“The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms.”

— Florian Cajori , book A History of Mathematics

Source: A History of Mathematics (1893), p. 161; Cited in: Moritz (1914, 263); Arithmetics

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Florian Cajori 13

American mathematician 1859–1930

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“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

“In the course of the fifteenth century, the sexagesimal division of the radius, in terms of which cords and goniometrical line-segments were expressed, was generally superseded, though not immediately replaced, by a decimal system of positional notation. Instead, mathematicians sought to avoid fractions by taking the Radius equal to a number of units of length of the form {\displaystyle 10^{n}} {\displaystyle 10^{n}}…The first to apply this method was the German astronomer Regiomontanus… the second half of the sixteenth and the first decades of the seventeenth century… observed of a gradual development of this method of Regiomontanus into a complete system of decimal positional fractions. Yet none of the steps taken by… writers is comparable in importance and scope with the progress achieved by Stevin in his De Thiende.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: Simon Stevin: Science in the Netherlands around 1600, 1970, p. 17-18

“The first period embraces the time between the first records of empirical determinations of the ratio of the circumference to the diameter of a circle until the invention of the Differential and Integral Calculus, in the middle of the seventeenth century. This period, in which the ideal of an exact construction was never entirely lost sight of, and was occasionally supposed to have been attained, was the geometrical period, in which the main activity consisted in the approximate determination of π by the calculation of the sides or the areas of regular polygons in- and circum-scribed to the circle. The theoretical groundwork of the method was the Greek method of Exhaustions. In the earlier part of the period the work of approximation was much hampered by the backward condition of arithmetic due to the fact that our present system of numerical notation had not yet been invented; but the closeness of the approximations obtained in spite of this great obstacle are truly surprising. In the later part of this first period methods were devised by which the approximations to the value of π were obtained which required only a fraction of the labour involved in the earlier calculations. At the end of the period the method was developed to so high a degree of perfection that no further advance could be hoped for on the lines laid down by the Greek Mathematicians; for further progress more powerful methods were required.”

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

Source: Squaring the Circle (1913), pp. 10-11

“In their almost miraculous insight, the founders of our country invented "federalism," a concept that is rooted in the rights of the individual.”

Walter Cronkite (1916–2009) American broadcast journalist

UN Address (1999)
Context: I suppose I'm preaching to the choir here. So let's not talk generalities but focus tonight on a few specifics of what the leadership of the World Federalist Movement believe must be done now to advance the rule of world law.
For starters, we can draw on the wisdom of the framers of the US Constitution in 1787. The differences among the American states then were as bitter as differences among the nation-states in the world today.
In their almost miraculous insight, the founders of our country invented "federalism," a concept that is rooted in the rights of the individual. Our federal system guarantees a maximum of freedom but provides it in a framework of law and justice.
Our forefathers believed that the closer the laws are to the people, the better. Cities legislate on local matters; states make decisions on matters within their borders; and the national government deals with issues that transcend the states, such as interstate commerce and foreign relations. That is federalism.
Today we must develop federal structures on a global level. We need a system of enforceable world law — a democratic federal world government — to deal with world problems.

“If the task of philosophy is to break the domination of words over the human mind […], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers […] I believe the cause of logic has been advanced already by the invention of this concept notation.”

Gottlob Frege book Begriffsschrift

Begriffsschrift (1879) Preface to the Begriffsschrift

“A large fraction of the flaws in software development are due to programmers not fully understanding all the possible states their code may execute in.”

John D. Carmack (1970) American computer programmer, engineer, and businessman

Quoted in "Functional programming in C++" http://gamasutra.com/view/news/169296/Indepth_Functional_programming_in_C.php

“It is impossible to over-emphasize the importance of al-Jazari's work in the history of engineering. Until modern times there is no other document from any cultural area that provides a comparable wealth of instructions for the design, manufacture and assembly of machines… Al-Jazari did not only assimilate the techniques of his non-Arab and Arab predecessors, he was also creative. He added several mechanical and hydraulic devices. The impact of these inventions can be seen in the later designing of steam engines and internal combustion engines, paving the way for automatic control and other modern machinery. The impact of al-Jazari's inventions is still felt in modern contemporary mechanical engineering [2].”

Donald Hill (1922–1994) British historian and engineer

Donald Hill, Studies in Medieval Islamic Technology,; as cited in: Salim Al-Hassani. " 800 Years Later: In Memory of Al-Jazari, A Genius Mechanical Engineer http://muslimheritage.com/article/800-years-later-memory-al-jazari-genius-mechanical-engineer," at muslimheritage.com, 2015.