### „If a and b yield C, but C is not equal to a+b, then we have emergence.“

— Varadaraja V. Raman American physicist 1932

page 313

Truth and Tension in Science and Religion

As quoted in Hermann Weyl, "Emmy Noether" (April 26, 1935) in Weyl's Levels of Infinity: Selected Writings on Mathematics and Philosophy (2012) p. 64.

— Varadaraja V. Raman American physicist 1932

page 313

Truth and Tension in Science and Religion

— Lloyd Alexander American children's writer 1924 - 2007

"The Flat-Heeled Muse", Horn Book Magazine (1 April 1965)

— Halldór Laxness, livro Kristnihald undir Jökli (bók)

Kristnihald undir Jökli (Under the Glacier/Christianity at Glacier) (1968)

The analytical method accepts as proven the most famous [ as known from Euclid ] symbolic use of equalities and proportions that are found in items such as:

1. The whole is equal to the sum of its parts.

2. Quantities being equal to the same quantity have equality between themselves. [a = c & b = c => a = b]

3. If equal quantities are added to equal quantities the resulting sums are equal.

4. If equals are subtracted from equal quantities the remains are equal.

5. If equal equal amounts are multiplied by equal amounts the products are equal.

6. If equal amounts are divided by equal amounts, the quotients are equal.

7. If the quantities are in direct proportion so also are they are in inverse and alternate proportion. [a:b::c:d=>b:a::d:c & a:c::b:d]

8. If the quantities in the same proportion are added likewise to amounts in the same proportion, the sums are in proportion. [a:b::c:d => (a+c):(b+d)::c:d]

9. If the quantities in the same proportion are subtracted likewise from amounts in the same proportion, the differences are in proportion. [a:b::c:d => (a-c):(b-d)::c:d]

10. If proportional quantities are multiplied by proportional quantities the products are in proportion. [a:b::c:d & e:f::g:h => ae:bf::cg:dh]

11. If proportional quantities are divided by proportional quantities the quotients are in proportion. [a:b::c:d & e:f::g:h => a/e:b/f::c/g:d/h]

12. A common multiplier or divisor does not change an equality nor a proportion. [a:b::ka:kb & a:b::(a/k):(b/k)]

13. The product of different parts of the same number is equal to the product of the sum of these parts by the same number. [ka + kb = k(a+b)]

14. The result of successive multiplications or divisions of a magnitude by several others is the same regardless of the sequential order of quantities multiplied times or divided into that magnitude.

But the masterful symbolic use of equalities and proportions which the analyst may apply any time is the following:

15. If we have three or four magnitudes and the product of the extremes is equal to the product means, they are in proportion. [ad=bc => a:b::c:d OR ac=b2 => a:b::b:c]

And conversely

10. If we have three or four magnitudes and the first is to the second as the second or the third is to the last, the product of the extremes is equal to that of means. [a:b::c:d => ad=bc OR a:b::b:c => ac=b2]

We can call a proportion the establishment of an equality [equation] and an equality [equation] the resolution of a proportion.“

— François Viète French mathematician 1540 - 1603

From Frédéric Louis Ritter's French Tr. Introduction à l'art Analytique (1868) utilizing Google translate with reference to English translation in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) Appendix

In artem analyticem Isagoge (1591)

— Gottlob Frege, Sense and reference

As cited in: M. Fitting, Richard L. Mendelsoh (1999), First-Order Modal Logic, p. 142. They called this Frege's Puzzle.

Über Sinn und Bedeutung, 1892

— Steve Sailer American journalist and movie critic 1958

How to Help the Left Half of the Bell Curve http://www.isteve.com/How_to_Help_the_Left_Half_of_the_Bell_Curve.htm, VDARE.com, July to September 2000

— Milton Friedman American economist, statistician, and writer 1912 - 2006

“Milton Friedman vs Free Lunch Advocate” https://www.youtube.com/watch?v=9Qe7fLL25AQ (1980s)

— Gottlob Frege, Sense and reference

The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a = b could not differ from a = a (i.e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing.

As cited in: M. Fitting, Richard L. Mendelsoh (1999), First-Order Modal Logic, p. 142. They called this Frege's Puzzle.

Über Sinn und Bedeutung, 1892

— John Rawls, livro A Theory of Justice

Fonte: A Theory of Justice (1971; 1975; 1999), Chapter II, Section 11, pg. 60

— Cyrus H. Gordon American linguist 1908 - 2001

let alone before the date of any known Hebrew text

Introduction

Adventures in the Nearest East (1957)

— Mignon McLaughlin American journalist 1913 - 1983

The Complete Neurotic's Notebook (1981), Love

— Howard Bloom American publicist and author 1943

Brace Yourself: The Five Heresies

The God Problem: How a Godless Cosmos Creates (2012)

— Willem de Sitter Dutch cosmologist 1872 - 1934

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

(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 British civil servant and academic 1861 - 1940

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)

— John Napier Scottish mathematician 1550 - 1617

Appendix, The relations of Logarithms & their natural numbers to each other

The Construction of the Wonderful Canon of Logarithms (1889)

— Mona Sahlin Swedish politician 1957

Mona Sahlin in an interview with the Swedish newspaper Göteborgs-Posten, October 22, 2000.

— Willem de Sitter Dutch cosmologist 1872 - 1934

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

— Proclus Greek philosopher 412 - 485

...

This... is what the the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales...

Proposition XV. Thereom VIII.

— John Wallis English mathematician 1616 - 1703

Fonte: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.I Of the variety of Elections, or Choice, in taking or leaving One or more, out of a certain Number of things proposed.