1643–1727 · England
Isaac Newton
Newton set calculus in motion — literally. Working in the
1660s and 1670s he developed what he called the “method of fluxions”:
rates of change and the inverse problem of recovering a quantity from its rate. He
then used it as the engine of the Principia (1687), deriving planetary
motion and gravity. His bitter priority dispute with Leibniz split European
mathematics for a century.
Why it lands in class: Newton is the answer to “why do we care about rates?”
Velocity, acceleration and force are the intuition students already have; the
derivative is the machinery that makes it precise.
1646–1716 · Germany
Gottfried Wilhelm Leibniz
Leibniz arrived at calculus independently of Newton and published
it first (1684). His lasting gift is the notation: the differentials
dy/dx, the integral sign ∫, and the word
“function.” His symbols were so well designed that we still use them
three centuries later.
Why it lands in class: when students use the chain rule or substitution, they are
thinking in Leibniz's language. That is not trivia — good notation is what
makes the operations feel inevitable rather than magic.
1789–1857 · France
Augustin-Louis Cauchy
Calculus worked before Cauchy, but it was not yet on solid ground.
He built the foundations: the derivative defined as a limit, continuity as a precise
property, and the convergence tests that still carry his name. His courses at the
École Polytechnique turned analysis into the rigorous subject it became.
Why it lands in class: the limit definitions students meet in their first week are
Cauchy's. He is the reason “approaches” can be made exact instead of
hand-waved.
1815–1897 · Germany
Karl Weierstrass
Weierstrass finished the job Cauchy started. He gave analysis the
formal limit definition — the arithmetic of “for every epsilon there is a
delta” — and in 1872 shocked the field with a function that is continuous
everywhere and differentiable nowhere. Both are still standard syllabus material.
Why it lands in class: his counterexample is the cleanest way to break the
assumption that “continuous” means “smooth.” Hand a strong
student that surprise and you have taught differentiability proper.
1718–1799 · Italy
Maria Gaetana Agnesi
Agnesi wrote Instituzioni analitiche (1748), the first
comprehensive calculus textbook and one of the clearest of its era — so clear
that it was translated and used across Europe. The curve named after her, the
“witch of Agnesi,” still appears in graphing exercises. Bologna offered
her a chair in mathematics, an honour almost unheard of for a woman at the time.
Why it lands in class: Agnesi is the reminder that calculus was never only one
story. She also gives students a curve with a memorable name and a tidy equation to
plot.
1707–1783 · Switzerland, Russia and Prussia
Leonhard Euler
The most prolific mathematician who ever lived, and the one whose
shorthand is now unavoidable. Euler gave us f(x), the sigma sign for a
sum, and the letter e; he found the exact sum of 1/n² (the Basel
problem); and eiπ + 1 = 0 ties
together five of the most famous constants there are. He kept publishing for years
after going blind.
Why it lands in class: every time a student writes f(x) or presses ex on a
calculator, they are using Euler’s notation. His exponential connects the
growth-and-decay problems in the syllabus to the calculus they lead into.
1826–1866 · Germany
Bernhard Riemann
Riemann turned the integral from an intuition into a definition.
The Riemann sum — slicing an area into rectangles and taking a limit
— is his, and it is still how textbooks introduce integration. In a short life he
also rebuilt geometry on new foundations, groundwork Einstein later used, and left the
Riemann hypothesis, perhaps the most famous open problem in mathematics.
Why it lands in class: the rectangles students draw under a curve are Riemann’s.
Showing them why the limit of those sums is the exact area is showing them the
definition itself, not a trick.
1850–1891 · Russia and Sweden
Sofia Kovalevskaya
Barred from university in her own country because she was a woman,
Kovalevskaya studied abroad, proved the Cauchy–Kovalevskaya theorem on
differential equations, and in 1884 became the first woman to hold a chair in
mathematics, at Stockholm. The recognition came only after doors had been shut in her
face, repeatedly.
Why it lands in class: she is the honest answer to “who does mathematics?”
— a person who did first-rank work while being told the work was not hers to do.
1736–1813 · Italy and France
Joseph-Louis Lagrange
Lagrange gave calculus the notation students still write: the prime,
f′(x), for a derivative. He also cast the mean value theorem in
the form we teach (“the mean-value form”) and built a whole mechanics on the
principle of least action, work that underlies modern physics. Napoleon made him a
count.
Why it lands in class: the prime is Lagrange’s, and the Mean Value Theorem —
the bridge between a derivative and an average rate of change — carries the
modern statement of his name.
1601–1665 · France
Pierre de Fermat
Fermat worked out how to find the greatest and least values of a
curve — and the tangents to it — decades before calculus had a name. His
“method of adequality” is the ancestor of setting a derivative to zero: at a
peak or a trough, the slope vanishes. A lawyer and judge by trade, he did mathematics
as a hobby and still left a mark on nearly every branch of it.
Why it lands in class: the rule “a maximum or minimum is where the derivative is
zero” starts with Fermat, and the famous “last theorem” is his too.
1685–1731 · England
Brook Taylor
Taylor’s name is on the expansion that turns a smooth function into
a sum of powers of x — the series that makes polynomial approximation possible.
He also invented integration by parts and opened a branch now called the calculus of
finite differences.
Why it lands in class: the Taylor series is how a calculator produces sin x,
cos x and ex. Showing students that a curve becomes a polynomial they
can add up turns “approximation” from a trick into a reason.
1698–1746 · Scotland
Colin Maclaurin
Maclaurin wrote the first systematic exposition of Newton’s
fluxions, in part to answer Bishop Berkeley’s charge that the new calculus rested
on shaky foundations. The Maclaurin series — a Taylor series centred at zero
— carries his name, and so does the integral test for the convergence of a
series.
Why it lands in class: the series students expand at x = 0 and the
convergence test they apply to 1/np are both his. He is also a neat story:
rigour arrived, in part, because a critic attacked.
1768–1830 · France
Joseph Fourier
Fourier asked how heat diffuses through a solid and answered with a
new kind of series: any periodic signal can be built from sines and cosines. That idea
— the Fourier series, and the Fourier transform grown from it — now
underpins signal processing, acoustics and image compression.
Why it lands in class: Fourier is the bridge from the trigonometry students already
know to the analysis behind sound, music and JPEGs. A square wave built out of sine
waves is a demonstration worth ten minutes.
1777–1855 · Germany
Carl Friedrich Gauss
Gauss worked in nearly every branch of the subject — number
theory, analysis, astronomy, geodesy, magnetism and differential geometry — and
is often called the Prince of Mathematicians. He showed that the curvature of a
surface is unchanged when it is bent without stretching (the theorema
egregium), groundwork that led, through Riemann, on to Einstein. He also gave us
the bell curve and the method of least squares.
Why it lands in class: the normal distribution, the error function and least-squares
fitting all carry his fingerprints, and his curvature theorem is the reason a flat map
of the Earth can never be faithful.
1630–1677 · England
Isaac Barrow
Barrow developed tangent methods that came close to the derivative
before calculus had a name, and he was the first to recognise that integration and
differentiation are inverse operations. He taught Newton at Cambridge and in 1669
resigned the Lucasian Chair so that Newton could take it.
Why it lands in class: the inverse relationship Barrow spotted is the Fundamental
Theorem of Calculus in all but name. He is also the honest reminder that even Newton
stood on other shoulders.
1661–1704 · France
Guillaume de l’Hôpital
L’Hôpital’s Analyse des infiniment petits
(1696) was the first textbook written on the differential calculus: tangents and maxima
and minima from infinitely small quantities, then points of inflection, curvature and
higher derivatives, and in Chapter 9 the rule that still carries his name for a limit
whose numerator and denominator both tend to zero. He acknowledged his debt to Leibniz
and the Bernoullis in his introduction, and he needed to: when Johann Bernoulli’s
course notes came to light in 1921 the book was seen to follow them closely, under an
arrangement between the two men that had left Bernoulli unable to complain at the time.
Editions kept appearing until 1781, and it set the pattern for the calculus books that
followed.
Why it lands in class: the 0/0 limits that l’Hôpital’s rule rescues
turn up in every first course, and the naming story is worth two minutes — the
name on a result is a piece of history, not a certificate of authorship.
1882–1935 · Germany and the United States
Emmy Noether
Noether’s theorem came out of a problem in physics. Invited to
Göttingen in 1915 to help Hilbert with the mathematics of relativity, she proved
that to every infinitesimal transformation of the Lorentz group there corresponds a
conservation theorem — in plain terms, a symmetry of a system comes with a
quantity that does not change. Einstein, in a letter to Hilbert, called it penetrating
mathematical thinking. She went on to reshape algebra, where her study of chain
conditions on ideals of rings is why the subject looks the way it does. Dismissed from
Göttingen in 1933 as a Jewish mathematician, she moved to Bryn Mawr in Pennsylvania
and died there two years later.
Why it lands in class: the conservation laws students take as facts — energy,
momentum — are the visible edge of Noether’s theorem. It is the cleanest
route from a symmetry a student can see in a formula to a statement about the physical
world.
Current · MIT, United States
Gilbert Strang
Strang came to MIT as an undergraduate, went to Oxford as a Rhodes
Scholar, took his doctorate at UCLA and has taught on the MIT faculty since. His
teaching reached far beyond one lecture hall: MIT OpenCourseWare carries his linear
algebra course and his seventeen-video Highlights of Calculus series, and many
universities teach from his textbooks.
