28

Documents like the one you linked to were not typed on a typewriter. When writing on typewriters, it was common to leave some space in the document in which the formulas could be inserted by hand. For professional publications, this was then given to a printer (the person, not the machine on your desk ;-)) who hot metal typeset the document. Printers had ...


25

See Earliest uses of mathematical symbols, which quotes F. Cajori, A History of Mathematical Notations, 2 volumes (1928-29) The use of z, y, x ... to represent unknowns is due to René Descartes, in his La géometrie (1637). Without comment, he introduces the use of the first letters of the alphabet to signify known quantities and the use of the last ...


22

Leibniz did use this notation for instance in his paper Supplementum geometriae practicae, Acta Eruditorum, April 1693, p. 179 (Google Books link):


20

According to Florian Cajori, A History of Mathematical Notations (1928 - Dover reprint), Vol II, page 128 : 498. It was Euler who first used the letter $i$ for $\sqrt -1$. He gave it in a memoir presented in 1777 to the Academy at St. Petersburg, and entitled "De formulis differentialibus etc.," but it was not published until 1794 after the death of Euler....


20

Notation $()$ is traditional, and $].[$ was introduced by Bourbaki. Much of the Bourbaki notations and terminology became standard, but English speaking people are the most conservative ones in this respect:-) (Recall the history of the metric system:-) Another example of the same is "injection", "surjection", "bijection". Many English authors still write ...


20

I just want to comment that the agreement on letters, by which we write $\frac d{dt}\mathbf L=\mathbf M$ for the law of angular momentum, must have come very late -- after 1964. As evidence, note that it is still written $\frac d{dt}\mathfrak N=\mathfrak M$ by Sommerfeld in Mechanik (1943, p.63); $\frac d{dt}\mathbf M=\mathbf L$ by Sommerfeld in Mechanics (...


19

It is really funny to read that in the beginning of 21-st century, some young people may think that journals and books printing had something to do with typewriters:-) If you look attentively at the page you scanned you will easily see that this is not a TeX font and not a typewriter. It is more beautiful. Before the middle 1990-th we lived in the "...


19

Before approximating roots Al-Samawal performs long division of 210 by 13 to five decimal places, not enough to notice that digits cycle after the sixth. And this is the problem with discovering it experimentally in general, rationals may have arbitrarily long periods (repetends), not to mention preperiods. According to Dickson, Al-Maridini in 15th century ...


18

"Radical" comes from Latin "radicalis", having roots, an equivalent "radix" was also commonly used earlier. While Rudolff did use the radix in 1525 his did not have the overbar on top, now called vinculum, Latin for bond. That innovation was added by Descartes in La Geometrie (1637). Before that all sorts of tricks had to be resorted to for grouping terms "...


16

Browsing the "original" historiographical" source for the "power" symbolisms , i.e. : Florian Cajori, A History of Mathematical Notations (1928, Dover reprint), page 335-on can be very istructive, showing how Descartes' choice was the last step of a complex process : Pietro Antonio Cataldi : $5$ $3(crossed)$ for $5x^3$ Joost Bürgi : with $vi$ on top of $8$...


15

We have price on the vertical axis because that's how Alfred Marshall (1890) drew his graphs in Principles of Economics. For better or worse, Principles was hugely influential. And so the present-day convention is Marshall's convention. As Humphrey (1992) writes: The Marshallian cross diagram bears Marshall's name because he gave it its most complete, ...


15

Yes, it is. In Giuseppe Peano's Arithmetices Principia (1889), the $\epsilon$ symbol is explained as follows (page x): Signum $\epsilon$ significat est. [The sign $\epsilon$ means is.] In his Principi di Logica matematica (1891), Peano gives the full explanation (page 3) : Per indicare la proposizione singolare « $x$ è un individuo ...


14

There was a related question on Math.SE, which Mauro Allegranza answered with reference to Cajori's classic History of Mathematical Notations (v.II, p.205). It is a great source and is freely available online. Surprisingly, it was not Leibniz, the notational lion of calculus, who introduced it. "A provisional, temporary notation $\Delta$ for differential ...


13

Yes, it has been: , or more stylized , the depression made by the tip of a Babylonian wedge shaped stylus on a clay tablet. When a circular stylus was used (rarely) the symbol was just $\bigcirc$. The earliest positional system was sexagesimal, with base 60, so it had cuneiform symbols for all digits from 1 to 59. Babylonians used it since before 2000 BC for ...


13

Although many problems that we now reduce to polynomial equations were solved since time immemorial early occurences are coached in verbal and/or geometric terms, and polynomials are not treated as separate items. For early occurences of geometric problems that lead (today) to quadratic equations see The origin of quadratic equation in actual practice. The ...


12

George Boole introduced the concept of empty set, or "nothing" as he called it, as the complement to the "universe" in his Mathematical Analysis of Logic (1847). His notations for them were somewhat boring, $0$ and $1$ respectively. Cantor wrote in 1880 "for the absence of points we choose the letter $O$". Frege, the founder of mathematical logic, ...


