„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 Cube
the 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><math>\begin{cases}y^2 = bx\\x^2 = ay\end{cases}</math></center>whence, as before,<center><math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</math></center>
Apollonius of Perga (1896)

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Thomas Heath Foto
Thomas Heath
englischer Mathematikhistoriker 1861 - 1940

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„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.

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„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.3
3) 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.

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„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 that
It 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

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