# Regula Pigri: How far does it date back?

Regula pigri, or rule of the lazy, is a method of multiplication of single-digit numbers. If one is to multiply 6 by 8, one would start by bending one's fingers like so:

B B B B S　　　　　　　　B B S S S

1. Add straight fingers on both hands, and set the sum for the 10s (1+3); 4_.
2. Multiply bent fingers, and set the product for the unit figure (4*2); 48.

It is said that people in the Medieval Period used this method so they didn't have to remember multiplication tables any more than 5*5. I want to verify this claim, but I could not get hold materials older than Robert Recorde (1618) by simply Googling for regula pigri.

Does anyone know earlier attestations of this method of hand calculation? Additionally, is Recorde the first one to set regula pigri down in Arabic numerals?

• There is a method called "Russian peasant multiplication". Commented Jul 7 at 13:33
• @nwr I think I understand the encoding now, based on the fact that the factors are $> 5$: Each factor is encoded as an implicit $5$ plus the number of straight fingers. Commented Jul 7 at 22:30
• @njuffa, thanks. Huswirt (1504) is definitely an older text that mentions this algorithm. How does he call the method, please? Commented Jul 10 at 13:07

The regula pigri from the question is a particular instance of what is called complementary multiplication in the literature, as it involves the decimal complement of the factors.

Johannes Tropfke, Geschichte der Elementarmathematik, Vol. 1, Leipzig: Veit & Comp. 1902 distinguishes the following variants on p. 44:

1. $$\ \ \ a \cdot b = 10a - a(10-b)$$
2. $$\ \ \ a \cdot b = 10 \cdot \left[a - (10 -b)\right] + (10-a) \cdot (10-b)$$
3. $$\ \ \ a \cdot b = (10- a)\cdot(10-b) + 10(a+b) - 100$$

Tropfke states that there are no indications that complementary multiplication originated in Arab or Indian mathematics. He follows Moritz Cantor in surmising a Roman origin, as such a method would flow naturally from Roman numerals involving the decimal complement, e.g. $$9 = \mathrm{IX}$$. He notes earliest written evidence from the 12th century and mentions that complementary multiplication was popular with authors of the 16th century.

Moritz Cantor, Vorlesungen über Geschichte der Mathematik 2nd ed., Vol. 1, Leipzig: B.G. Teuber 1894, pp. 491-492 writes about finger-assisted mental computations as described by various Roman authors. He sees continuity to wide-spread finger-assisted computation described in Medieval sources, and speculates that this could be the origin of complementary multiplication, claimed to still be in use in some rural communities in Wallachia in the 19th century.

Moritz Cantor, "Ueber einen Codex des Klosters Salem", Zeitschrift für Mathematik und Physik, Vol. 10, 1865, pp. 1-16 provides the complete Latin text of a manuscript liber algorizmi found in a collection of the University of Heidelberg but originally from Salem Abbey in Germany and dated to around 1200 or potentially slightly earlier based on calligraphic evidence. This includes a description of Tropfke's variant 2 of complementary multiplication on page 5.

The earliest printed source for complementary multiplication that I have been able to find a scan of is Johannes Huswirt, Enchiridion novus Algorismi. 2nd ed. Cologne: Heirs of H. Quentell 1504, p. 12. This is also a representative of Tropfke's variant 2.

Another printed source available as a scan is a book on mercantile arithmetic first published in 1489 of which I could only find a scan of an edition of 1508 provided by the Bavarian State Library. This describes Tropfke's variant 1, and (best I can tell) also variant 2. The uncertainty regarding use of variant 2 is due to me struggling with the old variant of German used in the textual description of the method.

Johannes Widmann, Behend und hüpsch Rechnung uff allen Kaufmannschafften, Pforzheim: Anßhelm 1508, p. 12

The same square arrangement of factors (left column) and their respective decimal complement (right column) that is shown in Huswirt's explanation of Tropfke's variant 2 also appears in the description of complementary multiplication in

Heinrich Schreiber (Grammateus), Ayn new kunstlich Buech, Vienna 1518