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 was first to note periodicity, in a particular example with sexagesimals. Leibniz observed in 1677 that expansion of $1/n$ is periodic for any base (he later added - relatively prime to $n$). From that time on patterns in repeating decimals (a.k.a. periodic fractions) were studied in the context of number theory.
The earliest mention that all repeating decimals are rationals comes late, surprisingly late, by Lambert in 1758, he justifies it by using the now familiar trick with the geometric series. In Euler's Elements of Algebra (1765), "one of the earliest books to set out algebra in the modern form... and one of Euler's few writings that are accessible to the general public" we find the same trick $9.999...=9+9/10+9/10^2+...=\frac{9}{1-1/10}=10$. However, according to the formal attitude of the 18th century, e.g. Euler's we also have $1+2+2^2+\dots=\frac{1}{1-2}=-1$, by definition. It took some time before Cauchy tied summing series to convergence. It is also clear from Euler's writings that he knew the converse, as probably did Leibniz and Lambert, that decimal expansion of every rational terminates or repeats, but they did not spell it out.
It seems surprising today but that rational numbers have decimal expansions that terminate or repeat, and that irrational numbers have non-terminating and non-repeating expansions are two distinct problems. This is because the connection between them relies on two things: that there are not just examples like $\sqrt{2}$ or $\pi$, but a species of things called "irrational numbers", and that completed, infinitely extending strings of decimal digits represent anything at all, and specifically their specimen. These realizations only crystallized in late 19th century. A symbolic year is 1872, when Cantor, Dedekind and Heine independently published their constructions of real numbers as a species. After 1872 acceptance was quick, a reply to a question in an 1889 issue of The Bizzarre shows that by then they were already commonly known as numbers having non-terminating and non-repeating decimal expansions.
In another book of Katz's the quoted passage is reproduced almost verbatim, except for one word:"one can potentially calculate an infinite decimal expansion of a number". This goes back to the view of Aristotle and Euclid that all infinity is only potential, not actual, not completed, which continued to exert heavy influence on mathematicians until late 19th century. Infinite strings of digits were mostly contemplated as processes, not finished objects. A notable exception was Stevin's De Thiende (1585), largely responsible for acceptance of the decimal system in Europe (it helped that Stevin was more engineer than mathematician). But rejection of actual infinities infected even the identification of repeating decimals with rationals, let alone of "lawless" strings with irrationals. This issue is deep seated, even today many students have Aristotelian intuitions of infinity in repeating decimals, according to Tall "interviews revealed that students continued to conceive of $0.999...$ as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are'".
It is interesting to compare this to a similar issue for continued fractions. Fowler argues that already Theaetetus (c. 417 – 369) had a heuristic understanding that simple terminating continued fractions correspond to commensurable ratios and periodic ones to ratios "commensurable in square" (rationals and quadratic irrationals in modern terms). But it was stated and proved explicitly only around the same time as for repeating decimals, by Euler in De Fractionlous Continious (1737) and Lagrange in Sur la Resolution des Equations Numeriques (1767), respectively.