„Proposition 13. The straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the moon move is less than double of the diameter of the moon, but has to it a ratio greater than that which 88 has to 45; and it is less than 1/9th part of the diameter of the sun, but has to it a ratio greater than that which 22 has to 225. But it has to the straight line drawn from the centre of the sun at right angles to the axis and meeting the sides of the cone a ratio greater than that which 979 has to 10125.“

p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)

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Aristarchos von Samos Foto
Aristarchos von Samos
griechischer Astronom, Vertreter des heliozentrischen Weltb…

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Aristarchus of Samos Foto

„Proposition 15. The diameter of the sun has, to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6.“

—  Aristarchus of Samos ancient Greek astronomer and mathematician

p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)
Variante: Proposition 10. The sun has to the moon a ratio greater than that which 5832 has to 1, but less than that which 8000 has to 1.

Aristarchus of Samos Foto
Aristarchus of Samos Foto

„Proposition 9. The diameter of the sun is greater than 18 times, but less than 20 times, the diameter of the moon.“

—  Aristarchus of Samos ancient Greek astronomer and mathematician

p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)
Variante: Proposition 7. The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth.

Aristarchus of Samos Foto
Aristarchus of Samos Foto
Aristarchus of Samos Foto
Aristarchus of Samos Foto
John Dee Foto
A.E. Housman Foto
Thomas Robert Malthus Foto

„The moon is not kept in her orbit round the earth, nor the earth in her orbit round the sun, by a force that varies merely in the inverse ratio of the squares of the distances.“

—  Thomas Robert Malthus British political economist 1766 - 1834

Quelle: An Essay on The Principle of Population (First Edition 1798, unrevised), Chapter XIII, paragraph 2, lines 19-22

James Bradley Foto

„If we suppose the distance of the fixed stars from the sun to be so great that the diameter of the earth's orbit viewed from them would not subtend a sensible angle, or which amounts to the same, that their annual parallax is quite insensible; it will then follow that a line drawn from the earth in any part of its orbit to a fixed star, will always, as to sense, make the same angle with the plane of the ecliptic, and the place of the star, as seen from the earth, would be the same as seen from the sun placed in the focus of the ellipsis described by the earth in its annual revolution, which place may therefore be called its true or real place.
But if we further suppose that the velocity of the earth in its orbit bears any sensible proportion to the velocity with which light is propagated, it will thence follow that the fixed stars (though removed too far off to be subject to a parallax on account of distance) will nevertheless be liable to an aberration, or a kind of parallax, on account of the relative velocity between light and the earth in its annual motion.
For if we conceive, as before, the true place of any star to be that in which it would appear viewed from the sun, the visible place to a spectator moving along with the earth, will be always different from its true, the star perpetually appearing out of its true place more or less, according as the velocity of the earth in its orbit is greater or less; so that when the earth is in its perihelion, the star will appear farthest distant from its true place, and nearest to it when the earth is in its aphelion; and the apparent distance in the former case will be to that in the latter in the reciprocal proportion of the distances of the earth in its perihelion and its aphelion. When the earth is in any other part of its orbit, its velocity being always in the reciprocal proportion of the perpendicular let fall from the sun to the tangent of the ellipse at that point where the earth is, or in the direct proportion of the perpendicular let fall upon the same tangent from the other focus, it thence follows that the apparent distance of a star from its true place, will be always as the perpendicular let fall from the upper focus upon the tangent of the ellipse. And hence it will be found likewise, that (supposing a plane passing through the star parallel to the earth's orbit) the locus or visible place of the star on that plane will always be in the circumference of a circle, its true place being in that diameter of it which is parallel to the shorter axis of the earth's orbit, in a point that divides that diameter into two parts, bearing the same proportion to each other, as the greatest and least distances of the earth from the sun.“

—  James Bradley English astronomer; Astronomer Royal 1693 - 1762

Miscellaneous Works and Correspondence (1832), Demonstration of the Rules relating to the Apparent Motion of the Fixed Stars upon account of the Motion of Light.

T.S. Eliot Foto
James Jeans Foto
Vitruvius Foto

„It is no secret that the moon has no light of her own, but is, as it were, a mirror, receiving brightness from the influence of the sun.“

—  Vitruvius, buch De architectura

Quelle: De architectura (The Ten Books On Architecture) (~ 15BC), Book IX, Chapter II, Sec. 3

Augustus De Morgan Foto

„In order to see the difference which exists between… studies,—for instance, history and geometry, it will be useful to ask how we come by knowledge in each. Suppose, for example, we feel certain of a fact related in history… if we apply the notions of evidence which every-day experience justifies us in entertaining, we feel that the improbability of the contrary compels us to take refuge in the belief of the fact; and, if we allow that there is still a possibility of its falsehood, it is because this supposition does not involve absolute absurdity, but only extreme improbability.
In mathematics the case is wholly different… and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us. In geometry, of the principles alluded to, those which are most commonly used are—
I. If a magnitude is divided into parts, the whole is greater than either of those parts.
II. Two straight lines cannot inclose a space.
III. Through one point only one straight line can be drawn, which never meets another straight line, or which is parallel to it.
It is on such principles as these that the whole of geometry is founded, and the demonstration of every proposition consists in proving the contrary of it to be inconsistent with one of these.“

—  Augustus De Morgan British mathematician, philosopher and university teacher (1806-1871) 1806 - 1871

Quelle: On the Study and Difficulties of Mathematics (1831), Ch. I.

Derek Parfit Foto
Marcus Aurelius Foto
E.E. Cummings Foto
Euclid Foto

„And the whole [is] greater than the part.“

—  Euclid, buch Elements

Καὶ τὸ ὅλον τοῦ μέρους μεῖζον
ἐστιν
Elements, Book I, Common Notion 8 (5 in certain editions)
Cf. Aristotle, Metaphysics, Book Η 1045a 8–10: "… the totality is not, as it were, a mere heap, but the whole is something besides the parts … [πάντων γὰρ ὅσα πλείω μέρη ἔχει καὶ μὴ ἔστιν οἷον σωρὸς τὸ πᾶν]"
Original: (el) Καὶ τὸ ὅλον τοῦ μέρους μεῖζον [ἐστιν].

John Dee Foto

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