12

The $r$ is for "radius", and in particular, describes the radial vector from the origin to the location described by the vector. This is sensible because some sort of polar or spherical coordinates are the most common for many physical applications, where the forces described have some sort of spherical symmetry, and point radially outward.


12

On page 10 of that book the author wrote The most important example of a ground field is the field of common rational numbers for which I use the freely invented symbol 9... where he uses the symbol in question instead of the 9 that I used above.


11

from "Earliest Uses of Various Mathematical Symbols" Infinity. The $\infty$ symbol was introduced by John Wallis (1616-1703) in 1655 in his De sectionibus conicis (On Conic Sections) as follows: Suppono in limine (juxta Bonaventurae Cavallerii Geometriam Indivisibilium) Planum quodlibet quasi ex infinitis lineis parallelis conflari: Vel ...


11

It helps to remember that there was no consensus notation for logarithms well into 20th century, with $\mathrm{l}\,x$, $\log x$, and $\mathrm{Log}\,x$ often used by different authors and in different senses. The superscript notation was not peculiar to the Netherlands, at least not originally. According to Cajori's History of Mathematical Notations: "...


11

Gibbs (1889, p. 140): $ \qquad \dfrac{d\,\log\mathrm V}{d\,\log p} = - \dfrac{d\,\log n}{d\,\log\lambda} $ Riemann (1868, p. 89): $ \qquad \dfrac{d^2y}{dx^2}-\dfrac1{\alpha\alpha}\dfrac{d^2y}{dt^2}=4\dfrac{d\smash[t]{\dfrac{dy}{d(x+\alpha t)}}}{d(x-\alpha t)} $ Hamilton (1843, p. 362): $ \quad\ \dfrac d{d\log t}\,(1+t)^{-m}\,\mathrm T_m $ Jacobi (1841, p....


10

The history of the idea underlying the short/long/synthetic division turned out to be far more complicated than I expected, somewhat reminiscent of the history of $0$, with no single inventor. According to the Angelfire timeline the modern long division symbol of English-speaking countries is first used in the 1888 teacher's edition of The Elements of ...


10

Here's some detail. From Herschel's 'On a Remarkable Application of Cotes's Theorem', Philosophical Transactions, 1813: ' Gauss wrote a review of this article in the Göttingsche Gelehrte Anzeigen (reprinted in Werke, vol. 4, p. 361): I will translate the last sentence: What the author says about the notation $\cos^2 A$, which some newer mathematical ...


10

Your guess is right: the notation $\mathfrak o$ goes back to Dedekind. If you get a copy of Dirichlet-Dedekind's Vorlesungen über Zahlentheorie and look in Dedekind's famous XI-th Supplement, which was the first systematic development of algebraic number theory, you'll see $\mathfrak o$ starting in section 170 when Dedekind defines Ordnung (= Order).


10

Yes and no. Peano originally used $\epsilon$ in Arithmetices Prinicipia Nova Methodo Exposita (1889), and stated that the symbol was an abbreviation for Latin est (is), apparently using a Greek letter for a Latin word. However, as Mauro Allegranza pointed out, in Principi di Logica Matematica (1891) he changes the script, and explains the use of $\varepsilon$...


10

I believe it’s because this function was used, and denoted “psi”, much before it got a name. Indeed, it looks like $(\log\circ\,\Pi)'$ and $(\log\circ\,\Gamma)'$ first occur in Euler (1755, pp. 797-801; 1769, p. 17), resp. Legendre (1810, p. 502), with no special name or notation. Surveys like Brunel (1886, p. 58; 1899, p. 162) or Jensen (1916, p. 140) ...


10

$ {\def\Target#1{\rlap{\smash{\label{#1}\phantom{\tag{#1}}}}}} {\def\BackUp{\raise{0.25em}{\Tiny{\boxed{\boldsymbol{\Uparrow} \hspace{-2px}}}}}} $tl;dr- It's unclear. The symbol $`` \hbar "$ itself wasn't anything new. Paul Dirac used it defining $\hbar \equiv \frac{h}{2 \pi}$ in a 1926 paper, but didn't explain the choice of the symbol. It might still be ...


9

I don't think that they had characters such as these in a typewriter. First off, that wasn't typewritten. Look at the "i"s in $\text{Variationsproblem}$ or the "fl" ligature in $\it\text{Kugelflächenfunction}$ (on the next page of the article). Typewriters use monospaced fonts, and they don't do ligatures. That article was typeset, printed in the venerable ...


9

You can see : Giuseppe Peano , Lezioni di Analisi Infinitesimale, 2 vols., 1893, page 17 : $$[f(x)]_{x=a}=f(a).$$ Not sure it is the earliest... but Peano was a prolific "inventor of notations". Regarding : how they express "$y$ under the condition that $x=2$" see e.g. page 34 [shortened] : let $y$ the natural logarithm of $x$ [...] and $f(x) = \...


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