…there is no prescribed route to follow to arrive at a new idea. You have to make the intuitive leap. But the difference is that once you’ve made the intuitive leap you have to justify it by filling in the intermediate steps. In my case, it often happens that I have an idea, but then I try to fill in the intermediate steps and find that they don’t work, so I have to give it up.

[A] quality of an inventor is imagination, because invention is a leap of the imagination from what is known to what has never been before.

[J.J.] Sylvester’s methods! He had none. “Three lectures will be delivered on a New Universal Algebra,” he would say; then, “The course must be extended to twelve.” It did last all the rest of that year. The following year the course was to be Substitutions-Théorie, by Netto. We all got the text. He lectured about three times, following the text closely and stopping sharp at the end of the hour. Then he began to think about matrices again. “I must give one lecture a week on those,” he said. He could not confine himself to the hour, nor to the one lecture a week. Two weeks were passed, and Netto was forgotten entirely and never mentioned again. Statements like the following were not unfrequent in his lectures: “I haven’t proved this, but I am as sure as I can be of anything that it must be so. From this it will follow, etc.” At the next lecture it turned out that what he was so sure of was false. Never mind, he kept on forever guessing and trying, and presently a wonderful discovery followed, then another and another. Afterward he would go back and work it all over again, and surprise us with all sorts of side lights. He then made another leap in the dark, more treasures were discovered, and so on forever.

[Technical courage means the] physician-scientist must be brave enough to adopt new methods. It is far too easy to learn one technique and then to repeat the same experiment over and over. In this fashion one can write many papers, receive large research grants, and remain solidly rooted in the middle of a scientific field. But the true innovator has the confidence to drop one set of experimental crutches and leap to another when he or she must move forward.

Natura non facit saltum or, Nature does not make leaps… If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).

A hundred devils leap into my body, if there be not more old drunkards than old physicians.

A natural law regulates the advance of science. Where only observation can be made, the growth of knowledge creeps; where laboratory experiments can be carried on, knowledge leaps forward.
[Attributed, probably incorrectly]

A problem is really a springboard for a leap into the unknown.

A scientist works largely by intuition. Given enough experience, a scientist examining a problem can leap to an intuition as to what the solution ‘should look like.’ ... Science is ultimately based on insight, not logic.

A space station will permit quantum leaps in our research in science, communications, in metals, and in lifesaving medicines which could be manufactured only in space.

After innumerable dynasties of giant creatures, after endless generations of fish and families of molluscs, man finally arrives, the degenerate product of a grandiose type, his mould perhaps broken by his Creator. Fired by his retrospection, these timid humans, born but yesterday, can now leap across chaos, sing an endless hymn, and configure the history of the universe in a sort of retrograde Apocalypse.

Among the sea-fishes many stories are told about the dolphin, indicative of his gentle and kindly nature…. It appears to be the fleetest of all animals, marine and terrestrial, and it can leap over the masts of large vessels.

As chemists, we must rename [our] scheme and insert the symbols Ba, La, Ce in place of Ra, Ac, Th. As nuclear chemists closely associated with physics, we cannot yet convince ourselves to make this leap, which contradicts all previous experience in nuclear physics.
Co-author with Fritz Strassmann, German chemist (1902-80)

As I looked down, I saw a large river meandering slowly along for miles, passing from one country to another without stopping. I also saw huge forests, extending along several borders. And I watched the extent of one ocean touch the shores of separate continents. Two words leaped to mind as I looked down on all this: commonality and interdependence. We are one world.

Could Hamlet have been written by a committee, or the “Mona Lisa” painted by a club? Could the New Testament have been composed as a conference report? Creative ideas do not spring from groups. They spring from individuals. The divine spark leaps from the finger of God to the finger of Adam, whether it takes ultimate shape in a law of physics or a law of the land, a poem or a policy, a sonata or a mechanical computer.

Creativity makes a leap, then looks to see where it is.

Deep beneath the surface of the Sun, enormous forces were gathering. At any moment, the energies of a million hydrogen bombs might burst forth in the awesome explosion…. Climbing at millions of miles per hour, an invisible fireball many times the size of Earth would leap from the Sun and head out across space.

Even a wise experiment when made by a fool generally leads to a false conclusion, but that fools’ experiments conducted by a genius often prove to be leaps through the dark into great discoveries.

Evolution: At the Mind's Cinema
I turn the handle and the story starts:
Reel after reel is all astronomy,
Till life, enkindled in a niche of sky,
Leaps on the stage to play a million parts.
Life leaves the slime and through all ocean darts;
She conquers earth, and raises wings to fly;
Then spirit blooms, and learns how not to die,-
Nesting beyond the grave in others' hearts.
I turn the handle: other men like me
Have made the film: and now I sit and look
In quiet, privileged like Divinity
To read the roaring world as in a book.
If this thy past, where shall they future climb,
O Spirit, built of Elements and Time?

