# Questions tagged [mathematics]

For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns.

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### Is there a 'lost calculus'?

Are there any 'lost' theorems of calculus that could be used to 'simplify' it? For example, are there ways to calculate derivatives without using limits, maybe by some forgotten methods in calculus?
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### Why is "Cardano's Formula" (wrongly) attributed to him?

Apparently, Cardano had learned a formula for solving cubic equations from Tartaglia, who had sworn him to secrecy, and in any event, not to publish it without giving Tartaglia due credit. Cardano ...
590 views

### When do we see for the first time the use of the Cartesian coordinates?

I want to see an exact image of the first use of the Cartesian plane. I guess it came the first time with Descartes.
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### Pythagoras vs. the idea of Pythagoras

Maybe we need some replies on current scholarly thinking. (Judging from some replies here, many of us are still using the myths current 100 years ago.) Is it true (as I have heard) that most, if not ...
1k views

### Was 18th century algebra more symbolic/formal than the modern conception?

I've found Lagrange's Sur la résolution des équations algébriques to be a very confusing and difficult read, and I think I'm starting to see why: it seems that Lagrange thinks of algebra in a much ...
2k views

### How were irrational numbers accepted by mathematicians?

What was behind accepting the existence of irrational numbers historically? Especially numbers that are not constructible on the real number line, say for example $\sqrt{2}$. Was it a (somewhat) ...
340 views

### Viète's Relevance and his Connection to Euler

Viète's equations are used in some proofs of the Basel problem, which was allegedly solved by Euler. Viète's equations include the following: given a polynomial, $$a_0 + a_1x+a_2x^2 + ... + a_nx^n$$ ...
2k views

### Historically, how did people define multiplication for negative numbers?

Which were the first mathematical developments to state that the product of two negative numbers is a positive number, and what was their justification for this choice? I am not interested in a modern ...
2k views

### What was the motivation for the development of modern, intrinsic, differential geometry?

I know that tensor calculus was developed around the same time as general relativity. Tensor calculus was the prime way to deal with geometric objects, based on expliciting all coordinates and doing ...
6k views

### Whose shoulders did Newton stand on?

In a letter to Robert Hooke in 1676, Newton wrote: If I have seen further it is by standing on the shoulders of giants. Do we know which giants Newton was referring to? And was he referring to a ...
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### Current ways of thinking in the History of Mathematics

As a research mathematician, working in number theory, who is interested in the history of his own field, I have done some reading in the History of Mathematics, particularly that of Ancient Greek and ...
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### Which came first, the natural logarithm or the base of the natural logarithm?

The natural logarithm function ($\ln x$) and the base of the natural logarithm function ($e$) are both extremely useful. They're also both closely related: $\ln (e^x)=x$, and $e^{\ln x}=x$. But which ...
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### Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?

The question is in the title, but allow me to provide some background. I’m aware that Leibniz introduced the word “function” into mathematics (around 1673) and that Johann Bernoulli or Euler ...
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### What was the notion of limit that Newton used?

I have read that the notion of limit became rigorous two centuries after the discover of calculus What Newton had in his mind regarding the notion of limit?
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### How was geometry historically used to solve polynomial equations?

I'm researching historical use of geometry to find solutions to polynomial equations. I'd like to ask for those familiar with this topic, could you describe the use of geometry by early mathematicians ...
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### The origin of quadratic equation in actual practice

I read that in ancient times the quadratic equation of this kind $$x^2+10x=39$$ had been solved long ago. I read that this kind of equation originated in the geometric question of "Given an area of 39,...
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### Introduction of $\imath$ and $\jmath$ notations for the imaginary unit

The imaginary unit is generally denoted $i$ or $\imath$. I have learned that the term imaginary ("imaginaires") was coined by R. Descartes in 1637, and the "i" notation was introduced by L. Euler (cf. ...
2k views

### When and how was the geometric understanding of gauge theories developed?

In theoretical physics, the modern perspective on gauge theory is that it is most elegantly described in the 'language' of differential geometry. I am interested in the history behind these ideas. ...
7k views

### What was the historical context of the development of Taylor series?

I knew about linear approximations, quadratic approximations and the use of Taylor polynomials to approximate a function. Furthermore, I was aware of other applications of Taylor polynomials and the ...
9k views

### What is the etymology behind sine, cosine, tangent, etc.?

I heard somewhere that it was actually a mistake in translation. What's the correct story?
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### Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...
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### Did Galileo's writings on infinity influence Cantor?

To what extent was Cantor motivated by Galileo's paradox? More generally, to what extent were late 19th century mathematicians motivated by, or even aware of, Galileo's paradox? This is an issue I've ...
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### When did it become understood that irrational numbers have non-repeating decimal representations?

I know that the notion of irrational number (in one form or another) goes back to the Pythagoreans, and therefore far predates the decimal system, and certainly the representation of non-integer ...
4k views

### When was zero actually introduced in mathematics?

Children learn counting things, naturally like, 1, 2, 3, ... and so on. Because it seems obvious to them. But, zero is something we need to teach them about. As far as my understanding goes zero was ...
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### Who invented short and long division?

