
Ahmes or Ahmose or A'hmose (ca. 1650 B.C.) was an ancient Egyptian
scribe who was given the task of copying a set of mathematical procedures.
His copy is called the 'Rhind Papyrus' after an Englishman who found
and preserved the papyrus in the 1800's.

Thales of Miletus (ca. 624 to 546 B.C.) was a preSocratic
Greek philosopher from Miletus in Asia Minor. Thales attempted to explain
natural phenomena without reference to mythology and was tremendously
influential in this respect. In mathematics, Thales used geometry to
solve problems such as calculating the height of pyramids and the
distance of ships from the shore. He is credited with the first use of
deductive reasoning applied to geometry, by deriving four corollaries
to
Thales' Theorem.
As a result, he has been hailed as the first true mathematician and
is the first known individual to whom a mathematical discovery has
been attributed.
Thales' Theorem :
Provided AC is a diameter, the angle at B is a constant right (90°) angle.

Pythagorus of Samos (ca. 570 to 495 B.C.) was a Greek philosopher,
mathematician, born on the island of Samos, and founder of the
religious movement called Pythagoreanism. Some have questioned whether
he contributed much to mathematics and natural philosophy. Many of
the accomplishments credited to Pythagoras may actually have been
accomplishments of his colleagues and successors. He is
known to most for his name being attached to the 'Pythagorean theorem'
 which was actually known to other cultures, such as China, independently.
It is probably appropriate that it be called the 'Pythagorean theorem'
rather than 'Pythagorus's theorem'.
The earliest known mention of Pythagoras's name in connection with the
theorem occurred five centuries after his death, in the writings of
Cicero and Plutarch.
A visual proof of the Pythagorean theorem.

Hippasus (ca. 500 B.C.) was a Pythagorean who showed that there
were irrational numbers  namely the square root of 2  the length of the
diagonal of the unit square. Hence he showed that the rationals are not
sufficient to describe all line lengths.
It is said that his fellow Pythagoreans were not very pleased about
this and held it against him  either by drowning or banishment.
They couldn't handle the truth!  to use a Jack Nicholson line from one of his
movies.

Zeno of Elea (ca. 490 to 430 B.C.) was a Greek philosopher known for
his paradoxes.

Hippocrates of Chios (ca. 470 to 410 B.C.) was a Greek mathematician
(geometer) and astronomer. The 'reductio ad absurdum' argument (or
proof by contradiction) has been traced to him. The major accomplishment
of Hippocrates is that he was the first to write a systematically
organized geometry textbook, called 'Stoicheia Elements', that is,
basic theorems, or building blocks of mathematical theory.
In the century after Hippocrates, at least four other mathematicians wrote
their own 'Elements', steadily improving terminology and logical structure.
In this way Hippocrates' pioneering work laid the foundation for Euclid's
'Elements' (ca. 325 BC) that was to remain the standard geometry textbook
for many centuries. Only a single, and famous, fragment of Hippocrates'
'Elements' is existent, embedded in the work of
Simplicius.

Plato (428 to 348 B.C.) was a Greek philosopher who
founded an 'academy' in Athens. Above the gate of his academy: "No one
ignorant of geometry can enter here." (That sounds exclusionary for a
teacher. The quote probably should have read "can leave here".)
"Plato is important in the history of mathematics largely for his role
as inspirer and director of others, and perhaps to him is due the
sharp distinction in ancient Greece between arithmetic (in the sense of
the theory of numbers) and logistic (the technique of computation)."

Eudoxus of Cnidus (ca. 408 to 355 B.C.) was a Greek astronomer, mathematician,
scholar and student of Plato. Eudoxus is considered by some to be the greatest
of classical Greek mathematicians, and in all antiquity, second only to
Archimedes. He rigorously developed Antiphon's method of exhaustion,
a precursor to the integral calculus which was also used in a masterly
way by Archimedes in the following century. In applying the method,
Eudoxus proved such mathematical statements as: areas of circles are to
one another as the squares of their radii, volumes of spheres are to
one another as the cubes of their radii, the volume of a pyramid is
onethird the volume of a prism with the same base and altitude, and
the volume of a cone is onethird that of the corresponding cylinder.

Menaechmus (ca. 380 to 320 B.C.) was a Greek mathematician and geometer
who revealed facts about
conic sections (ellipse, parabola, hyperbola  derived from cones).

Aristaeus the Elder (ca. 370 to 300 B.C.) was a Greek mathematician
who worked on conic sections. He was a contemporary of Euclid, though
probably older. Pappus (see below) gave Aristaeus great credit for a work
entitled 'Five Books concerning Solid Loci' which was used by Pappus but
has been lost. Aristeus may have also authored the book 'Concerning the
Comparison of Five Regular Solids'. This book has also been lost.
We know of it through a reference by the Greek mathematician Hypsicles.

Autolycus of Pitan (ca. 360 to 290 BC) was a Greek astronomer,
mathematician, and geographer. Autolycus' surviving works include
a book on spheres entitled 'On the Moving Sphere' and another
'On Risings and Settings' of celestial bodies. His book on spheres
gives indications of what theorems were well known in his day.

Euclid (ca. 325 to 265 B.C.) is known for his book
'Elements of Geometry', which deduced many mathematical principles
from a small set of
'axioms'. For 2,000 years that book had no peer as an
introduction to and reference work on geometry.
Euclid knew of and presented what is now called the
'Fundamental Theorem of Arithmetic': Every whole number can be
written as a product of prime numbers. The 'Elements' also showed
that the number of primes is infinite and included a proof of the
'Pythagorean theorem'.

Archimedes (287 to 212 B.C.) was a Greek mathematician, physicist,
engineer, inventor, and astronomer. He showed many surprising geometric relationships,
such as: For a sphere inside a minimal containing cylinder, the surface area
of the sphere is the same as that of the cylinder. He also made scientific
discoveries concerning centersofgravity and floating bodies. He approximated
the value of pi via inscribed and circumscribed regular polygons about
a circle.
He devised war engines to repel Romans who besieged Syracuse, Greece.
The Romans eventually took the city and sent soldiers to get Archimedes,
to meet with the Roman governor.
Archimedes, who often drew figures in the sand, told the soldiers "Don't
disturb my circles!" (probably in Greek, eh?). When he ("the Sand Reckoner")
tried to stop the soldiers, they killed him, perhaps thinking that he was attacking
(or insulting) them. (This is one of the many interesting tales of the deaths of
mathematicians. See Hippasus above.)

Eratosthenes of Cyrene (ca. 276 to 195 B.C.) Greek mathematician, geographer,
poet, astronomer, and music theorist. He invented a system of
latitude and longitude. He was the first person to calculate the
circumference of the earth (apparently believed the earth was a sphere,
not flat). He also proposed a simple algorithm for finding prime numbers.
This algorithm is known in mathematics as the
'Sieve of Eratosthenes'. The sieve of Eratosthenes is one of a number of
prime number sieves.

Apollonius of Perga (ca. 262 to 192 B.C.) was a Greek geometer and astronomer
known for his writings on conics.

Hypsicles (ca. 190 to 120 B.C.) was a Greek mathematician and astronomer
known for authoring 'On Ascensions' and Book XIV of Euclid's Elements.
The book continues Euclid's comparison of regular solids inscribed in spheres.

Hipparchus of Nicaea (190 to 120 B.C.) was a Greek astronomer, geographer,
and mathematician. He is considered the founder of trigonometry but is most
famous for his incidental discovery of precession of the equinoxes.
He is known to have compiled the first trigonometric table  a table of
values of the
trigonometric chord of angles. (This chord is sometimes written as 'crd'.)

Theodosius of Bithynia (ca. 160 to 100 B.C.) was a Greek astronomer and
mathematician who wrote the 'Sphaerics', a book on the geometry of the sphere.
Two other works by Theodosius have survived: 'On Habitations', describing
the appearances of the heavens at different climes, and 'On Days and Nights',
a study of the apparent motion of the Sun.

We switch to A.D. here  or as they say nowadays C.E.

Heron_of_Alexandria (ca. 10 to 70 A.D.) was a Greek mathematician
and engineer. Heron described a method of iteratively computing the
square root. Today, though, his name is most closely associated with
Heron's Formula for finding the area of a triangle from its side lengths.
The imaginary number, or imaginary unit, is also noted to have been first
observed by Hero (or Heron) while calculating the volume of a pyramidal frustum.

Claudius Ptolemy (83 to 161 A.D.) was a GrecoRoman writer of Alexandria,
known as a mathematician, astronomer, geographer, and astrologer.
He lived in the city of Alexandria in the Roman province of Egypt, wrote in
Greek, and held Roman citizenship. He believed in an Earthcentered universe
and provided a means of (roughly) predicting the motions of heavenly bodies
based on circles as their paths  or
'epicycles', where an epicycle is a
circle whose center follows a circular path. Ptolemy presented a useful
tool for astronomical calculations in his 'Handy Tables', which tabulated
all the data needed to compute the positions of the Sun, Moon and planets,
the rising and setting of the stars, and eclipses of the Sun and Moon.

Diophantus (ca. 210 to 290 A.D.), sometimes called "the father of algebra",
was an Alexandrian Greek mathematician and the author of a series of books
called Arithmetica, many of which are now lost. These texts deal with solving
algebraic equations.

Pappus of Alexandria (ca. 290 to 350 A.D.) was one of the last
great Greek mathematicians of Antiquity, known for his Synagoge or
Collection (c. 340), and for Pappus's Theorem in projective geometry.

Serenus of Antinouplis (ca. 300 to 360 A.D.) was a Greek mathematician who
came from Antinouplis, a city in Egypt founded by Hadrian. He wrote two works
entitled 'On the Section of a Cylinder' and 'On the Section of a Cone'.
In the preface of 'On the Section of a Cylinder', Serenus states that his
motivation for writing this work was that "many persons who were students
of geometry were under the erroneous that the oblique section of a cylinder
was different from the oblique section of a cone known as an ellipse,
whereas it is of course the same curve." The work consists of 33 propositions.

Theon_of_Alexandria (ca. 335 to 405 A.D.) was a GrecoEgyptian
scholar and mathematician who lived in Alexandria, Egypt. He edited and
arranged Euclid's Elements and wrote commentaries on works by Euclid and
Ptolemy. His daughter
Hypatia
also won fame as a mathematician, but she was murdered by a Christian mob.

Here we enter the Dark Ages of Europe, when many Christians suspected
mathematics to be the work of the devil. For the next 1,100 years or so,
mathematical works are preserved and advanced elsewhere, notably in the
Middle East and India. Some of the ancient Alexandrian texts (that were not
burned by the Romans ca. 48 B.C.  or Christians ca. 392 A.D.  or
Muslims ca. 640 A.D.) are translated into Arabic, and, much later,
back into Greek and Latin.

Aryabhata (475 to 550? A.D.) was an Indian astronomer and mathematician.
He worked on the approximation for pi, equations, and series. He found that
'the sum of cubes is the square of the sum' : 1 + 2^3 + 3^3 + ... + n^3
= ( 1 + 2 + 3 + ... + n )^2 . (It blows my mind that this is true for all
positive integers n.)
Also, he found that the ratio of the circumference
to the diameter of a circle is 62832/20000 = 3.1416, which is accurate to
five significant figures.

Brahmagupta (598 to 665 A.D.) was an Indian mathematician and astronomer.
He wrote out a set of rules to compute with the number zero.
It took a while for people to recognize that having the number zero is quite
helpful in many ways  if only as a place holder in representations of large
numbers.

AlKhwarizmi (AlKhowarizmi, Mohammed ibn Musa) (ca. 780 to 850 A.D.)
was a Persian mathematician,
astronomer and geographer during the Abbasid Empire, a scholar in the
House of Wisdom in Baghdad. He wrote a book 'Kitab wa al jabr
wa al muqabalah' (The Book of Shifting and Balancing) which described
six types of
quadratic equation and solved them in a methodical manner.
His works introduced the decimal positional number system to the Western
world. The words 'al jabr' are the source of the English word 'algebra'.
A very influential person.

Albtegnius alBattani (ca. 858 to 929 A.D.) was an Arab Muslim
astronomer, astrologer, and mathematician. He introduced a number
of trigonometric relations, and he authored a book of astronomical tables
that was frequently quoted by many medieval astronomers, including Copernicus.
One of his bestknown achievements was the determination of the solar year
as being 365 days, 5 hours, 46 minutes and 24 seconds.

Habash alHasib (Ahmed ibn Abdallah alMervazi) (ca. 870 A.D.)
was a Persian astronomer, geographer, and mathematician. He flourished
in Baghdad and died in Samarra, Iraq. In 830, using the notion of "shadow"
 equivalent to our tangent in trigonometry  he compiled a table of
such shadows which seems to be the earliest of its kind. He also
introduced the cotangent, and produced the first tables for it.
He provided estimates of the circumference and diameter of the earth,
moon, and sun  and the minmax distances between the earth and the moon
and the distance between the earth and the sun.

Abul alWafa (940 to 998 A.D.) was a Persian mathematician and
astronomer who worked in Baghdad. He made important innovations in
spherical trigonometry, and his work on arithmetics for businessmen
contains the first instance of using negative numbers in a medieval
Islamic text. He is also credited with compiling the tables of sines
and tangents at 15' intervals. He also introduced the sec and cosec functions.
He established several trigonometric identities such as sin(a ± b) in
their modern form, where the Ancient Greek mathematicians had expressed
the equivalent identities in terms of chords. He also discovered the
law of sines for spherical triangles: sin A/sin a = sin B/sin b = sin C/sin c
where A, B, C are the sides and a, b, c are the opposing angles.

Omar Khayyam (1048 to 1131 A.D.) was a Persian philosopher,
mathematician, astronomer and poet. He also wrote treatises on
mechanics, geography, mineralogy, music, and Islamic theology. He
provided an algebraic method of solving certain
cubic equations, while others, such as Menaechmus, provided
geometric methods.

Gherardo of Cremona (1114? to 1187) was an Italianborn translator
of scientific books from Arabic into Latin. He worked in Toledo, Spain
and obtained the Arabic books in the libraries at Toledo. Some of the
books had been originally written in Greek and were unavailable in
Greek or Latin in Europe at the time. Gerard of Cremona is the most
important translator among the Toledo School of Translators who
invigorated medieval Europe in the twelfth century by transmitting
medieval Arabic and ancient Greek knowledge in astronomy, medicine
and other sciences, by making the knowledge available in the Latin
language. One of Gerard's most famous translations is of Ptolemy's
Almagest from Arabic texts found in Toledo.

Leonardo of Pisa (Leonardo Fibonacci) (1170 to 1250 A.D.) was an
Italian mathematician. Fibonacci is best known to the modern
world for the spreading of the HinduArabic numeral system in Europe,
primarily through his composition in 1202 of 'Liber Abaci' (Book of Calculation),
and for a number sequence named the Fibonacci numbers after him, which
he did not discover but used as an example in the 'Liber Abaci'.

Nicole Oresme (13201325? to 1382) was a French theologian.
He wrote influential works on economics, mathematics, physics,
astrology and astronomy, philosophy, and theology. He was Bishop of Lisieux,
a translator, and a counselor of King Charles V of France.
He is known for developing the first proof of the divergence of the
harmonic series  1 + 1/2 + 1/3 + 1/4 + ... Several centuries later,
the Bernoulli brothers did not know of his proof and came up with their own.

Ulugh Begh (Mirza Mohammad Taraghay bin Shahrukh) of Samarkand
(currently in Uzbekistan)
(1393 to 1449) was a Timurid (Middle East) ruler as well as an
astronomer, mathematician and sultan. Ulugh Beg was notable for his
work in astronomyrelated mathematics, such as trigonometry and
spherical geometry. He built the great Ulugh Beg Observatory in
Samarkand between 1424 and 1429. It was considered by scholars to
have been one of the finest observatories in the Islamic world at
the time and the largest in Central Asia. He wrote accurate
trigonometric tables of sine and tangent values correct to at least
eight decimal places.

George of Trebizond (1396 to 1486) was a Greek philosopher and scholar,
who migrated to Italy and became one of the pioneers of the Renaissance.
His numerous works consisted of translations from Greek into Latin and
original essays in Greek. He wrote a commentary on Ptolemy's Almagest.

Nicolaus Cusanus (1401 to 1464) was a German philosopher, theologian,
jurist, and astronomer. Most of Nicholas of Cusa's mathematical ideas
can be found in his essays, 'De Docta Ignorantia' (Of Learned Ignorance),
'De Visione Dei' (On the Vision of God) and 'On Conjectures'. He also wrote
on squaring the circle in his mathematical treatises. His astronomical views
evince complete independence of traditional doctrines. The earth is a star
like other stars, is not the centre of the universe, is not at rest,
nor are its poles fixed. The celestial bodies are not strictly spherical,
nor are their orbits circular. Had Copernicus been aware of these assertions,
he would probably have been encouraged by them to publish his own monumental work.

Leone Battista Alberti (1404 to 1472 A.D.), an Italian architect and artist,
raised the ideas of vanishing point (a term coined later) and perspective
in his treatise, 'Della Pittura' ('On Painting'). It is claimed that that
treatise contained the first scientific study of perspective.

Georg von Peurbach (also Purbach, Peurbach, Purbachius,
his real surname is unknown) (1423 to 1461) was an Austrian astronomer,
mathematician, and instrument maker.
Peurbach has been called the father of mathematical and observational
astronomy in the West. He began to work up Ptolemy's Almagest and
Alhazen's 'On the Configuration of the World'. He replaced Ptolemy's
chords with the sines from Arabic mathematics, and calculated tables
of sines for every minute of arc for a radius of 600,000 units. This was
the first transition from the duodecimal to the decimal system.
Perubach taught in Vienna. His most famous pupil was
Johann Müller of Königsberg, later known as Regiomontanus. Upon the death
of Peurbach, Regiomontanus finished several of the works started by Purbach.

