The Measurers:
a Flemish Image of Mathematics in the Sixteenth Century

The Measurers

#### The Painting

Two versions of The Measurers are known - the painting in the Museum of the History of Science and a drawing in the collection of the Ecole nationale supérieure des Beaux-Arts in Paris. The drawing has the title Les Mesures and is attributed to the school of Frans Floris. The painting has formerly been attributed to Hendrik van Balen, who, like Floris, was an artist in Antwerp. The attribution to van Balen is now denied on stylistic grounds, but Antwerp in the late sixteenth century is still judged to be the origin of the painting. The relationship between the drawing and the painting remains to be established, but the former is a finer and more delicate work. There are few significant differences between the contents, but one additional feature of the painting is the amorous couple towards the rear of the tavern interior behind the instrument maker. The most likely relationship seems to be that the painting derives from the drawing or from some other version as yet unknown. The single word 'Horatius' is on the lower border of the painting and this is taken to be a reference to the words of the Roman poet Horace: 'Est modus in rebus' - 'There is measure in (all) things'

See catalogue number 1

#### The Instrument Maker

Central to the picture and crucial to all the activity it presents is the instrument maker. This is unusual. Except as the subject of an individual portrait, generally shown holding a pair of compasses, the instrument maker is scarcely ever seen. In allegorical accounts of mathematics his instruments are taken for granted and their production unacknowledged. Here, by contrast, the maker occupies centre stage as he bends to the classic operation of practical mathematics - using the compasses ('dividers' in modern terminology) to scribe what are probably projected lines on a plane surface, perhaps in this case the plate of a sundial. A rule or a square lies across his lap, a small rule is held in his belt and a small pair of compasses is stuck in his hat.

Surrounding the maker are the products of his trade, apparently drawn from individual examples. Moving clockwise from his left foot there is first a geometrical square with a pivoted alidade or rule, used for surveying, together with another pair of dividers. The use of the square with equally divided sides for measuring angles as ratios (in trigonometrical terms as tangents), instead of in degrees, was particularly associated with the practice of surveying (see figure 19). The equal-arm balance is of the type used by an apothecary. The polyhedral sundial was a particular challenge and a product of which a maker might well be especially proud, showing his ability to project hour lines on to faces with a wide variety of orientations. Dialling was one of the most cultivated of the mathematical sciences and often an occasion for ingenuity and ostentation. The rule and square nearby had more everyday uses, but beside the dial is a quadrant with a plumb-line, which may well represent an horary quadrant for telling the time from the altitude of the sun. No hour-lines are represented, however, so that it may be a simple surveying instrument for altitude measurement.

Two complementary instruments for building are shown, both with plumb-lines, one serving as a level, the other a perpendicular. There is another rule - perhaps a gauging rule - and a further, prominently displayed pair of dividers. Finally a pair of adjustable scribers lies in front of a globe. The outlines on the latter are not clear but are probably meant to indicate land masses on a terrestrial globe.

Although there are a great many more instruments known from the period, this is a fair representation of the more common examples. It is notable that both astronomy (except in relation to dialling) and navigation (except perhaps for the globe and just possibly the quadrant) are not represented, though these were two of the most prestigious among the mathematical sciences. In fact a globe was not a useful sea- going instrument and there were more sophisticated altitude measuring instruments for mariners than the quadrant, so these two cannot really be seen as exceptions. This is certainly an everyday, almost domestic, view of the usefulness of mathematics. The sundial might be seen as a slight extravagance but in the context of this picture it serves rather to remind us how common an interest was taken in the practice of dialling.

See catalogue numbers 13 to 23

#### Music

Music was one of the four mathematical sciences of the quadrivium, the more advanced part of the medieval university curriculum, the other subjects being arithmetic, geometry and astronomy. In The Measurers, however, we see, as with the other aspects of practical mathematics, a rather more domestic and everyday representation. The adult, while beating time with his left hand, holds the music open for the child to follow the line of notes with his finger. A drum, with a pipe lying across a single drumstick, rests on the ground. This probably represents the pipe and tabor combination used for rustic dance and so emphasizes again the commonplace and the familiar. While mathematics is useful, it is not always serious; we will later see that it has a place also in the tavern.

