It could not have been possible to build the longships – or make their sails – without an arithmetic and geometric capability. Even the most visually and spatially capable person must, at some stage, record an idea, or communicate with lesser mortals, or experiment with imagery. Imagine: you have set a keel that measures 59 and a half feet long and you want the thwarts to be 40 inches apart, each thwart being 8 inches wide. How many thwart pieces will you need and how wide shall you make the largest?. Now explain that to an apprentice.
This is the world of the engineering mind. It is not that of the abstract mathematician, and neither is it the world of the fine artist. A good engineer will build a serviceable prototype and then test it. He will then measure its performance, record how he has made it, then build another, adjusting where he perceives the need. If he is a genius, he might get it nearly right on the fifth iteration. The mathematician won’t know where to start. The fine artist won’t know where to end.
Tricks that the shipwright and sailmaker might borrow
The rear of the Hunterston Buckle
Underlying image from the Catalogue of the National Museum of Antiquities of Scotland (Google)
Unfortunately, most historical and archaeological scholars of the past emerged from personal backgrounds that were far removed from those of skilled tradesmen in manual fabrication and construction. This problem is referred to in a masters’ thesis entitled Broaching the subject: the geometry of Anglo-Saxon composite brooches by Anna Luella Isbell of the University of Iowa in 2015. In it she reveals the use of squares and circles, and this implies the use of compasses to inscribe arcs – from which perpendicular alignments and right angles could have been draughted. However, her work was concerned with circular disc brooches and their decoration, and she presumably had not the incentive to consider more adventurous geometric constructions using other tools.
The internal shape of the Hunterston buckle loop is clearly a true ellipse, with a perihelion of one sixth the major axis in length. The shape would have been easily constructed on a wooden board with two pins and a length of thread, 8X long, between them.
So the trick of describing an ellipse was well known in the seventh century when the Hunterston Buckle was made – probably by an Angle metalworker in Northumbria. But it was also known in the ninth century in Scandinavia. The common pair of shoulder brooches shown here in Norway were also elliptical, but with a perihelion of one eighth the major axis.
Isbell also refers to the need of a straightedge by the Anglo-Saxon craftsmen. I would go further and point to the need for the Scandinavian craftspeople to understand how they might measure distance. The Værne Kloster Spur, and the Søllested Harnes Bow, and very many other artefacts show symmetry in their design. To create such things, a craftsman needs to measure one side to replicate it on the other. In the spacing of nails or runes or other repetitive markings, he must also have the ability to divide a line into equal divisions. Admittedly, this can be done by continual subdivision by eye, but this becomes problematic with the odd numbers: five, seven and nine.
Speaking of the numeral five, it commonly occurs in snowflakes – now there’s a thought. And how do you draw a pentangle without a protractor?
Perhaps the most obvious need for arithmetic and geometric capability is in the skill of weaving, particularly in the complicated twills that the viking women were capable of.