For a dying man it is not a difficult decision [to agree to become the world's first heart transplant] … because he knows he is at the end. If a lion chases you to the bank of a river filled with crocodiles, you will leap into the water convinced you have a chance to swim to the other side. But you would not accept such odds if there were no lion.

For forty-nine months between 1968 and 1972, two dozen Americans had the great good fortune to briefly visit the Moon. Half of us became the first emissaries from Earth to tread its dusty surface. We who did so were privileged to represent the hopes and dreams of all humanity. For mankind it was a giant leap for a species that evolved from the Stone Age to create sophisticated rockets and spacecraft that made a Moon landing possible. For one crowning moment, we were creatures of the cosmic ocean, an epoch that a thousand years hence may be seen as the signature of our century.

Great Empedocles, that ardent soul,
Leapt into Etna and was roasted whole.

Have you ever watched an eagle held captive in a zoo, fat and plump and full of food and safe from danger too?
Then have you seen another wheeling high up in the sky, thin and hard and battle-scarred, but free to soar and fly?
Well, which have you pitied the caged one or his brother? Though safe and warm from foe or storm, the captive, not the other!
There’s something of the eagle in climbers, don’t you see; a secret thing, perhaps the soul, that clamors to be free.
It’s a different sort of freedom from the kind we often mean, not free to work and eat and sleep and live in peace serene.
But freedom like a wild thing to leap and soar and strive, to struggle with the icy blast, to really be alive.
That’s why we climb the mountain’s peak from which the cloud-veils flow, to stand and watch the eagle fly, and soar, and wheel... below...

Heroes of physics, Argonauts of our time
Who leaped the mountains, who crossed the seas …
You have confirmed in uncomfortable places
What Newton knew without leaving his study.

Hieron asked Archimedes to discover, without damaging it, whether a certain crown or wreath was made of pure gold, or if the goldsmith had fraudulently alloyed it with some baser metal. While Archimedes was turning the problem over in his mind, he chanced to be in the bath house. There, as he was sitting in the bath, he noticed that the amount of water that was flowing over the top of it was equal in volume to that part of his body that was immersed. He saw at once a way of solving the problem. He did not delay, but in his joy leaped out of the bath. Rushing naked through the streets towards his home, he cried out in a loud voice that he had found what he sought. For, as he ran, he repeatedly shouted in Greek; “Eureka! Eurekal I’ve found it! I’ve found it!”

How often people speak of art and science as though they were two entirely different things, with no interconnection. An artist is emotional, they think, and uses only his intuition; he sees all at once and has no need of reason. A scientist is cold, they think, and uses only his reason; he argues carefully step by step, and needs no imagination. That is all wrong. The true artist is quite rational as well as imaginative and knows what he is doing; if he does not, his art suffers. The true scientist is quite imaginative as well as rational, and sometimes leaps to solutions where reason can follow only slowly; if he does not, his science suffers.

If we make a couple of discoveries here and there we need not believe things will go like this for ever. An acrobat can leap higher than a farm-hand, and one acrobat higher than another, yet the height no man can overleap is still very low. Just as we hit water when we dig in the earth, so we discover the incomprehensible sooner or later.

Incandescent carbon particles, by the tens of millions, leap free of the log and wave like banners, as flame. Several hundred significantly different chemical reactions are now going on. For example, a carbon atom and four hydrogen atoms, coming out of the breaking cellulose, may lock together and form methane, natural gas. The methane, burning (combining with oxygen), turns into carbon dioxide and water, which also go up the flue. If two carbon atoms happen to come out of the wood with six hydrogen atoms, they are, agglomerately, ethane, which bums to become, also, carbon dioxide and water. Three carbons and eight hydrogens form propane, and propane is there, too, in the fire. Four carbons and ten hydrogens—butane. Five carbons … pentane. Six … hexane. Seven … heptane. Eight carbons and eighteen hydrogens—octane. All these compounds come away in the breaking of the cellulose molecule, and burn, and go up the chimney as carbon dioxide and water. Pentane, hexane, heptane, and octane have a collective name. Logs burning in a fireplace are making and burning gasoline.

It is slower, but better, to creep along the downward path that to leap over the cliff.

It’s worth remembering that it is often the small steps, not the giant leaps, that bring about the most lasting change.

Knowledge is necessary too. A child with great intuition could not grow up to become something worthwhile in life without some knowledge. However there comes a point in everyone’s life where only intuition can make the leap ahead, without knowing precisely how.

Medicinal discovery,
It moves in mighty leaps,
It leapt straight past the common cold
And gave it us for keeps.

Nothing is accomplished all at once, and it is one of my great maxims, and one of the most completely verified, that Nature makes no leaps: a maxim which I have called the law of continuity.
[Referring to the gradual nature of all change from an initial state, through a continuous series of intermediate stages, to a final state.]

Owing to their [minor planets or asteroids] small size; … The force of gravity on their surfaces must be very small. A man placed on one of them would spring with ease 60 feet high, and sustain no greater shock in his descent than he does on the Earth from leaping a yard. On such planets giants may exist; and those enormous animals which here require the buoyant power of water to counteract their weight, may there inhabit the land.

