The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established.... The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.
Thomas Little HeathRelated topics
body book case diagonal geometry great incommensurable number proof proving pythagorean ratio side square thus till universal well means aristotle absurdum PythagorasRelated quotes
You are not traditionally beautiful; and you know it. We women do. But what most people mean by beauty is really a kind of aesthetic acceptability, not so much character as a lack of it, a set of features and lineaments that hide their history, that suggest history itself does not exist. But the template by which we recognize the features and forms in the human body that cause the heart to halt, threatening to spill us over into the silence of death-that is drawn on another part of the soul entirely...But all sing, chant, hymn the history of the body, if only because we all know how people regard bodies that deviate from the lauded and totally abnormal norm named beauty. Most of us would rather not recognize such desires in ourselves and thus avoid all contemplation of what the possession of such features means about the lives, the bodies, the histories of others, preferring instead to go on merely accepting the acceptable. But that is not who I am.
Samuel R. Delany
I could go on, but the point is clear: religions make explicit claims about reality-about what exists and happens in the universe. These claims involve the existence of gods, the number of such gods (polytheism or monotheism), their character and behavior (usually loving and beneficent, but, in the case of Hindu and ancient Greek gods, sometimes mischievous or malevolent), how they interact with the world, whether or not there are souls or life after death, and, above all, how the deities wish us to behave-their moral code.
These are empirical claims, and although some may be hard to test, they must, like all claims about reality, be defended with a combination of evidence and reason. If we find no credible evidence, no good reasons to believe, then those claims should be disregarded, just as most of us ignore claims about ESP, astrology, and alien abduction. After all, beliefs important enough to affect you for eternity surely deserve the closest scrutiny.
Jerry Coyne
God's greatness lies in forgiving, in showing mercy, and in this greatness He is greater than the heart which condemns itself. Behold, this is that greatness of God which we have to speak of especially within the sanctuary; for here we know God differently, more closely – from the other side, if one may say so – that out there, where doubtless He is revealed, is to be known in His works, whereas here He is to be known as He has revealed Himself, as He would be known by Christians. The tokens of God's greatness can be known in nature everyone can behold with wonder, or rather there is no special token, for the works themselves are the tokens; thus every one can see the rainbow and must wonder when he sees it. But the token of God's greatness in showing mercy exists only for faith; this token is in fact the Sacrament. God's greatness in nature is manifest, but God's greatness in showing mercy is a secret which has to be believed.
Soren Kierkegaard
We are... bound to attach the greatest importance to the preliminary step taken by Lavoisier, who is even more justly called the father of modern chemistry than Kepler is called the father of modern astronomy. The exact claims of Lavoisier to this important place in the history of chemistry have been variously stated: ...since his time, and greatly through his labours, the quantitative method has been established as the ultimate test of chemical facts; the principle of this method being the rule that in all changes of combination and reaction, the total weight of the various ingredients-be they elementary bodies or compounds-remains unchanged. The science of chemistry was thus established upon an exact, a mathematical basis. By means of this method Lavoisier, utilising and analysing the results gained by himself and others before him, notably those of Priestley, Cavendish, and Black, succeeded in destroying the older theory of combustion, the so-called phlogistic theory.
Antoine Lavoisier
Rachel was looking into the mirror at an angle of 45°, and so had a view of the face turned toward the room and the face on the other side, reflected in the mirror; here were time and reverse-time, co-existing, cancelling one another exactly out. Were there many such reference points, scattered throughout the world, perhaps only at nodes like this room which housed a transient population of the imperfect, the dissatisfied; did real time plus virtual or mirror-time equal zero and thus serve some half-understood moral purpose? Or was it only the mirror world that counted; only a promise of a kind that the inward bow of a nose-bridge or a promontory of extra cartilage at the chin meant a reversal of ill fortune such that the world of the altered would thenceforth run on mirror-time; work and love by mirror-light and be only, till death stopped the heart's ticking (metronome's music) quietly as light ceases to vibrate, an imp's dance under the century's own chandeliers....
Thomas Pynchon