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In the book 'Problem Solving and Number Theory' I read

The law of quadratic reciprocity was discovered for the first time, in a complex form, by L. Euler who published it in his paper entitled “Novae demonstrationes circa divisores numerorum formae $xx + nyy$ .”

When and who introduced the notation $x^2$ ? What is the name for this notation? ( Not scientific, is it? )

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See Earliest known use of symbols of mathematics in the "Operations" page, under "Exponents". First use of modern notation for positive integers seems to date back to Descartes's Geometry, except for squares. For negative integers and fractions, Newton. I believe it is called "exponential notation". –  Arturo Magidin Oct 26 '11 at 20:04
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You should check the book by FLorian Cajori about the history of notations. It has a whole chapter on the history of exponents! –  Mariano Suárez-Alvarez Oct 26 '11 at 20:10
    
"And... Descartes tended not to use 2 as an exponent, however, usually writing $aa$ rather than $a^2$, perhaps because $aa$ occupies no less space than $a^2$." - That's it, then. ( If you create an answer, I'll accept it. Thank you ). –  ndroock1 Oct 26 '11 at 20:15
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Cajori's book is available on archive.org, the relevant section on exponents starts here. –  t.b. Oct 26 '11 at 20:18
    
Just had a quick browse on Google ebooks. The book by Florian Cajori ( two volumes actually ) is a gem! - Thanks for mentioning. –  ndroock1 Oct 26 '11 at 20:22
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up vote 12 down vote accepted

According to this page the earliest known use of integers to represent repeated multiplication is by Nicole Oresme in the mid 1300s. However, he didn't use a raised integer notation. The rest of this answer is taken from that page.

Nicolas Chuquet used raised integers in 1484, though for him $12^3$ was a shorthand for $12x^3$.

In 1636 James Hume used roman numerals as exponents, e.g. for $12^3$ he would have written $12^\textrm{iii}$, but apart from that minor distinction he was essentially using modern notation.

Rene Descartes used raised arabic numericals as exponents in 1637, with the exception that he tended to write $xx$ rather than $x^2$, though he would still write $x^3$, $x^4$ etc. He wrote:

...$aa$ ou $a^2$ pour multiplier à par soiméme; et $a^3$ pour le multiplier encore une fois par $a$, et ainsi à l'infini.

which roughly translates as

...$aa$ or $a^2$ to multiply by itself, and $a^3$ to multiply again by $a$, and so ad infinitum.

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I don't know specifically who, but I recall that the notion was already invented during Euler's time.

It was just conventional to write $xx$ instead of $x^2$, i.e. one would write $x, xx, x^3, x^4, \ldots$. This is probably similar to why we write $f',f'', f^{(3)}, f^{(4)}, \ldots$ for the notation of a derivative.

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So in Euler's time it was convention to write $xx$ instead of $x^2$ ? –  ndroock1 Oct 26 '11 at 20:09
    
Using $xx$ lasted for a long time. Even as late as Riemann's "Über die Anzahl der Primzahlen unter einen gegebenen Grösse" [Wikipedia]. –  kahen Oct 26 '11 at 22:22
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In their modern form, exponents were introduced by Descartes in the early $1630$s, at the same time as $x$. There are numerous precursor forms of the exponent.

Although Descartes used the notation $x^n$ for $n \ge 3$, he ordinarily used $xx$ instead of our $x^2$. The notation of Descartes was fairly quickly widely adopted, with England as usual being more cautious. The form $x^2$ was used by some people, the form $xx$ by others. Euler used both. I believe he used $x^2$ far more often than $xx$. Maybe he thought $xx$ looked nice in a title.

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