References on Mathematics of the Vikings

Question

I know maybe this question is more appropriate to the community History of Science and Mathematics, and I have already posted it there. But, since it is considerably smaller than the Mathematics community, it has reached very few people (around 49 views within almost one and a half day). Due to this, I think I can ask that question here, where I can reach more people and, then, increase the chances of getting satisfactory answers... (I didn't get any answer there and therefore I'll deleted the question, leaving it only here)

So, let's go to the question:

We know Vikings were good, among other things, in building ships, so they must have had some knowledge in astronomy, geography, maybe, math, and so on.

Is there any "good" reference (rather a book on the academic literature, which explores it in a more detailed and exhaustive way, instead of only "sites on the web") on the Mathematics of the Vikings? I need as much references on this subject as I can get... (I know this task can be difficult, and that such a reference MIGHT NOT EXIST, since Vikings could have had not a written culture, leaving no recording of the math they had possibly developed).

Maybe a reference which is about Viking Science instead of Viking Math would help.

Thank you!

Answer 1

It’s not at all my field, but as a hobby I’ve been fighting my way through Old Norse literature in the original for many years, and along the way I’ve picked up some information on what primary sources of various types exist. Very little of what I’m about to mention is in English, but it may serve as a starting point for searching.

N. Beckman and Kr. Kålund, Alfræði íslenzk: Islandsk encyklopædisk litteratur: II. Rímtöl [Encyclopaedic literature on the calendar], Copenhagen: S.L. Møllers, 1914-16, has both a long introductory discussion (in Swedish) and editions of the three extant Rímtöl (calendrical treatises): Rímtöl (1150), composed probably by Bjarni Bergþórsson inn tölvísi; Rímtöl II, composed during the last quarter of the $13$th century; and Rímtöl III, composed during the $14$th century; it is available here [PDF]. (I took the dates from here. Unfortunately, I don’t know of an English translation of either the introductory material or the texts.

The previous link mentions two other relevant original texts, Algorismus from the early $14$th century and Odda tala, composed by Oddi Helgason during the $13$th century, exists in three versions: GkS 1812 4to, Hauksbók and Rímbegla. Algorismus is the oldest known mathematical treatise in a Scandinavian language, surviving in the $14$th century Hauksbók. This PDF has a brief discussion of it in Norwegian and a translation into Norwegian, and this site gives it in modern Icelandic spelling with a discussion in Icelandic; a diplomatic edition of Hauksbók can be found here.

I did find an interesting paper, ‘Ethnomathematics at the Margin of Europe — A Pagan Calendar’ [PDF], by Kristín Bjarnadóttir, which appeared in the Proceedings of the $11$th International Congress on Mathematical Education.

Finally, this page is a brief article by Þorsteinn Vilhjálmsson, ‘Time-reckoning in Iceland before literacy’, that appeared in Clive L.N. Ruggles (ed.), Archaeoastronomy in the $1990$s, Loughborough, UK: Group D Publications, $1991$, pp. $69$-$76$; if nothing else, the references may be useful.

Answer 2

Viggo Brun, a Norwegian professor of mathematics, wrote a short book Regnekunsten i det gamle Norge (The Art of Calculating in Old Norway) in 1961. He gives examples from the Gulating Law and the Frostating Law involving calculations of fines, in particular when various relatives of killed persons were to be properly compensated. More distant relatives would get a fraction of the amount closer relatives were entitled to. The laws were codified in the 13th century, but are surely based on older traditions.