# „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.“

1548 - 1620

### „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

### „[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.“

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

### „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
p.465

### „::1 2 3 4 5 6 7 8 9“

—  Beth Anderson American neo-romantic composer 1950

### „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

### „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

### „Vieta: 1QC - 15QQ + 85C - 225Q + 274N, aequator 120. Modern form:x^6 - 15x^4 + 85x^3 - 225x^2 + 274x = 120</center“

—  David Eugene Smith American mathematician 1860 - 1944
p.430

### „The writer must be four people: 1) The nut, the obsede 2) The moron 3) The stylist 4) The critic. 1 supplies the material; 2 lets it come out; 3 is taste; 4 is intelligence.“

—  Susan Sontag American writer and filmmaker, professor, and activist 1933 - 2004

### „[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
p.461

### „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} ][[File:HeathsApolloniusPgxxColor. jpg|thumb|Doubling the Cubethe 2 solutions of Menaechmus]]and 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. The point P where the two parabolas intersect is given by\begin{cases}y^2 = bx\\x^2 = ay\end{cases}whence, as before,\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</center“

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

### „The stages of human development are to strive for: (1) Besitz [Possession] (2) Wissen [Knowledge] (3) Können [Ability] (4) Sein [Being]“

—  Erwin Schrödinger Austrian physicist 1887 - 1961
Writings of August 1918, quoted in A Life of Erwin Schrödinger (1994) by Walter Moore

### „Take a unit, halve it, halve the result, and so on continually. This gives—1 1&frasl;2 1&frasl;4 1&frasl;8 1&frasl;16 1&frasl;32 1&frasl;64 1&frasl;128 &c.;Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c...; We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2 is always equal to the last term added.... We say that—1, 1 + 1&frasl;2, 1 + 1&frasl;2 + 1&frasl;4, 1 + 1&frasl;2 + 1&frasl;4 + 1&frasl;8, &c.; &c.;is a series of quantities which continually approximate to the limit 2. Now the truth is, these several quantities are fixed, and do not approximate to 2.... it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, "let x be a quantity which continually approximates to the limit 2," we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for x which shall be as near to 2 as we please.“

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

### „A productive mistake is: (1) made in the service of mission and vision; (2) acknowledged as a mistake; (3) learned from; (4) considered valuable; (5) shared for the benefit of all.“

—  Pete Seeger American folk singer 1919 - 2014
p. 90

### „Framework 2 is connected with the creativity and vitality of your world. In your terms, the dead waken in Framework 2 and move through it to Framework 3, where they can be aware of their reincarnational identities and connections with time, while being apart from a concentration on earth realities. In those terms, the so-called dead dip in and out of earth probabilities by travelling through Framework 2, and into those probabilities connected with earth realities. Some others may wind up in Framework 4, which is like Framework 2, except that it is a creative source for other kinds of realities not physically oriented at all and outside of, say, time concepts as you are used to thinking about them. In a way impossible to describe verbally, some portion of each identity also resides in Framework 4, and in all other Frameworks.“

—  Jane Roberts American Writer 1929 - 1984
p. 139

### „These seven stages we shall name as follows:1. Mixture2. Gestation3. Expansion4. Age of Conflict5. Universal Empire6. Decay7. Invasion“

—  Carroll Quigley American historian 1910 - 1977
Chapter 5, Historical Change in Civilizations, p. 146

### „[is] a term coined in Pearl (1985) to emphasize three aspects: (1) the subjective nature of the input information; (2) the reliance on Bayes' conditioning as the basis of updating information; (3) the distinction between causal and evidential models of reasoning, a distinction that underscores Thomas Bayes' paper of 1763“

—  Judea Pearl Computer scientist 1936
p. 14