Johann Muller of Konigsberg (later known as Regiomontanus) (1436 to 1476)
was a German mathematician, astronomer, astrologer, translator, instrument
maker and Catholic bishop. His work on arithmetic and algebra,
'Algorithmus Demonstratus', was among the first containing symbolic algebra.
In 'Epytoma in almagesti Ptolemei', he critiqued the translation of
Almagest by George of Trebizond, pointing out inaccuracies. Later
Nicolaus Copernicus would refer to this book as an influence on his own work.
He founded the world's first scientific printing press and in 1472 he
published the first printed astronomical textbook, the 'Theoricae novae
Planetarum' of his teacher Georg von Peurbach.
A prolific author, Regiomontanus was internationally famous in his lifetime.
Despite having completed only a quarter of what he had intended to write,
he left a substantial body of work.

Christopher_Columbus (1451 to 1506) was an Italian explorer,
navigator, and colonizer, born in the Republic of Genoa, in today's
northwestern Italy. His dates are placed here mainly to compare with the
dates of mathematicians listed here, thus giving some sense of the state
of mathematics and astronomy at the time of Columbus's voyages.
Columbus learned Latin, as well as Portuguese and Castilian,
and read widely about astronomy, geography, and history, including the
works of Claudius Ptolemy.

Scipione del Ferro (1465 to 1526) was an Italian mathematician who
provided an algebraic solution to cubic equations of the form x^3 + c x = d
(the 'depressed cubic equation').

Albrecht Durer (1471 to 1528) was a German painter, engraver,
printmaker, mathematician, and theorist from Nuremberg. His highquality
woodcuts (nowadays often called Meisterstiche or "master prints")
established his reputation and influence across Europe when he was
still in his twenties, and he has been conventionally regarded as
the greatest artist of the Northern Renaissance ever since.
Dürer's work on geometry is called the 'Four Books on Measurement'
('Underweysung der Messung mit dem Zirckel und Richtscheyt' or
'Instructions for Measuring with Compass and Ruler'). The first book
focuses on linear geometry. Dürer's geometric constructions include
helices, conchoids and epicycloids. He also draws on Apollonius,
and Johannes Werner's work on conics of 1522. The second book moves
onto the construction of regular polygons. The third book applies
these principles of geometry to architecture, engineering and typography.
The fourth book completes the progression of the first and second by
moving to threedimensional forms and the construction of polyhedra.
Here Dürer discusses the five Platonic solids, as well as seven
Archimedean semiregular solids, as well as several of his own invention.
In all these, Dürer shows the objects as
nets. Finally, Dürer discusses
the 'construzione legittima', a method of depicting a cube in two
dimensions through linear perspective. It was in Bologna that Dürer
was taught (possibly by
Luca Pacioli or
Bramante) the principles of
linear perspective, and evidently became familiar with the
'costruzione legittima' in a written description of these principles
found only, at this time, in the unpublished treatise of
Piero della Francesca. He was also familiar with the
'abbreviated construction' as described by
Alberti and the
geometrical construction of shadows, a technique of
Leonardo da Vinci.
Although Dürer made no innovations in these areas, he is notable as
the first Northern European to treat matters of visual representation
in a scientific way, and with understanding of Euclidean principles.
In addition to these geometrical constructions, Dürer discusses in
this last book an assortment of mechanisms for drawing in perspective
from models and provides woodcut illustrations of these methods that
are often reproduced in discussions of perspective.

Nicolas Copernicus (1473 to 1543) was a Polish astronomer and mathematician
who provided much evidence that a
suncentered universe explained many phenomena that were not explained
well by an earthcentered universe. His theory was made public in
his book 'De Revolutionibus Orbim Coelestium' ('On the Revolutions of
the Heavenly Orbs'), which is often credited as being the beginning of the
socalled 'Scientific Revolution'. Some claim that the book was not published
until he was near death because he knew it could be dangerously controversial
 as Galileo was to find out. (It seems it took Newton to raise
mathematics out of the Dark Ages.)

Ferdinand Magellan (ca. 1480 to 1521) was a Portuguese explorer.
Magellan's expedition of 15191522 became the first expedition to sail
from the Atlantic Ocean into the Pacific Ocean, and the first to cross
the Pacific. His expedition completed the first circumnavigation of
the Earth, although Magellan himself did not complete the entire voyage,
being killed during the Battle of Mactan in the Philippines.
His dates are placed here mainly to compare with the
dates of mathematicians listed here, thus giving some sense of the state
of mathematics and astronomy at the time of Magellan's voyages.
Magellan no doubt used astronomical (navigation) information from
Ptolomy or Copernicus or Arabian astronomersandcartographers, either
directly from translations of their works or indirectly via 'commentaries'
or reworking of those works.

Michael Stifel (1487 to 1567) was a German monk and mathematician.
He was an Augustinian who became an early supporter of Martin Luther.
Stifel was later appointed professor of mathematics at Jena University.
Stifel's most important work, 'Arithmetica integra' (1544), contained
important innovations in mathematical notation. It has the first use
of multiplication by juxtaposition (with no symbol between the terms)
in Europe. He is the first to use the term "exponent". The book contains
a table of integers and powers of 2 that some have considered to be an
early version of a logarithmic table. Further topics dealt with in
the 'Arithmetica integra' are negative numbers (which Stifel calls
'numeri absurdi') and sequences.

Francesco Maurolico (1494 to 1575) was a Greek mathematician
and astronomer of Sicily. The proof by 'mathematical induction'
had been used implicitly for centuries by Greek, Indian, and Arab
mathematicians, but Maurolico first stated it formally. Basically:
If S is a statement about integers:
 Show that S is true for n=1.
 Show that if S is true for n=k, then it is also true
for n=k+1.
 Then one can declare that S is true for all positive integers.
Throughout his lifetime, Maurolico made contributions to the fields
of geometry, optics, conics, mechanics, music, and astronomy. He edited
the works of classical authors including Archimedes, Apollonius, Autolycus,
Theodosius and Serenus. He also composed his own unique treatises on
mathematics.

Niccolo Tartaglia (1500 to 1557) was an Italian mathematician,
engineer (designing fortifications), a surveyor, and a bookkeeper
from the then Republic of Venice. He provided an algebraic solution to
cubic equations of the form x^3 + b x^2 = d. Tartaglia is also known
for having given an expression (Tartaglia's formula) for the volume
of a tetrahedron (including any irregular tetrahedra). This is a
generalization of Heron's formula for the area of a triangle.

Gerolamo Cardano (1501 to 1576) was an Italian Renaissance
mathematician, physician, astrologer and gambler. He wrote more than
200 works on medicine, mathematics, physics, philosophy, religion,
and music. His gambling led him to formulate elementary rules in
probability, making him one of the founders of the field. He was a
teacher of Lodovico Ferrari (see below).

Pedro Nunes (1502 to 1578) was a Portuguese mathematician,
cosmographer, and professor. Nunes, considered to be one of the
greatest mathematicians of his time, is best known for his
contributions in the technical field of navigation, which was
crucial to the Portuguese period of discoveries. He was the
first to understand why a ship maintaining a steady course would
not travel along a great circle, the shortest path between two
points on Earth, but would instead follow a spiral course, called
a loxodrome. The later invention of logarithms allowed Leibniz
to establish algebraic equations for the loxodrome. In his
'Treaty defending the sea chart', Nunes argued that a nautical
chart should have its parallels and meridians shown as straight
lines. Yet he was unsure how to solve the problems that this caused:
a situation that lasted until Mercator developed the projection
bearing his name. The Mercator Projection is the system which
is still used. Nunes also solved the problem of finding the day
with the shortest twilight duration, for any given position,
and its duration. This problem per se is not greatly important,
yet it shows the geometric genius of Nunes as it was a problem
which was independently tackled by Johann and Jakob Bernoulli
more than a century later with less success. They could find a
solution to the problem of the shortest day, but failed to
determine its duration, possibly because they got lost in
the details of differential calculus which, at that time,
had only recently been developed. The achievement also shows
that Nunes was a pioneer in solving maxima and minima problems,
which became a common requirement only in the next century
using differential calculus.

Gemma Frisius (1508 to 1555) was a physician, mathematician,
cartographer, philosopher, and instrument maker. He created
important globes, improved the mathematical instruments of
his day and applied mathematics in new ways to surveying and navigation.
In 1533, he described for the first time the method of triangulation
still used today in surveying. Twenty years later, he was the first
to describe how an accurate clock could be used to determine longitude.
His students included Gerardus Mercator (who became his collaborator),
Johannes Stadius, John Dee, Andreas Vesalius and Rembert Dodoens.

Gerhardus Mercator (1512 to 1594) was a 'Belgian/German' cartographer,
philosopher and mathematician. He is best known for his work in
cartography, in particular the world map of 1569 based on a new
projection which represented sailing courses of constant bearing
as straight lines. He was the first to use the term 'Atlas' for
a collection of maps.

Georg Joachim Rhaeticus (1514 to 1576) was a mathematician,
cartographer, navigationalinstrument maker, medical practitioner,
and teacher from the Austrian area. He is perhaps best known for
his trigonometric tables and as Nicolaus Copernicus's sole pupil.
He facilitated the publication of Copernicus's 'De revolutionibus
orbium coelestium' ('On the Revolutions of the Heavenly Spheres').
Rheticus produced the first publication of sixfunction
trigonometric tables (although the word trigonometry was not yet
coined). A student,
Valentin Otto,
oversaw the hand computation of approximately 100,000 ratios
to at least ten decimal places.
When completed in 1596, the volume, 'Opus palatinum de triangulus',
filled nearly 1,500 pages. Its tables were accurate enough to be
used in astronomical computation into the early twentieth century.

Peter_Ramus (or Pierre de la Ramée) (1515 to 1572) was an
influential French humanist, logician, and educational reformer.
A Protestant convert, he was killed during the
St. Bartholomew's Day Massacre. He was also known as a
mathematician, a student of
Johannes Sturm.
He had students of his own. He corresponded with
John Dee on mathematics, and at one point recommended
to Elizabeth I that she appoint Dee to a university chair.
His emphasis on technological applications and engineering mathematics
was coupled to an appeal to nationalism (France was well behind Italy,
and needed to catch up with Germany).

Lodovico Ferrari (1522 to 1565) was an Italian mathematician.
Ferrari aided Cardano on his solutions for quadratic equations and
cubic equations, and was mainly responsible for the solution of
quartic equations that Cardano published.
(Ferrari died of white arsenic poisoning, allegedly murdered by
his greedy sister. Another interesting death.)

Abraham Ortelius (1527 to 1598) was a Flemish cartographer and
geographer, generally recognized as the creator of the first modern
atlas, the 'Theatrum Orbis Terrarum' ('Theatre of the World').
He is also believed to be the first person to imagine that the
continents were joined together before drifting to their present positions.

Rafael Bombelli (1530? to 1573) was an Italian mathematician.
Born in Bologna, he is a central figure in the understanding of
imaginary numbers (a name used later). Bombelli had the foresight
to see that imaginary numbers were crucial and necessary to
solving quartic and cubic equations. Bombelli explained arithmetic
with complex numbers (a name used later). He was careful to
point out that real parts add to real parts, and imaginary parts
add to imaginary parts. Bombelli felt that none of the works on
algebra by the leading mathematicians of his day provided a careful
and thorough exposition of the subject. Instead of another
convoluted treatise that only mathematicians could comprehend,
Rafael decided to write a book on algebra that could be understood
by anyone. His text would be selfcontained and easily read by
those without higher education. The book that he wrote in 1572,
was entitled 'L'Algebra'. Another contribution: Bombelli used a
method related to continued fractions to calculate square roots.

Cornelius Gemma (1535 to 1577?) was a physician, astronomer
and astrologer, and the oldest son of cartographer and
instrumentmaker Gemma Frisius (see above). As an astronomer,
Gemma is significant for his observations of a lunar eclipse
in 1569 and of the 1572 supernova appearing in Cassiopeia.
His predictions for 1561 provided detailed information on
every lunar phase, and most planetary aspects and phases of
fixed stars in relation to the sun, with a thoroughness that
surpassed the predictions of his contemporaries. In his medical
writings, in 1552, he published the first illustration of a
human tapeworm. He remained committed to astrologic medicine,
however, and believed that astral conjunctions generated disease.
Gemma died around 1578 in an epidemic of the plague, to which
a third of the population at Leuven also succumbed.
He was only in his midforties.

Clavius (1538 to 1612) was a German Jesuit mathematician and
astronomer who modified the proposal of the modern Gregorian calendar
after the death of its primary author, Luigi Lilio. His math works
included a 'Commentary on Euclid' (1574), 'Geometrica Practica' (1604),
and 'Algebra' (1608). His mathematical works (in 5 volumes) are
available online.

Francois Viete (Latin: Franciscus Vieta) (1540 to 1603) was
a French mathematician whose work in algebraic equations was an important step
towards modern algebra, due to its innovative use of letters as
parameters in equations. He is also known for discovering the first
infinite product in the history of mathematics  a product which was
is equal to pi.

Ludolf van Ceulen (1540 to 1610) was a GermanDutch mathematician
who taught fencing and mathematics. Ludolph spent a major
part of his life calculating the numerical value of the mathematical
constant pi, using essentially the same methods as those employed by
Archimedes some seventeen hundred years earlier. He published a
20decimal value in 1596, later expanding this to 35 decimals.

Tycho Brahe (1546 to 1601) was a Danish nobleman known for his
accurate and comprehensive astronomical and planetary observations.
Tycho was well known in his lifetime as an astronomer and alchemist.
He refuted the Aristotelian belief in an unchanging celestial realm.
His precise measurements indicated that "new stars," (stellae novae,
now known as supernovae) lacked the parallax expected in sublunar
phenomena, and were therefore not "atmospheric" tailless comets as
previously believed, but were above the atmosphere and moon.
Using similar measurements he showed that comets were also not
atmospheric phenomena, as previously thought. But he did not believe
the earth could orbit the sun, because he could not detect any
parallax effect in observing the stars. He was the last of
the major naked eye astronomers, working without telescopes for
his observations.
Tycho worked to combine the geocentric Copernican system with the
heliocentric Ptolemaic system into his own model of the universe.
In his "geoheliocentric" system, the sun, moon, and stars circle
a central Earth, while the five planets orbit the Sun. Although
Tycho's planetary model was discredited within the next 100 to 150
years, his astronomical observations were an essential contribution
to the scientific revolution.

John Napier (1550 to 1617) was a Scottish landowner,
mathematician, physicist, and astronomer. He used logarithms (exponents)
to reduce multiplication and division to addition and subtraction.
The process of multiplying two numbers:
 Look up their logarithms in a logarithm table.
 Add the two logarithms.
 Look up the antilogarithm of that logarithm in an antilogarithm table.

Jobst Burgi (1552 to 1632) was a Swiss clockmaker, a maker
of astronomical instruments, and a mathematician. Among his
major inventions were the
crossbeat escapement, and the
remontoire, two mechanisms which improved the accuracy of
mechanical clocks of the time by orders of magnitude.
This allowed clocks to be used, for the first time, as scientific
instruments, with enough accuracy to time the passing of stars
(and other heavenly bodies) in the crosshairs of telescopes to
start accurately charting stellar positions.
Besides clocks, he also made mechanized celestial globes, and
he made sextants for Kepler (see below). He invented logarithms
independently of John Napier, since his method is distinct
from Napier's. There is evidence that Bürgi arrived at his
invention as early as 1588, six years before Napier began
work on the same idea. By delaying the publication of his
work to 1620, Bürgi lost his claim for priority in historic
discovery. Bürgi was also a major contributor to
prosthaphaeresis, a technique for computing products
quickly using trigonometric identities, which predated logarithms.

Luca Valerio (1553 to 1618) was an Italian mathematician.
He developed ways to find volumes and centers of gravity of
solid bodies using the methods of Archimedes. He corresponded
with Galileo Galilei and was a member of the Accademia dei Lincei.

Edward Wright (1558 to 1615) was an English mathematician and
cartographer noted for his book 'Certaine Errors in Navigation' (1599),
which for the first time explained the mathematical basis of the
Mercator projection, and set out a reference table giving the
linear scale multiplication factor as a function of latitude,
calculated for each minute of arc up to a latitude of 75°.
This was in fact a table of values of the integral of the secant
function, and was the essential step needed to make practical
both the making and the navigational use of Mercator charts.
(Mercator had not explained his method.)
A skilled designer of mathematical instruments, Wright made
models of an
astrolabe and a
pantograph,
and a type of
armillary sphere. In the 1610 edition
of 'Certaine Errors', he described inventions such as the
"searing" that enabled mariners to determine the magnetic
variation of the compass, the sun's altitude and the time
of day in any place if the latitude was known; and a device
for finding latitude when one was not on the meridian using
the height of the pole star.
Wright translated John Napier's pioneering 1614 work which
introduced the idea of logarithms from Latin into English.
Wright's work influenced, among other persons, Dutch astronomer
and mathematician
Willebrord Snellius;
Adriaan Metius, the
geometer and astronomer from Holland; and the English mathematician
Richard Norwood, who calculated the length of
a degree on a great circle of the earth (367,196 feet ;
111,921 m) using a method proposed by Wright.
John Collins, in "Navigation by the Mariners
Plain Scale New Plain'd" (1659), stated that Mercator's chart
ought "more properly to be called Wright's chart".

Thomas Harriot (1560 to 1621) was an English astronomer,
mathematician, ethnographer, and translator.
After his graduation from Oxford in 1580, Harriot was first
hired by Sir Walter Raleigh as a mathematics tutor. He used
his knowledge of astronomy/astrology to provide navigational
expertise, help design Raleigh's ships, and serve as his
accountant. Prior to his expedition to the Americas with
Raleigh (to Roanoke Island, 1585), Harriot wrote
a treatise on navigation. In addition, he made efforts
to communicate with Manteo and Wanchese, two Native Americans
who had been brought to England. Harriot deciphered a phonetic
alphabet to transcribe their Carolina Algonquian language.
As a scientific adviser during the voyage, Harriot was asked
by Raleigh to find the most efficient way to stack cannon balls
on the deck of the ship. His ensuing theory about the
closepacking of spheres shows a striking resemblance to
atomism and modern atomic theory, which he was later accused of
believing. His correspondence about optics with Johannes Kepler,
in which he described some of his ideas on spherestacking,
later influenced
Kepler's conjecture on 'spherepacking'.
Halley's Comet in 1607 turned Harriot's attention towards astronomy.
In early 1609 he bought a "Dutch trunke" (telescope), invented
in 1608, and his observations were amongst the first uses of
a telescope for astronomy. Harriot is now credited as the
first astronomer to draw an astronomical object after
viewing it through a telescope. He drew a map of the Moon on
July 26, 1609, preceding Galileo by several months.
The observatory in the campus of the College of William and
Mary (Williamsburg, Virginia) is named in Harriot's honour.
Harriot apparently died of a cancer that started on his lip.
Harriot's accomplishments remain relatively obscure because
he did not publish any of his results and also because many of
his manuscripts have been lost. Those that survive are sheltered
in the British Museum and in the archives of the Percy family
at Petworth House (Sussex) and Alnwick Castle (Northumberland).