I am curious who came up with algorithms that we use today to manually solve mathematical division problems, such as short or long division; how were they established or standardized that way and why?...
619 views

### Cartesian coordinate system in Newton's work

In the english translation of Newton's work "Enumeratio linearum tertii ordinis" by C.R.M. Talbot, we can see in a figure the depiction of a Cartesian coordinate system pretty much as we know it today:...
562 views

### When were the concepts of pure and applied Mathematics introduced?

I know that there are no standard definitions for pure and applied mathematics however I would like to know who first considered them as two separate entities, I have seen people mention it was around ...
13k views

### What is the difference between Calculus of Newton and that of Leibniz?

Are there any differences between the study of Calculus done by Newton as compared to that done by Leibniz? If yes, please mention point by point.
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### How did Napier come to invent logarithms?

What was Napier's original logic, leading to his invention of logarithms? In other words, how did Napier, using the mathematics that was available at that time, derive them?
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### Why did the ancient Greeks originally become interested in conic sections?

How much is known, or can be conjectured, about why the Greeks originally became interested in the somewhat arbitrary construction of intersecting a plane with a cone? The folklore that I've heard is ...
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### Why were geometers dissatisfied with the parallel postulate?

Euclid himself already treats it with gloves, it has an unusually precise formulation, and is not used in the first 28 propositions of the Elements. Why? Did he doubt it? It's not like Euclid was a ...
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### Who gets credit for the real numbers?

If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more ...
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### Origin of 360 degrees?

This is by far one of the most challenging and popular HSM questions on the Net. Proofs are, countless discussions about it in math forums. The answers only led to two theories, which Wikipedia does a ...
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### When did people start viewing a matrix as a linear transformation between two vector spaces?

The notion of a matrix appeared far ahead of that of a vector space. So when did people start considering a matrix as a linear transformation between two vector spaces?
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### Who discovered the power rule for derivatives?

Who discovered the general rule for differentiating polynomials, in particular that the derivative of $x^n$ is $n x^{n-1}$, and when? I appreciate the answer may not be a clear-cut individual and year,...
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### Historical roots of the justification for the rule for multiplication of negative numbers

As a follow up question with respect to : Who wrote down minus times minus is equal to plus? and to : Historically, how did people define multiplication for negative numbers?, it can be interesting to ...
4k views

### Irrationality of the square root of 2

We know that Pythagoreans in Ancient Greece discovered that the square root of two is an irrational number. Why was that discovery historically significant? What value was that knowledge to the ...
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### Why is the radical symbol $\sqrt{}$ called "radical"?

This question arose in a conversation with a teacher who was introducing square roots to her students. I know from the website Earliest Uses of Symbols of Operation that the symbol $\sqrt{}$ has its ...
1k views

### Did Poincaré say that set theory is a disease?

This question has been discussed on several sites including MathOverflow but with not definite result. Presumely HSE is best suited. Jeremy Gray denies that Poincare said, "Later generations will ...
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### Has a digit ever been used to represent the number "10"?

Ten is special to humans, as there are 10 fingers on two hands, and fingers are still the basic counting medium for people. So, was there any digit representing the number "10" in a positional system ...
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### Was there a very early culture that's number system was 12-based, like ours is 10-based?

There are several uses of 12 in some old systems of measurement. Some of them make sense given current context (There are 12 lunar cycles per year), however some of them seem to be arbitrarily chosen. ...
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### What led to the fall of Göttingen?

Göttingen was the place in which many important mathematicians such as Riemann worked. It was also one of the main locations for the development of quantum theory in the twenties (e.g. Heisenberg, ...
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### What is the origin of polynomials and notation for them?

This may be quite a broad question, but lately I've been wondering about the history behind polynomials. Nowadays these are pretty much the simplest kind of functions to work with, but I'd like to ...
3k views

### What was Euler's motivation for introducing $i$ for $\sqrt{-1}$?

[Mauro Allegranza has answered the question of who introduced the notation $i$ (Euler, followed later by Gauss), so I have changed the title. I have also edited the question in other ways to make it ...
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### Who introduced the Principle of Mathematical Induction for the first time?

Can you tell me the name of the mathematician, who introduced the Principle of Mathematical Induction for the first time? (with reliable source). Please don't say De Morgan because I have read the ...
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### Who discovered the covering homomorphism between SU(2) and SO(3)?

Who discovered this? It is quite nontrivial and very important in quantum mechanics.
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### Who invented the Leibnitz notation $\frac{d^2y}{dx^2}$ for the *second* derivative?

This MSE question made me wonder where the Leibnitz notation $\frac{d^2y}{dx^2}$ for the second derivative comes from. It does not arise immediately as the obvious generalization of $\frac{dy}{dx}$. ...
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### Why did mathematicians not see that $f_n(x)=x^n$ is a counterexample to Cauchy's "theorem" about limits of continuous functions?

In 1821 Cauchy claimed that the limit of a sequence of continuous functions is continuous. In 1826 Abel gave a complicated trigonometric counterexample. When we teach analysis courses, we usually give ...