a frog when it's tapped by a standard-issue magic wand.

In my view, Euler's tranquil temperament, fairness, and generosity were integral to his greatness as a mathematician and scientist- he was never inclined to waste time and energy engaging in petty one-upmanship (like his mentor, Johann Bernoulli, who was known for getting into the eighteenth-century version of flame wars with his older brother, mathematician Jakob Bernoulli, and even with his own son, Daniel, over technical disputes), brooding about challenges to his authority (like Newton), or refusing to publish important findings because of the fear that they might be disputed (like Gauss).

Quand mon cerveau excite dans mon ame la sensation d'un arbre ou d'une maison, je prononce hardiment, qu'il existe réellement hors de moi un arbre ou une maison, dont je connois même le lieu, la grandeur ou d'autres propriétés. Ainsi ne trouve-t-on ni homme ni bête qui doutent de cette vérité. Si un paysan en vouloit douter ; s'il disoit, par exemple, qu'il ne croyait pas que son baillif existe, quoiqu'il fut devant lui, on le pretendroit pour un fou et cela avec raison : mais dès qu'un philosophe avance de tels sentimens, il veut qu'on admire son esprit et ses lumières, qui surpassent infiniment celles du peuple.

In 1994, Karl Sims was doing experiments on simulated organisms, allowing them to evolve their own body designs and swimming strategies to see if they would converge on some of the same underwater locomotion strategies that real-life organisms use.5, 6, 7 His physics simulator—the world these simulated swimmers inhabited—used Euler integration, a common way to approximate the physics of motion. The problem with this method is that if motion happens too quickly, integration errors will start to accumulate. Some of the evolved creatures learned to exploit these errors to obtain free energy, quickly twitching small body parts and letting the math errors send them zooming through the water.

Multiplying an infinitesimal times an infinitely large number yields a finite number. There's no analogue to this rule in regular arithmetic. However, it accords with the intuitive idea that when the infinitely large is pitted against the infinitely small, the two basically cancel each other out in a titanic clash, and after the dust clears a finite number remains.

Reality is based on unobservable mathematical points (singularities). That’s the secret of existence. What’s at the centre of a black hole? – a singularity. What was the Big Bang? – a singularity event. What is the Big Crunch? – when spacetime returns to a singularity. What is light made of? – photonic singularities (immaterial and dimensionless; according to Einstein’s special theory of relativity, photons have no mass, are maximally length contracted to zero, and time has stopped for them). The whole universe is made of light. It comes from light and returns to light. Light is all about points – singularities. Light is the basis of thought, the basis of mind, and the basis of matter. Everything is derived from light, and light is nothing but mathematical points defined by the generalised Euler Formula, and it creates the visible world via Fourier mathematics.

Why is there something rather than nothing? Only because something can exist as nothing – via the mathematical capacity to express “ nothing ” in non-zero terms, e.g. e iπ + 1 = 0. In other words, wherever you see “ nothing ” (zero), you might in fact be confronting e iπ + 1 (“something” ), without knowing it. Only mathematics has this unique capacity to the ground state of the universe. The compulsory ground state of the universe is zero, the minimum value possible. There is no sufficient reason for any arbitrary non-zero number to serve as the ground state.

Now be honest-wouldn't you have expected e^i*pi to be (a) gibberish aling the lines of "elephant inkpie," or, if it were mathematically meaningful, to be (b) an infinitely complicated irrational number? Indeed, e^i*pi is a transcendental number raised to an imaginary transcendental power. And if (b) were the case, surely e^i*pi would not compute no matter how much computer power were available to try to pin down its value. As you know, neither (a) nor (b) is true, because e^i*pi = -1. (I suspect the fact that both (a) and (b) are provably false is the reason that Benjamin Peirce, the nineteenth-century mathematician, found Euler's formula (or a closely rekated formula) "absolutely paradoxical.") In other words, when the three enigmatic numbers are combined in this form, e^i*pi , they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers. It's as if greenish-pink androids rocketing toward Alpha Centauri in 2370 had hit a space time anomaly and suddenly found themselves sitting in a burger joint in Topeka, Kansas, in 1956. Elvis, of course , was playing on the jukebox.