Thomas Fincke (1561 to 1646) was a Danish mathematician
and physicist, and a professor at the University of Copenhagen
for more than 60 years. His lasting achievement is found in
his book 'Geometria rotundi' (1583), in which he introduced the
modern names of the trigonometric functions tangent and secant.

Adriaan_van_Roomen (1561 to 1615) was a Flemish mathematician.
He met Kepler (see below), and discussed with François Viète two questions about
equations and tangencies. He then spent some time in Italy, particularly
with Clavius in Rome in 1585. After 1610 he tutored mathematics in Poland.
He worked in algebra, trigonometry and geometry; and on the decimal
expansion of pi.

Henry Briggs (1561 to 1630) was an English mathematician notable for
changing the original logarithms invented by John Napier into common (base 10)
logarithms. He published Napier's logarithms as simplified tables
and helped them gain acceptance among the scientific and academic communities.

Bartholomaeus Pitiscus (1561 to 1613) was a 16thcentury German
trigonometrist, astronomer and theologian who first coined the word
Trigonometry. Pitiscus is sometimes credited with inventing the
decimal point, the symbol separating integers from decimal fractions,
which appears in his trigonometrical tables and was subsequently
accepted by John Napier in his logarithmic papers (1614 and 1619).
Pitiscus edited 'Thesaurus mathematicus' (1613) in which he improved
the trigonometric tables of Georg Joachim Rheticus.

Galileo Galilei (1564 to 1642) was an Italian physicist, mathematician,
and astronomer. He provided rules for how a body falls under
gravity. For example: If a body is dropped from rest, the distance it falls
varies with the square of the time. Other discoveries by Galileo :
 The first thermometer.
 Used the telescope to discover the moons of Jupiter.
 Used the telescope to show that the Moon has mountains and
valleys.
 A pendulum swings to and fro in the same time, regardless of
the angle through which it swings. (This is about as counterintuitive
as the the fact that a heavy body and a light body fall at the
same speed. Aristotle believed the latter  but, unlike Galileo,
Aristotle was loathe to put his 'logic' to experimental test.)
Also Galileo was a leading supporter of Copernicus's Suncentered system
 which led to his spending his last years under house arrest due to
the influence of the Catholic Church.

Johannes Kepler (1571 to 1630) was a German mathematician,
astronomer and astrologer. Using copious data on planetary movements, he
provided three laws about the motion of planets. The first and most revolutionary:
Planets travel in ellipses rather than circles. Isaac Newton was later able
to deduce the 3 Kepler laws based on his laws of motion and of gravity.
Also, Kepler showed that the square, triangle and hexagon are the only regular
polygons that can be used to cover a plane without leaving any gaps. This is
known as a
'tesselation' or 'tiling' of the plane.

William Oughtred (1574 to 1660) was an English mathematician.
After John Napier invented logarithms, and Edmund Gunter created
the logarithmic scales (lines, or rules) upon which slide rules
are based, it was Oughtred who first used two such scales sliding
by one another to perform direct multiplication and division;
and he is credited as the inventor of the slide rule in 1622.
Gunter required the use of a pair of dividers, to lay off distances
on his rule; Oughtred made the step of sliding two rules past each
other to achieve the same ends.
Oughtred also introduced the "×" symbol for multiplication as
well as the abbreviations "sin" and "cos" for the sine and
cosine functions.
Oughtred published 'Clavis Mathematicae' (The Key to Mathematics)
in 1631. It became a classic, reprinted in several editions, and
used by Wallis and Isaac Newton (see below) amongst others.
It was not ambitious in scope, but aimed to represent current
knowledge of algebra concisely. It argued for a less verbose
style of mathematics, with a greater dependence on symbols.

Mathurin Jousse
(ca. 1575? to 1645) was a French inventor
and a technician in iron works. He was also said to be an
"engineer and architect of the town of de La Flèche". It is
said that he carried out several repair/construction jobs of,
large and small, for the town and the Jesuit college there,
such as repair of the large clock on the SaintThomas bell tower.
Jousse was curious about science and techniques. He possessed scientific
instruments, some he made himself, and a rich library in which
there were many books of arithmetic, geometry and astronomy.
His treatise "Secret d'architecture", entirely devoted to
stereotomy, appeared in La Flèche in 1642. It is said that
he knew Francois Derand (see below), a student at the Jesuit college in
La Fleche who also wrote a text on stereotomy.
 Bartholomäus Souvey, a.k.a. Bartolomeo Sovero (ca.1577 to 1629)
was a Swiss mathematician whose life and works are not well
documented. He is mentioned on page 47 of the Hofmann
history book (see top of this page), in relation to studies
of a logarithmic series and studies of the hyperbola.

Johann Faulhaber (1580 to 1635) was a German mathematician.
Born in Ulm, Faulhaber trained as a weaver and later took the role of
a surveyor of the city of Ulm. He collaborated with Johannes Kepler and
Ludolph van Ceulen (see above). Besides his work on the fortifications of cities
(notably Basel and Frankfurt), Faulhaber built water wheels in his
home town and geometrical instruments for the military. Faulhaber
supervised the first publication of Henry Briggs's logarithms in Germany.
Faulhaber's major math contribution involved calculating the sums of powers
of integers. Jacob Bernoulli makes references to Faulhaber in his
'Ars Conjectandi'. Faulhaber made a major impression on Descartes
and influenced his thinking. In 1631, Faulhaber published 'Academia Algebra'.

Edmund Gunter (1580 to 1626) was an English mathematician, of Welsh descent.
In 1619, Gunter was appointed professor of astronomy in Gresham College, London.
This post he held till his death. In 1620, he published his 'Canon triangulorum'.
With Gunter's name are associated several useful inventions, descriptions of which
are given in his treatises on the Sector, Crossstaff, Bow, Quadrant and other
instruments. In 1624, Gunter published a collection of his mathematical works.
It was entitled 'The description and use of sector, the crossstaffe, and other
instruments for such as are studious of mathematical practise'. It was a manual not
for cloistered university fellows but for sailors and surveyors in real world.
It was written, and published, in English not Latin.

Willebrord Snellius van Roijen 'Snell' (1581 to 1626) was a Dutch astronomer
and mathematician. In the west, especially the English speaking countries,
his name has been attached to the law of refraction of light for several
centuries, but it is now known that this law was first discovered by
Ibn Sahl in 984. The same law was also investigated by Ptolemy and
in the Middle Ages by Witelo, but due to lack of adequate mathematical
instruments (trigonometric functions) their results were saved as
tables, not functions. In 1615, he planned and carried into practice
a new method of finding the radius of the earth, by determining
the distance of one point on its surface from the parallel of latitude
of another, by means of triangulation. His work 'Eratosthenes Batavus'
("The Dutch Eratosthenes"), published in 1617, describes the method.
Snellius also produced a new method for calculating pi  the first
such improvement since ancient times.

Artus de Lionne (1583 to 1663)

Gregoire de SaintVincent (1584 to 1667) was a Flemish Jesuit and
mathematician. SaintVincent discovered that the area under a rectangular
hyperbola ( i.e. a curve given by xy = k ) is the same over [a,b] as
over [c,d] when a/b = c/d. This discovery was fundamental for
the development of the theory of logarithms and an eventual recognition
of the natural logarithm (whose series representation was discovered by
Nicholas Mercator (not to be confused with Gerardus Mercator the cartographer),
but was only later recognized as a log of base e).
The stated property allows one to define a function A(x) which is the
area under said curve from 1 to x, which has the property that
A(xy) = A(x)+A(y). Since this functional property characterizes
logarithms, it has become mathematical fashion to call such a
function A(x) a logarithm. In particular when we choose the rectangular
hyperbola xy = 1, one recovers the natural logarithm.
To a large extent, recognition of de SaintVincent's achievement in
quadrature of the hyperbola is due to his student and coworker
Alphonse Antonio de Sarasa, with
Marin Mersenne (see below) acting as catalyst.
It was in attempting to
'square the circle' that SaintVincent made
these discoveries.

Isaac Beeckman (1588 to 1637) was a Dutch philosopher and scientist.
Rejecting Aristotle, Beeckman developed the concept that matter is
composed of atoms. In 1618, he became a teacher and friend of René Descartes.
He convinced Descartes to devote his studies to a mathematical approach to
nature. When Descartes returned to the Dutch Republic in the autumn of
1628, Beeckman also introduced him to many of Galileo's ideas. In his time,
Beeckman was considered to be one of the most educated men in Europe.
For example, he had deeply impressed Mersenne, despite their opposing
religious views.

Marin Mersenne (1588 to 1648), a French priest, conjectured that all
numbers of the form 2^n  1, where n is prime, are prime. It turns out that
2^11  1 is 23 x 89, and hence not prime. If it were true, we would have a
simple formula for finding infinitely many primes.
For n = 2,3,5,7, the Mersenne numbers are 3, 7, 31, 127  all prime.
Note that Euclid (ca. 325265 B.C.) showed that there are infinitely many
primes, but he did not find a rule for generating a succession of them.

Thomas Hobbes (1588 to 1679) was an English philosopher, best known
today for his work on political philosophy. His mathematics contributions
are not very significant, but he got into a long dispute with John Wallis,
who was second only to Isaac Newton as the leading English mathematician
of their age.

Richard Norwood (1590 to 1665) was an English mathematician, diver,
and surveyor. In 1616, he was sent to survey the islands of Bermuda
(also known as the Somers Isles). He was (around 1630 to 1640) a teacher
of mathematics in London. Between 1633 and 1635, he personally measured,
partly by chain and partly by pacing, the distance between London and
York, making corrections for all the windings of the way, as well as for
the ascents and descents. He also, from observations of the sun's altitude,
computed the difference of latitude of the two places, and so calculated
the length of a degree of the meridian. His result was some 600 yards
too great; but it was the nearest approximation that had then been made
in England. (See Edward Wright above.) Isaac Newton noted Norwood's work
in his 'Principia Mathematica'. Norwood is credited with founding Bermuda's
oldest school, Warwick Academy, in 1662. He died at Bermuda in 1675,
aged about eightyfive, and was buried there. He has been called
"Bermuda's outstanding genius of the seventeenth century" (in spite of
some alleged collusion with the governor during the 1616 survey)  probably
by Bermudans eager for a noted Bermudan.
 H. Bond (ca. 1649) A conjecture by Bond is mentioned in
a work of R. Norwood, 1657. (Ref: Page 47 of Hofmann.)

Francois Derand (ca. 1590 to 1644) was a French Jesuit architect.
In 1643, he published "L'architecture des voûtes", a treatise on
stereotomy (cutting and assembling blocks into complex structures)
that is considered his masterwork.

Girard Desargues (1591 to 1661) was a French mathematician and engineer,
and is considered one of the founders of projective geometry.
Desargues' theorem and the
Desargues graph are named in his honor.

Pierre Gassendi (1592 to 1655) was a French philosopher, priest,
scientist, astronomer, and mathematician. In 1645, he accepted the
chair of mathematics in the Collège Royal in Paris, and lectured
for several years. In 1648, illhealth compelled him to give up
his lectures at the Collège Royal. He had published criticisms
of Decartes' views, but around this time he became reconciled
to Descartes, after years of coldness. In his dispute with Descartes,
Gassendi apparently held that the evidence of the senses remains the
only convincing evidence; yet he maintains, as is natural from
his mathematical training, that the evidence of reason is
absolutely satisfactory.

Wilhelm Schickard (1592 to 1635), a German astrologer, designed a machine
that could perform the four basic arithmetic operations  addition, subtraction,
multiplication, and division. Unfortunately, before it was completed, it was
destroyed in a fire, but sufficient details survived for the machine
to be successfully reconstructed in the 1950's.

Albert Girard (1595 to 1632) was a Frenchborn mathematician.
He gave the inductive definition for the Fibonacci numbers. He was the
first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric
functions in a treatise. Girard was the first to state, in 1632,
that each prime of the form 4n+1 was the sum of two squares. Girard also
showed how the area of a spherical triangle depends on its interior angles.
The result is called
Girard's theorem.

Jacobus_Golius (born Jacob van Gool) (1596 to 1667) was a Dutch mathematician
and a student of Arabic and Eastern languages. Golius taught mathematics to
the French philosopher René Descartes, and later corresponded with him.
It is therefore highly probable that he was able to read to him parts of
the mathematical Arabic texts he had started to collect, among others
on the Conics.

Rene Descartes (1596 to 1650), a French physicist, physiologist, and
mathematician, was the first to systematically develop the idea
of analytic (or Cartesian, or coordinate) geometry  think 'graph paper'.
To some extent, modern philosophy began with his "Cogito ergo sum"  "I think,
therefore I am."

Bonaventura Cavalieri (1598 to 1647) was an Italian mathematician.
He is known for his work on the problems of optics and motion,
work on the precursors of infinitesimal calculus, and the introduction
of logarithms to Italy. 'Cavalieri's principle' in geometry partially
anticipated integral calculus. 'Cavalieri's principle' states
that the volumes of two objects are equal if the areas of their
corresponding crosssections are in all cases equal.
The same principle had been previously discovered by Zu Gengzhi (480525)
of China. It was a significant step on the way to modern infinitesimal
calculus.

Adriaan Vlacq (1600 to 1667) was a Dutch book publisher and author
of mathematical tables. Vlacq published a table of logarithms from 1 to
100,000 to 10 decimal places in 1628 in his 'Arithmetica logarithmica'.
This table extended Henry Briggs' original tables which only covered
the values 120,000 and 90,001 to 100,000.

Forimond de Beaune (1601 to 1652) was a French jurist and mathematician,
and an early follower of René Descartes. His 'Tractatus de limitibus aequationum'
was reprinted in England in 1807. In it, he finds upper and lower bounds for
the solutions to quadratic equations and cubic equations, as simple functions
of the coefficients of these equations. His 'Doctrine de l'angle solide' and
'Inventaire de sa bibliothèque' were also reprinted, in Paris in 1975.
Another of his writings was 'Notae breves', the introduction to a 1649
edition of Descartes' 'La Géométrie'.

Pierre de Fermat (1601 to 1665) conjectured that 2^2^n + 1, where n is any
positive integer, is prime. It turns out that for n=5, the result can be factored
into 641 x 7,600,417.
For n = 0,1,2,3,4, the Fermat numbers are 3, 5, 17, 257, and 65,537  all prime.
Up to 2008, these are the only Fermat numbers that have been found to be prime.
Fermat traded letters with Pascal on a gambling problem  how stakes should be
divided in an interrupted game  a certain kind headstails game. Fermat's solution,
which was refined by Pascal, was to look to the future rather than the past. Instead
of considering the points that had been played, they looked at the points that
would be played if the game were allowed to continue. The theory of probability
is said to have begun with this gambling problem.

Pierre de Carcavy (1600 or 1603? to 1684) was French and a member of the
Académie Royal des Sciences, 16661684. He had many friends,
including Huygens, Fermat, and Pascal, and carried on an extensive
correspondence. He was instrumental in passing information among
these mathematicians.

Antoine de Laloubère (Antonius Lalovera) (1600  1664) was a
French mathematician who gained the respect of Wallis.

Jacques de Billy (1602 to 1679) was a French Jesuit mathematician.
From 1629 to 1668, he taught mathematics at several Jesuit colleges.
Billy maintained a correspondence with the mathematician Pierre de Fermat.
A couple of Billy's pupils were Jacques Ozanam and Claude Gaspard
Bachet de Méziriac. Billy produced a number of results in number theory
which have been named after him. Billy's mathematical works include
'Diophantus Redivivus'. In the field of astronomy, he published several
astronomical tables. Billy was one of the first scientists to reject
the role of astrology in science. He also rejected old notions about
the malevolent influence of comets.

Gilles Persone de Roberval (1602 to 1675), a French mathematician,
was born at Roberval, Oise, near Beauvais, France. His name was
originally Gilles Personne or Gilles Personier, that of Roberval,
by which he is known, being taken from the place of his birth.
He was appointed to the chair of philosophy at Gervais College in 1631,
and two years later to the chair of mathematics at the Royal College
of France. A condition of tenure attached to this chair was that the
holder should propose mathematical questions for solution, and should
resign in favour of any person who solved them better than himself;
but, notwithstanding this, Roberval was able to keep the chair about
40 years, till his death.
Roberval was one of those mathematicians who,
just before the invention of the differential and integral calculus,
occupied their attention with problems which are only soluble, or
can be most easily solved, by some method involving limits or
infinitesimals, which would today be solved by calculus. He worked on
the quadrature of surfaces and the cubature of solids, which he
accomplished, in some of the simpler cases, by an original method
which he called the "Method of Indivisibles"; but he lost much of
the credit of the discovery as he kept his method for his own use,
while Bonaventura Cavalieri published a similar method which he
independently invented.
Another of Roberval's discoveries was a very general method of
drawing tangents, by considering a curve as described by a moving
point whose motion is the resultant of several simpler motions.
He also discovered a method of deriving one curve from another,
by means of which finite areas can be obtained equal to the areas
between certain curves and their asymptotes. To these curves,
which were also applied to effect some quadratures, Evangelista
Torricelli gave the name "Robervallian lines."
He invented a special kind of balance, the Roberval Balance.
A Roberval balance with balanced masses.
Note that the masses don't need to be centered on the plates.

