# „[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" d, and in order to determine the curvature… we must express N, or equivalently V, to which it is assumed proportional, in terms of d. …from the second of formulae (3) and… (4)… to the approximation here adopted, 5)V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + …);…plotting N against… d and comparing… with the formula (5), it should be possible operationally to determine the "curvature" K.“

Geometry as a Branch of Physics (1949)

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##### Howard P. Robertson
amerikanischer Mathematiker und Physiker 1903 - 1961

## Ähnliche Zitate

### „All the light which is radiated… will, after it has traveled a distance r, lie on the surface of a sphere whose area S is given by the first of the formulae (3). And since the practical procedure… in determining d is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius d, it follows…4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + …);whence, to our approximation 4)d = r (1- \frac{K r^2}{6} + …), orr = d (1 + \frac{K d^2}{6} + …).</center“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

Geometry as a Branch of Physics (1949)

### „The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces… The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + …),V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + …).“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

Geometry as a Branch of Physics (1949)

### „The solution of (1), which represents a homogeneous manifold, may be written in the form:ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)where \kappa = \sqrt \frac{\lambda}{3}. If we consider \rho as determining distance from the origin… and \tau as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world…“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

"On Relativistic Cosmology" (1928)

### „The Rule of Three, or Golden Rule of Arithmeticall whole Numbers. Be the three termes given 2 3 4. …To finde their fourth proporcionall Terme: that is to say, in such Reason to the third terme 4, as the second terme 3, is to the first terme 2 [Modern notation: \frac{x}{4} = \frac{3}{2}]. …Multiply the second terme 3, by the third terme 4, & giveth the product 12: which dividing by the first terme 2, giveth the Quotient 6: I say that 6 is the fourth proportional terme required.“

—  Simon Stevin Flemish scientist, mathematician and military engineer 1548 - 1620

Disme: the Art of Tenths, Or, Decimall Arithmetike (1608)

### „The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive.“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" $R = \frac{1}{K^\frac{1}{2}}$ which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing...
Geometry as a Branch of Physics (1949)

### „The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" R = \frac{1}{K^\frac{1}{2}} which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing…“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

Geometry as a Branch of Physics (1949)

### „The words "routine analyses" are used to denote the analyses performed by laboratories, frequently attached to industrial plants, and distinguished by the following characteristics: (1) All the analyses or measurements of the same kind, for example, are designed to measure the sugar content in beets or to determine the coordinates of a star. (2) The analyses are carried out day after day using the same methods and the same instruments. (3) While all the analyses are of the same kind, the quantity n varies from time to time, where n represents some small number, 2, 3, 4, 5.“

—  Jerzy Neyman Polish statistician 1894 - 1981

p. 46 of "On a statistical problem arising in routine analyses and in sampling inspections of mass production." http://www.jstor.org/stable/2235624 The Annals of Mathematical Statistics 12, no. 1 (1941): 46–76.

### „It is picked out from numbers progressing in continuous proportion. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical one which advances by unequal and proportionally increasing or decreasing intervals. Arithmetical progressions: 1, 2, 3, 4, 5, 6, 7, &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c, Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1.“

—  John Napier Scottish mathematician 1550 - 1617

The Construction of the Wonderful Canon of Logarithms (1889)

### „A new point is determined in Euclidean Geometry exclusively in one of the three following ways:Having given four points A, B, C, D, not all incident on the same straight line, then(1) Whenever a point P exists which is incident both on (A, B) and on (C, D), that point is regarded as determinate.(2) Whenever a point P exists which is incident both on the straight line (A, B) and on the circle C(D), that point is regarded as determinate.(3) Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate.The cardinal points of any figure determined by a Euclidean construction are always found by means of a finite number of successive applications of some or all of these rules (1), (2) and (3). Whenever one of these rules is applied it must be shown that it does not fail to determine the point. Euclid's own treatment is sometimes defective as regards this requisite.In order to make the practical constructions which correspond to these three Euclidean modes of determination, correponding to (1) the ruler is required, corresponding to (2) both ruler and compass, and corresponding to (3) the compass only.…it is possible to develop Euclidean Geometry with a more restricted set of postulations. For example it can be shewn that all Euclidean constructions can be carried out by means of (3) alone…“