See catalogue numbers 24 to 27

#### Weighing

In the sixteenth century Antwerp became one of the most important commercial centres of Europe. At a crossroads of maritime and trans-continental trade, it attracted many merchants from Germany, Italy, France, Spain and England. Although its economic fortunes faltered in the political unrest towards the end of the century, the citizens of Antwerp must have been used to seeing large balances and weights such as are represented here.

The equal-arm balance, known since antiquity, was built on a large scale for heavy goods; for lesser tasks a small unequal-arm balance, such as a steelyard, might be used, except where delicacy and accuracy were essential, as when weighing precious metal. An index arm is just visible in the painting; when a balance is achieved its position coincides with that of the suspension rod. Weight is thus taken with a level beam, not as was sometimes the case, with scales tipped towards the side of the goods being measured. The livre used at Antwerp for general commerce was equivalent to 470.2 grammes, at Brussels 467.7 grammes and at Louvain 469.25 grammes.

See catalogue number 28

#### Gauging

The population of Antwerp increased with the city's commercial expansion and the brewing industry, which had formerly catered only for local demand, grew into an export trade. The term 'gauging', which has come to refer to all manner of precision measurement, was originally applied to assessing the volume of liquid in a cask or barrel and was an operation generally associated with excise officers. The basic measurement was made by dipping a graduated rod into the wine or ale, but, given the particular shape of a barrel and the use of different sizes for different purposes, the graduation of such gauging rods and their proper application required expertise and came to form a specialized branch of practical geometry. In the painting the scene is clearly a tavern and the gauging rod is held vertically through the opening in a horizontal barrel.

See catalogue numbers 29 to 31

#### Surveying

Two surveyors are shown, one measuring bearings or azimuths with a horizontal instrument mounted on a staff, the other taking altitudes with a hand-held quadrant. The horizontal instrument may alternatively be a surveyor's cross for laying out lines at particular angles - the detail of the painting is not precise. In any case the two co-ordinates are treated separately; altazimuth instruments were known at this time but were little used. On the ground in the drawing, though not in the painting, is a pole, the traditional surveyor's instrument for measuring length. The use of angle-measuring instruments was a relatively recent innovation in surveying practice, dependent on the introduction of more sophisticated geometry. It was Gemma Frisius who had first proposed that, through the technique of triangulation, a great deal of routine linear measurement with poles could be avoided. A base-line could be established and the positions of all visible features secured by taking their bearings from either end of this line (figures 29 and 30). The appropriate instrument for such work would be some form of simple theodolite (see figure 31).

See catalogue numbers 32 and 33

#### Measuring Grain

Dry measure was taken by volume in open vessels of standard size, such as the one being filled by the man and smoothed flat by the woman. The latter is using a cylindrical 'strike' to level the grain with the top edge of the vessel. There had been a long conflict between the common practice of measuring by the 'heaped' bushel and the more accurate method of 'striking' to the level of the edge. If heaped measure was used, the dimensions of the vessel and not only its volume were important, as the diameter affected the size of the heap. The traditional Antwerp measure was the viertel, at 77 litres equivalent to a little more than twice an English bushel. It comprised 4 meukens, or 56 pots, or 112 pintes or 224 upers. The viertel is a good example of the diversity of standards. The Antwerp viertel for oats, at 96.25 litres, was different from that for grain in general, 77 litres; in Brussels the equivalent viertels were 12.87 litres and 12.19 litres, and in Louvain 8.75 litres and 7.5 litres.

See catalogue numbers 34 and 35

#### Measuring Cloth

The cloth trade in Antwerp employed 1600 people in 1564 (Isacker & Uytven, p. 96) and the main import was of English woollen cloth, which was treated and dyed before being sold on to customers throughout Europe. There was also an important luxury trade in silk. It is natural, therefore, that in The Measurers length measurement is shown applied to cloth. A writer at the beginning of the sixteenth century records that five Flemish ells (the ell being the traditional weaver's measure) are equivalent to three English and that the Flemish ell is three quarters of an English yard (Connor, 1987). The English ell was 11/4 yards. This is not entirely consistent with the value given by Doursther for the ell or aune used at Antwerp of 695 mm - the aune de Brabant, comprising 16 tailles. A further rule, with brass ends, together with a pair of scissors, lies in front of the woman measuring cloth.

See catalogue numbers 36 to 39

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