Bernard Frenicle de Bessy (1605 to 1675) was a French (amateur) mathematician,
who wrote numerous mathematical papers, mainly in number theory and
combinatorics. He is best remembered for 'Des quarrez ou tables magiques',
a treatise on magic squares published posthumously in 1693, in which
he described all 880 essentially different normal magic squares of
order 4. The
Frénicle standard form, a standard representation of
magic squares, is named after him. He solved many problems created
by Fermat and also discovered the cube property of the number 1729,
later referred to as a
taxicab number.
He corresponded with the Descartes, Huygens, Mersenne, and Fermat,
who was his personal friend.

Honore Fabri (1606? to 1688) was a French Jesuit theologian,
mathematician, and physicist. For for six years mathematics at
the Jesuit college at Lyons, attracting many pupils.

Vincentio Reinieri (1606 to 1647) was an Italian mathematician
and astronomer. He was a friend and disciple of Galileo Galilei.

Evangelista Torricelli (1608 to 1647) was an Italian physicist
and mathematician, best known for his invention of the
barometer.
After Galileo's death on 8 January 1642, Grand Duke Ferdinando II
de' Medici asked him to succeed Galileo as the grandducal
mathematician and professor of mathematics at the University of Pisa.
In this role, he solved some of the great mathematical problems
of the day, such as finding a
cycloid's area and center of gravity.
He also designed and built a number of telescopes and simple
microscopes. Several large lenses, engraved with his name,
are still preserved in Florence.
Torricelli also discovered
Torricelli's Law, regarding the speed of a fluid flowing out
of an opening, which was later shown to be a particular case of
Bernoulli's principle.
Torricelli gave the first scientific description of the cause of wind:
"... winds are produced by differences of air temperature, and
hence density, between two regions of the earth."
Torricelli is also famous for the discovery of the
Torricelli's trumpet (also  perhaps more often  known as
Gabriel's Horn) whose surface area is infinite, but whose volume
is finite. This was seen as an "incredible" paradox by many at
the time, including Torricelli himself, and prompted a fierce
controversy about the nature of infinity.
Cycloid genererated by a fixed point on a rolling circle.

Giovanni Alfonso Borelli (1608 to 1679) was an
Italian physiologist, physicist, and mathematician. He contributed
to the modern principle of scientific investigation by continuing
Galileo's custom of testing hypotheses against observation.
Trained in mathematics, Borelli also made extensive studies of
Jupiter's moons, the mechanics of animal locomotion and, in
microscopy, of the constituents of blood. He also used microscopy
to investigate the
stomatal movement of plants, and undertook
studies in medicine and geology. During his career, he enjoyed
the patronage of Queen Christina of Sweden, who had also been
exiled to Rome for converting to Catholicism.
Borelli's major scientific achievements are focused around
his investigation into biomechanics. The American Society of
Biomechanics uses the Borelli Award as its highest honour for
research in the area. Borelli is also considered to be the
first man to consider a selfcontained underwater breathing
apparatus along with his early submarine design.
Borelli also had interests in physics, specifically the
orbits of the planets. Borelli's measurements of the orbits
of satellites of Jupiter are mentioned in Volume 3 of
Newton's Principia.

Jan Mikolaj Smoguleski (1610 to 1656) was a Polish nobleman,
politician, scholar, and Jesuit missionary credited with
introducing logarithms to China. He also taught astronomy and mathematics
in China and was much respected by Chinese scholars. His fame as a
scholar and teacher spread, and in 1653 he was invited by the
Shunzhi Emperor to his court in Beijing. After further travels
in Asia, he died in China.

John Pell (1611 to 1685) was an English mathematician.
Diophantine equations was a favorite subject with Pell. He lectured
on them when he taught for a while in Amsterdam. He is now best remembered,
erroneously, for the indeterminate equation x^2  n * y^2 = 0, where
n is a nonsquare integer, which is known as Pell's equation.
Pell's equation. (See Brouncker below.) Its solutions may be used to
accurately approximate the square root of n by rational numbers of the form x/y.

Antoine Arnauld (1612 to 1694)  'le Grand' as contemporaries
called him, to distinguish him from his father  was a French Roman
Catholic theologian, philosopher, and mathematician.
He wrote a book 'New Elements' (about 1660, printed 1667) that was
intended to be a first step on the path to reform in introductory
instruction in geometry. One of his contemporaries described him as
the Euclid of the 17th century.

Claude Clersellier (1614 to 1684) was a French government
employee (with a large dowry) who admired Descartes and
published many of Descartes' papers, and translated many
into French.

Phillipe de la Hire (1614 to 1718) was a French mathematician
and astronomer. La Hire wrote on graphical methods, 1673; on
conic sections , 1685; a treatise on
epicycloids , 1694; one on
roulettes , 1702; and, lastly, another on
conchoids , 1708. His works on conic sections and epicycloids
were founded on the teaching of Desargues, of whom he was a
pupil. Not surprisingly then, la Hire did work on
mathematics of perspective geometry.
Generation of a 'roulette' curve.
Generation of a (repeating, periodic) 'epicycloid' curve.

Frans van Schooten Jr. (1615 to 1660) was a Dutch mathematician
who is most known for popularizing the analytic geometry of René Descartes.
In 1646, he published a collection of Vieta's writings.
Van Schooten's father was a professor of mathematics at Leiden,
having Christiaan Huygens, Johann van Waveren Hudde, and René de Sluze
as students. In 1646, he inherited his father's position and
one of his pupils, Huygens. Van Schooten's 1649 Latin translation of
and commentary on Descartes' 'Géométrie' was valuable in that it
made the work comprensible to the broader mathematical community,
and thus was responsible for the spread of analytic geometry to the world.
A later 2volume edition (of 1659 and 1661) was the edition that
Gottfried Leibniz and Isaac Newton knew. Van Schooten was one of
the first to suggest, in exercises published in 1657, that the
analytic geometry ideas be extended to threedimensional space.
Van Schooten's efforts also made Leiden the centre of the mathematical
community for a short period in the middle of the seventeenth century.

John Wallis (1616 to 1703) was an English mathematician
who is given partial credit for the development of infinitesimal
calculus. His works were read by Newton, and they had great influence
on Newton. Between 1643 and 1689, Wallis served as chief cryptographer
for Parliament and, later, the royal court. He was appointed in 1649
to be the Savilian Chair of Geometry at Oxford University, where he
lived until his death in 1703.
Wallis made significant contributions to algebra, trigonometry,
geometry, and the analysis of infinite series.
In 1655, Wallis published a treatise on conic sections in which
they were defined analytically. This was the earliest book in
which these curves are considered and defined as curves of the
second degree. It helped to remove some of the perceived difficulty
and obscurity of René Descartes' work on analytic geometry.
Wallis had an interesting view of the 'infinitesimal' lines that
make up a plane: In his 'Treatise on the Conic Sections', Wallis
wrote, "I suppose any plane (following the 'Geometry of Indivisibles'
of Cavalieri) to be made up of an infinite number of parallel lines,
or as I would prefer, of an infinite number of parallelograms
of the same altitude; (let the altitude of each one of these be an
infinitely small part of the whole altitude, ... and the altitude
of all to make up the altitude of the figure." This view was probably
useful in his work on quadrature (areas under curves).
In 'Arithmetica Infinitorum',published in 1656, he developed the
standard notation for powers, extending the notation (and rules of
multiplication) from positive integers to negative integers, zero,
and rational numbers. Then he went on to extend
Cavalieri's quadrature formula to rational exponents and
negative exponents (except 1). But he was unable to find the area
under the circular curve y = sqrt(1  x^2) by applying that area formula
for powers of x, because he did not have a way of writing that
square root of a binomial as a sum of powers of x. It was Newton who
later came up with a generalization of the
binomial theorem to exponents that are rational numbers
(like onehalf)  as well as exponents that are negative numbers.
Then Newton was able to devise a sum (an infinite series) for the
area under a circle (or its quadrant), and thus a series for pi.
But Wallis was able to devise an infinite product for pi 
one that was an improvement over the infinite product that was found
earlier by Vieta (see above). The
Wallis product had the advantage that it was easier to compute
because it involved only integers, whereas the Vieta product involved
square roots (an infinite number of them).

Sir Jonas Moore (1617 to 1679) was an English mathematician, surveyor,
Ordnance Officer and patron of astronomy. In later life, his wealth and
influence as Surveyor General of the Ordnance enabled him to become a
patron and principal driving force behind the establishment of
the Royal Observatory, Greenwich. By 1650 he was an established
mathematics teacher and published his first book, 'Moores Arithmetick'.
In the 1660's, he was involved in some large scale surveying projects.

Alfonso Anton de Sarasa (1618 to 1667) was a Jesuit mathematician
from Flanders (Belgium) who contributed to the understanding of
logarithms, particularly as areas under a hyperbola.

Henry Oldenberg (1618 to 1677) was born in Bremen, Germany. He trained
in theology. He came to London in 1653, as a diplomat and settled in England
where his lifelong patron was
Robert Boyle. After the
Restoration, he became an early member (original fellow) of the
Royal Society (founded in 1660), and served as its first secretary along
with John Wilkins, maintaining an extensive network of scientific contacts
through Europe. He also became the founding editor of the 'Philosophical
Transactions of the Royal Society'. Oldenburg began the practice of
sending submitted manuscripts to experts who could judge their quality
before publication. This was the beginning of both the modern scientific
journal and the practice of peer review. 'Philosophical Transactions of
the Royal Society' continues today and is the longest running scientific
journal in the world.

Michelangelo Ricci (1619 to 1682) was an Italian mathematician
and a Cardinal of the Roman Catholic Church. He studied mathematics under
Benedetto Castelli who had been a student of Galileo Galilei.
Ricci studied the maxima of functions of the form x^m * (a  x)^n and
tangents to curves with equation y^m = k * x^n, using methods that are
an early form of induction. He also studied
spirals (1644) and
cycloids (1674) and recognised that the study of tangents and the
calculation of areas are reciprocal operations (anticipating the
derivatives and antiderivatives of the soon to be developed calculus).
Ricci is also known for his correspondence with Torricelli, Vincenzo Viviani,
and Rene de Sluze (listed on this page).

Carlo Roberto de Cammillo Dati (1619 to 1676) was a Florentine nobleman,
a disciple of Galileo (15641642) and, in his youth, an acquaintance of
Evangelista Torricelli (16081647). Dati authored many scientific works,
including the "Lettera ai Filaleti della vera storia della cicloide e della
famosissima esperienza dell'argento vivo" [Letter to the Filaleti regarding
the true story of the cycloid and the wellknown sterling silver experience]
(Florence, 1663) which discussed a 1644 experiment by Torricelli.
Dati was a founder of the
Accademia del Cimento, and was Secretary of the
Accademia della Crusca.

William Brouncker (1620? to 1684) was an English mathematician who
introduced Brouncker's formula (a continued fraction for pi/4), and was the first
President of the Royal Society.
His mathematical work concerned in particular the calculations of the lengths of
the parabola and cycloid, and the quadrature of the hyperbola, which requires approximation
of the natural logarithm function by infinite series. He was the first European to
solve what is now known as
Pell's equation  x^2  n * y^2 = 1. He was the first in England to take interest in
generalized continued fractions and, following on the work of John Wallis, he provided
development in the generalized continued fraction of pi.

Nicolaus Mercator (1620? to 1687)  not to be confused with Gerardus Mercator,
the cartographer  was a mathematician born in Germany who, after 1641, lived
in the Netherlands, Denmark, France, and England. He taught mathematics in London (16581682).
In 1666 he was elected a Fellow of the Royal Society. He designed a marine chronometer for
Charles II, and designed and constructed the fountains at the Palace of Versailles (16821687).
Mathematically, he is most well known for his treatise 'Logarithmotechnica' on logarithms,
published in 1668. In this treatise, he described the
Mercator series, also independently discovered by Gregory SaintVincent (1584 to 1667),
see above.

Abbe Jean Picard (1620 to 1682) was a French astronomer and priest. He was the first
person to measure the size of the Earth to a reasonable degree of accuracy in a survey
conducted in 166970. Picard was the first to attach a telescope with crosswires (developed by
William Gascoigne) to a
quadrant, and one of the first to use a micrometer screw on his instruments.
The quadrant he used to determine the size of the Earth had a radius of 38 inches
and was graduated to quarterminutes. The
sextant he used to find the meridian had a radius
of six feet, and was equipped with a
micrometer to enable minute adjustments. These equipment
improvements made the margin of error only ten seconds, as opposed to Tycho Brahe's four
minutes of error. This made his measurements 24 times as accurate. Isaac Newton was to use
this level of accuracy in his theory of universal gravitation.
Picard collaborated and corresponded with many scientists, including Isaac Newton,
Christiaan Huygens, Ole Rømer, Rasmus Bartholin, Johann Hudde, and Giovanni Cassini.
These correspondences led to Picard's contributions to areas of science outside the field
of geodesy, such as the aberration of light, or his discovery of mercurial phosphorescence
upon his observance of the faint glowing of a barometer. This discovery led to Newton's
studies of light's visible spectrum.
Picard also developed what became the standard method for measuring the
right ascension of a celestial object.
In this method, the observer records the time at which the object
crosses the observer's
meridian. Picard made his observations using the precision pendulum
clock that Dutch physicist Christiaan Huygens had recently developed.

Claude François Milliet Dechales (1621 to 1678) was a French Jesuit priest and
mathematician. He published a translation of the works of Euclid, though of lesser
quality than that of
Gilles Personne de Roberval (16021675). Louis XIV had him appointed professor
of hydrography in Marseille where he taught navigation and military engineering.
He then moved to the Trinity College in Lyon in 1674, where he simultaneously taught
philosophy (4 years), mathematics (6 years) and theology (5 years). He published
in Lyon his 'Cursus seu Mondus Matematicus' (1674).

R. Fr. deSluse (1622 to 1685) was a Walloon (Belgian) mathematician and
churchman. Aside from mathematics he also produced works on astronomy, physics,
natural history, general history and theological subjects related to his work
in the Church. He corresponded with the mathematicians and intellectuals of the day.
His correspondents included Blaise Pascal, Christiaan Huygens, John Wallis, and
Michelangelo Ricci. Sluse contributed to the development of calculus and this
work focuses upon spirals, tangents, turning points and points of inflection.
He and Johannes Hudde found algebraic algorithms for finding tangents, minima and
maxima that were later utilized by Isaac Newton. These algorithms greatly improved
upon the complicated algebraic methods of Pierre de Fermat and René Descartes,
who themselves had improved upon Roberval's kinematic, but geometric, nonalgorithmic
methods of determining tangents. He wrote numerous tracts, and, in particular,
discussed at some length spirals and points of inflexion. The
'Conchoid of de Sluze' is named after him. He is described by John Wallis in his
'Algebra' as "a very accurate and ingenious person." Several of his works were included
in the Transactions of the Royal Society, e.g. his method of drawing tangents to
geometrical curves.

Vincenzo Viviani (1622 to 1703) was an Italian mathematician and scientist.
He was a pupil of Torricelli and a disciple of Galileo.
In 1639, at the age of 17, he was an assistant of Galileo Galilei in Arcetri.
He remained a disciple until Galileo's death in 1642. From 1655 to 1656, Viviani
edited the first edition of Galileo's collected works.
After Torricelli's 1647 death, Viviani was appointed to fill his position at
the Accademia dell'Arte del Disegno in Florence. Viviani was also one of the first
members of the Grand Duke's 'Accademia del Cimento' (Academy of Experiments), when
it was created a decade later.
In 1660, Viviani and Giovanni Alfonso Borelli conducted an experiment to determine
the speed of sound. Timing the difference between the seeing the flash and hearing
the sound of a cannon shot at a distance, they calculated a value of 350 meters
per second (m/s), considerably better than the previous value of 478 m/s obtained
by Pierre Gassendi.
Upon his death, Viviani left an almost completed work on the resistance of solids,
which was subsequently completed and published by Luigi Guido Grandi.
In 1737, the Church finally allowed Galileo to be reburied in a grave with an
elaborate monument. The monument that was created in the church of Santa Croce was
constructed with the help of funds left by Viviani for that specific purpose.
Viviani's own remains were moved to Galileo's new grave as well. (Another interesting
mathematical 'death story'.)

John Newton (1622 to 1678) was an English mathematician and astronomer.
He was the author of several works on arithmetic and astronomy, designed to
facilitate the use of decimal notation and logarithmic methods. He was also
an advocate of educational reform in grammar schools; he protested against
the narrowness of the system which taught Latin and nothing else to boys
ignorant of their mother tongue; and complained that hardly any grammarschool
masters were competent to teach arithmetic, geometry, and astronomy.
With the object of supplying the means of teaching a wider and more practical
curriculum, he wrote schoolbooks on these subjects, and also on logic and rhetoric.

Blaise Pascal (1623 to 1662), a French mathematician, at just 21 years of
age designed a machine that could add and subtract. It was efficient enough to
be commercially produced, under the name of a Pascaline.
Pascal's triangle was known for many centuries before Pascal himself lived.
In China it is known as Yanghui's triangle; in Iran as Khayyam's triangle
(see Khayyam above); in Italy as Tartaglia's triangle (see Tartaglia above).
This triangle of integers arises in expanding binomials, like (x + y)^n.
Pascal was the first to use the triangle of integers to calculate probabilities.

Niklaus Bernoulli (1623 to 1708) was the father of Jacob and Johann Bernoulli,
famous mathematicians  who started a lineage of Bernoulli mathematicians (see below,
around 1654 and thereafter).

Stefano degli Angeli (1623 to 1697) was an Italian mathematician, philosopher, and
Jesuat. From 1654 to 1667 he devoted himself to the study of geometry, continuing
the research of Cavalieri and Evangelista Torricelli based on the method of 'indivisibles'.
He then moved on to mechanics, where he often found himself in conflict with Giovanni
Alfonso Borelli (1608 to 1679) and Giovanni Riccioli (1598 to 1671).
James Gregory (1638 to 1675) studied under Angeli from 1664 until 1668 in Padua.
The Jesuits essentially declared war on the method of infinitesimals
(a.k.a. indivisibles) and on the Jesuats. On 6 December 1668, Pope Clement IX issued a
brief suppressing the Jesuati order, and Angeli never published on inifinitesimals again.
This is an example of the suppresion on the Continent that seemed to set the stage for
Wallis, Barrow, Newton, and others in England to employ infinitesimals to great advantage.