First, let me frame what I'm calling beautiful. It's not simply the equation's neat little string of symbols. Rather, it's the entire nimbus of meaning surrounding the formula, including its funneling of many concepts into a statement of stunning brevity, its arresting combination of apparent simplicity and hidden complexity, the way its derivation bridges disparate topics in mathematics, and the fact that it's rich with implications, some of which weren't apparent until many years after it was proved to be true. I think most mathematicians would agree that the equation's beauty concerns something like this nimbus.

Chinese philosopher Tsang Lap-Chuen is a leading modern exponent of the idea that the sublime involves this kind of experience. In The Sublime: Groundwork towards a Theory, published in 1998, he wrote that the sublime "evokes our awareness of our being on the threshold from the human to that which transcends the human; which borders on the possible and the impossible; the knowable and the unknowable; the meaningful and the fortuitous; the finite and the infinite." In his view, there is no single essential common property possessed by sublime works or sublime natural objects, nor is there a single emotional state evoked by all of them. But he argues that there's a common thread in experiences of the sublime, which is that they take us "to the limit of some human possibility.

EULER CALCULATED WITHOUT APPARENT EFFORT, as men breathe, or as eagles sustain themselves in the wind” (as Arago said), is not an exaggeration of the unequalled mathematical facility of Leonard Euler (1707-1783)

There’s nothing mysterious about souls. They are simply mathematical singularities outside space and time. They are autonomous Fourier frequency domains. They are Leibnizian monads and Cartesian minds. They are the quintessence of ontological mathematics and the basis of everything. We can even write a precise equation for souls = singularities = minds = monads, namely, the God Equation, which is just the generalised Euler Formula, the centrepiece of mathematical analysis and fundamental to physics.

There’s nothing mysterious about souls. They are simply mathematical singularities outside space and time. They are autonomous Fourier frequency domains. They are Leibnizian monads and Cartesian minds. They are the quintessence of ontological mathematics and the basis of everything. We can even write a precise equation for souls = singularities = minds = monads, namely, the God Equation, which is just the generalised Euler Formula, the centrepiece of mathematical analysis and fundamental to physics.

Finally, in 1748, Euler published the explicit formula in his book Introductio in Analysis Infinitorum.

Euler’s constant, which is γ = 0.577215664901532 …. After π and e, γ is perhaps the most important mathematical constant not appearing in elementary arithmetic.

The commonplace idea that math is a manmade language is comically dumb. We know what manmade languages are like: English, German, French, Spanish, Japanese, and so on. None of these has any resemblance whatsoever to math. How can a human being invent π or e? How can a human pluck Euler’s Formula out of nothing? The reason why people are so keen to say that math is manmade is because if they’re wrong then the converse is true: man is mathmade! That is, of course, exactly the position of ontological mathematics, and it represents the highest possible wisdom

Consider Euler’s identity: eiπ + 1 = 0. Here we have the exact situation where a something – the expression on the left – is exactly equal to zero (nothing). The expression on the left is not nonexistence. It has properties, capacities, potentialities, the ability to interact with others of its kind, indeed, to contribute to the entire infinite system of mathematics, which, in the end, reduces to nothing but a set of infinite tautologies expressing 0 = 0. Since we know that eiπ + 1 = 0, we can substitute 0 for eiπ + 1, leaving 0 = 0. So, here we have something whose essence is to exist and yet be nothing. This is true of the whole of ontological mathematics, and it’s the only system of which this is true, hence it’s the only true system.

Consider Euler’s identity: eiπ + 1 = 0. Here we have the exact situation where a something – the expression on the left – is exactly equal to zero (nothing). The expression on the left is not nonexistence. It has properties, capacities, potentialities, the ability to interact with others of its kind, indeed, to contribute to the entire infinite system of mathematics, which, in the end, reduces to nothing but a set of infinite tautologies expressing 0 = 0. Since we know that eiπ + 1 = 0, we can substitute 0 for eiπ + 1, leaving 0 = 0. So, here we have something whose essence is to exist and yet be nothing. This is true of the whole of ontological mathematics, and it’s the only system of which this is true, hence it’s the only true system.

Euler’s formula – although deceptively simple – is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed.