Difficulty: Introductory | Prerequisites: None (this is background/context material for MATH 104)
Big Picture
This material covers the historical origins of calculus: who developed it, what they published, and how the subject was placed on rigorous mathematical footing. It sits at the very beginning of a Calculus 1 course and provides context for the tools and notation you will use all semester. You do not need prior calculus knowledge, but you should be comfortable with the idea that mathematics has evolved over centuries through competing approaches and published works. If your course skips this section on exams, it still helps to know why the notation looks the way it does and where the core ideas came from.
TL;DR
Calculus was developed independently by Newton and Leibniz in the late 1600s. Newton called it "fluxions"; Leibniz published first and gave us much of the notation we still use. The subject was later made rigorous through formal axioms, definitions, and the resolution of philosophical objections to infinitesimals.
Mathematical Physics
A branch of applied mathematics that uses mathematical methods to solve problems in physics, the study of matter, motion, energy, and force.
Think of it as the toolkit that lets physicists turn real-world phenomena into equations they can solve.
Cycloid
The curve traced by a point on the rim of a circular wheel as it rolls along a straight line without slipping. It is an example of a roulette, a curve generated by one curve rolling on another.
In simple terms, imagine sticking a pen to the edge of a bicycle tyre and watching the line it draws as the wheel rolls forward.
Principia Mathematica (Philosophiae Naturalis Principia Mathematica)
A three-volume work by Isaac Newton, first published 5 July 1687, in Latin. It sets out Newton's laws of motion (the foundation of classical mechanics), Newton's law of universal gravitation, and a derivation of Kepler's laws of planetary motion.
In simple terms, this is the single most important book in the history of physics, the one that unified motion on Earth and in the heavens under one set of laws.
Method of Fluxions
A book by Isaac Newton, completed in 1671 but not published until 1736. In it he described what we now call differential calculus. "Fluxions" was Newton's own word for derivatives.
Think of it as Newton's private manuscript on calculus that the world only saw decades after he wrote it.
Nova Methodus pro Maximis et Minimis
The first published work on calculus, by Gottfried Leibniz, appearing in the Acta Eruditorum in 1684. It is considered the birth of infinitesimal calculus as a public, shared discipline.
In simple terms, this is the paper that introduced calculus to the wider mathematical community, using notation closer to what we still write today.
Science of Fluxions
Isaac Newton's own name for what we now call calculus.
Think of it as the original brand name for the subject: Newton called it fluxions, Leibniz called it calculus, and Leibniz's name stuck.
Infinitesimal
A quantity that is infinitely small, not zero but smaller than any positive real number. Early calculus relied on infinitesimals before limits provided a rigorous replacement.
In simple terms, it is a number so tiny it is practically zero, yet not quite zero, used to reason about rates of change and areas under curves.
Mathematical Rigour
The practice of adhering absolutely to logical constraints and predefined rules in mathematical reasoning. In proofs, every step must follow from previous steps with no gaps or hand-waving.
Think of it as the difference between "this seems right" and "this is provably, airtight correct."
Foundations (of Calculus)
The rigorous development of calculus from precise axioms and definitions, replacing the informal reasoning about infinitesimals that Newton and Leibniz originally used.
In simple terms, this is the project (carried out mainly in the 1800s) that put calculus on solid logical ground so mathematicians could trust every result.
Ghosts of Departed Quantities
A phrase coined by Bishop George Berkeley in The Analyst (1734) to mock infinitesimals. Berkeley argued they were logically incoherent: "They are neither finite Quantities nor Quantities infinitely small, nor yet nothing."
Think of it as the most famous criticism of early calculus, the complaint that eventually forced mathematicians to replace infinitesimals with limits.
Isaac Newton (1643–1727)
Developed calculus independently, calling it the "Science of Fluxions."
Completed Method of Fluxions in 1671, though it was not published until 1736, well after his death.
Published Principia Mathematica in 1687, applying his new mathematics to physics: laws of motion, universal gravitation, and planetary orbits.
Newton's approach to calculus was geometric and physical. He thought in terms of quantities that "flow" (change continuously), and their rates of flow (fluxions).
Gottfried Wilhelm Leibniz (1646–1716)
Developed calculus independently of Newton.
Published Nova Methodus pro Maximis et Minimis in 1684, making it the first published work on calculus.
Leibniz introduced the notation that survived: the integral sign (∫), the d-notation for differentials (dx, dy), and the product/quotient structures we still use.
The priority dispute between Newton and Leibniz over who invented calculus first is one of the most famous quarrels in the history of science.
Maria Gaetana Agnesi (1718–1799)
Italian mathematician, philosopher, theologian, and humanitarian.
The first woman to write a mathematics textbook and the first woman appointed as a mathematics professor at a university.
Wrote one of the earliest and most complete works on finite and infinitesimal analysis in 1748 (Instituzioni analitiche ad uso della gioventu italiana).
Known for the "Witch of Agnesi," a curve studied in her textbook (the name comes from a mistranslation).
Bishop George Berkeley (1685–1753)
An Irish philosopher, not a mathematician, who published The Analyst in 1734.
Criticised the logical foundations of calculus, calling infinitesimals "ghosts of departed quantities."
His critique was not an attack on the results of calculus (which he accepted as useful) but on the reasoning used to derive them.
Berkeley's objections pushed later mathematicians to develop the rigorous foundations of calculus using limits instead of infinitesimals.
The Problem with Early Calculus
Newton and Leibniz both relied on infinitesimals to build calculus, but neither could define them precisely. An infinitesimal was treated as a nonzero quantity when needed for division, then treated as zero when convenient for simplification. This logical inconsistency worked in practice (the results were correct) but troubled mathematicians and philosophers.
Berkeley's Critique
In The Analyst (1734), Berkeley pointed out that if you divide by a quantity and then set it to zero, you have contradicted yourself. His phrase "ghosts of departed quantities" captured the absurdity: the infinitesimal has "departed" (been set to zero) yet its ghost (the ratio it produced) remains. Berkeley did not reject the usefulness of calculus, only its logical justification.
The Move Toward Rigour
The resolution came in the 1800s, primarily through the work of Cauchy, Weierstrass, and others who replaced infinitesimals with the epsilon-delta definition of a limit. This is the "foundations" of calculus: rebuilding every theorem from precise axioms so that no step relies on hand-waving about infinitely small quantities.
This is why your Calculus 1 course begins with limits. The limit concept is the patch that fixed the logical hole Berkeley identified.
Why Notation Matters
Leibniz's notation (dy/dx, ∫) won out over Newton's dot notation (ẏ) because it was more flexible and easier to manipulate algebraically. When you write dy/dx in your course, you are using Leibniz's framework. When your physics course writes ẍ for acceleration, that is Newton's.
Students often think Newton published calculus before Leibniz. He did not. Leibniz published first (1684). Newton developed the ideas earlier but kept them largely private until Principia (1687), and Method of Fluxions was not published until 1736.
Students often assume the Newton-Leibniz dispute means one of them copied the other. The historical consensus is that they developed calculus independently.
Students sometimes confuse "fluxions" with "flux" from physics. Newton's fluxions are rates of change (derivatives), not the same concept as electromagnetic flux or fluid flux.
Students often think infinitesimals were proven wrong. They were not. Berkeley's critique was about logical justification, not about whether calculus produced correct results. In the 20th century, Abraham Robinson showed infinitesimals can be made rigorous through nonstandard analysis.
⚠️ Know which major work belongs to which mathematician: Principia and Method of Fluxions are Newton; Nova Methodus is Leibniz. Exam questions love matching works to authors.
⚠️ Be able to explain what "ghosts of departed quantities" refers to and who said it (Berkeley, The Analyst, 1734). This is a favourite short-answer or fill-in-the-blank question.
⚠️ Understand why calculus needed to be put on rigorous foundations: the informal use of infinitesimals was logically inconsistent, even though it produced correct results.
⚠️ Know that Maria Gaetana Agnesi was the first woman to write a mathematics textbook and the first woman appointed as a university mathematics professor. Her 1748 work covered both finite and infinitesimal analysis.
⚠️ The date 1684 (Leibniz's publication) and 1687 (Newton's Principia) are commonly tested. Leibniz published on calculus first; Newton published Principia (which used calculus for physics) three years later.
True or False: Newton published his calculus work before Leibniz.
False. Leibniz published in 1684; Newton's Method of Fluxions was not published until 1736.
Fill in the blank: Berkeley called infinitesimals "________ of departed quantities."
Ghosts.
True or False: A cycloid is the curve traced by a point on the rim of a rolling wheel.
True.
Fill in the blank: Newton's own term for calculus was the "Science of ________."
Fluxions.
True or False: Maria Gaetana Agnesi was the first woman to be appointed a mathematics professor at a university.
True.
Q: What is the Principia Mathematica, and why is it significant?
A: A three-volume work by Isaac Newton, published in 1687. It sets out Newton's laws of motion, the law of universal gravitation, and derives Kepler's laws of planetary motion. It is regarded as one of the most important works in the history of science.
Q: Who published the first work on calculus, and when?
A: Gottfried Leibniz, in 1684, with Nova Methodus pro Maximis et Minimis, published in the Acta Eruditorum.
Q: What did Bishop Berkeley mean by "ghosts of departed quantities"?
A: Berkeley was criticising the use of infinitesimals in calculus. He argued they were logically incoherent: treated as nonzero for division, then discarded as zero. The "ghost" is the result that remains after the quantity itself has been set to zero.
Q: What is the difference between Newton's and Leibniz's approaches to calculus?
A: Newton thought in terms of physical motion and "fluxions" (rates of flow). Leibniz developed a more algebraic, symbolic approach with the dy/dx and integral notation we use today. Both arrived at equivalent results independently.
Q: What does "foundations of calculus" refer to?
A: The rigorous development of calculus from precise axioms and definitions, replacing the informal reasoning about infinitesimals. This was achieved mainly in the 19th century through the epsilon-delta definition of limits.
Q: Why is Maria Gaetana Agnesi notable in the history of mathematics?
A: She was the first woman to write a mathematics handbook and the first woman appointed as a mathematics professor at a university. Her 1748 work was one of the most complete treatments of finite and infinitesimal analysis of its time.
This material connects directly to limits, the next major topic in MATH 104. The epsilon-delta definition of a limit is the modern answer to Berkeley's critique of infinitesimals. Understanding why limits were needed gives you context for the formal definitions you will encounter next.
It also connects to derivatives and integrals, which are the core of the course. Newton's fluxions are what we now call derivatives; Leibniz's notation (dy/dx, ∫) is how you will write them. The historical context explains why calculus has two parallel notations.
More broadly, this material connects to mathematical proof and rigour, which matters throughout any mathematics degree. The story of calculus shows that correct results can come from shaky reasoning, and that making the reasoning airtight is a separate (and important) project.
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