—  E. W. Hobson British mathematician 1856 - 1933

Quelle: Squaring the Circle (1913), pp. 7-8

### „Although Cardan reduced his particular equations to forms lacking a term in x^2, it was Vieta who began with the general formx^3 + px^2 + qx + r = 0and made the substitution x = y -\frac{1}{3}p, thus reducing the equation to the formy^3 + 3by = 2c.He then made the substitutionz^3 + yz = b, or y = \frac{b - z^2}{z},which led to the formz^6 + 2cz^2 = b^2,a sextic which he solved as a quadratic.“

—  David Eugene Smith American mathematician 1860 - 1944

Quelle: History of Mathematics (1925) Vol.2, p.465

### „Consider an event, for example the outburst if a nova… Suppose this event is observed from two stars in line with the nova, and suppose further that the two stars are moving uniformly with respect to each other in this line. Let the epoch at which these stars passed by each other be taken as the zero of time measurement, and let an observer A on one of the stars estimate the distance and epoch of the nova outburst to be x units of length and t units of time, respectively. Suppose the other star is moving toward the nova with velocity v relative to A. Let an observer B on the star estimate the distance and epoch of the nova outburst to be x' units of length and t' units of time, respectively. Then the Lorentz formulae, relating x' to t', arex' = \frac {x-vt}{\sqrt{1-\frac{v^2}{c^2}}}; \qquad t' = \frac {t-\frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}These formulae are… quite general, applying to any event in line with two uniformly moving observers. If we let c become infinite then the ratio of v to c tends to zero and the formulae becomex' = x - vt; \qquad t' = t.“

—  Gerald James Whitrow British mathematician 1912 - 2000

p, 125
The Structure of the Universe: An Introduction to Cosmology (1949)

### „The discovery of Hippocrates amounted to the discovery of the fact that from the relation(1)\frac{a}{x} = \frac{x}{y} = \frac{y}{b}it follows that(\frac{a}{x})^3 = [\frac{a}{x} \cdot \frac{x}{y} \cdot \frac{y}{b} =] \frac{a}{b}and if a = 2b, [then (\frac{a}{x})^3 = 2, and]a^3 = 2x^3.The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations(2)x^2 = ay, y^2 = bx, xy = ab[or equivalently…y = \frac{x^2}{a}, x = \frac{y^2}{b}, y = \frac{ab}{x} ]Doubling the Cubethe 2 solutions of Menaechmusand the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2).Let AO, BO be straight lines placed so as to form a right angle at O, and of length a, b respectively. Produce BO to x and AO to y.The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i. e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i. e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have\begin{cases}y^2 = b. ON = b. PM = bx\\ and\\ xy = PM. PN = ab\end{cases}whence\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a.“

—  Thomas Little Heath British civil servant and academic 1861 - 1940

The point P where the two parabolas intersect is given by<center>$\begin{cases}y^2 = bx\\x^2 = ay\end{cases}$</center>whence, as before,<center>$\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.$</center>
Apollonius of Perga (1896)

