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A previous question credits François Viète with introducing the well-known longhand method for the computation of square roots in digit-by-digit manner. This method is related to the binomial theorem.

However, in a German source ("Kleine Enzylopädie Mathematik". Leipzig, Bibliographisches Institut 1965) I came across a remark that the German mathematician Michael Stifel was already aware of this method in the 16th century.

Google provides a full scan of the book "Die Coss Christoffs Rudolffs, Mit schönen Exempeln der Cosz. Durch Michael Stifel Gebessert vnd sehr gemehrt" of 1571. This is a book on algebra written originally by Christoph Rudolff that was improved and expanded by Stifel.

The fourth chapter of the book describes the computation of square roots and cube roots. Starting at Fol. 42 the procedure for taking square roots is demonstrated, using the computation of $\sqrt{72352036}$ as an example. What is described definitely appears to be the longhand square root algorithm taught in school until a few decades ago, before the advent of affordable electronic calculators.

Wikipedia does not have much on Rudolff, but notes that "He introduced the radical symbol (√) for the square root." Since Stifel's additions to Rudolff's "Coss" of 1515 seem to be carefully indicated in the 1571 edition, it would seem that the longhand square root method was already described in Rudolff's original, but I have not managed to track down a scan of that yet. It is also not clear whether Rudolff invented the method. As Stifel's edition makes no such claim, I think it is possible that the algorithm predates Rudolff and possibly the 16th century.

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Whenever you wonder about who first introduced something mathematical into Western Europe, Leonardo of Pisa aka Fibonacci should be the first person you think of. Fibonacci's 1202 work Liber Abaci has a chapter dedicated to the finding of square and cube roots, with an introduction referencing Euclid and Al-Khwarizmi, and then launching into a demonstration of finding the square root of 10.

Google Books has a preview of L. E. Sigler's translation, with the relevant section here.

It can be difficult to parse Fibonacci's notation-free verbiage (F. calls "first" what we would call the "last" digit and vice versa), but if you read the chapter introduction you see it explained that it's based on the fact that $(a+b)^2 = a^2 +2ab+b^2$ (just as the modern method is) and as you go through the chapter (I used my printed copy) there are several examples of square root extractions with diagrams that extract the roots digit by digit.

Incidentally, Fibonacci does not instruct the reader to "mark off pairs of digits starting from the right" as an initial setup step. Instead, he tells us to count the digits (his examples are all of integers) and understand that when a number has $2n$ or $2n-1$ digits, the square root has $n$ digits (in its integer part). And if $2n-1$ digits, the leftmost group has only one digit.

That isn't an actual difference in the method; it's is just another way of wording the initial step, and the difference in wording is not integral to the method.

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    $\begingroup$ "Liber Abaci" was my guess when the question arose in a comment on Software Engineering Stackexchange. However, the description in the linked translation at Google Books does not look like the classical longhand method to me; at least not in any form I recognize. $\endgroup$ – njuffa Nov 30 '16 at 1:38
  • $\begingroup$ @njuffa I've expanded my answer a little but you seem to be introducing a new requirement that isn't apparent in the question. $\endgroup$ – Spencer Nov 30 '16 at 2:39
  • $\begingroup$ I appreciate the clarification, based on which this does seem to be equivalent to the longhand method I referred (and linked) to; I introduced no new requirements. Google Books doesn't give me access to the entire text, which makes it difficult for me to assess the description. One characteristic part of the longhand method is that numbers are partitioned into pairs of digits starting at the right, I didn't see that. Another characteristic is the computation of the next partial remainder via subtraction of $(20a+b)b$, which I also couldn't identify. Both are clear in Stifel's text. $\endgroup$ – njuffa Nov 30 '16 at 3:37
  • $\begingroup$ @njuffa It is a preview of a copyrighted translation and has a limted amount of viewing made available. The examples that follow in the printed book take pairs of digits starting at the left (I think you meant that anyway) with an implicit $0$ for a number with an odd number of digits. It can be confusing because F. calls "first" what we would call the "last" digit and vice versa. $\endgroup$ – Spencer Nov 30 '16 at 4:01
  • $\begingroup$ I understand why Google imposes viewing restrictions on works still under copyright. I did mean partitioning the number into digit pairs starting from its right end (for the integer examples used here; more generally, starting at the decimal point) and working to the left. The left-most group then comprises one or two digits. Since we just need that left-most group to determine the first digit of the result, by consulting a (mental) table of $n^{2}$, for $n=1,\cdots,9$, that is a perfectly fine way to get started. $\endgroup$ – njuffa Nov 30 '16 at 4:13

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