Cultural context: Renaissance in Europe (beyond Italy) started in 16^{th} century
Hieronymus Bosch 14501516 and Pieter Bruegel the Elder 1525–1569: Netherlandish painters

Erasmus of Rotterdam 1466–1536 Dutch Renaissance humanist, Catholic priest, social critic, teacher, and theologian

Albrecht Dürer 1471–1528: painter, printmaker and theorist of the German Renaissance

Thomas More,^{ }1478–1535 English lawyer, social philosopher, writer, statesman and humanist, in 1516 Utopia is published

Protestant Reformation: 1517 Martin Luther The NinetyFive Theses, 1518 Zwingli, 1533 Calvin, 1547 Church of England

CounterReformation: Spanish Inquisition since 1478, in 1572 St. Bartholomew's Day massacre, since 1540 Jesuit order

Arithmetic, Number theory and discrete Math

1536 Hudalrichus Regius finds the fifth perfect number (previously known were 6, 28, 496 and 8128). The number 2^{12}(2^{13}  1) = 33550336 is the first perfect number to be discovered since ancient times. 1544 Stifel publishes Arithmetica integra which contains binomial coefficients and the notation +, , √.

1555 J. Scheybl gives the sixth perfect number 2^{16}(2^{17}  1) = 8589869056 but his work remains unknown until 1977.

1563 G.Cardan writes his book Liber de Ludo Aleae on games of chance but it would not be published until 1663.

1575 Maurolico publishes Arithmeticorum libri duo which contains examples of inductive proofs.

1603 P.Cataldi found the 6^{th} and 7^{th} perfect numbers, with p=17,19. The next one was found only by Euler 130 years later.

Application of Math in Astronomy and mechanics

1541 Rheticus publishes his trigonometric tables and the trigonometrical parts of Copernicus's work.

1543 Copernicus publishes De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres): Sun (not the Earth) is at rest in the centre of the Universe. Copernican Revolution was the beginning of Scientific Revolution of 15^{th}16^{th} centuries.

1571 Viète 15401603 Canon Mathematicus a math introduction to his astronomy treatise: trigonometry with tables. In Analytical Art invented word “coefficient” and notation: variablesconsonants and coefficients vowels, he knew the binomial formula.

1586 Stevin publishes De Beghinselen der Weeghconst containing the theorem of the triangle (parallelogram) of forces.

1595 Clavius writes Novi calendarii romani apologia that helped Pope Gregory XIII to introduce what is now called the Gregorian calendar.

Geometry

1551 Recorde translates and abridges Euclid's Elements as The Pathewaie to Knowledge.

Real Numbers

1585 Stevin publishes De Thiende in which he presents an elementary and thorough account of decimal fractions.

1590 Cataldi uses continued fractions in finding square roots;

1593 Van Roomen calculates π to 16 decimal places.

Galileo Galilei 15641642 "father of science": astronomer, physicist, engineer, philosopher, and mathematician, famous for defending heliocentrism, “the universe … is written in language of math”, considered coordinate system and parabola as the ideal trajectory 1589 falling unequalweightballs from Leaning Tower of Pisa; 1610 telescope observations in Sidereus Nuncius (Message from the stars): Jupiter satellites, Venus phases, rings of Saturn, Sun spots, Milky Way; 1623 writes about Copernicus, 1632 new book

1600 William Gilbert “De Magnete” “Father of electricity and magnetism”
Francis Bacon 1561–1626 philosopher, statesman, scientist, jurist, orator, writer, “father of empiricism” (method of scientific revolution), inductive methodology, improvement of mankind, reforms in Law, 1627 “New Atlantis” utopian novel, knighted in 1603
1609 Johannes Kepler 1571–1630: Astronomia nova (New Astronomy); the orbits are elliptic, word “focus”, 1596 3 laws of planetary motion, 5 platonic solids and distances
1628 William Harvey: circulary rout of blood
1661 Robert Boyle: “Sceptical Chymist” principles of modern chemistry
1665 Robert Hooke: “Micrographia” microscopic observation of cellular structure, word “cell” is introduced; 1660 law of elasticity

Scientific Revolution of 15^{th}16^{th} centuries (after Copernicus)
John Napier 15501617: invention of logarithms in 1614, Table of Logarithms; 1624:
14digit log table

Thomas Hariote 15601621: recognized negative and imaginary roots, (xa)(xb)… has no other roots

Marin Mersenne 15881648: theologian, philosopher, mathematician, music theorist, "father of acoustics".[1] Mersenne, "the center of the world of science and mathematics during the first half of the 1600s." In 1635 he set up the informal Académie Parisienne which had nearly 140 correspondents including astronomers, philosophers and mathematicians and was the precursor of the Académie des sciences established by Colbert in 1666; numbers M_{n} = 2^{n} − 1 are called Mersenne numbers: Mersenne primes ( M_{2}=3, M_{3}=7, M_{5}=31, M_{7} =127, but not M_{11}=2047=23×89) are related to the perfect numbers.
Girard Desargues 1591–1661: mathematician and engineer, a founders of projective geometry.^{ }Desargues' theorem

Albert Girard 15951632: “A new discovery of Algebra” 1629 the Fund. Theorem of Algebra, symmetric Functions, sum of powers of roots: A, A22B, A33AB+3C, etc., stated “sum of two squares” theorem of Fermat, 1626 abbreviations sin, cos, tan; the area of a spherical triangle.

Bonaventura Cavalieri 1598–1647: optics, motion, Cavalieri's principle in geometry anticipated infinitesimal and integral calculus calculus, introduced logarithms in Italy
Evangelista Toricelli 1608–1647: physicist and mathematician, inventor of the barometer, “method of Indivisibles”

Henry Oldenburg 16191677: German theologian, diplomat, a natural philosopher, the creator of scientific peer review, one of the foremost intelligencers of Europe of 17^{th} century, created the first scientific journal in 1665 “ Philosophical Transactions of the Royal Society”

John Wallis 1616 1703 Math: infinitesimal calculus, chief cryptographer for Parliament and the
Royal court, introduced the symbol for infinity and 1/∞ for an infinitesimal.
Christiaan Huygens 16291695: Dutch mathematician, astronomer, physicist (mechanics, optics), probabilist and horologist
Isaac Barrow 16301677: teacher of Newton, infinitesimal calculus, fundamental theorem of calculus (slope/area)
Robert Hooke 16351703 natural philosopher, architect and polymath, "England's Leonardo"; Law of elasticity,, etc.

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