JeanBaptiste du Hamel (1624 to 1706) was a French cleric and natural philosopher of
the late seventeenth century, and the first secretary of the Academie Royale des Sciences.
At the age of eighteen (1642), he published an explanation of the work of Theodosius of Bithynia
called 'Sphériques de Théodose', to which he added a treatise on trigonometry.
He published two of his works, the 'Astronomia Physica' and 'De Meteoris et Fossibilus'
in 1660 (at age 36), both of which analyze and compare ancient theories with Cartesianism.
The Académie des Sciences was officially founded at the end of 1666. Over time, he did
less and less scientific writing. By 1700, his scientific work was minimal, and his
increasing attention directed toward the church and religion.

John Collins (1625 to 1683) was an English mathematician. He is most known for his
extensive correspondence with leading scientists and mathematicians such as Giovanni
Alfonso Borelli, Gottfried Leibniz, Isaac Newton, and John Wallis. His correspondence
provides details of many of the discoveries and developments made in his time, and shows
his activity as an 'intelligencer'. He helped forward many important publications. To him
was due the printing of
Isaac Barrow's 'Optical and Geometrical Lectures', as well as of his editions of
'Apollonius' and 'Archimedes'; of
John Kersey's 'Algebra',
Thomas Branker's translation of
Rhonius's 'Algebra', and
Wallis's 'History of Algebra'.
He took an active part
in seeing
Jeremiah Horrocks's 'Astronomical Remains' through the press.

Pietro Mengoli (1625 to 1686) was an Italian mathematician and clergyman
from Bologna, where he studied with Bonaventura Cavalieri at the University of
Bologna, and succeeded him in 1647. He remained as professor there for the
next 39 years of his life. In 1644, it was Mengoli who first posed the famous
Basel problem, solved in 1735 by
Leonhard Euler. He wrote a paper in 1650 in which
he proved that the sum of the alternating harmonic series is equal to the natural
logarithm of 2. He also proved that the harmonic series does not converge, and
provided a proof that Wallis' product for pi is correct.

Gerard Kinckhuysen (1625 to 1666) was a Dutch mathematician. In 1661, he published at
Haerlem an introduction to algebra written in Dutch. It became known, even in England, for
the clarity and compactness of its presentation. There was an attempt in England (by Nicolaus
Mercator, Collins, and Newton), in 1670, to translate it into Latin.

Erhard Weigel (1625 to 1699) was a German mathematician, astronomer and philosopher.
From 1653 until his death, he was professor of mathematics at
Jena University. He was the teacher of Leibniz in summer 1663, and other notable students.
He also worked to make science more widely accessible to the public, and what would today
be considered a populariser of science.

Johan de Witt (1625 to 1672) was a key figure in Dutch politics in the mid17th century,
when its sea trade was flourishing.Besides being a statesman, Johan de Witt also was an
accomplished mathematician. In 1659, he wrote "Elementa Curvarum Linearum" as an appendix
to Frans van Schooten's translation of René Descartes' "La Géométrie". In this, De Witt
derived the basic properties of quadratic forms, an important step in the field of linear
algebra. In 1671, his 'Waardije van Lyfrenten naer Proportie van Losrenten' was published
('The Worth of Life Annuities Compared to Redemption Bonds'). This work combined the
interests of the statesman and the mathematician. His negligence of the Dutch land army
(as the Dutch regents focused only on merchant vessels, thinking they could avoid war)
proved disastrous when the Dutch Republic suffered numerous early defeats in 1672.
In the hysteria that followed the effortless invasion by an alliance of three countries,
he and his brother Cornelis de Witt were blamed and lynched in The Hague.
(Another interesting death of a mathematician.)

Jean Dominique Cassini (Giovanni Domenico Cassini) (1625 to 1712) was an Italian
mathematician, astronomer, astrologer and engineer. Cassini is known for his work
in the fields of astronomy and engineering. Cassini discovered four satellites of
the planet Saturn and noted the division of the rings of Saturn; the Cassini Division
was named after him. Giovanni Domenico Cassini was also the first of his family to
begin work on the project of creating a topographic map of France. The Cassini spaceprobe,
launched in 1997, was named after him and became the fourth to visit Saturn and the
first to orbit the planet.
In 1672 he sent his colleague Jean Richer to Cayenne, French Guiana, while he himself
stayed in Paris. The two made simultaneous observations of Mars and, by computing
the parallax, determined its distance from Earth. This allowed for the first time
an estimation of the dimensions of the solar system: since the relative ratios of
various sunplanet distances were already known from geometry, only a single absolute
interplanetary distance was needed to calculate all of the distances.
Cassini was also the first to make successful measurements of longitude by the method
suggested by Galileo, using eclipses of Jupiter's moons as a clock.

Johannes Hudde (1628 to 1704) was a burgomaster (mayor)
of Amsterdam between 1672  1703, a mathematician, and
governor of the Dutch East India Company. From 1654 to 1663, he worked under
van Schooten, and, with his fellow students, Johan de Witt and Hendrik van Heuraet,
they translated Descartes's "La Géométrie" into Latin. Each of the students added to
the work. Hudde's contribution consisted of a study on maxima and minima and a
theory of equations. Hudde corresponded with Baruch Spinoza and Christiaan Huygens,
Johann Bernoulli, Isaac Newton and Leibniz. Newton and Leibniz mention Hudde many
times and used some of his ideas in their own work on infinitesimal calculus.

Christian_Huygens (1629 to 1695) was a prominent Dutch mathematician and scientist.
He is known particularly as an astronomer, physicist, probabilist and horologist
(clock designermaker). His work included early telescopic studies of the rings of
Saturn and the discovery of its moon Titan, the invention of the pendulum clock
and other investigations in timekeeping. He published major studies of mechanics
and optics, and pioneered work on games of chance.

Richard Delamain (ca. 1630) was an English mathematician,
known for works on the circular slide rule and sundials.

Isaac Barrow (1630 to 1677) was an English Christian theologian and mathematician
who is generally given credit for his early role in the development of infinitesimal
calculus; in particular, for the discovery of the
fundamental theorem of calculus. His work centered on the properties of the tangent.
Isaac Newton was a student of Barrow's, and Newton went on to develop calculus in a
nearmodern form. In 1663, Barrow was selected as the first occupier of the Lucasian
chair at Cambridge. During his tenure of this chair, he published two mathematical works
of great learning and elegance, the first on geometry and the second on optics.
In 1669, he resigned his professorship in favour of Isaac Newton. After that time,
he devoted himself to matters theological rather than mathematical.

John Locke (1632 to 1704) was an English philosopher and physician,
widely regarded as one of the most influential writers of the 1600's.
His contributions to 'social contract' theory are reflected in the
United States Declaration of Independence. He felt that something must
be capable of being tested repeatedly and that nothing is exempt
from being disproven. This corresponded with what was happening in
physics ('natural philosophy') as Galileo was testing previous conceptions
of motion and gravity. Locke's ideas about 'natural rights' are today
considered quite revolutionary for that period in English history.
He was deeply receptive to arguments for political and religious
tolerance and the necessity of the separation of church and state.
His arguments concerning liberty and the social contract later
influenced the written works of Alexander Hamilton, James Madison,
Thomas Jefferson, and other Founding Fathers of the United States.
Thomas Jefferson wrote: "Bacon, Locke and Newton... I consider them
as the three greatest men that have ever lived, without any exception, ...".

Christopher Wren (1632 to 1723) is one of the most highly
acclaimed English architects in history. He was accorded responsibility
for rebuilding 52 churches in the City of London after the Great Fire
in 1666. Wren was a notable anatomist, astronomer, geometer, and
mathematicianphysicist, as well as an architect. He was a founder
of the Royal Society (president 168082), and his scientific work
was highly regarded by Isaac Newton and Blaise Pascal.
In London, he did experiments involving determining longitude through
magnetic variation and through lunar observation to help with navigation,
and helped construct a 35 foot (11 meter) telescope with Sir Paul Neile.
Wren also studied and improved the microscope and telescope. He had also
been making observations of the planet Saturn from around 1652 with the
aim of explaining its appearance. In mathematics, in 1658, he found the
length of an arc of the cycloid using an exhaustion proof based on
dissections to reduce the problem to summing segments of chords of
a circle which are in geometric progression. Wren studied mechanics
thoroughly, especially elastic collisions and pendulum motions.
He performed studies in meteorology, and in 1662 he invented the
tipping bucket rain gauge and, in 1663, designed a "weatherclock"
that would record temperature, humidity, rainfall and barometric pressure.
A working weather clock based on Wren's design was completed by Robert
Hooke in 1679. Wren contributed was optics. He published a description of
an engine to create perspective drawings and he discussed the grinding of
conical lenses and mirrors. Out of this work came another of Wren's important
mathematical results, namely that the hyperboloid of revolution is a
ruled surface. These results were published in 1669. In subsequent years,
Wren continued with his work with the Royal Society, although after the
1680s his scientific interests seem to have waned: no doubt his architectural
and official duties absorbed more time. It was a challenge posed by Wren that
led to Newton's 'Principia'. Robert Hooke had theorised that planets, moving
in vacuo, describe orbits around the Sun because of a rectilinear inertial
motion by the tangent and an accelerated motion towards the Sun. Wren's
challenge to Halley and Hooke, for the reward of a book worth thirty shillings,
was to provide, within the context of Hooke's hypothesis, a mathematical theory
linking the Kepler's laws with a specific force law. Halley took the problem
to Newton for advice, prompting the latter to write a ninepage answer,
'De motu corporum in gyrum', which was later to be expanded into the Principia.
Whereas Hooke had many good ideas, Newton had the background to provide the
mathematical theory.

Hendrik van Heuraet (1633 to 1660?) was a Dutch mathematician also
known as Henrici van Heuraet. He dealt with areas bounded by curves ('integration'),
and was author of 'Epistola de Transmutatione Curvarum Linearum in Rectus'
[On the Transformation of Curves into Straight Lines] (1659). From 1653,
he studied at Leiden University where he interacted with Frans van Schooten,
Johannes Hudde, and Christiaan Huygens. After 1659, his trail is lost.

Petrus (Pieter) van Schooten (1634 to 1679) was the halfbrother of Frans
van Schooten, and was appointed to the chair Frans occupied at his death in 1660,
at the Engineering School at Leiden, becoming the third van Schooten to hold
this chair in succession. Pieter van Schooten continued in this position until
his death in 1679.

Robert Hooke (1635 to 1703) was an English 'natural philosopher' and architect.
At one time he was simultaneously the curator of experiments of the Royal Society,
a member of its council, Gresham Professor of Geometry, and Surveyor to the City
of London after the Great Fire of London (in which capacity he appears to have
performed more than half of all the surveys after the fire). he was employed as
an assistant to Thomas Willis and to Robert Boyle, for whom he built the vacuum
pumps used in Boyle's gas law experiments. He built some of the earliest Gregorian
telescopes and observed the rotations of Mars and Jupiter. In 1665, he inspired
the use of microscopes for scientific exploration with his book, 'Micrographia'.
Based on his microscopic observations of fossils, Hooke was an early proponent
of biological evolution. He investigated the phenomenon of refraction, deducing
the wave theory of light, and was the first to suggest that matter expands when
heated and that air is made of small particles separated by relatively large distances.
He performed pioneering work in the field of surveying and mapmaking.
He hypothesized that gravity follows an inverse square law which governs the motions
of the planets, an idea which was subsequently developed by Isaac Newton.
Much of Hooke's scientific work was conducted in his capacity as part of the
household of Robert Boyle or as curator of experiments of the Royal Society,
a post he held from 1662.

James Gregory (1638 to 1675) was a Scottish mathematician and astronomer.
He described an early practical design for the reflecting telescope  the Gregorian
telescope  and made advances in trigonometry, discovering infinite series
representations for several trigonometric functions. In his book 'Geometriae
Pars Universalis' (1668), Gregory gave both the first published statement and
proof of the fundamental theorem of the calculus (stated from a geometric point
of view, and only for a special class of the curves considered by later versions
of the theorem), for which he was acknowledged by Isaac Barrow. James Gregory
discovered the diffraction grating by passing sunlight through a bird feather
and observing the diffraction pattern produced. In particular, he observed the
splitting of sunlight into its component colours  this occurred a year after
Newton had done the same with a prism and the phenomenon was still highly controversial.
Gregory, an enthusiastic supporter of Newton, later had much friendly correspondence
with him and incorporated his ideas into his own teaching.
About a year after assuming the Chair of Mathematics at Edinburgh (~1674), James
Gregory suffered a stroke while viewing the moons of Jupiter with his students.
He died a few days later at the young age of 36. He was the uncle of mathematician
David Gregory.

Nicolas Malebranche (1638 to 1715) was a French Oratorian and
rationalist philosopher. Malebranche wrote on the laws of motion,
a topic he discussed extensively with Leibniz. He also wrote on
mathematics and, although he made no major mathematical discoveries
of his own, he was instrumental in introducing and disseminating
the contributions of Descartes and Leibniz in France. Malbranche
introduced l'Hôpital to Johann Bernoulli, with the ultimate result
being the publication of the first textbook in infinitesimal calculus.

The links below are under construction. Some may be removed.
Some will have notes added.

Francois de Raynaud (Regnauld) (ca. 1670)
(student of Honor Fabri)

Yoshisuke_Matsunaga (ca. 1639 to 1744) Calculated pi to
51 decimal places.

Georg Mohr (1640 to 1697 ca. 1672) was a Danish mathematician.
He travelled in the Netherlands, France, and England.

Jacques Ozanam (1640 to 1718) was a French mathematician.
He gave up theology after four years of study and began
to give free private instruction in mathematics at Lyon.
Later, as the family property passed entirely to his elder brother,
he was reluctantly driven to accept fees for his lessons.
In 1670, he published trigonometric and logarithmic tables
more accurate than the existing ones of Ulacq, Pitiscus, and Briggs.
He secured an invitation to settle in Paris. There he enjoyed
prosperity and contentment for many years. He married, had a large
family, and derived an ample income from teaching mathematics to
private pupils, chiefly foreigners. His mathematical publications
were numerous and well received. 'Récréations mathématiques et physiques'
(published 1694) was later translated into English and is
well known today. He was elected a member of the Académie des
Sciences in 1701. The death of his wife plunged him into
deep sorrow, and the loss of his foreign pupils through
the War of the Spanish Succession reduced him to poverty.
He died in Paris in 1718.

Seki Takakazu (1642 to 1708) was a Japanese mathematician.
He has been described as Japan's "Newton." He created a new
algebraic notation system, and also, motivated by astronomical
computations, did work on infinitesimal calculus and Diophantine
equations. A contemporary of Gottfried Leibniz and Isaac Newton,
Seki's work was independent.

Isaac Newton (1642 to 1727) is known for developing/discovering/elucidating
 differential calculus
 integral calculus
 the Fundamental Theorem of Calculus, which shows the relationship
between differentiation and integration
 the 3 'Newtonian' laws of motion
 the law of gravity  any two bodies in the universe attract each
other as the inverse of the square of the distance apart (of the centers)
 the precession of the equinoxes

Olaus Roemer (1644 to 1710) was a Danish astronomer who in 1676 made
the first quantitative measurements of the speed of light.

Andreas_Werckmeister (1645 to 1706) was a German organist, music theorist,
and composer. He wrote on mathematics and music  in particular, on
tuning systems.

Gottfried Leibniz (1646 to 1716), a German mathematician, designed a
machine that could perform the four basic arithmetic operations  addition,
subtraction, multiplication, and division. He was therefore rather scornful of
the limited functions of the Pascaline (see Pascal above). Unfortunately, the
carry mechanism was faulty and, hence, the machine was never produced in quantity.
Leibniz is known as the codiscoverer or independent discoverer of the
'infinitesimal' calculus  along with Isaac Newton. Leibniz published first.

John Flamsteed (1646 to 1719) was an English astronomer and the first
Astronomer Royal. He catalogued over 3000 stars. His star catalogue
tripled the number of entries in Tycho Brahe's sky atlas.

Denis Papin (1647 to 1712) was a French physicist,
mathematician and inventor, best known for his pioneering
invention of the steam digester, the forerunner of the
steam engine, and of the pressure cooker.

Giovanni Ceva (1647 to 1734) was an Italian mathematician
widely known for proving
Ceva's theorem in elementary geometry. But it was proven much
earlier by
Yusuf AlMu'taman ibn Hud, an eleventhcentury king of Zaragoza.
Giovanni's brother, Tommaso Ceva was also a wellknown poet and mathematician.

Tommaso Ceva (1648 to 1737) was an Italian mathematician

Joseph Raphson (1648? to 1715?) was an English mathematician
known best for the NewtonRaphson method.

Count E.W. von Tschirnhaus (1651 to 1708)

Michel Rolle (1652 to 1719) was a French mathematician.
He is best known for Rolle's theorem (1691), and he deserves
to be known as the coinventor in Europe of Gaussian elimination (1690).

Joseph Sauveur (1653 to 1716) was a French mathematician
and physicist. He was a professor of mathematics and in 1696
became a member of the French Academy of Sciences.

Pierre Varignon (1654 to 1742) was a French mathematician.

Jacob Bernoulli (1654 to 1705)  Law of Large Numbers

Jacques Bernoulli (1655 to 1705)  radius of curvature ~1692

Edmond Halley (ca. 1656 to 1742)

Charles Renee Reynaud (1656 to 1728)

Bernard de Fontanelle (1657 to 1757)

Jean Christophe Fatio (1659 to 1720)

David Gregory (1659 to 1708) was a Scottish mathematician
and astronomer. He was professor of mathematics at the University
of Edinburgh, Savilian Professor of Astronomy at the University
of Oxford, and a commentator on Isaac Newton's Principia.
He was a nephew of astronomer and mathematician James Gregory.

Marquis Guillaume de L'Hospital (1661 to 1704) Textbook on differential calculus ~1696

Louis Carre (1663 to 1711) was a French mathematician
and member of the French Academy of Sciences. He was the
author of one of the first books on integral calculus.

