History of Mathematics

The area under the curve and the number π

The third book of the Instituzioni deals with integral calculus: areas under curves, volumes, lengths. The versiera gives a surprising example. The curve never touches the $x$-axis; it stretches out without end on both sides. Even so, the region between the curve and the axis has a finite area. A well-known antiderivative With $a = 1$, the curve is $y = \frac{1}{1 + x^2}$. We need a function whose derivative is $\frac{1}{1 + x^2}$. It exists and is an old acquaintance: the arctangent. If $u = \arctan x$, then $x = \tan u$ and

October 29, 2026 Read

Quaternions: the formula on the bridge

After showing that complex numbers are couples of real numbers and that their multiplication rotates the plane, Hamilton wanted to do the same in space. He was looking for triplets $a + bi + cj$ with a multiplication that respected the law of the moduli: the size of the product should be the product of the sizes. He tried for years, above all in 1843, so much so that his children asked him every morning whether he could multiply triplets yet.

October 27, 2026 Read

The von Neumann architecture: the EDVAC report

The ENIAC, finished in 1945 at the University of Pennsylvania, was the fastest electronic computer in the world: it did five thousand additions a second with almost eighteen thousand valves. But to change calculation one had to rewire cables and switches on a panel, a job of days done by a team of programmers. Two programmers operating the ENIAC’s control panel, at the Moore School of the University of Pennsylvania, around 1946.United States Army · Wikimedia Commons · public domain The ENIAC’s engineers, J. Presper Eckert and John Mauchly, were already planning the next machine, the EDVAC, in which the program would be kept in memory. Von Neumann began attending the meetings and, in June 1945, wrote a text that organised the group’s ideas in an abstract, almost biological language, speaking of “organs” and “neurons”.

October 26, 2026 Read

The universal machine: the program is data too

Each Turing machine does one thing only. The one that adds 1 adds 1; the one that writes $\frac17$ writes $\frac17$. To do a different calculation, you would need a different machine. In 1936 that was normal: mechanical calculators, punched-card tabulators and tide-predicting machines were built for one task, and changing task meant changing machine or rewiring. In the same paper, Turing showed that this is not necessary. A machine that reads machines The table of rules of any machine can be written as text on a tape. Turing then described a special machine, $U$, the universal machine, which does the following:

October 18, 2026 Read

Napier's bones: multiplying with rods

Logarithms were meant for the great calculations of astronomy. For everyday sums Napier invented something else, published in 1617, the year of his death, in a little book called Rabdologia, from the Greek rhabdos, rod, and lógos, calculation: calculation with rods. The rods Each rod, or “bone”, because the most expensive ones were made of ivory, carries at the top a digit from 0 to 9 and, below it, its multiples from 1 to 9, in nine small squares cut by a diagonal. The tens go above the diagonal and the units below. The rod for 7, for example, shows $0/7$, $1/4$, $2/1$, $2/8$, $3/5$, $4/2$, $4/9$, $5/6$, $6/3$.

October 17, 2026 Read

Galois imaginaries: finite fields

In June 1830, aged 18, Galois published a short paper, On the theory of numbers, in a Paris scientific bulletin. In it he invented new numbers, which he called imaginaries, for a world in which only finitely many numbers exist. For more than a century the paper looked like a curiosity. Today it is inside every mobile phone. Counting with remainders In clock arithmetic we count only with the remainders of division by a fixed number. With the number 5 there are only $0, 1, 2, 3, 4$, and the sums “wrap around”:

October 16, 2026 Read

Dividing the lemniscate with ruler and compass

In 1694 Jakob Bernoulli studied a curve shaped like a figure eight lying on its side, which he called lemniscus, a ribbon bow. With its foci at the points $(\pm\frac1{\sqrt2},0)$, it is the set of points whose distances to the two foci have product $\frac12$. Its equation is $$(x^2+y^2)^2=x^2-y^2,$$or, in polar coordinates, $r^2=\cos2\theta$. A century and a half later Abel proved about it a theorem that seems to have come out of Euclid’s book: for which numbers $n$ can the lemniscate be divided into $n$ arcs of equal length using only ruler and compass?

October 16, 2026 Read

The Nash bargaining solution

A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc. — John Nash, The Bargaining Problem (1950) In 1948, still an undergraduate at the Carnegie Institute of Technology, Nash took a course in international economics, the only economics course of his life. He came out of it with a question: when two people negotiate, how do they share the gains from a deal? Economists thought the answer depended on each side’s psychology, skill and bluffing, and that there was nothing to say in general. Nash showed that, if the deal must satisfy a few reasonable requirements, there is only one possible answer. The paper, written before he arrived at Princeton, was published in 1950 in the journal Econometrica.

October 15, 2026 Read

Maxwell's equations: from Faraday's lines of force to vectors

I was at first almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well. — Michael Faraday, letter to Maxwell (25 March 1857) Until 1820, electricity and magnetism were separate subjects. Electricity made sparks and raised the hair on your head; magnetism made compasses point north. That year, during a lecture at the University of Copenhagen, the Dane Hans Christian Oersted connected a battery to a wire and saw a nearby magnetic needle move. Electric current produced magnetism.

October 14, 2026 Read

Four squares are enough

Take any positive whole number and try to write it as a sum of perfect squares, $0, 1, 4, 9, 16, 25, \ldots$, using as few terms as possible: $$\begin{aligned} 5&=4+1,\\ 6&=4+1+1,\\ 7&=4+1+1+1,\\ 30&=25+4+1,\\ 2026&=2025+1=45^2+1^2. \end{aligned}$$Some numbers are squares, others need two, others three. The number 7 needs four: there is no other way. Could some number need five? In 1770 Lagrange proved that no number does: Every positive whole number is a sum of four squares of integers.

October 14, 2026 Read

The exponential calculus: differentiating x to the power x

Leibniz’s calculus knew how to differentiate powers with a fixed exponent, like $x^3$ or $x^{1/2}$: the derivative of $x^n$ is $nx^{n-1}$. But in 1694, in a letter to Leibniz, Johann Bernoulli put a strange creature on the table: the expression $$y=x^x,$$in which the base and the exponent vary at the same time. The rule for $x^n$ does not work, because the exponent is not fixed. Nor does the rule for $a^x$, because the base is not fixed. In 1697 Johann published in the Acta Eruditorum the “principles of the exponential calculus”, with the method for handling these powers, which he called percurrent, because the exponent runs through values.

October 13, 2026 Read

The Sophie Germain identity: factoring what looks prime

Some numbers look prime and are not. 629, for example, is not even, does not end in 5 and is not a multiple of 3. But $629=5^4+4$, and that form gives it away: it factors by an identity that bears Sophie Germain’s name. The identity For any numbers $a$ and $b$, $$\begin{aligned}a^4+4b^4={}&\left(a^2+2b^2+2ab\right)\\&\times\left(a^2+2b^2-2ab\right).\end{aligned}$$The trick for finding it is completing the square, as in the quadratic formula. The sum $a^4+4b^4$ is almost the square of $a^2+2b^2$; only the middle term, $4a^2b^2$, is missing. Adding and subtracting that term,

October 13, 2026 Read