“A standard of reference for the arrangement of the stars may be had by comparing their distribution to a certain properly modified equality of scattering. The equality which I propose does not require that the stars should be at equal distances from each other, nor is it necessary that all those of the same nominal magnitude should be equally distant from us.”

— William Herschel

Source: Sir William Herschel: His Life and Works (1880), Ch.4 "Life and Works" from a memoir, published (1817).

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William Herschel 36

German-born British astronomer, technical expert, and compo… 1738–1822

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“An equal scattering of the stars may be admitted in certain calculations; but when we examine the milky way, or the closely compressed clusters of stars… this supposed equality of scattering must be given up.”

William Herschel (1738–1822) German-born British astronomer, technical expert, and composer

p, 125
Astronomical Observations relating to the Construction of the Heavens... (1811)

“Society should treat all equally well who have deserved equally well of it, that is, who have deserved equally well absolutely. This is the highest abstract standard of social and distributive justice; towards which all institutions, and the efforts of all virtuous citizens, should be made in the utmost degree to converge.”

John Stuart Mill book Utilitarianism

Source: Utilitarianism (1861), Ch. 5

“On symbolic use of equalities and proportions. Chapter II.
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 (1540–1603) French mathematician

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)

“Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.”

Archimedes book On the Equilibrium of Planes

Book 1, Postulate 1.
On the Equilibrium of Planes

“Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the Law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights.”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART I: THIS WORLD, Chapter 9. Of the Universal Colour Bill
Context: The Art of Sight Recognition, being no longer needed, was no longer practised; and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon to be considered superfluous, and fell into disrespect and neglect even at our University. The inferior Art of Feeling speedily experienced the same fate at our Elementary Schools.... Year by year the Soldiers and Artisans began more vehemently to assert — and with increasing truth — that there was no great difference between them and the very highest class of Polygons, now that they were raised to an equality with the latter, and enabled to grapple with all the difficulties and solve all the problems of life, whether Statical or Kinetical, by the simple process of Colour Recognition. Not content with the natural neglect into which Sight Recognition was falling, they began boldly to demand the legal prohibition of all "monopolizing and aristocratic Arts" and the consequent abolition of all endowments for the studies of Sight Recognition, Mathematics, and Feeling. Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the Law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights.

“If we arrive at an equation containing on each side the same term but with different coefficients, we must take equals from equals until we get one term equal to another term. But, if there are on one or on both sides negative terms, the deficiencies must be added on both bides until all the terms on both sides are positive. Then we must take equals from equals until one term is left on each side.”

Diophantus Arithmetica

As quoted by Thomas Little Heath, Diophantos of Alexandria: https://books.google.com/books?id=ABkPAAAAIAAJ A Study in the History of Greek Algebra (1885)
Arithmetica (c. 250 AD)

“But now was turning my desire and will,
Even as a wheel that equally is moved,
The Love which moves the sun and the other stars.”

Dante Alighieri book Paradiso

Canto XXXIII, closing lines, as translated by Henry Wadsworth Longfellow
The Divine Comedy (c. 1308–1321), Paradiso
Context: As the geometrician, who endeavours
To square the circle, and discovers not,
By taking thought, the principle he wants,Even such was I at that new apparition;
I wished to see how the image to the circle
Conformed itself, and how it there finds place;But my own wings were not enough for this,
Had it not been that then my mind there smote
A flash of lightning, wherein came its wish. Here vigour failed the lofty fantasy:
But now was turning my desire and will,
Even as a wheel that equally is moved, The Love which moves the sun and the other stars.

“How should we like it were stars to burn
With a passion for us we could not return?
If equal affection cannot be,
Let the more loving one be me.”

W. H. Auden (1907–1973) Anglo-American poet

The More Loving One (1957)

“For Prudence, is but Experience; which equal time, equally bestows on all men, in those things they equally apply themselves unto.”

Thomas Hobbes book Leviathan

The First Part, Chapter 13, p. 60
Leviathan (1651)

“Equal justice and equal opportunity are ideals we should seek, but we should recognize that humans administer the ideals and that humans do not have equal ability.”

Frank Herbert (1920–1986) American writer

Dune Genesis (1980)
Context: In the beginning I was just as ready as anyone to fall into step, to seek out the guilty and to punish the sinners, even to become a leader. Nothing, I felt, would give me more gratification than riding the steed of yellow journalism into crusade, doing the book that would right the old wrongs.
Reevaluation raised haunting questions. I now believe that evolution, or deevolution, never ends short of death, that no society has ever achieved an absolute pinnacle, that all humans are not created equal. In fact, I believe attempts to create some abstract equalization create a morass of injustices that rebound on the equalizers. Equal justice and equal opportunity are ideals we should seek, but we should recognize that humans administer the ideals and that humans do not have equal ability.

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William Herschel 36

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