[The] structural theory is of extreme simplicity. It assumes that the molecule is held together by links between one atom and the next: that every kind of atom can form a definite small number of such links: that these can be single, double or triple: that the groups may take up any position possible by rotation round the line of a single but not round that of a double link: finally that with all the elements of the first short period [of the periodic table], and with many others as well, the angles between the valencies are approximately those formed by joining the centre of a regular tetrahedron to its angular points. No assumption whatever is made as to the mechanism of the linkage. Through the whole development of organic chemistry this theory has always proved capable of providing a different structure for every different compound that can be isolated. Among the hundreds of thousands of known substances, there are never more isomeric forms than the theory permits.

Art has a double face, of expression and illusion, just like science has a double face: the reality of error and the phantom of truth.

Because a child of one doubles its age after the passage of a single year, it can be said to be aging rapidly.

Circles to square and cubes to double
Would give a man excessive trouble.
The longitude uncertain roams,
In spite of Whiston and his bombs.

For all these years you were merely
A smear of light through our telescopes
On the clearest, coldest night; a hint
Of a glint, just a few pixels wide
On even your most perfectly-framed portraits.
But now, now we see you!
Swimming out of the dark - a great
Stone shark, your star-tanned skin pitted
And pocked, scarred after eons of drifting
Silently through the endless ocean of space.
Here on Earth our faces lit up as we saw
You clearly for the first time; eyes wide
With wonder we traced the strangely familiar
Grooves raked across your sides,
Wondering if Rosetta had doubled back to Mars
And raced past Phobos by mistake –
Then you were gone, falling back into the black,
Not to be seen by human eyes again for a thousand
Blue Moons or more. But we know you now,
We know you; you’ll never be just a speck of light again.

He that borrows the aid of an equal understanding, doubles his own; he that uses that of a superior elevates his own to the stature of that he contemplates.

I think each individual is never a plane but a polyhedron. Naturally, whenever a ray of light falls on a face, a vertex, an edge of this polyhedron; the arc that it reflects is undoubtedly variable, very complex and single or multicoloured. I don’t believe in plane men, I think we’re all multiple. We don’t have a double life, we have a multiple life. However, it is no less true that we’re thought to have a common denominator. I think I am or I aspire to be an honest man that tries not to bother too many people in this valley of tears.

If the earth’s population continues to double every 50 years (as it is now doing) then by 2550 A.D. it will have increased 3,000-fold. … by 2800 A.D., it would reach 630,000 billion! Our planet would have standing room only, for there would be only two-and-a-half square feet per person on the entire land surface, including Greenland and Antarctica. In fact, if the human species could be imagined as continuing to multiply further at the same rate, by 4200 A.D. the total mass of human tissue would be equal to the mass of the earth.

If you take a number and double it and double it again and then double it a few more times, the number gets bigger and bigger and goes higher and higher and only arithmetic can tell you what the number is when you decide to quit doubling.

It is not always possible to know what one has learned, or when the dawning will arrive. You will continue to shift, sift, to shake out and to double back. The synthesis that finally occurs can be in the most unexpected place and the most unexpected time. My charge ... is to be alert to the dawnings.

Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence”.

Once in a cycle the comet
Doubles its lonesome track.
Enriched with the tears of a thousand years,
Aeschylus wanders back.

One day a math-deficient frog was sitting on a tiny lily pad in a big pond—a lily pad that doubled in size each night—and on this day the pad covered just one-eighth of the pond. The frog still saw the vast majority of his beloved water and so was unconcerned. Then, just three days later, he woke to find the pond had vanished while he slept.

Population, when unchecked, goes on doubling itself every twenty-five years, or increases in a geometrical ratio. … The means of subsistence, under circumstances the most favorable to human industry, could not possibly be made to increase faster than in an arithmetical ratio.

Talking to the Monbuttu was very laborious as the conversation required double interpretation. I spoke in Arabic to a Niam Niam and he spoke in his language to a Niam Niam who understood Monbuttu or to a Monbuttu who understood Niam Niam.

The power of my [steam] engine rises in a geometrical proportion, while the consumption of fuel has only an arithmetical ratio; in such proportion that every time I added one fourth more to the consumption of fuel, the powers of the engine were doubled.

The Sun is no lonelier than its neighbors; indeed, it is a very common-place star,—dwarfish, though not minute,—like hundreds, nay thousands, of others. By accident the brighter component of Alpha Centauri (which is double) is almost the Sun's twin in brightness, mass, and size. Could this Earth be transported to its vicinity by some supernatural power, and set revolving about it, at a little less than a hundred million miles' distance, the star would heat and light the world just as the Sun does, and life and civilization might go on with no radical change. The Milky Way would girdle the heavens as before; some of our familiar constellations, such as Orion, would be little changed, though others would be greatly altered by the shifting of the nearer stars. An unfamiliar brilliant star, between Cassiopeia and Perseus would be—the Sun. Looking back at it with our telescopes, we could photograph its spectrum, observe its motion among the stars, and convince ourselves that it was the same old Sun; but what had happened to the rest of our planetary system we would not know.

What vexes me most is, that my female friends, who could bear me very well a dozen years ago, have now forsaken me, although I am not so old in proportion to them as I formerly was: which I can prove by arithmetic, for then I was double their age, which now I am not.