“And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second.”

— John Napier

Appendix, The relations of Logarithms & their natural numbers to each other
The Construction of the Wonderful Canon of Logarithms (1889)

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John Napier46

Scottish mathematician 1550–1617

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“If a first sine be multiplied into a second producing a third, the Logarithm of the first added to the Logarithm of the second produces the Logarithm of the third. So in division, the Logarithm of the divisor subtracted from the Logarithm of the dividend leaves the Logarithm of the quotient.”

John Napier (1550–1617) Scottish mathematician

Appendix, The relations of Logarithms & their natural numbers to each other
The Construction of the Wonderful Canon of Logarithms (1889)

“As a square number is known to be the product of a number multiplied by itself, so every polygonal number, multiplied by one number and added to another, both of which depend upon the number of its angles, produces a square number. I shall prove this, and shall show also how from a given side to find its polygon and conversely. Some auxiliary propositions must first be proved.”

Diophantus Arithmetica

As quoted by James Gow, A Short History of Greek Mathematics https://books.google.com/books?id=9d8DAAAAMAAJ (1884) <br class="br">Arithmetica (c. 250 AD)

“The first myth of management is that it exists. The second myth of management is that success equals skill.”

Robert Heller (1932–2012) British magician

Robert Heller cited in : Jonathon Green (1984) The Cynic's Lexicon: A Dictionary of Amoral Advice. p. 92

“If a is any definite number, then all numbers of the system R fall into two classes, A1 and A2, each of which contains infinitely many individuals; the first class A1 comprises all numbers a1 that are < a, the second class A2 comprises all numbers a2 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A1, A2 is such that every number of the first class A1 is less than every number of the second class A2.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

“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)

“Elections serve two purposes. The first, and obvious, purpose is to accurately choose the winner. But the second is equally important: to convince the loser.”

Bruce Schneier (1963) American computer scientist

American Elections Are Too Easy to Hack. We Must Take Action Now, Schneier, Bruce, 2018-04-18, The Guardian, 2018-08-12 https://www.theguardian.com/commentisfree/2018/apr/18/american-elections-hack-bruce-scheier, <br class="br">Elections

“The second part of the book… contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity.”

E. W. Hobson (1856–1933) British mathematician

A Treatise on Plane Trigonometry https://books.google.com/books?id=_ktLAAAAMAAJ (1891) Preface

“The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio.”

James Whitbread Lee Glaisher (1848–1928) English mathematician and astronomer

Source: Encyclopedia Britannica, 9th Edition; Article “Logarithms.”; Reported in Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book, (1914) : On the invention of logarithms

“Ralegh's own circle, which turned Christopher Marlowe into a rationalist, was dominated by… Thomas Hariot. For navigation is dependent on astronomy; it went hand in hand with the new speculations about the world and the solar system… in turn, the voyages of the great navigators inspired the literature of Elizabethan England. The worlds of art and science and the physical world unfolded then together. It was not by accident that the first table of logarithms was published within a few years of the First Folio.”

Jacob Bronowski (1908–1974) Polish-born British mathematician

"Sense and Sensibility"
The Common Sense of Science (1951)

“L. Walras first formulated the state of the economic system at any point of time as the solution of a system of simultaneous equations representing the demand for goods by consumers, the supply of goods by producers and the equilibrium condition that supply equal demand on every market.”

Gérard Debreu (1921–2004) French economist and mathematician

Arrow, Kenneth J., and Gerard Debreu. " Existence of an equilibrium for a competitive economy http://cowles.econ.yale.edu/P/cp/p00b/p0087.pdf." Econometrica: Journal of the Econometric Society (1954): p. 265

“And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second.”

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John Napier46

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