# Were decimal fractions known in Europe before Stevin?

It is commonly[1,2] held that Simon Stevin introduced the decimal number system with the decimal point (at least in Europe) in his 1585 book De Thiende. However, in della Porta's book Magia Naturalis, Book 17, Chapter 16, there is a clear reference to a decimal point as 360.0.0.0. You can see the 1658 English translation here and the 1619 Latin version. I do not have access to the original 1558 version. Are these post factum additions of the 17th century (possibly by della Porta himself) or is it truly an anticipation of the decimal system?

[1] J.T. Derrese, G Varden Berghe, "The Wonderful World of Simon Stevin" Chapter 3; page 58: "It is clear that Stevin regarded De Thiende as his own invention and that he was very proud of it."

[2] Dirk Jan Struik, "A Source Book in Mathematics, 1200-1800", page 7, "The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende"

• No, it is not "commonly held". Stevin promoted decimals in Europe, and he was influential. But he did not invent them, nor even claimed to do so. They were imported to Europe from Islamic Middle East long before Stevin, and described in Fibonacci's Liber Abaci back in 1202, for example. What Stevin did pioneer was identifying irrationals with infinite decimal fractions and treating them as "proper" numbers, contrary to the Greek tradition, see How were irrational numbers accepted by mathematicians? – Conifold Oct 26 at 7:42
• What I meant is the decimal system with decimal point. Can you please give the page where Fibonacci uses a decimal point? As far as I can tell he used fractions as in $\frac{1}{10}\frac{2}{10}\frac{3}{10}4$ to represent our $4.321$. – Chrystomath Oct 26 at 10:06
• Are you asking about inventing decimals or notational conventions for them? Using a point hardly counts as "invention", and Wikipedia already has an article on the history of the decimal separator. – Conifold Oct 26 at 10:19
• My question is about the decimal point. The Wikipedia article nowhere mentions della Porta, or Stevin for that matter! – Chrystomath Oct 26 at 10:31
• OK, the decimal system was in use already. But only for calculations involving integers. Here we are talking about an extended use of this system with fractional places for tenths, hundredths and so on. (Of course there were previous notations with place value 1/60 and 1/(60)^2 and so on: degrees minutes seconds). – Gerald Edgar Oct 26 at 13:38

For a history of decimal fractions see Smith's History Of Mathematics, vol II, pp. 238ff. In the Middle East, Smith gives credit to al-Kashi (c.1400), but the relevant algorithms, in a table notation, appear already in al-Samawal (c.1150), see Katz's History of Mathematics, 7.2.3.

Notationally, the fractional separator was initially a bar placed over the digit of units, which was later replaced by a point, possibly for ease of printing. Pellos used the decimal point in print already in 1492, but Cardano still used a bar in 1539. However, neither Pellos nor Cardano appreciated the algorithmic significance of the separator in calculations. According to Smith, the first man to do so in Europe was Rudolff, who performed a sample calculation in Exempel Büchlin (1530). He also used the bar rather than the decimal point, and his work was not appreciated at the time. Smith does not mention della Porta, but his use was likely purely notational, a la Pellos.

The first comprehensive explanation of decimal fraction calculations in Europe is indeed due to Stevin's De Thiende (1585), and after him they became more common. But their final triumph only came after the invention of logarithms by Napier. However, Stevin did not use either the bar or the point, and his notation was, in fact, quite unwieldy: he wrote the power of $$1/10$$ in a circle after each digit (see facsimile in Smith, p.243). The propagation of the modern notation was due to Bürgi, Kepler, Beyer and Napier. Bürgi used a dot or a comma, and his example was followed. Napier did not use the decimal point in his original publication of the logarithmic tables in 1614, but it appears in their English translation by Wright (1616), and Napier adopted it in Rabdologie (1617). Kepler and Beyer used both comma and $$','',''',''''$$ placed over digits (as in the ancient sexagesimal notation) in 1616. Here is from Smith:

"Another influence leading to the invention of the decimal fraction was the rule for dividing numbers of the form $$a\cdot10^n$$, attributed by Cardan (1539) to Regiomontanus... Borghi (1484) elaborates this rule, but it appears in its most interesting form in the rare arithmetic of Pellos (1492), who unwittingly made use of the decimal point for the first time in a printed work (p. 239). The use of the dot before and after integers had been common in the medieval manuscripts, as in the case of Chuquet's work already mentioned, but its use to separate the integer from what is practically a decimal fraction is first seen here. Later writers commonly used a bar for this purpose, as was the case with Rudolff (1530; see page 241), Cardan (1539), Cataneo (1546), and various other writers... Pellos, however, did not recognize the significance of the decimal point, as is evident from the facsimile on page 239, and no more did Cardan appreciate the significance of the bar that he used for the same purpose.

[...] The first man who gave evidence of having fully comprehended the significance of all this preliminary work seems to have been Christoff Rudolff, whose Exempel- Büchlin appeared at Augsburg in 1530. In this work he solved an example in compound interest, and used the bar precisely as we should use a decimal point today (see page 241). If any particular individual were to be named as having the best rea- son to be called the inventor of decimal fractions, Rudolff would seem to be the man, because he apparently knew how to operate with these forms as well as merely to write them, as various predecessors had done. His work, however, was not appreciated, and apparently was not understood, and it was not until 1585 that a book upon the subject appeared.

The first to show by a special treatise that he understood the significance of the decimal fraction was Stevin, who published a work upon the subject in Flemish, followed in the same year (1585) by a French translation. This work, entitled in French La Disme, set forth the method by which all business calculations involving fractions can be performed as readily as if they involved only integers. Stevin even went so far as to say that the government should adopt and enforce the use of the decimal system, thus anticipating the modern metric system. He was the first to lay down definite rules for operating with decimal fractions, and his treatment of the subject left little further to be done except to improve the symbolism... The improvement in the symbolism was due largely to Bürgi, Kepler, and Beyer, and to the English followers of Napier... It is unquestionably true that the invention of logarithms had more to do with the use of decimal fractions than any other single influence."

• I also thought at first that della Porta's use was purely notational. But actually it is the result of a calculation of a square root, as in the root of 172800 is 415.6.9. Since della Porta wasn't a mathematician as such, perhaps Italians were already using decimal fractions in 1558? – Chrystomath Oct 27 at 6:40
• @Chrystomath There was a trick for roots, where the number was multiplied by an even power of 10 and then the answer divided by 10 to half that power, to avoid fractions. It was known in Europe as early as Hispalensis (c.1140), and Ries included it into his Rechnung (1522), see Smith, p.236. You'll have to inspect della Porta's writing closer to see if he had algorithms for calculations on fractions or simply used the trick and wrote answers with a point. – Conifold Oct 27 at 7:44