### „Suppose we a certain Number of things exposed, different each from other, as a, b, c, d, e, &c.; The question is, how many ways the order of these may be varied? as, for instance, how many changes may be Rung upon a certain Number of Bells; or, how many ways (by way of Anagram) a certain Number of (different) Letters may be differently ordered?Alt.1,21) If the thing exposed be but One, as a, it is certain, that the order can be but one. That is 1.2) If Two be exposed, as a, b, it is also manifest, that they may be taken in a double order, as ab, ba, and no more. That is 1 x 2 = 2. Alt.33) If Three be exposed; as a, b, c: Then, beginning with a, the other two b, c, may (by art. 2,) be disposed according to Two different orders, as bc, cb; whence arise Two Changes (or varieties of order) beginning with a as abc, acb: And, in like manner it may be shewed, that there be as many beginning with b; because the other two, a, c, may be so varied, as bac, bca. And again as many beginning with c as cab, cba. And therefore, in all, Three times Two. That is 1 x 2, x 3 = 6.Alt.34) If Four be exposed as a, b, c, d; Then, beginning with a, the other Three may (by art. preceeding) be disposed six several ways. And (by the same reason) as many beginning with b, and as many beginning with c, and as many beginning with d. And therefore, in all, Four times six, or 24. That is, the Number answering to the case next foregoing, so many times taken as is the Number of things here exposed. That is 1 x 2 x 3, x 4 = 6 x 4 = 24.5) And in like manner it may be shewed, that this Number 24 Multiplied by 5, that is 120 = 24 x 5 = 1 x 2 x 3 x 4 x 5, is the number of alternations (or changes of order) of Five things exposed. (Or, the Number of Changes on Five Bells.) For each of these five being put in the first place, the other four will (by art. preceeding) admit of 24 varieties, that is, in all, five times 24. And in like manner, this Number 120 Multiplied by 6, shews the Number of Alternations of 6 things exposed; and so onward, by continual Multiplication by the conse quent Numbers 7, 8, 9, &c.;6) That is, how many so ever of Numbers, in their natural Consecution, beginning from 1, being continually Multiplied, give us the Number of Alternations (or Change of order) of which so many things are capable as is the last of the Numbers so Multiplied. As for instance, the Number of Changes in Ringing Five Bells, is 1 x 2 x 3 x 4 x 5 = 120. In Six Bells, 1 x 2 x 3 x 4 x 5 x 6 = 120 x 6 = 720. In Seven Bells, 720 x 7 = 5040. In Eight Bells, 5040 x 8 = 40320, And so onward, as far as we please.“

—  John Wallis English mathematician 1616 - 1703

Quelle: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.II Of Alternations, or the different Change of Order, in any Number of Things proposed.

### „1. Zero is a number.2. The successor of any number is another number.3. There are no two numbers with the same successor.4. Zero is not the successor of a number.5. Every property of zero, which belongs to the successor of every number with this property, belongs to all numbers.“

—  Giuseppe Peano Italian mathematician 1858 - 1932

As expressed in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
Peano axioms

### „Counting is the most simple and primitive of narratives -- 1 2 3 4 5 6 7 8 9 10 -- a tale with a beginning, a middle and an end and a sense of progression -- arriving at a finish of two digits -- a goal attained, a dénouement reached.“

—  Peter Greenaway British film director 1942

Fear of Drowning By Numbers

### „Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^{2^n} + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^{2^5} + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation.“

—  Florian Cajori, buch A History of Mathematics

Quelle: A History of Mathematics (1893), p. 180; also cited in Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914) pp. 156-157. https://books.google.com/books?id=G0wtAAAAYAAJ&pg=PA156

### „[Zuanne de Tonini] da Coi… impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but… without any explanation. At any rate, the two cubics x^3 + ax^2 = c and x^3 + bx = c could now be solved. The reduction of the general cubic x^3 + ax^2 + bx = c to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types x^3 = ax^2 + c and x^3 + ax^2 = c by substituting x = y + \frac{1}{3}a and x = y - \frac{1}{3}a respectively, and transformed the type x^3 + c = ax^2 by the substitution x = \sqrt[3]{c^2/y}, thus freeing the equations of the term x^2. This completed the general solution, and he applied the method to the complete cubic in his later problems.“

—  David Eugene Smith American mathematician 1860 - 1944

Quelle: History of Mathematics (1925) Vol.2, p.461

### „4-5-1? Never. It was always 4-3-3 for me as a player and as manager, just like [Frank] Rijkaard at Barcelona. With 4-3-3, it's much easier to make combinations going forward. With only one forward, who is he going to pass to? Who is he going to make combinations with? Football is about having the best offensive play possible. I always like to play offensive football and nobody will convince me otherwise.“

—  Johan Cruyff Dutch association football player 1947 - 2016

In an interview with Lawrence Donegan of The Guardian https://www.theguardian.com/football/2005/oct/01/sport.comment6 (1 October 2005).

### „[T]he space constant K… "curvature" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space.“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

Geometry as a Branch of Physics (1949)

### „Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption… We must therefore examine the relation between this astronomer's "distance" d… and the distance r which appears as an element of the geometry.“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

Geometry as a Branch of Physics (1949)