“In general, the rate of evaporation (m) of a substance in a high vacuum is related to the pressure (p) of the saturated vapor by the equation m=\sqrt{\frac{M}{2\pi RT}}p. Red phosphorus and some other substances probably form exceptions to this rule.”

— Irving Langmuir

Irving Langmuir, "The Constitution and Fundamental Properties of Solids and Liquids. Part I. Solids.", Journal of the American Chemical Society, September 5, 1916

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Irving Langmuir 2

American chemist and physicist 1881–1957

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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 (1802–1829) Norwegian mathematician

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

“[M]isunderstanding of probability may be the greatest of all general impediments to scientific literacy.”

Stephen Jay Gould book Dinosaur in a Haystack

"Happy Thoughts on a Sunny Day in New York City", p. 9
Dinosaur in a Haystack (1995)

“Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed… all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a^{m} and a^{n}, which is a^{m+n} when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a+b)^{n}, determined by the principles of arithmetical algebra, when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for (a+b)^n, when n is general both in form and value.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. vi-vii
A Treatise on Algebra (1842)

“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 (1860–1944) American mathematician

Source: 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 (1860–1944) American mathematician

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

“E = mc2 really applies only to isolated bodies at rest. In general, when you have moving bodies, or interacting bodies, energy and mass aren't proportional. E = mc2 simply doesn't apply. …For moving bodies, the correct mass-energy equation is
E=\frac {mc^2} {\sqrt{1-\frac{v^2} {c^2}}}
where v is the velocity. For a body at rest (v=0), this becomes E = mc2. …we must consider the special case of particles with zero mass… examples include photons, color gluons, and gravitons. If we attempt to put m = 0 and v = c in our general mass-energy equation, both the numerator and denominator on the right-hand-side vanish, and we get the nonsensical relation E = 0/0. The correct result is that the energy of a photon can take any value. …The energy E of a photon is proportional to the frequency f of the light it represents. …they are related by the Planck-Einstein-Schrödinger equation E = hf, where h is Plank's constant.”

Frank Wilczek (1951) physicist

when the velocity <math>v</math> approaches the speed of light c, the denominator approaches 0 thus E approaches infinity, unless m = 0.
Source: The Lightness of Being – Mass, Ether and the Unification of Forces (2008), Ch. 3, p. 19 & Appendix A

“I often think to myself that it's not I but someone else who is P. M. and is the recipient of those continuous marks of respect and affection from the general public who called in Downing Street or at the station to take off their hats and cheer. And then I go back to the House of Commons and listen to the unending stream of abuse of the P. M., his faithlessness, his weakness, his wickedness, his innate sympathy with Fascism and his obstinate hatred of the working classes.”

Neville Chamberlain (1869–1940) Former Prime Minister of the United Kingdom

Letter to Hilda Chamberlain (28 May 1939), quoted in Maurice Cowling, The Impact of Hitler. British Politics and British Policy. 1933-1940 (Cambridge: Cambridge University Press, 1975), p. 293.
Prime Minister

“But the nonexistent apples will still fall toward the nonexistent ground at a meaningless rate of 9.8 m/s2.”

Eliezer Yudkowsky (1979) American blogger, writer, and artificial intelligence researcher

Quantum Non-Realism http://lesswrong.com/lw/q5/quantum_nonrealism/ (May 2008)
Context: The nature of "reality" is something about which I'm still confused, which leaves open the possibility that there isn't any such thing. But Egan's Law still applies: "It all adds up to normality." Apples didn't stop falling when Einstein disproved Newton's theory of gravity. Sure, when the dust settles, it could turn out that apples don't exist, Earth doesn't exist, reality doesn't exist. But the nonexistent apples will still fall toward the nonexistent ground at a meaningless rate of 9.8 m/s2.

“It is remarkable of the simple substances that they are generally in some compound form.”

Robert Chambers (publisher, born 1802) book Vestiges of the Natural History of Creation

Source: Vestiges of the Natural History of Creation (1844), p. 35
Context: It is remarkable of the simple substances that they are generally in some compound form. Thus oxygen and nitrogen, though in union they form the aerial envelope of the globe, are never found separate in nature. Carbon is pure only in the diamond. And the metallic bases of the earths, though the chemist can disengage them, may well be supposed unlikely to remain long uncombined, seeing that contact with moisture makes them burn. Combination and re-combination are principles largely pervading nature. There are few rocks, for example, that are not composed of at least two varieties of matter, each of which is again a compound of elementary substances. What is still more wonderful with respect to this principle of combination, all the elementary substances observe certain mathematical proportions in their unions. It is hence supposed that matter is composed of infinitely minute particles or atoms, each of which belonging to any one substance, can only (through the operation of some as yet hidden law) associate with a certain number of the atoms of any other.

“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.”

Thomas Little Heath (1861–1940) British civil servant and academic

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)

“In general, the rate of evaporation (m) of a substance in a high vacuum is related to the pressure (p) of the saturated vapor by the equation m=\sqrt{\frac{M}{2\pi RT}}p. Red phosphorus and some other substances probably form exceptions to this rule.”

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Irving Langmuir 2

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