John Craig (1663 to 1731 ca. 1687) was a Scottish
mathematician and theologian.

Takebe Kenko (or Takebe Katahiro) (1664 to 1739)

Nicolas Fatio de Duiller (1664 to 1753)

Antoine Parent (1666 to 1716) was a French mathematician,
born at Paris and died there, who wrote in 1700 on
analytical geometry of three dimensions. His works were
collected and published in three volumes at Paris in 1713.

John Harris (ca. 1666 to 1719)

Abraham de Moivre (1667 to 1754)

Jean Bernoulli (Johann) (1667 to 1748)

Giovanni Girolamo Saccheri (1667 to 1733) was an Italian Jesuit
priest, scholastic philosopher, and mathematician.

Luigi Guido Grandi (1671 to 1742) was an Italian monk,
priest, philosopher, mathematician, and engineer.

George Cheyne (16711743) was a pioneering physician,
early protopsychologist, philosopher and mathematician.

John Keill (1671 to 1721) was born in Edinburgh,
Scotland, and was primarily a mathematician and a disciple
of Isaac Newton. He studied at Edinburgh University,
under David Gregory, and obtained his bachelors degree
in 1692 with a distinction in physics and mathematics.

William Jones (1675 to 1749) was a Welsh mathematician,
most noted for his proposal for the use of the Greek letter pi
to represent the ratio of the circumference of a circle to
its diameter. He was a close friend of Sir Isaac Newton and
Sir Edmund Halley. In November, 1711 he became a Fellow of
the Royal Society, and was later its VicePresident.

Humphry Ditton (1675 to 1715) was an English mathematician.
He studied theology, but on the death of his father, he devoted
himself to the study of mathematics. Through the influence of
Isaac Newton he was elected mathematical master in Christ's Hospital.
He did work on perspective.

Jacopo Riccati (1676 to 1754) was an Italian mathematician,
born in Venice. He is now remembered for the Riccati equation.

Antonio Schinella Conti (1677 to 1749) was an Italian
historian, mathematician, philosopher and physicist. He was
known as Abbé Conti (in Italian, Abate Conti) and is famous
for having been the intermediary, in England in 171516,
in the LeibnizNewton calculus controversy.

Jacques Cassini (1677 to 1756) was a French astronomer,
son of the famous Italian astronomer Giovanni Domenico Cassini.

Pierre Rémond de Montmort (1678 to 1719) was a French mathematician.
De Montmort is known for his book on probability and games of chance.
Another of de Montmort's interests was the subject of finite differences.

Charles Hayes (16781760) was an English mathematician
and chronologist, author of an early book on the method of fluxions.

Jakob Hermann (1678 to 1733) was a mathematician who
worked on problems in classical mechanics. He appears to
have been the first to show that the LaplaceRungeLenz
vector is a constant of motion for particles acted upon
by an inversesquare central force. He received his
initial training from Jacob Bernoulli and was a distant
relative of Leonhard Euler.

Ephraim Chambers (ca. 1680 to 1740) Wrote 'Cyclopaedia'.

John Colson (1680 to 1760) Translated several of Newton's
works into English.

Gabriele Manfredi (1681 to 1761) was an Italian mathematician who
undertook important work in the field of calculus.

Roger Cotes (1682 to 1716) was an English mathematician,
known for working closely with Isaac Newton by proofreading
the second edition of his famous book, the Principia, before
publication. He also invented the quadrature formulas known
as NewtonCotes formulas and first introduced what is known
today as Euler's formula. He was the first Plumian Professor
at Cambridge University from 1707 until his death.

Count Giulio Carlo di Fagnano (1682 to 1766) was an Italian mathematician.
He was probably the first to direct attention to the theory of elliptic integrals.

AmédéeFrançois Frézier (1682 to 1773)

Savin (ca. 1701) (need info and link)

Guisnee (ca. 1705) (need info and link)

Chamberlayne (ca. 1718) (need info and link)

Burkhardt (ca. 1721) (need info and link)

Bragelonge (ca. 1730) (need info and link)

Brook Taylor (1685 to 1731), an English mathematician, coined the
phrase "vanishing point". His treatise 'Linear Perspective' (1715) expounded
the principle of vanishing points and was of value to artists.
This is the same Taylor who originated Taylor's theorem, a formula important
in differential calculus, which relates a function to its derivatives by
means of a power series.

Nicolaus I Bernoulli (1687 to 1759)

Robert Simson (1687 to 1768 ca. 1756) A Scottish
mathematician and professor of mathematics at the
University of Glasgow. In 1756, appeared, both in Latin
and in English, the first edition of his Euclid's 'Elements'.
This work, which contained only the first six and the
eleventh and twelfth books, and to which, in its English
version, he added the Data in 1762, was for long the
standard text of Euclid in England.

Christlieb von Clausberg (1689 to 1751) Died in Copenhagen. (need more info and link)

Christian Goldbach (1690 to 1764) conjectured that every even number
above two is the sum of two primes. Although 'The Goldbach Conjecture' has
been verified (as of 2008) up to a very large number, general proof is still
elusive.

Heinrich? Kuhn (1690? to 1769?)
A teacher in Danzig. In 1750, was
the first to give the square root of minus one a
geometric picture, analogous to the geometric interpretation
of the negative reals. (need more info and a link)
Mentioned in 'A History of Mathematics' by Cajori.

James Stirling (1692 to 1770) was a Scottish mathematician.
The Stirling numbers, Stirling permutations, and Stirling's
approximation are named after him. He also proved the correctness
of Isaac Newton's classification of cubics

Henry Pemberton (1694 to 1771) was an English physician
and man of letters. He became Gresham Professor of Physic,
and edited the third edition of Principia Mathematica.

Voltaire (1694 to 1778)

Nicolaus II Bernoulli (1695 to 1726)

Henri Pitot (1695 to 1771) was a French hydraulic engineer
and the inventor of the pitot tube. The
Pitot theorem
of plane geometry is named after him.

Colin MacLaurin (1698 to 1746) was a Scottish mathematician.
The Maclaurin series, a special case of the Taylor series,
is named after him.

George Campbell (ca. 1700 to 1766) Involved in a dispute
with
Colin MacLaurin (1698  1746) over complex roots.

Samuel Klingenstierna (1698 to 1765) Swedish mathematician

Maupertuis (1698 to 1759) Often credited with having invented
the
principle of least action.

Pierre Bonger (1698 to 1758) (need more info and link)

Daniel Bernoulli (1700 to 1782)

William Braikenridge (also Brakenridge) (1700? to 1762) was
a Scottish mathematician and cleric, a Fellow of the Royal Society
from 1752.In geometry the
BraikenridgeMaclaurin theorem was independently discovered
by Colin Maclaurin. It occasioned a priority dispute after
Braikenridge published it in 1733.

Edmund Stone (ca. 1700 to 1768) Son of a gardener of the
Duke of Argyll.
Selftaught.

Charles Marie de la Condamine (1701 to 1774)

Thomas Bayes (1701 to 1761) An English
mathematician and Presbyterian minister, known for
having formulated a specific case of the theorem that
bears his name:
Bayes's theorem. His friend Richard Price edited
and presented this work in 1763, after Bayes' death,
as 'An Essay towards solving a Problem in the Doctrine of
Chances'.

Georg Wolfgang Krafft (1701 to 1754) When Euler became
the professor of mathematics in St. Petersburg in 1733, Krafft
took over Euler's previous position, that of professor of
physics.

Gabriel Cramer (1704 to 1752)
Swiss mathematician  namesake of
Cramer's rule.

Louis Goudin (1704 to 1760) A French
astronomer and member of the French Academy of Sciences.
He worked in Peru, Spain, Portugal and France.

Johann Andreas Segner (1704 to 1777) Was a Hungarianborn
scientist. He was born in the Kingdom of Hungary, in the
former Hungarian capital city of Pozsony (today Bratislava).
In 1735, Segner became the first professor of mathematics
at the University of Göttingen, a position created for him.
One of the bestknown scientists of his age, Segner was a
member of the academies of Berlin, London, and Saint Petersburg.
Segner produced the first proof of Descartes' rule of signs.
He was the first scientist to use the reactive force of water
and constructed the first waterjet, the Segner wheel,
which resembles one type of modern lawn sprinkler.
Historians of science remember him as the father of the
water turbine.

Jean Castillon [aka Giovanni Salvemini] (17041791)
Was as an Italian mathematician and astronomer. In 1745,
he was elected to the Royal Society. In 1765, Frederick
the Great appointed him 'Astronomer Royal' of the
Observatory of Berlin. Succeeding JosephLouis Lagrange,
he was appointed Director of the Mathematics Section of
the Berlin Academy, a role he held until his death.
He studied conic sections, cubic equations and problems
of artillery. Among his latest publications mathematics
note He is also known for the "Castillon's problem".

Leonhard Euler (1707 to 1783)  super prolific mathematician.
 solved the 7 bridges of Konigsberg problem  led to graph theory
 V + F = E + 2  the formula connecting the vertices, faces, and edges
of a solid plyhedron.
 The number 'e' is such that the gradient of y=e^x is equal to itself.
 e^(i x pi) = 1

Vincenzo Riccati (1707 to 1775)

Giordano Riccati (1709 to 1790)

Eustachio Zanotti (17091782) Was a noted astronomer
and mathematician and was the son of Giampietro Zanotti,
an Italian painter and art historian. (need a better link)

Thomas Simpson (1710 to 1761) Wrote popular textbooks,
including works on 'fluxions'.

Patrick_Murdoch (ca. 1710? to 1774) Part of Murdoch's 1746
'Newtoni genesis curvarum per umbras' (Newton's generation
of curves by shadows) deals with perspective.

François Jacquier (1711 to 1788) A French priest.
In 173942, François Jacquier and another French priest,
Thomas LeSeur (17xx? to 17xx?), produced, with the
assistance of
JeanLouis Calandrini (17031758), an extensively
annotated version of the
1726 3rd edition of Newton's 'Principia'. Sometimes this
is referred to as the 'Jesuit edition'. It was much used, and
reprinted more than once in Scotland during the 19th century.

David Hume (1711 to 1776) Scottish philosopher,
historian, economist.

Johann Samuel König (1712 to 1757) Had disagreements
with Euler on the principle of least action.

Christian Ehrenfried Eschenbach (17121788)
belongs to the forerunners of scholars of legal medicine in Germany.
As a principal reelected 11 times and dean of the medical faculty
at Rostock University, he defended academic positions in difficult
times. His bibliography comprises numerous text books, e.g. on surgery,
anatomy, pathology and obstetrics as well as various fields of mathematics.

Alexis Claude Clairaut (1713 to 1765) See
Clairaut's theorem and
Clairaut's equation and
Clairaut's relation.

Jean Paul de Gua (1713 to 1785) A French mathematician
who published in 1740 a work on analytical geometry in
which he applied it, without the aid of differential calculus,
to find the tangents, asymptotes, and various singular points
of an algebraic curve.

Denis Diderot (1713 to 1784) Best known for
serving as cofounder, chief editor, and contributor to
the Encyclopédie along with Jean le Rond d'Alembert (below).
First volume was published in 1751.

Cesar Francois Cassini (1714 to 1784)
French astronomersurveyorcartographer.

Arima Yoriyuki (1714 to 1783) Japanese mathematician.
In 1766, he found the a rational approximation of pi,
correct to 29 digits  the ratio of two 15 digit integers.

Barmann (ca. 1745) (more info and link needed)

Jean d'Alembert (1717 to 1783) French mathematician,
mechanician, physicist, philosopher, and music theorist.
D'Alembert's formula for obtaining solutions to the wav
equation is named after him. The wave equation is sometimes
referred to as D'Alembert's equation.

Francesco Riccati (1718 to 1791) A son of Jacopo Ricatti (above).
Italian architect. (more info and better link needed)

Maria Gaetana Agnesi (1718 to 1789) An Italian
mathematician and philosopher. She is credited with
writing the first book discussing both differential and
integral calculus and was an honorary member of the
faculty at the University of Bologna.

Matthew Stewart (1719 to 1785) Published
Stewart's theorem in 1746.

Abraham Gotthelf Kastner (1719 to 1800) German mathematician
known for his textbooks.

John Landen (1719 to 1790 ca. 1771) An English
mathematician. Landen's capital discovery is that of the
Landen's transformation for the expression of the arc
of an hyperbola in terms of two elliptic arcs. His researches
on elliptic functions are of considerable elegance, but
their great merit lies in the stimulating effect which
they had on later mathematicians. He also showed that the
roots of a cubic equation can be derived by means of
the infinitesimal calculus. He lived a very retired life,
and saw little or nothing of society. When he did mingle
in it, his dogmatism and pugnacity caused him to be
generally shunned.

Fridericus Guilielmus De Oppel (1720 to 1769) Wrote
'Analysis Triangulorum' (1746). (need more info and link)

Joseph Torelli (1721 to 1781) was an Italian mathematician.
His edition of the collected works of Archimedes was printed
at Oxford in 1792. The preparation of this work was a
labour performed, among many other pursuits, during most of his life.

Ravelli (ca. 1751) (need more info and a link)

John Lawson (1723 to 1779) An English mathematician.
Published some works on tangencies.

JeanÉtienne Montucla (1725 to 1799) Wrote a history of
math. The first part was published in 1758.

Johann Heinrich Lambert (1728 to 1777) A Swiss
mathematician, physicist, philosopher and astronomer.
He is best known for proving the irrationality of pi.

Fredric Mallet (1728 to 1797) Was one of
Klingenstierna's (above, 1698) most devoted students.
Sorted through Klingenstierna's manuscripts for
eventual publication. (need a better link)

Charles Bossut (1730 to 1814) Was a French mathematician.
His works include 'Traité élémentaire d'hydrodynamique' in 1771,
'Traité élémentaire de méchanique statique' in 1772,
'Cours de mathématiques' in 1781, and 'Histoire générale
des mathématiques' in 1810.

Étienne Bezout (1730 to 1783) A French
mathematician. Wrote 'Théorie générale des équations
algébriques', published at Paris in 1779, which,
in particular, contained much new and valuable matter
on the theory of elimination and symmetrical functions
of the roots of an equation. He used determinants in
a paper in the 'Histoire de l'académie royale', 1764,
but did not treat the general theory.

Martin Johan Wallenius (1731 to 1773) All five squarable
lunes
were given in a dissertation by Martin Johan Wallenius in 1766.
(need a better link)

Girolamo Saladini (17311813) Saladini was a
student of Vincenzo Riccati (above), and was a coauthor
with Riccati of the two volume book 'Institutiones Analyticae'
(Vol 1, 1765; Vol 2, 1767) which contains the formulas for
the addition and subtraction of hyperbolic functions
as well as other, now standard, formulas analogous to
those for the circular (sin, cos) functions. They also computed
the derivatives of sinh x and cosh x. (need a better link)

Gian Francesco Malfatti (1731 to 1807) was an Italian
mathematician. He studied in Bologna where his mentors
included Vincenzo Riccati, F. M. Zanotti and G. Manfredi.
Malfatti posed the problem of carving three circular columns
out of a triangular block of marble, using as much of
the marble as possible, and conjectured that three
mutuallytangent circles inscribed within the triangle
would provide the optimal solution. These tangent circles
are now known as Malfatti circles after his work, despite
the earlier work of Japanese mathematician Ajima Naonobu
and of his countryman Gilio di Cecco da Montepulciano on
the same problem  and despite the fact that the conjecture
was later proven false. Several triangle centers derived
from these circles are also named after both Ajima and
Malfatti. Additional topics in Malfatti's research
concerned quintic equations, and the property of the
lemniscate of Bernoulli that a ball rolling down an
arc of the lemniscate, under the influence of gravity,
will take the same time to traverse it as a ball rolling down
a straight line segment connecting the endpoints of the arc.

Francois Daviet de Foncenex (1734 to 1799) In 1759, he attempted a proof of
the fundamenatal theorem of algebra  as did Euler in 1749,
Lagrange in 1772, and Laplace in 1795. (need a better link)

Achille_Pierre Dionis du Sejour (1734 to 1794)
was a French astronomer and mathematician. He wrote on
algebraic curves as well as on astronomy (planets and comets).

William Wales (1734? to 1798) Made astronical
observations and calculations on the 2nd voyage of Captain
James Cook.

AlexandreThéophile Vandermonde (1735 to 1796)
A French musician (violinist), mathematician and chemist
who worked with Bézout and Lavoisier. His name is now
principally associated with determinant theory in mathematics.

Joseph Louis Lagrange (1736 to 1813) An Italian
mathematician and astronomer. He made significant
contributions to all fields of analysis, number theory,
and both classical and celestial mechanics. In 1766,
Lagrange succeeded Euler as the director of mathematics
at the Prussian Academy of Sciences in Berlin, Prussia,
where he stayed for over twenty years.

Edward Waring (1736 to 1798) An English mathematician who
was elected in 1760 as
Lucasian Professor of Mathematics at Cambridge,
holding the chair until his death. In 1762, he published
the full 'Miscellanea Analytica', mainly devoted to the
theory of numbers and algebraic equations. See
Waring's (number theory) problem and
Waring's prime number conjecture.

Erland Samuel Bring (1736 to 1798) was a Swedish
mathematician. At Lund University, he wrote eight volumes
of mathematical work in the fields of algebra, geometry,
analysis and astronomy, including 'Meletemata quaedam
mathematematica circa transformationem aequationum
algebraicarum' (1786). Bring developed a transformation
to simplify a quintic equation to the form x^5 + px + q = 0.

Jacques Antoine Joseph Cousin (1739 to 1800)
Wrote 'Lecons de Calcul Differentiel et de Calcul Integral'
(1777). His main position was as professor for mathematics
and experimental physics at the College Royale, which he
held from 1769 until his death. (need a better link)

Anders Johan Lexell (1740 to 1784)
was a FinnishSwedish astronomer, mathematician, and
physicist who spent most of his life in Russia.
Lexell made important discoveries in
polygonometry and celestial mechanics. He contributed
to spherical trigonometry with new and interesting solutions,
which he took as a basis for his research of comet and
planet motion. His name was given to a theorem of
spherical triangles. Lexell was one of the most prolific
members of the Russian Academy of Sciences at that time,
having published 66 papers in 16 years of his work there.
In later life (circa 1783), Lexell became very attached to
Leonhard Euler, who lost his sight in his last years but
continued working using his elder son Johann Euler to read
for him. Lexell helped Leonhard Euler greatly, especially
in applying mathematics to physics and astronomy. He helped
Euler to write calculations and prepare papers.