Why it lands in class: Highlights of Calculus covers the arc of a first course
in seventeen short videos — worth showing in class, and worth sending to the
student who missed a week.
Current · Lamar University, United States
Paul Dawkins
Dawkins teaches at Lamar University, and since 2003 he has put his
course notes on the web as Paul’s Online Notes: a complete, free set
covering Algebra, Calculus I, II and III and Differential Equations, with practice
problems and full solutions. They are written to be usable by anyone, not only his own
students — each course assumes nothing beyond the standard prerequisites.
Why it lands in class: this is the second explanation a student needs at eleven at
night — worked examples, practice with answers, a list of common errors and a
short guide to studying math, all free in one place.
Current · Macalester College, United States
David Bressoud
Bressoud is DeWitt Wallace Professor Emeritus at Macalester College,
a former president of the Mathematical Association of America and a former director of
the Conference Board of the Mathematical Sciences. His research asks the practical
question — what actually happens in a calculus classroom. He led
Characteristics of Successful Programs in College Calculus, a five-year NSF
study of mainstream Calculus I, then Progress through Calculus, which examined
the Precalculus-to-Calculus II sequence and what universities can change to help
students succeed, and he ran the project that produced The Role of Calculus in the
Transition from High School to College Mathematics, the report of a 2016 workshop.
Why it lands in class: if you want evidence rather than folklore about why students pass
or stall in first-year calculus, this is the body of work to start from — and his
monthly column keeps it readable.
Current · 3Blue1Brown, United States
Grant Sanderson
Sanderson makes the 3Blue1Brown videos, a channel whose whole aim
is to explain mathematics visually. It grew out of side projects he built while
studying mathematics and computer science at Stanford; after graduating he spent a
year at Khan Academy producing videos, articles and exercises, mostly in multivariable
calculus, and since the end of 2016 the channel and its companion site have been his
main work. The animation engine behind the videos, Manim, is his too.
Why it lands in class: the visual explanations build an idea from a picture before
the algebra arrives, which is what the student who can do the steps but cannot say
what a derivative means actually needs.
Current · Wolfram Research, United States
Eric Weisstein
Weisstein studied physics and astronomy at Cornell and Caltech and
took his doctorate at Caltech in 1996. In 1995 he began putting a large collection of
mathematical facts — accumulated since his teens — onto the early internet,
work that helped define how reference material would later be presented online. He
joined Wolfram in 1999 and MathWorld was unveiled at mathworld.wolfram.com that same
year. Since then he has worked on Wolfram|Alpha, where he maintains the majority of
the mathematical domains together with the graph, polyhedron and unit data
collections.
Why it lands in class: MathWorld is the reference you reach for when a student asks
what a Fresnel integral or a curl actually is — its calculus and analysis section
runs from the mean value theorem to divergence, each entry with the notation and the
conditions spelled out.
Current · Khan Academy, United States
Sal Khan
Khan is founder and CEO of Khan Academy, a nonprofit whose stated
mission is to provide a free, world-class education to anyone, anywhere. Its course
library includes a complete Calculus 1 course — eight units running from limits
and continuity through the derivatives rules, integrals, differential equations and
their applications — built from video lessons, practice sets and teacher tools,
alongside the rest of its maths, science and test-prep material.
Why it lands in class: it is free at the point of use, so the student who missed the
lesson can rewatch the same idea the same evening, and a teacher can assign it without
a purchase order.
Current · University of Arizona, United States
Deborah Hughes Hallett
Hughes Hallett is Professor of Mathematics at the University of
Arizona and a member of its graduate faculty, and Adjunct Professor of Public Policy at
the Harvard Kennedy School. She graduated from Cambridge and has taught in Turkey and
in Brunei. Her work is on strategies to improve the teaching of mathematics: with
Andrew Gleason at Harvard she organised the Calculus Consortium for Higher Education
and started a foundation to promote innovative curriculum and pedagogy. She has served
on committees for the National Academy of Sciences, is a fellow of the American
Association for the Advancement of Science, and is the author or co-author of seven
books.
Why it lands in class: how the subject is taught is itself a research question, and
she has spent a career on it and built an international consortium around the
answer.
Current · Mathispower4u, United States
James Sousa
Sousa created Mathispower4u, a library of video tutorials that work
through examples on camera. The collection is organised course by course, and Calculus
I, II and III each have their own video library — alongside arithmetic, algebra,
geometry, trigonometry, statistics, linear algebra and differential equations —
so a student can go straight to the topic they are missing and watch the whole problem
from the first line to the answer.
Why it lands in class: one video, one worked example, at the student’s own pace.
That is the shape of help a stuck student actually uses.
Historic entries are drawn from the MacTutor
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