I came across an algebra problem book written in 1899 for students of Dar al-Fonun ([dɒːɾolfʊˈnuːn], meaning, "polytechnic college",) the only modern educational institute in Iran at the time (established in 1851). This period marks a significant transition in algebra in Iran, where, algebra gradually shifted from being expressed in words to using symbols. There was no public education system as such, so most books intended for the very few readers outside Dar al-Fonun typically only covered topics up to quadratic equations.
However, this particular book, written by a graduate of Dar al-Fonun at the request of his teacher, contains a collection of 1,000 algebra problems. The book does not provide solutions, only the answers. However, at the start of the book and each section, there are usually the formulas needed or very short explanations of the methods. Towards the end of the book, there are a couple of cubic equations, the solutions to which are given first as continued fractions and then as decimals. The author notes that since the method has been discussed in his algebra book, there’s no need for further explanation. However, the algebra book does not exist (if it ever existed at all).
There was a tradition of attempting to solve cubic equations in Iran, starting with Khayyam, and there are records of Islamic/Iranian mathematicians providing numerical solutions to cubic equations. However, there is no known record of using continued fractions for this purpose, suggesting that this method was likely learned from elsewhere. I am curious about where this knowledge might have originated. Here are three (out of five) examples provided in the book. $$x^3-2=0$$
$$ x = 1 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{5 + \cfrac{1}{1 + \cfrac{1}{1 + \dots}}}}}=1.25992...$$ $$x^3-15x^2+63x-50=0$$ $$x = 1 + \cfrac{1}{35 + \cfrac{1}{1 + \cfrac{1}{1+\dots}}}=1.02803 $$
Notice that if we write $1.02803$ as a continued fraction, we get $[1; 35, 1, 2, \dots]$ rather than $[1; 35, 1, 1, \dots]$. This suggests the possibility that the author derived the continued fraction independently of the decimal solution. This becomes even more evident in the following equation, where the author provides a continued fraction that does not match the decimal solution:
$$x^3 - 12x^2 + 57x - 94 = 0$$ $$ x = 3 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{5 + \cfrac{1}{3 + \dots}}}} = 3.36216\ldots $$ $$ (3.36216 = [3; 2, 1, 3, 5, \dots]) $$
Could the author have approached the continued fraction form of the solution independently of the decimal form? If so, how? Was there any similar work in Europe? Whether he had access to such work, and how, is another matter. Or, did he simply make a mistake or simplification in converting decimal solutions to continued fractions? If this was the case, what was the most popular or well-known work in Europe at the time that might have been the source? Knowing of a similar work in Europe could help trace how and by whom this knowledge traveled to Iran.