Stanislaus Wydra (1741 to 1804) was a professor at
Prague University. He wrote an 88 page book titled
'Elementa Calculi Differentialis et Integralis' (1783).
He was a
pupil, and later biographer, of
Joseph Stepling. (Wydra/Vydra needs a better link)

Carl Hindenburg (1741 to 1808) was a German
mathematician born in Dresden. His work centered
mostly on combinatorics and probability.
Hindenburg cofounded the first German mathematical
journals. Between 1780 and 1800, he was involved at
different times with the publishing of four different
journals all relating to mathematics and its applications.
One of Hindenberg's best students, according to Donald
Knuth, is Heinrich August Rothe. Another student,
Johann Karl Burckhardt, published the book 'Theorie der
Kettenbrüche' after being encouraged by Hindenberg to
work on continued fractions. He also influenced
Christian Kramp's work in combinatorics.

Antonio Cagnoli (1743 to 1816) was an Italian
astronomer, mathematician and diplomat. He worked in
Paris and Verona. He set up an observatory in Verona 
which was
damaged by Bonaparte's cannons, but Bonaparte
made amends with money and the gift a very accurate clock.

Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet
(1743 to 1794) was a French philosopher, mathematician,
and early political scientist whose Condorcet method
in voting tally selects the candidate who would beat
each of the other candidates in a runoff election.
Condorcet was one of the first to systematically
apply mathematics in the social sciences. In 1786, Condorcet
worked on ideas for the differential and integral calculus,
giving a new treatment of infinitesimals  a work
which was never printed.

Caspar Wessel (1745 to 1818) A NorwegianDanish
mathematician and cartographer. In 1799, Wessel was
the first person to describe the geometrical interpretation
of complex numbers as points in the complex plane.

Gaspard Monge (1746 to 1818) Was a French mathematician,
the inventor of descriptive geometry (the mathematical
basis of technical drawing), and the father of differential
geometry. During the French Revolution he served as the
Minister of the Marine, and was involved in the reform
of the French educational system, helping to found the
École Polytechnique.

Reuben Burrow (1747 to 1792) An English mathematician
and orientalist. Initially a teacher, he was later appointed
astronomerroyal at the Royal Greenwich Observatory.
He later conducted research in India, becoming one of the
first members of the Asiatic Society. He was also interested
in ancient geometry, as he has proved by his book on
Apollonius: 'A Restitution of the Geometrical Treatise of
Apollonius Pergæus on Inclinations' (1779), and was
curious to investigate the mathematical treatises in
ancient Hindu and other Oriental literature. He later
published 'Hindoo Knowledge of the Binomial Theorem'.

Jean Dominque IV Cassini (1748 to 1845) French astronomer.

D'Amondans Charles de Tinseau (1748 to 1822)
Graduated as a military engineer in 1771 and later became
a mathematician after becoming a student of
Monge (above).

??? Rowing (ca. 1770) (need more info and link)

Pierre Simon de Laplace (1749 to 1827) A French
mathematician and astronomer whose work was pivotal
to the development of mathematical astronomy and statistics.
He summarized and extended the work of his predecessors
in his fivevolume 'Mécanique Céleste' (Celestial Mechanics)
(17991825). This work translated the geometric study of
classical mechanics to one based on calculus, opening up
a broader range of problems. In statistics, the Bayesian
interpretation of probability was developed mainly by Laplace.
In France, the normal distribution is called the Laplacian
distribution. (In Germany, it is called the Gaussian
distribution.)

Lorenzo Mascheroni (1750 to 1800) was an
Italian mathematician. In his 'Geometria del Compasso'
(1797), he proved that any geometrical construction,
which can be done with compass and straightedge,
can also be done with compasses alone. However, the
priority for this result (now known as the MohrMascheroni
theorem) belongs to the Dane Georg Mohr, who had
previously published a proof in 1672. In his
'Adnotationes ad calculum integrale Euleri' (1790),
he published a calculation of what is now known as
the EulerMascheroni constant, usually denoted as gamma.

Simon Antoine Jean L'Huilier (1750 to 1840)
was a Swiss mathematician of French Hugenot descent.
He is known for his work in mathematical analysis and
topology, and in particular the generalization of
Euler's formula for planar graphs.

AdrienMarie Legendre (1752 to 1833) A French mathematician.
Legendre polynomials and Legendre transformation are named
after him. Legendre is known as the author of 'Éléments de
géométrie' which was published in 1794 and was the leading
elementary text on the topic for around 100 years. This
text greatly rearranged and simplified many of the propositions
from Euclid's Elements to create a more effective textbook.

Lazar Nicolas Carnot (1753 to 1823) was a French politician,
engineer, and mathematician. The BordaCarnot equation of
fluid dynamics and Carnot's theorem in plane geometry
are named after him.

Captain William Lambton (1753 to 1823) was a British
soldier, surveyor, and geographer. He was involved in
surveying the boundary between Canada and the United States.
He also did surveying in India.

Jean Baptiste Meusnier (1754 to 1793) A French
mathematician, engineer and Revolutionary general.
He is best known for
Meusnier's theorem on the curvature of surfaces.
He also discovered the
helicoid. He worked with Lavoisier on the
decomposition of water and the evolution of hydrogen.

Jurij Vega (1754 to 1802) was a Slovene mathematician,
physicist and artillery officer. Vega published a series of
books of logarithm tables. His major work was 'Thesaurus
Logarithmorum Completus' (Treasury of all Logarithms)
that was first published 1794. Over the years, Vega wrote
a four volume textbook 'Vorlesungen über die Mathematik'
(Lectures about Mathematics). Volume I appeared in 1782
when he was 28 years old, Volume II in 1784, Volume III
in 1788 and Volume IV in 1800. His textbooks also contain
interesting tables: for instance, in Volume II one can
find closed form expressions for sines of multiples of
3 degrees, written in a form easy to work with. In 1789,
Vega achieved a world record when he calculated pi to 140
places, of which the first 126 were correct. Vega had improved
John Machin's formula from 1706.

Nicolas Fuss (1755 to 1826) was a Swiss mathematician,
living most of his life in Russia. He moved to Saint
Petersburg to serve as a mathematical assistant to
Leonhard Euler from 17731783, and remained there until
his death. He contributed to spherical trigonometry,
differential equations, the optics of microscopes and
telescopes, differential geometry, and actuarial science.
He also contributed to Euclidean geometry, including
the problem of Apollonius.

A. Giordano (ca. 1785) "... a Neapolitan lad A. Giordano,
who was only 16 but who had shewn marked mathematical ability
..." (need more info and a link)

Theodor von Schubert (1758 to 1825) became
a tutor of mathematics and astronomy.
In 1785, he became an assistant of the Russian Academy
of Sciences as a geographer, and by June 1789 he was
a full member. In 1803, he became head of the astronomical
observatory of the Academy. Wrote some texts on astronomy.

Louis Francois Antoine Arbogast (1759 to 1803) was a French
mathematician. He wrote on series and the derivatives
known by his name. His notion of using discontinuous functions
to integrate partial differential equations became important
in Cauchy's more rigorous approach to analysis. He conceived
the calculus as operational symbols. The formal algebraic
manipulation of series investigated by Lagrange and Laplace
in the 1770s has been put in the form of operator equalities
by Arbogast by 1800.

Paolo Ruffini (1765 to 1822) was an Italian mathematician and philosopher.
By 1788 he had earned university degrees in philosophy, medicine/surgery,
and mathematics. Among his work was an incomplete proof (AbelRuffini theorem)
that quintic (and higherorder) equations cannot be solved by radicals (1799),
and Ruffini's rule which is a quick method for polynomial division.
Ruffini also made contributions to group theory in addition to probability
and quadrature of the circle.

Sylvestre François Lacroix (1765 to 1843) was a French
mathematician. He displayed a particular talent for
mathematics, calculating the motions of the planets by
the age of 14. In 1793 he became examiner of the Artillery
Corps, replacing PierreSimon Laplace in the post. By 1794,
he was aiding his old instructor, Gaspard Monge, in creating
material for a course on descriptive geometry. In 1799, he
was appointed professor at the École Polytechnique. Lacroix
produced most of his texts for the sake of improving his
courses. During his career he produced a number of
important textbooks in mathematics. Translations of these
books into the English language were used in British
universities, and the books remained in circulation for
nearly 50 years.

Thomas Malthus (1766 to 1834), an Englishborn economist, used arithmetic
and geometric progressions to describe the growth of population.
Arithmetic progressions and
geometric progressions had been known since the time of the Greeks.

Joseph Fourier (1768 to 1830) was a French mathematician
and physicist. Best known for initiating the investigation of
Fourier series and their applications to problems of heat transfer
and vibrations. (Any wave form can be made by combining
sine and cosine functions.)

Jean Robert Argand (1768 to 1822) was a gifted amateur
mathematician. In 1806, while managing a bookstore in Paris,
he published the idea of geometrical interpretation of complex
numbers known as the Argand diagram. Argand is also renowned
for delivering a proof of the 'fundamental theorem of algebra'
in his 1814 work 'Réflexions sur la nouvelle théorie d'analyse'
(Reflections on the new theory of analysis). It was the first
complete and rigorous proof of the theorem, and was also the
first proof to generalize the fundamental theorem of algebra
to include polynomials with complex coefficients.

Jean Nicolas Pierre Hachette (1769 to 1834), French mathematician,
was born at Mézières, where his father was a bookseller. His labours
were chiefly in the field of descriptive geometry and 3 dimensions, with its
application to the arts and mechanical engineering. It was left
to him to develop the geometry of Monge, and to him also is due
in great measure the rapid advancement which France made soon
after the establishment of the École Polytechnique in the
construction of machinery.

Heinrich August Rothe (1773 to 1842) was a German
mathematician, a professor of mathematics at Erlangen.
The RotheHagen identity, a summation formula for binomial
coefficients, appeared in Rothe's 1793 thesis. It is named
for him and for the later work of Johann Georg Hagen.
The same thesis also included a formula for computing the
Taylor series of an inverse function from the Taylor series
for the function itself, related to the Lagrange inversion
theorem. In the study of permutations, Rothe was the
first to define the inverse of a permutation, in 1800.
He developed a technique for visualizing permutations
now known as a Rothe diagram.

Nathaniel Bowditch (1773 to 1838) was an early
American mathematician remembered for his work on
ocean navigation. He is often credited as the founder
of modern maritime navigation. His book 'The New
American Practical Navigator', first published in 1802,
is still carried on board every commissioned U.S. Naval vessel.

Louis Poinsot (1777 to 1859) added two new regular solids to the five
Platonic solids and the two 'stellated' regular solids pointed out by Kepler
 for a total of nine.
 the five Platonic solids (tetrahedron, octahedron, cube, dodecahedron,
icosahedron)
 small stellated dodecahedron  attributed to Kepler and, before Kepler,
to artist Paolo Uccello (13971475)
 great stellated dodecahedron  attributed to Kepler and, before Kepler,
to artist Wenzel Jamnitzer (15081585).
 great dodecahedron (found by Poinsot)
 great icosahedron (found by Poinsot)
Augustin Cauchy (17891857 ; see below) proved that these are the only regular solids.

Carl Friedrich Gauss (1777 to 1855) devised a method of constructing a
17sided polygon using straight edge and compass only. Greek mathematicians
had been able to constrocuh polygons of 3 and 5 sides with straight edge and
compass only. Gauss wanted a 17sided polygon on his grave, but the stonemason
refused because it would simply look like a circle.
Going well beyond this, Gauss showed that a regular nsided polygon can be
so constructed if n is a Fermat prime (2^2^k + 1, for an integer k). The
next Fermat primes after 17 are 257 and 65,537. In 1832, F.J. Richelot,
who taught mathematics at the University of Konigsberg, showed how to construct
a 257sided polygon.
In 1894, J. Hermes wrote out the construction of a regular 65,537 sided polygon.
It had taken him 10 years and took up 200 pages. 'Unfortunately, it is likely
to contain a mistake.'
Gauss proved the
'Fundamental Theorem of Algebra': A polynomial equation of
degree n has n roots, in the complex plane. The theorem provides an example
of where the theory of complex numbers is more complete than that of the real
numbers.

Christoph Bernoulli (1782 to 1863)

Charles Julien Brianchon (1783 to 1864) was a French mathematician
and chemist. He entered into the École Polytechnique in 1804
and studied under Monge, graduating first in his class in 1808,
after which he took up a career as a lieutenant in Napoleon's
artillery. Later, in 1818, Brianchon became a professor in the
Artillery School of the Royal Guard at Vincennes. Brianchon is
best known for his proof of
Brianchon's theorem (1810).

Friedrich Bessel (1784 to 1846)

Charles Xavier Thomas de Colmar (1785 to 1870), a Frenchman who served in the
French army, in 1820 produced the first really successful commercial calculating
machine. (See Schickard, Pascal, and Leibniz above.) By the 1860's, the machine
was commercially successful. Under different guises, it continued to be produced
for well over 100 years  to around 1920? or 1960?

Claude Louis Marie Henri Navier (1785 to 1836)

Jean Victor Poncelet (1788 to 1867)

Augustin Cauchy (1789 to 1857) provided rigorous definitions of the
concept of continuity of curves and their derivatives. He put the
field of 'real and complex analysis' on firm ground.

Jean Francois Champollion (1790 to 1832)

Abel (ca 1824)

August Mobius (1790 to 1868) conceived of the Mobius strip, which has
just one edge and one surface.

George Everest (1790 to 1866)

Charles Babbage (1791 to 1871) designed a 'difference engine' and an
'analytic engine'  motivated by his thought: "I am thinking that these
tables [logarithms] might be calculated by steam."
The 'difference engine' had a fairly precise task  to calculate and print
out the values of certain functions. Known values of the required function
were entered and the machine then calculated other values (thus the 'difference').
The 'analytic engine' was like a modern computer, except that it worked
mechanically rather than electronically. It was programmable, in that it could
carry out any mathematical task. It could branch and it could loop.
The analytic engine was never actually built. Babbage kept changing his mind
about what he wanted and quarreled with everyone working with him. The British
government withdrew from the project in 1842. The secretary of the Royal Astronomical
Society wrote: "We got nothing for our 17,000 pounds but Mr. Babbage's grumblings.
We should at least have had a clever toy for our money."

Nicolai Lobachevsky (1792 to 1856) developed a system of geometry
in which Euclid's 5th postulate (on parallel lines) no longer holds.
(See Janos Bolyai.)

Ernst Weber (1795 to 1878)

Jakob Steiner (1796 to 1863)

Gustav Theodor Fechner (1801 to 1887)

Janos Bolyai (1802 to 1850) developed a system of geometry
in which Euclid's 5th postulate (on parallel lines) no longer holds.
(See Nicolai Lobachevsky.)

Neils Abel (1802 to 1829), a Norwegian, in 1821, showed that there is no
algebraic formula for the roots of the general quintic equation.

William Rowan Hamilton (1805 to 1865) developed quaternions, which can be
used to model positions in four dimensions.

Thomas Kirkman (1806 to 1895) posed a problem in combinatorics called
'Kirkman's schoolgirl problem'. Sudoku puzzles are an example of these
types of combinatoric problem.

Joseph Liouville (1809 to 1882), in 1844, constructed the first
transcendental numbers  numbers that were not the solutions of algebraic
equations (composed of polynomials).

Benjamin Peirce (1809 to 1880)

Evariste Galois (1811 to 1832) developed a theory of groups of permutations
of solutions to polynomial equations which was used to show that, although
quadratic, cubic, and quartic equations had a general solution in the form of
an algegraic formula, quintic (and higher) equations do not have such a general
solution.
'Group theory' became a major area of mathematics. It has been extened to many
other structures besides permutations and is used in nuclear and quantum physics.

Johann Gustav Bernoulli (1811 to 1863)

Pierre Wantzel (1814 to 1848)  Constructions with straight edge and
compass only (like the 3 Greek problems of trisecting the angle, squaring the
circle, and doubling the cube) were shown to be equivalent to using the four
basic operations of arithmetic ( +  x / ) and taking square roots. The way was
then clear to show that the Greek problems were insoluble. If a problem
involves a length which cannot be built up from 1 by + ,  , x , / , and squre
root, then it cannot be constructed and the problem cannot be solved.

George Boole (1815 to 1864) set up a system to codify logical argument
as a form of algebra. This is used in the design of computer circuitry.

Karl Wierstrass (1815 to 1897) is called, by some, "the father of
modern analysis". He put calculus on more solid ground by introducing
more rigor in definitions and proofs.

John Tyndall (1820 to 1893)

Hermann Ludwig Helmholz (1821 to 1894)

Charles Hermite (1822 to 1901) showed in 1873 that the number 'e' is
transcendental.

Francis Galton (1822 to 1911) develped the 'correlation coefficient'.

Jule Antoine Lissajous (1822 to 1880)

Bernhard Riemann (1826 to 1866)  The Riemann Hypothesis is a conjecture
about the zeros, in the complex plane, of an infinite series. If this hypothesis
can be proved to be true, it will provide the solution to many other unsolved
problems.

James Clerk Maxwell (1831 to 1879) devised a set of equations summarizing
the behavior of electricity and magnetism.

Richard Dedekind (1831 to 1916)

Alvan Graham Clark (1832 to 1897)

Charles Lutwidge Dodgson  alias Lewis Carroll (1832 to 1898) besides
writing 'Alice in Wonderland', was a mathematician who devised a paradox
in which a tortoise traps Achilles into an infinite regress of implication.
There is a story that the British Queen Victoria (18191901; reigned 18371901)
was so delighted by the Alice books that she asked Dodgson to send her his
next publication. She was surprised, and probably disappointed, to receive
'An Elementary Treatise on Determinants'.

Alexander Henry Rhind (1833 to 1863) acquired the Rhind
Papyrus. (See 'Ahmes', an Egyptian scribe, at the top of
this list.)

