# „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.“

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)

Übernommen aus Wikiquote. Letzte Aktualisierung 3. Juni 2021. Geschichte
##### Thomas Heath
englischer Mathematikhistoriker 1861 - 1940

## Ähnliche Zitate

### „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

### „In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation x^2 + ax + b = 0 he placed u + z for x. He then hadu^2 + (2z + a)u +(z^2 + az + b) = 0.He now let 2z + a = 0, whence z = -\frac{1}{2}a,and this gaveu^2 - \frac{1}{4}(a^2 - 4b) = 0.u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.andx = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}.</center“

—  David Eugene Smith American mathematician 1860 - 1944

Quelle: History of Mathematics (1925) Vol.2, p.449

### „[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

### „Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type x^4 + 2gx^2 + bx = c, wrote it as x^4 + 2gx^2 = c - bx, added gx^2 + \frac{1}{4}y^2 + yx^2 + gy to both sides, and then made the right side a square after the manner of Ferrari. This method… requires the solution of a cubic resolvent.Descartes (1637) next took up the question and succeeded in effecting a simple solution… a method considerably improved (1649) by his commentator Van Schooten. The method was brought to its final form by Simpson“

—  David Eugene Smith American mathematician 1860 - 1944

1745
Quelle: History of Mathematics (1925) Vol.2, p.469

### „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 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)

### „Hippocrates also attacked the problem of doubling the cube. …Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i. e. finding x, y such that a:x=x:y=y:b, where a, b are the two given straight lines. It is easy to see that, if a:x=x:y=y:b, then b/a = (x/a)3, and, as a particular case, if b=2a, x3=2a3, so that the side of the cube which is double of the cube of side a is found.“

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

Achimedes (1920)

### „People then form their judgment on what they think of X and Y.“

—  Kim Stanley Robinson American science fiction writer 1952

John Boone
Red Mars (1992)
Kontext: The only part of an argument that really matters is what we think of the people arguing. X claims a, Y claims b. They make arguments to support their claims with any number of points. But when their listeners remember the discussion, what matters is simply that X believes a and Y believes b. People then form their judgment on what they think of X and Y.

### „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 differential equation of the first order\frac {dy}{dx} = f(x, y)… prescribes the slope \frac {dy}{dx} at each point of the plane (or at each point of a certain region of the plane we call the field")…. a differential equation of the first order… can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines…. It leaves open the choice between the two possible directions in the line of a given slope. Thus… we should say specifically "direction of an unoriented straight line" and not merely "direction."“

—  George Pólya Hungarian mathematician 1887 - 1985

Mathematical Methods in Science (1977)

### „In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space.“

—  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)

### „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.

### „The purpose of the mathematical theory of statistics is to deal with the relationship between 2 or more variable quantities without assuming that one is a single-valued mathematical function of the rest. The statistician does not think a certain x will produce a single-valued y; not a causative relation but a correlation. The relationship between x and y will be somewhere within a zone and we have to work out the probability that the point (x,y) will lie in different parts of that zone. The physicist is limited and shrinks the zone into a line. Our treatment will fit all the vagueness of biology, sociology, etc. A very wide science.“

—  Karl Pearson English mathematician and biometrician 1857 - 1936

As quoted by E.S. Pearson, Karl Pearson: An Appreciation of Some Aspects of his Life and Work (1938) and cited in Bernard J. Norton, "Karl Pearson and Statistics: The Social Origins of Scientific Innovation" in Social Studies of Science, Vol. 8, No. 1, Theme Issue: Sociology of Mathematics (Feb.,1978), pp. 3-34.

### „\frac {dy}{dx} = \frac {\omega^2x}{g}…The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form\frac {dy}{dx}…Leibniz's notation …is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation… need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. …it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure… it brought mathematics into disrepute… some of the best minds… such as… Berkeley, complained that calculus is incomprehensible. …\frac {dy}{dx} is the limit of a ratio of dy to dx… Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio… and multiply by dx to achieve the separation of variables. We get{dy} = \frac {\omega^2x}{g}xdx“

—  George Pólya Hungarian mathematician 1887 - 1985

Mathematical Methods in Science (1977)

### „In many cases it is a matter for decision and not a simple matter of fact whether x understands y; and so on.“

—  John Rogers Searle American philosopher 1932

Minds, Brains and Programs (1980)

### „[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

Geometry as a Branch of Physics (1949)

### „math> (\left|x\right\rang \left|y\right\rang- \left|y\right\rang \left|x\right\rang) … was my first lesson in quantum mechanics, and in a very real sense my last, since the rest is mere technique, which can be learnt from books.“

—  John Clive Ward British-Australian nuclear physicist 1924 - 2000

J. C. Ward, Memoirs of a Theoretical Physicist (Optics Journal, Rochester, 2004).

### „Proof that the penis is a much later development than the female vulva is found in the evidence that the male himself was a late mutation from an original female creature. For man is but an imperfect female. Geneticists and physiologists tell us that the Y chromosome that produces males is a deformed and broken X chromosome — the female chromosome. All women have two X chromosomes, while the male has one X derived from his mother and one Y from his father. It seems very logical that this small and twisted Y chromosome is a genetic error — an accident of nature, and that originally there was only one sex — the female.“

—  Elizabeth Gould Davis, buch The First Sex

The First Sex, ch. 1 - Woman and the Second Sex (1971).

### „Lety5 - ay4 + by3 - cy2 + dy - e = 0be the general equation of the fifth degree and suppose that it can be solved algebraically,—i. e., that y can be expressed as a function of the quantities a, b, c, d, and e, composed of radicals. In this case, it is clear that y can be written in the formy = p + p1R1/m + p2R2/m +…+ pm-1R(m-1)/m,m being a prime number, and R, p, p1, p2, etc. being functions of the same form as y. We can continue in this way until we reach rational functions of a, b, c, d, and e. [Note: main body of proof is excluded]…we can find y expressed as a rational function of Z, a, b, c, d, and e. Now such a function can always be reduced to the formy = P + R1/5 + P2R2/5 + P3R3/5 + P4R4/5, where P, R, P2, P3, and P4 are functions or the form p + p1S1/2, where p, p1 and S are rational functions of a, b, c, d, and e. From this value of y we obtainR1/5 = 1/5(y1 + α4y2 + α3y3 + α2y4 + α y5) = (p + p1S1/2)1/5,whereα4 + α3 + α2 + α + 1 = 0.Now the first member has 120 different values, while the second member has only 10; hence y can not have the form that we have found: but we have proved that y must necessarily have this form, if the proposed equation can be solved: hence we conclude thatIt is impossible to solve the general equation of the fifth degree in terms of radicals.It follows immediately from this theorem, that it is also impossible to solve the general equations of degrees higher than the fifth, in terms of radicals.“

—  Niels Henrik Abel Norwegian mathematician 1802 - 1829

A Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree (1824) Tr. W. H. Langdon, as quote in A Source Book in Mathematics (1929) ed. David Eugene Smith