JF Ptak Science Books Post 2011

There is an interesting side note to this blog's series on the histories of holes and dots--a mathematical aspect involving decimal points, decimal notation and placeholders. This is exclusive of the number zero, however, which is an entirely different topic.

The book that this beautifully-illustrated counting board (below) is found is in Gregor Reisch's (1467-1525) *Margarita Philosophica* (1503) and depicts (amidst much else in the greatly humanist volume) representations of the mathematicians Boethius and Pythagoras working math problems on the given tools of their day. The tools on the right seem to be circles, but they're not--they're counting stones, and for our intents and purposes here, they shall be dots, and in the history of dots in math and business reckoning they have had a strong and long life.

We can see in his expression that Boethius, on the left, is rather enjoying himself, knowing the superiority of his system of counting, which was the the Hindu-Arabic number notation--he definitely has a sly, self-appreciating smile on his face. Pythagoras, working with the old counting table, definitely looks worried, or at least unhappy, unsettled. Never mind that Pythagoras (570-495 b.c.e., none of whose works exist in the original, another sort of entry in our Blank History category) was at a definite disadvantage in the calculating department, being dead and all that for hundreds of years before the Arabic notation was more widely introduced in the West, probably being introduced by Pisano/Fibonnaci in the 12th century. But it does fall to Boethius, the smirker, to have introduced the digits into Europe for the very first time, deep into the history of the Roman Empire, in the 6th century.

The numerical stand-ins in the Reisch book with which Pythagoras worked were blank, coin-like slugs used as placeholders, and would be used in place of rocks or pebbles or whatever other material was at hand. It is interesting to note that the Latin expression, "calculos ponere", which basically means "to calculate"or "to compute", is more literally translated into "to set counters" or "to place pebbles" (upon a counting board) or to set an argument^{2}, which is exactly what some of the Roman daily reckoners would do at their work. And also used, in this case, by the unhappy Pythagoras.

The foundation for the .14159... that comes to the right of the integer 3 in pi is a relatively recent idea in the history of the maths--at least so far as the represrntation of the ideas in numbers and the decimal point is concerned.

Simon Stevin (1548-1620) introduced the idea of decimal numbers in his 36-page *De Thiende* ('The Art of Tenths"^{1}) in 1585, an idea that replaced much more cumbersome earlier methods of representation. So, the number 3.14159 would be written in the Stevein notation as (where in this case numbers enclosed by brackets, i.e. "[9]" would have been represented in print as a 9 within a circle) 3[0]1[1]4[2]1[3]5[4]9[5]. It is also seen here:

[Source: math Words, here.]

[Full text available here.]

The importance of the introduction of this idea is difficult to underestimate, according to many and by example the *The Princeton Companion to Mathematics *by Timothy Gowers:

*The Flemish mathematician and engineer Simon Stevin is remembered for*

his study of decimal fractions. Although he was not the first to use

decimal fractions (they are found in the work of the tenth-century

Islamic mathematician al-Uqlidisi),it was his tract De Thiende (“The tenth”), published in 1585 and translated into English (as Disme: The Art of Tenths, or Decimall Arithmetike Teaching ) in 1608, that led to their widespread adoption in

Europe. Stevin, however, did not use the notation we use today. He drew

circles around the exponents of the powers of one tenth: thus he wrote

7.3486 as 7�3�4�8�6�4. In De Thiende* Stevin not only demonstrated how *

decimal fractions could be used but also advocated that a decimal system

should be used for weights and measures and for coinage.

This idea would be further developed by Bartholomeus Pitiscus (1561-1613) who was the first to introduce the decimal point in 1612^{3}. It was a far more robust and simple was of dealing with decimal notation than anything that had come before.

Notes:

1. *Decimal arithmetic: Teaching how to perform all computations whatsoever by whole numbers without fractions, by the four principles of common arithmetic: namely, addition, subtraction, multiplication, and division. *

2. The Reisch book is remarkable: it is basically a Renaissance encyclopedia of general knowledge, divided into twelve books: grammar, dialectics, rhetoric, arithmetic, music, geometry, astronomy, physics, natural history, physiology, psychology, and ethics.

3. Pitiscus was also the first to introduce the term "trigonometry" earlier in 1595 in a highly important and influential work he produced in 1595.