John William Strutt (Lord Rayleigh) (1842 to 1919)

Georg Cantor (1845 to 1918) proved that the rational numbers can be
counted. Developed many other results on countability or noncountability
('infinities') of number systems.

Arnold Buffum Chace (1845 to 1932)

Julius Plucker (1801 to 1868) was a German
mathematician and physicist. He made fundamental
contributions to the field of analytical geometry and
was a pioneer in the investigations of cathode rays
that led eventually to the discovery of the electron.
He also vastly extended the study of Lamé curves.
In 1865, Plücker invented what was known as 'line geometry'
in the nineteenth century. In 'projective geometry',
Plücker coordinates refer to a set of homogeneous
coordinates introduced initially to embed the set of
lines in three dimensions as a quadric in five dimensions.

Friedrich Ludwig Gottlobb Frege (1848 to 1925)
was a German mathematician, logician and philosopher.
He is considered to be one of the founders of modern
logic and made major contributions to the foundations
of mathematics.

Felix Klein (1849 to 1925) conceived of the Klein bottle  a twisted
shape that has no inside or outside, and exists only in four dimensions.
It can be made by joining two Mobius strips edge to edge, in four dimensions.

Oliver Heaviside (1850 to 1925)

Sonia Kovalevsky (1850 to 1891)

Ferdinand von Lindemann (1852 to 1939) proved that pi is transcendental
 that is, not a solution of an algebraic equation. This showed that the
ancient Greek problem of 'squaring the cirle' (with straight edge and
compass only) was impossible.

Elisha Scott Loomis (1852 to 1940) was an American teacher,
mathematician, genealogist, writer and engineer. Probably his
bestknown work is his book 'The Pythagorean Theorem', in
which he collected, classified, and discussed more than
340 proofs.

Egraf Stepanovich Fedorov (1853 to 1919) showed that precisely 17
wallpaper patterns exist, where a wallpaper pattern is a regular arrangement
of shapes, repeated across a plane. If all possible combinations of translations,
rotations, and reflections are considered, there are essentially
17 different types of pattern.

Henri Poincare (1854 to 1912) was a founder of the field of topology.
It is said that "Topologists think that a doughnut is the same as a coffee cup."
Topologists are not content with objects in two or three dimensions.
They extend the definitions of topology into four or more dimensions.

Aleksandr Lyapunov (1857 to 1918)  the Central Limit Theorem  the mean
of a large enough sample is approximately normal  that is, if enough data
is taken, the normal distribution can be used to calculate probabilities.

Giuseppe Peano (1858 to 1932) constructed a 'spacefilling curve 
a curve which is continuous,
no breaks in it, which passes through every point of a square  even though
a square, unlike a line, is two dimensional and has nonzero area.

Max Planck (1858 to 1947)  a quantum physicist

Florian Cajori (1859 to 1930) was an American
historian of mathematics.

D'Arcy Wentworth Thompson (1860 to 1948) was a
Scottish biologist, mathematician, and classics scholar.
A pioneering mathematical biologist, he is mainly
remembered as the author of the 1917 book 'On Growth and Form'.
The book pioneered the scientific explanation of morphogenesis,
the process by which patterns are formed in plants and animals.

David Eugene Smith (1860 to 1944) was an American
mathematician, educator, and editor. He became a professor
of mathematics at Teachers College, Columbia University
(1901) where he remained until his retirement in 1926.
Smith became president of the Mathematical Association of
America in 1920. He wrote a large number of publications of
various types: He was editor of the Bulletin of the
American Mathematical Society; contributed to other
mathematical journals; published a series of textbooks;
translated Felix Klein's 'Famous Problems of Geometry',
Fink's 'History of Mathematics', and the 'Treviso Arithmetic'.
He edited Augustus De Morgan's 'A Budget of Paradoxes' (1915)
and wrote several textbooks on teaching mathematics.

Alfred North Whitehead (1861 to 1947) was an English
mathematical logician and philosopher.

David Hilbert (1862 to 1943), like Peano, devised a spacefilling curve.
In a famous speech in 1900, he identified a set of unsolved mathematical problems
to be attacked in the next century. The Riemann hypothesis and Goldbach's conjecture
were on his list and they were still unsolved as of 2008  over a century later.
To solve a Hilbert problem has become a high ambition for any mathematician.

Hermann Minkowski (1864 to 1909) was a
LithuanianGerman mathematician. He used geometrical methods
to solve problems in number theory, mathematical physics,
and the theory of relativity.

Jacques Hadamard (1865 to 1963), like Charles de la Vallee Poussin,
proved that the density of the prime numbers thins out in a logarithmic fashion.

Charles de la Vallee Poussin (1866 to 1962), like Jacques Hadamard, proved
that the density of the prime numbers thins out in a logarithmic fashion.

Theodore_Andrea_Cook (1867 to 1928) wrote
'The Curves of Life: Being an Account of Spiral Formations
and Their Application to Growth in Nature, to Science and
to Art: with the special reference to the manuscripts of
Leonardo da Vinci'(1914). According to Cook,
Mark Barr, an American mathematician, in about 1909, gave
the golden ratio the name of phi, the first Greek letter in the
name of Phidias, the Greek sculptor who lived around 450 BC

Helge von Koch (1870 to 1924) developed the snowflake curve, which is
jagged everywhere.

Ernst Zermelo (1871 to 1953) was a German logician and
mathematician, whose work has major implications for the
foundations of mathematics. He is known for his role in
developing ZermeloFraenkel axiomatic set theory and his
proof of the wellordering theorem, which originally
involved use of the 'Axiom of Choice'.

Bertrand Russell (1872 to 1970)  Russell's paradox threatened the
foundations of mathematics

Gino Loria (1862 to 1954) was an Italian mathematician
and historian of mathematics. Loria did research on
projective geometry, special curves and rational
transformations in algebraic geometry, and elliptic functions.
He is best known as a historian of mathematics. He wrote a
history of mathematics and was especially concerned with
the history of mathematics in Italy and among the ancient Greeks.

Edmund Georg Hermann Landau (1877 to 1938) was a German
mathematician who worked in the fields of number theory and
complex analysis. In 1903, Landau gave a much simpler proof
than was then known of the prime number theorem and later
presented the first systematic treatment of analytic number
theory in the 'Handbuch der Lehre von der Verteilung der
Primzahlen'.

Albert Einstein (1879 to 1955)  'nuf said.

Oswald Veblen (1880 to 1960) was an American
mathematician, geometer and topologist, whose work
found application in atomic physics and the theory of
relativity. He proved the Jordan curve theorem in 1905.
The Jordan Curve Theorem states that a continuous
closed curve cuts the plane into two distinct parts. There
are so many 'pathological' curves, such as the snowflake
curve and spacefilling curves, that a complete proof is
very tricky.

Lipot Fejer (1880 to 1959) was a Hungarian mathematician.
In 1911, Fejér was appointed to the chair of mathematics
at the University of Budapest and he held that post until
his death. He was the thesis advisor of mathematicians
such as John von Neumann, Paul Erdos, George Pólya and
Pál Turán. Fejér's research concentrated on harmonic
analysis and, in particular, Fourier series. Fejér
collaborated to produce important papers, such as
a major work with Frigyes Riesz in 1922 on conformal
mappings (specifically, a short proof of the Riemann
mapping theorem).

Liutzen (L.E.J.) Brouwer (1881 to 1966) was a Dutch
mathematician and philosopher, a graduate of the
University of Amsterdam, who worked in topology, set theory,
measure theory and complex analysis. He worked with
fixed point theorems  in particular, the 'hairy ball theorem'
 if every point on a sphere is assigned a direction along
the sphere, there will be at least one point with no direction.

Emmy Noether (1882 to 1935) was an influential German
mathematician known for her groundbreaking contributions
to abstract algebra and theoretical physics. Described
by Pavel Alexandrov, Albert Einstein, Jean Dieudonné,
Hermann Weyl, Norbert Wiener and others as the most
important woman in the history of mathematics, she
revolutionized the theories of rings, fields, and algebras.
In physics, Noether's theorem explains the fundamental
connection between symmetry and conservation laws.

Tobias Dantzig (1884 to 1956) was a Baltic German
Russian American mathematician and the author of 'Number:
The Language of Science  A critical survey written for the
cultured nonmathematician' (1930) and 'Aspects of Science'
(1937).

George Polya (1887 to 1985)  Hungarian  emigrated to the U.S. in 1940 
In mathematics education, wrote a set of principles for solving mathematical
(and other) problems  published in 'How to Solve It' (1957).

Aleksandr Friedmann (1888 to 1925) was a Russian and
Soviet physicist and mathematician. He is best known for
his pioneering theory that the universe is expanding,
governed by a set of equations he developed now known as
the Friedmann equations.

Abraham Frankel (1891 to 1965)  set theory  Zermelo and Frankel
put set theory on a secure axiomatic foundation, clarifying what is a set
and what is not.

Stefan Banach (1892 to 1945)  Given three solids, there is a plane
cutting each of the three solids exactly in half. A later, more general
result is called the StoneTukey theorem, which states:
Given n shapes in ndimensional space, there is an (n  1) dimensional plane
that cuts each of the shapes in half.
Because there can be very 'hairy' shapes, this proof can be expected to be
difficult  like the Jordan Curve Theorem (see Osvald Veblen).

Mauritz Cornelius Escher (1898 to 1972) was a Dutch
graphic artist. He is known for his often mathematically
inspired woodcuts, lithographs, and mezzotints. These
feature impossible constructions, explorations of infinity,
architecture, and tessellations.

Dietrich Mahnke (1884 to 1939) was a German philosopher
and historian of mathematics. Mahnke's work in the history
of mathematics focussed primarily on Leibniz's development
of the infinitesimal calculus. At the time of his death,
Mahnke was editing of volume of Leibniz's mathematical
correspondence. This project was then taken over by
Joseph Ehrenfried Hofmann. Mahnke was killed in a car accident.

Karl Menger (1902 to 1985) was an American mathematician.
He is credited with
Menger's theorem. He worked on mathematics of algebras,
algebra of geometries, curve and dimension theory, etc.
Moreover, he contributed to game theory and social sciences.
He has been associated with
'the traveling salesman problem' :
A traveling salesman must visit a number of cities. What is
the most efficient route?

Oskar Morgenstern (1902 to 1977) was a Germanborn
economist. In collaboration with mathematician John
von Neumann, he founded the mathematical field of
'game theory' and its application to economics.

John von Neumann (1903 to 1957) was a Hungarian pure
and applied mathematician. He made contributions in many
areas of mathematics and physics, including game theory.
He worked on the Manhattan project, which
was set up in 1942 to develop atomic bombs.

George Stibitz (1904 to 1995), a Bell Labs researcher,
in 1937 built a machine, on his kitchen table,
that did binary arithmetic. He is known for his work in
the 1930s and 1940s on the realization of Boolean logic
digital circuits using electromechanical relays as the
switching element. (See Claude Shannon, below.)

George Gamow (1904 to 1968) was a theoretical physicist
and cosmologist  notably an early advocate and developer
of Lemaître's Big Bang theory. He discovered a theoretical
explanation of alpha decay via quantum tunneling, and
worked on radioactive decay of the atomic nucleus,
star formation, stellar nucleosynthesis and Big Bang
nucleosynthesis (which he collectively called
nucleocosmogenesis), and molecular genetics. In his middle
and late career, Gamow focused more on teaching, and
became well known as an author of popular books on science,
including 'One Two Three ... Infinity'.

Marian Rejewski (1905 to 1980) a Polish mathematician who succedded in
breaking the basic Enigma code of the Germans near the start of world war II.

Tommy Flowers (1905 to 1998), a telecommunications and electronics expert,
using his experience of the British telephone system, proposed the 'Colossus'
electronic computer for decoding topsecret German messages.

Albert W. Tucker (1905 to 1995) dealt with the prisoner's dilemma, the most famous
example in game theory: Two suspects are arrested. Should each one stay silent
or confess to the crime?

Alexandr Osipovich Gelfond (1906 to 1968) was a Soviet
mathematician. His most famous result is the Gelfond theorem:
If alpha and beta are algebraic numbers (with alpha not equal
0 or 1), and if beta is not a real rational number, then any
value of alphatothebeta is a transcendental number.
Before Gelfond's work, only a few numbers such as e and pi
were known to be transcendental. After his work, an infinite
number of transcendentals could be easily obtained. Some of
them are named in Gelfond's honor: 2^sqrt(2) is known as
the GelfondSchneider constant, and e^pi is known as
Gelfond's constant.

Kurt Godel (1906 to 1978) showed the 'consistency' of 'the
Continuum Hypothesis'  the hypothesis that there is no
infinity between the infinities of natural and real numbers.
That is, he showed you cannot disprove it. (See Paul Cohen.)

John Mauchly (1907 to 1980) helped develop ENIAC, the first, generalpurpose
electronic computer. (See Eckert below.)

Alan Turing (1912 to 1954) described machines that define what is
and what is not possible in terms of computing. He built on what Rejewski
had done to break the Enigma code of the Germans on a daily basis, as the
code was changed each day, during world war II.

Paul Erdos (1913 to 1996) wrote an astonishing number of mathematical
papers (1,500), wandering from country to country with only a suitcase,
collaborating with other mathematicians.
The Erdos number is named after him. It is a measure of collaboration with Erdos.
1 means direct collaboration. The smaller the Erdos number of a mathematical
collaborator, the more closely the collaborator is connected with Erdos.

Claude Shannon (1916 to 2001) in 1937 showed how to design circuits
for binary arithmetic.

Charles Frederick Mosteller (1916 to 2006) was one of
the most eminent statisticians of the 20th century. He was
the founding chairman of Harvard's statistics department,
from 1957 to 1971. Mosteller wrote over 50 books and over
350 papers, with over 200 coauthors. He was an avid fan of
the Boston Red Sox and attended the games in the run up
to the World Series. He conducted perhaps the first academic
investigation of baseball after his favorite team, the
Boston Red Sox, lost the 1946 World Series. He has been
mentioned with 'the Secretary Problem'  choosing the
best secretary from a pool of candidates.

Edward Norton Lorentz (1917 to 2008) was an American
mathematician and meteorologist, and a pioneer of chaos
theory. He discovered the strange attractor notion and
coined the term 'butterfly effect'.

J. Presper Eckert (1919 to 1995) helped develop ENIAC, the first, generalpurpose
electronic computer. (See Mauchly above.)

Rene Thom (1923 to 2002)  Catastrophe theory deals with slight changes
that can lead to great changes. Thom sought to find a mathematical way
to analyze catastrophic changes.

Benoit Mandelbrot (1924 to 2010) did work on fractals.

Wolfgang Haken (1928 to ?), with Kenneth Appel, proved the 4 Color Conjecture
 with the help of about 1,200 hours on a computer.

Kenneth Appel (1932 to 2013), with Wolfgang Haken, proved the 4 Color Conjecture
 with the help of about 1,200 hours on a computer.

Roger Penrose (1931 to ?), an English physicist, invented several
'tesselations' of the plane, built up from two quadrilaterals known
as 'kites' and 'darts', due to the shape of those quadrilaterals.
These tesselations are not periodic: they do not repeat themselves
and any two parts of the tesselation are essentially different from
each other.

Paul Cohen (1934 to 2007) showed the 'independence' of 'the
Continuum Hypothesis'  the hypothesis that there is no infinity
between the infinities of natural and real numbers. That is, he showed
the continuum hypothesis is undecidable  it can be neither proved
nor disproved from the standard ZF (ZermeloFraenkel) axioms.
(See Kurt Godel above.)

Robert May (1938 to ?) is an Australian scientist who studied
chemical engineering and theoretical physics (B.Sc. 1956) and
received a Ph.D. in theoretical physics in 1959. He has provided
a model for population growth that takes limited resources into account.
Reference: His 1976 paper "Simple mathematical models with very
complicated dynamics".

John Horton Conway (1937 to ?) is a British mathematician
active in the theory of finite groups, knot theory, number theory,
combinatorial game theory and coding theory. He has also contributed
to many branches of recreational mathematics, notably the invention
of the cellular automaton called 'the Game of Life', which has been
implemented as a computer program.

Yuri Matiyasevich (1947 to ?) showed that if there is a rule for finding
the elements of a set, the search for the elements can be carried out by
an algebraic formula
 and no matter how complicated the rules are for listing the terms, the
rules can be expressed just in therms of +, , and x. In short, every recursively
enumerable set is Diophantine.
In the 17th century two formulae were suggested for listing primes: 2^n  1 and
2^2^n + 1. In fact, the primes are a recursively enumerable set. Just look at each
number in turn and if it is a prime, write it down in the list.
Matiyasevich realized that there must be a Diophantine formula for listing the
primes but he thought it would be too complicated to write down. But J. P. Jones
found such a formula in 1976. See J.P. Jones.

James P. Jones ( ? to ?) in 1976 found a formula to list the primes. This
used 26 variables, which was very convenient, otherwise he would have had
to extend the alphabet. See page 195 of 'The Little Book of Mathematical Principles'
mentioned in the Introduction at the top of this page.

Stephen Cook (1939 to ?)  If P = NP were shown to be true, the
secrecy of codes (encryption) would be under threat. P = Polynomial ;
NP = Near Polynomial. P = NP means if a problem is 'tractable' with a lucky
guess (NP), then it is tractable without such a guess (P).

Andrew Wiles (1953 to ?) proved Fermat's conjecture that there are no
solutions in positive integers of x^n + y^n = z^n, if n is larger than 2.
x=3, y=4, z=5 is a solution for the case when n=2.
The case for n=3 was proved in 1770  for n=5 in 1825  for n=7 in
1839. (Only prime number need to be considered.) Then all cases up to
n=100 were proved, but no 'complete' solution was found until 1994.

Thomas C. Hales (1958 to ?) provided a proof of the conjecture on the
most efficient way to stack spheres (which is the way grocers stack their
oranges). Hale's proof was so complicated that it could not be done by hand.
Like the 4 Color Conjecture, it needed very many cases to be checked by a
computer, and so, as with the 4 Color proof, it has been reluctantly accepted
by mathematicians.