Psychologists pay lip service to the scientific method, and use it whenever it is convenient; but when it isn’t they make wild leaps of their uncontrolled fancy.…

Spirituality leaps where science cannot yet follow, because science must always test and measure, and much of reality and human experience is immeasurable.

Such is the tendency of the human mind to speculation, that on the least idea of an analogy between a few phenomena, it leaps forward, as it were, to a cause or law, to the temporary neglect of all the rest; so that, in fact, almost all our principal inductions must be regarded as a series of ascents and descents, and of conclusions from a few cases, verified by trial on many.

Talent jogs to conclusions to which genius takes giant leaps.

That’s one small step for a man, one giant leap for mankind.
[Transmission received and heard as “That’s one small step for man… one giant leap for mankind.”]

The [Moon] surface is fine and powdery. I can kick it up loosely with my toe. It does adhere in fine layers like powdered charcoal to the sole and sides of my boots. I only go in a small fraction of an inch, maybe an eighth of an inch, but I can see the footprints of my boots and the treads in the fine sandy particles.
[First report, immediately after stepping on to the Moon and saying “That's one small step for (a) man; one giant leap for mankind.”]

The creative element in the mind of man … emerges in as mysterious a fashion as those elementary particles which leap into momentary existence in great cyclotrons, only to vanish again like infinitesimal ghosts.

The experimental investigation by which Ampere established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the 'Newton of Electricity'. It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics.

The experimental investigation by which Ampère established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the “Newton of Electricity”. It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics.

The intellect has little to do on the road to discovery. There comes a leap in consciousness, call it intuition or what you will, and the solution comes to you and you don’t know why or how.

The mind can proceed only so far upon what it knows and can prove. There comes a point where the mind takes a higher plane of knowledge, but can never prove how it got there. All great discoveries have involved such a leap

The most ominous conflict of our time is the difference of opinion, of outlook, between men of letters, historians, philosophers, the so-called humanists, on the one side and scientists on the other. The gap cannot but increase because of the intolerance of both and the fact that science is growing by leaps and bounds.

The point to remember is that a giant leap into space can be a giant leap toward peace down below.

The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity... the rigid electron is no working hypothesis, but a working hindrance. Approaching Maxwell's equations with the concept of the rigid electron seems to me the same thing as going to a concert with your ears stopped up with cotton wool. We must admire the courage and the power of the school of the rigid electron which leaps across the widest mathematical hurdles with fabulous hypotheses, with the hope to land safely over there on experimental-physical ground.

The shrewd guess, the fertile hypothesis, the courageous leap to a tentative conclusion—these are the most valuable coin of the thinker at work.

Tragically isolated, imprisoned in his own “self,” man has made a desperate effort to “leap beyond his shadow,” to embrace the external world. From this effort was born science….

Use now and then a little Exercise a quarter of an Hour before Meals, as to swing a Weight, or swing your Arms about with a small Weight in each Hand; to leap, or the like, for that stirs the Muscles of the Breast.

We all remember the fairy tales of science in our infancy, which played with the supposition that large animals could jump in the proportion of small ones. If an elephant were as strong as a grasshopper, he could (I suppose) spring clean out of the Zoological Gardens and alight trumpeting upon Primrose Hill. If a whale could leap from the water like a trout, perhaps men might look up and see one soaring above Yarmouth like the winged island of Laputa.

We find it a law of our state of being that where only observation can be made the growth of knowledge creeps; where experiment can be made knowledge leaps forward.

We have here spoken of the prediction of facts of the same kind as those from which our rule was collected. But the evidence in favour of our induction is of a much higher and more forcible character when it enables us to explain and determine cases of a kind different from those which were contemplated in the formation of our hypothesis. The instances in which this has occurred, indeed, impress us with a conviction that the truth of our hypothesis is certain. No accident could give rise to such an extraordinary coincidence. No false supposition could, after being adjusted to one class of phenomena, so exactly represent a different class, when the agreement was unforeseen and contemplated. That rules springing from remote and unconnected quarters should thus leap to the same point, can only arise from that being where truth resides.

Whatever Nature undertakes, she can only accomplish it in a sequence. She never makes a leap. For example she could not produce a horse if it were not preceded by all the other animals on which she ascends to the horse’s structure as if on the rungs of a ladder. Thus every one thing exists for the sake of all things and all for the sake of one; for the one is of course the all as well. Nature, despite her seeming diversity, is always a unity, a whole; and thus, when she manifests herself in any part of that whole, the rest must serve as a basis for that particular manifestation, and the latter must have a relationship to the rest of the system.

When in doubt, make a fool of yourself. There is a microscopically thin line between being brilliantly creative and acting like the most gigantic idiot on earth. So what the hell, leap.

While there is still much to learn and discover through space exploration, we also need to pay attention to our unexplored world here on earth. Our next big leap into the unknown can be every bit as exciting and bold as our pioneering work in space. It possesses the same “wow” factor: alien worlds, dazzling technological feats and the mystery of the unknown.