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More discoveries have arisen from intense observation of very limited material than from statistics applied to large groups. The value of the latter lies mainly in testing hypotheses arising from the former. While observing one should cultivate a speculative, contemplative attitude of mind and search for clues to be…
Man does not limit himself to seeing; he thinks and insists on learning the meaning of phenomena whose existence has been revealed to him by observation. So he reasons, compares facts, puts questions to them, and by the answers which he extracts, tests one by another. This sort of control, by means of reasoning and…
230(231-1) ... is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for; as they are merely curious without being useful, it is not likely that any person will attempt to find a number beyond it.
We know from X-ray diffraction studies that a strand of DNA is 1.5 nanometers (1.5 x 10 to the -9 meters) in radius. Assume a cylinder 1 meter long (the arm's length) with a radius of 1.5 nanometers and work out the volume (length x pi r-squared). A typical animal cell is about 8 micrometers (8 x 10 to the -6 meters)…
It seems to me that your Reverence and Signor Galileo act prudently when you content yourselves with speaking hypothetically and not absolutely, as I have always understood that Copernicus spoke. To say that on the supposition of the Earth's movement and the Sun's quiescence all the celestial appearances are explained…
These estimates may well be enhanced by one from F. Klein (1849-1925), the leading German mathematician of the last quarter of the nineteenth century. 'Mathematics in general is fundamentally the science of self-evident things.' ... If mathematics is indeed the science of self-evident things, mathematicians are a…
It is, I believe, justifiable to make the generalization that anything an organic chemist can synthesize can be made without him. All he does is increase the probability that given reactions will 'go.' So it is quite reasonable to assume that given sufficient time and proper conditions, nucleotides, amino acids…
Laplace’s cool confidence in avowing atheistical sentiments was legendary. The story was told that after Napoleon had inspected a copy of Laplace’s Systéme du Monde, he challenged the astronomer about his beliefs. ‘Monsieur Laplace! Newton has frequently spoken of God in his book. I have already gone over yours, and I…
There is another approach to the extraterrestrial hypothesis of UFO origins. This assessment depends on a large number of factors about which we know little, and a few about which we know literally nothing. I want to make some crude numerical estimate of the probability that we are frequently visited by…
The mathematician Freeman Dyson, of the Institute for Advanced Study, offers a scheme in which the planet Jupiter is broken down piece by piece, transported to the distance of the Earth from the Sun, and reconstructed into a spherical shell – a swarm of individual fragments revolving about the Sun. The advantage of…
On the title page of this chapter is shown the message. It is etched on a 6-inch by 9-inch gold-anodized aluminum plate, attached to the antenna support struts of Pioneer 10. The expected erosion rate in interstellar space is sufficiently small that this message should remain intact for hundreds of millions of years…
The name is symbolic. The first pale roots and shoots of a germinating seed look so tiny and fragile; it's hard to believe it can grow into a big tree. Yet there is so much life force in that seed that the roots can work their way through boulders to reach the water, and the shoot can work its way through cracks in a…
It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and…
Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these…
Science and technology multiply around us. To an increasing extent they dictate the languages in which we speak and think. Either we use those languages, or we remain mute.
Foreshadowings of the principles and even of the language of [the infinitesimal] calculus can be found in the writings of Napier, Kepler, Cavalieri, Pascal, Fermat, Wallis, and Barrow. It was Newton's good luck to come at a time when everything was ripe for the discovery, and his ability enabled him to construct…
In the mathematics I can report no deficience, except it be that men do not sufficiently understand this excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent…
Imagine, say, that you wanted to travel from New York to Madrid, two cities that are at almost the same latitude. If the earth were flat, the shortest route would be to head straight east. If you did that, you would arrive in Madrid after traveling 3,707 miles. But due to the earth's curvature, there is a path that on…
Calculus is one of the most important parts of mathematics. It is fundamental to all of modern science. How could one part of mathematics be of such central importance? It is because calculus gives us the tools to study rates of change and motion. All analytical subjects, from biology to physics to chemistry to…
The principles of logic and mathematics are true universally simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying…