How accurate was Viking Age weighing in Sweden?
Sperber, Erik
Fornvännen 157-166
http://kulturarvsdata.se/raa/fornvannen/html/1988_157
Ingår i: samla.raa.se
How accurate was Viking Age weighing in Sweden?
By Erik Sperber
Sperber, E. 1988. How accurate was Viking Age weighing in Sweden? Fornvännen 83. Stockholm.
From the dimensions of a few of the many Viking Age balances found in Sweden, mostly in a fragmentary state, their sensitivity at different loads could be calcu- lated. For three balances the calculations could be expcrimentally corroboraied.
The most common lype, having a balance beam of about 100—120 mm, was shown lo give 1 mm deflexion of lhe needle lip for a differential load of 0.2—0.4 gat nearly zero total load. At a load of 35 gon each pan this deviation was caused by an extra load ol 0.8—1.2 g, i.e. 2—3% of the unilateral load. Twenly-four weights from the Bandlunde, Gotland, lind weighing about 1.5 g were shown to have been manufactured with ±0.08 g standard deviation from the average value. A balance fragment from the same find had a sensitivity of 0.2 g per 1 mm deflexion. The precision of the weights and the balance thus correspond well.
Erik Sperber, Odenvägen 4, S-151 60 Södertälje, Sweden.
T h e presence of a large number ol Viking Age balances and weights in Sweden raises the ques- tion, how accurate was weighing in the com- merce of this period.
T h e balances, in particular, lend themselves to a much more penetrating study than has been made earlier. A few of them are still func- tioning. As to the weights , corrosion has added a great deal to the variance of the weight sets.
Yet the recent very well preserved find at Band- lunde, Gotland, gives us new possibilities. Its study has m e r d y begun.
Trade in Sweden, of course, was always part offrade in Europé. Some preliminary studies of the weighing accuracy in ancient times there- fore have been included in the present artide.
The balances found in Sweden.
The balances found in Sweden are mostly of the collapsible type. They are very similar, if not identical, in construction and could well derive from a common " f ä c t o r y " or handicraft cen- ter. Similar collapsible beam balances are known to have existed in the late Roman and Byzantine Empires and have also come to light from contemporary Egypt (Kisch 1966).
Berg and Ottosson (1984) listed 115 balances,
mostly in a fragmentary state, found in Swe den. It is widdy accepted that these balances were used for weighing the silver, and some- times gold, coins and piéces that often appear together with the balances in contemporary hoards. In the times in question, it was neces- sary to weigh not only the silver piéces, but also the coins, which often differed widdy in weight.
T h e standard of the mints was so low that coins intended to represent the same value some- times differed in weight by a factor of four even for new coins.
A few of the balances are so well preserved that they can still be used for their original pur- pose. Three of them are kept in the Museum of National antiquities in Stockholm, others are on exhibition in other museums or in private hands.
Earlier weighing experiments
Weighing experiments to estimate the sensi- tivity or — as it turned out — the lack of sensi- tivity of Viking Age balances have previously been carried out by the Finnish physicist G. G.
Hällström (1841) and the Swedish archaeologist T. J. Arne (1918).
Hällström put all the ancient weights avail-
Furmannen 83(1988)
158 E. Sperber
a b l e to h i m o n t h e p a n s of h i s b a l a n c e a n d found o u t t h a t it c o u l d n o t d i s c r i m i n a t e b e t w e e n 100.3 a n d 102.1 g (diff. 1.8 % ) . H e also i n d i c a t e d t h e f e a t u r e of t h e b a l a n c e r e s p o n s i b l e for t h e lack of sensitivity. H ä l l s t r ö m ' s r e s u l t s will b e f u r t h e r d i s c u s s e d l å t e r in this a r t i d e .
A r n e t e s t e d t h e b a l a n c e from V å r d i n g e w h i c h is a v a i l a b l e also t o t h e p r e s e n t a u t h o r . H e u s e d a n o t h e r m e t h o d . W i t h n o o t h e r l o a d , h e p u t m i l l i g r a m w e i g h t s o n o n e of t h e p a n s u n t i l a deflexion c o u l d b e o b s e r v e d . H e h a d to a d d 0.45 to 0.50 g to r e a c h this p o i n t . ( A r n e d i d n o t specify t h e total l o a d . M y m e a s u r e m e n t s , h o w - ever, c l e a r l y show t h a t it m u s t have b e e n close t o zero. In a d d i t i o n , A r n e w o u l d p r o b a b l y h a v e specified a n y o t h e r l o a d . )
Experimental estimation
of the sensitivity of a functioning balance
T h r e e b a l a n c e s from t h e V i k i n g A g e w e r e avail- a b l e for t h e e s t i m a t i o n of t h e i r sensitivity, t h e s e n s i t i v i t y b e i n g d e f m e d a s the overweight on one of the pans which causes a deflexion of the needle 's tip hy 1 mm from zero. O b v i o u s l y t h e v a l u e 1 m m m a y b e d i s c u s s e d . A s m a l l e r d e v i a t i o n still c o u l d b e u s e d . O n t h e o t h e r h a n d , t h e rest p o i n t of t h e n e e d l e is o b s c u r e d by t h e h o l d e r of t h e b a l a n c e a n d in m o s t cases c a n n o t b e easily o b s e r v e d .
In o u r e x p e r i m e n t s n o a t t e m p t w a s m a d e t o r e a c h e x a c t l y 1 m m . I n s t e a d , a n u m b e r of over- w e i g h t s w e r e u s e d a n d t h e c o r r e s p o n d i n g tip d e v i a t i o n s m e a s u r e d in m m . T h e m e a s u r e - m e n t s w e r e t r a n s f o r m d i n t o t h e t a n g e n t of t h e a n g l e of t h e d e v i a t i o n from t h e h o r i z o n t a l a n d t h e n p l o t t e d . (Fig. 1.) F r o m t h e set of s t r a i g h t lines o b t a i n e d , t h e r e l a t i o n t g a / g r a m over- w e i g h t c o u l d b e c a l c u l a t e d . T h e z e r o p o i n t of t h e b a l a n c e c o u l d b e c a l c u l a t e d too, b u t is of m i n o r i n t e r e s t for t h e p r e s e n t i n v e s t i g a t i o n . T h e r e s u l t s of m e a s u r e m e n t s of t h e t h r e e b a - l a n c e s a n d a n i n e t e e n t h c e n t u r y b a l a n c e a r e given in T a b l e 1.
T h e s e n s i t i v i t y of all t h e t h r e e a n c i e n t b a - l a n c e s at z e r o l o a d t u m s o u t t o b e 0.2 to 0.4 g l o r
1 m m deflexion. At h i g h e r l o a d s t h e s e n s i t i v i t y d e c r e a s e s a n d for very h i g h l o a d s , u p t o 50 t o 100 g o n e a c h p a n , t h e difference b e t w e e n t h e t w o p a n s m u s t b e 2 t o 3 % t o b e a b l e t o b e d e - t e c t e d .
tg oc
fig. 1. Weighing wilh the balance from Vårdinge pa- rish, Sö. ( Kervveight on one pan vs. tlie tangent ol ihe angle between lhe beam and the horizontal. In the shaded area the pointer of the balance cannot easily be observed. — Vägning med balansvåg lian Vår- dinge sn. So ( S H M 15115). Övervikt på ena vågskå- len jämför) med utslagel uttryckt som tångens (or ut- slagsvinkeln. Inom del skuggade området kan v 'iså- rens läge mindre lätt observeras.
Fig. 2. Important dimensions and definitions of a Vi- king Age balance.
a support point — punkt d ä r balken un- derstöds
suspension points of the pans — vågskå- larnas upphängningspunktcr
center of gravity of the beam — balkens tyngdpunkt
length of the pointer — visarens längd C lo / hall lhe length of the beam — halva
balklängden~/ m m
= di m m distance from the support point to lhe center of gravity — avstånd från balkens upphängningspunkt till dess tyngdpunkt
= dj m m distance from the support point to lhe straight line connecting the two suspension points ol the pans — av- stånd frän balkens upphängningspunkt till den linje som förenar vågskålarnas upphängningspunkter
b a to d c to e;
a lo b
a to b
For mannen 83 (1938)
Viking Age weighing 159
Table 1. T h e sensitivity ofbalances: calculated and found. — Vågarnas känslighet: beräknad resp. funnen.
1:1 Balances in working order. — A n v ä n d b a r a vågar.
Sensitivity Balance
number
g/tga Load,g
0 2 x 2 0 2 x 50
0 2 x 2 2 2 x 4 2 0 2 x 2 0 2 x 40
0 2 x 1 7 2 x 3 5 2 x 30
calculated 12.2 19.8 31.8 13.3 20.1 27.8 5.2 12.1 18.2 0.99 2.65 4.42 5.93
lound 12.7 21.4 33.2 12.2 9.3 12.7 13.3 22.8 3.8 7.8 14.4 1.19 2.47 4.95 6.48
foun 0.4 0.7 1.1 0.3 0.4 (1.7 0.2 0.5 0.8 0.03 0.07 0.15 0,18
g/1 mm deflexion % of load found SHM 15115
Sö, Värdingc
SHM 26039 Up, Hållnas
SHM 6819 Öl?
I*ltli century
3.4 2.2
1.8 1.7
2.4 2.(1
0.4 0,4 0.4
1:2 Balances and fragments nm in working order. — Fj användbara vågar och fragment.
Sellsim ilc Balance
number 1 .ond.
L;0 2 x 1 3 2 x 35
0
2 x 35 2 • 100 2 x 500
0 2 x . l 3 2 x 35
II
2 x 1 3 2 x 35
g / t g a calculated
1.3 8.5 16 1!»
27 43 1 Kl
5
'i
15 4 8 15
g/I mm deflexion '< ol load i .iii ul.ned call ulan il SHM MIH
(in.-Visby
SHM 1304 Öl?
MIM 12960 (Jo. Hellvi
Bandlundc Go, Burs
0.2 0.4 0.8 0.3 0.5 0.7 2 0.2 0.3 0.5 0.2 0.3 0.6
3 2
1.4 (1.7 0.5
2 1.4
2 1.6
The ailculalion of lhe sensitivity of a balance
T h e s e n s i t i v i t y of a s y m m e t r i c a l two a r m ba- l a n c e like t h o s e u s e d a b o v e c a n a l s o b e c a l c u - lated from its m a i n d i m e n s i o n s . T h e s e c h a r a c - teristic d i m e n s i o n s a r e e x p l a i n e d in t h e l e g e n d of Fig. 2, w h i c h d o e s not show t h e p a n s .
T h e b a s i c e q u a t i o n for a b a l a n c e at e q u i l i - b r i u m , u s i n g t h e s y m b o l s of Fig. 2 is:
7', " / . - T , % • • • • • • E q . (1)
7', a n d T = weight of e a c h p a n , l o a d i n c l u d e d .
/, a n d / , = l e n g t h of t h e b a l a n c e a r m s , i.e. t h e
d i s t a n c e b e t w e e n t h e p o i n t s c a n d e a n d f a n d c
rvrminm 83(1988)
160 E . Sperber
r e s p e c t i v d y . If b o t h a r m s a r e e q u a l , t h e e q u a - t i o n teils u s t h a t t h e l o a d s a r e e q u a l too. If t h e a r m s a r e n o t e q u a l , useful r e s u l t s m a y still b e o b t a i n e d by e x c h a n g i n g t h e r i g h t a n d t h e left l o a d . T h e ( g e o m e t r i c ) a v e r a g e of t h e t w o w e i g h - i n g s will give t h e t r u e w e i g h t .
O f t h e b a l a n c e s m e a s u r e d , i n d u d i n g t h o s e in I r a g m e n t s , t h e a r m s of e a c h p r o v e d t o b e e q u a l w i t h i n o n e p e r c e n t o r so. T h u s , i n e q u a l i t y d o e s n o t s e e m t o h a v e b e e n a m a j o r p r o b l e m .
If t h e r e is a n o v e r w e i g h t o n o n e of t h e p a n s , t h e a b o v e e q u a t i o n d o e s n o t fit. T h e b a l a n c e will r e m a i n in a n e w e q u i l i b r i u m p o s i t i o n at a n a n g l e a from t h e h o r i z o n t a l . T h e p r o b l e m is i l l u s t r a t e d in Fig. 3 , w i t h t h e a d d i t i o n a l a s - s u m p t i o n is m a d e t h a t t h e t w o s u s p e n s i o n p o i n t s of t h e p a n s a n d t h e s u p p o r t p o i n t of t h e b e a m a r e all s i t u a t e d o n t h e s a m e s t r a i g h t l i n e . T h i s c o n d i t i o n is m e t w i t h by all b a l a n c e s u s e d for c h e m i c a l a n a l y s i s s i n c e a b o u t A.D. 1800. Its i m p o r t a n c e w a s s t r e s s e d e.g., by G a h n a n d Ber- z e l i u s in Lärobok i kemien ( B e r z e l i u s 1818). T h e o v e r w e i g h t , in this c a s e , will b e c o u n t e r b a l a n - c e d by t h e w e i g h t of t h e b e a m :
df s i n a
-q = l ' c o s a ' p
IL
d r i E q . (2) In o t h e r w o r d s , t h e deflexion is g r e a t e r w h e n t h e b e a m is longer, a n d s m a l l e r if t h e b e a m is h e a v y o r if t h e c e n t e r of g r a v i t y is f u r t h e r away.
W i t h t h e V i k i n g A g e b a l a n c e s t h e t h r e e s u s - p e n s i o n a n d s u p p o r t p o i n t s w e r e n e v e r s i t u a t e d un a s t r a i g h t line, t h u s m e e t i n g this i m p o r t a n t c o n d i t i o n for a n a n a l y t i c a l b a l a n c e . For o u r cal- c u l a t i o n s , this m e a n s t h a t t h e d i s t a n c e d, ( F i g . 2) w a s different from z e r o a n d in fact c o n s i d c r - a b l e , m o s t l y 6 t o 10 m m . W h e n this d, t e r m is t a k e n i n t o c o n s i d e r a t i o n , o u r formula b e c o m e s , after a slight r e a r r a n g e m e n t :
p d,-q d j - T o r tgcx =
tga I. I E q . ( 3 )
T h e p o i n t s , w e i g h t s a n d d i s t a n c e s of t h i s for- m u l a c a n b e m e a s u r e d o n m o s t b a l a n c e s , in s o m e c a s e s w i t h s o m e difficulty. T h e c e n t e r of gravity, for i n s t a n c e , is i n d i s t i n c t a n d its site h a s to b e a p p r o x i m a t e d . D e s p i t e t h e e r r o r i n t r o - d u c e d h e r e , it will b e p o s s i b l e to e s t i m a t e t h e s e n s i t i v i t y of t h e b a l a n c e s t o t h e n e a r e s t 10—
20 % , a n a c c u r a c y sufficient for o u r p u r p o s e of a t t a i n i n g a m o m e n t a r y p i c t u r e af w e i g h i n g ,
polnl ol suspension
w
q grams weight of the beam — vägbalkens tyngd / ' / ' g r a m s weight of each pan i n d u d i n g chains and load — vikt av en vågskål med kedjor och belastning
/' grams overweight on one of the pans — över- vikt på endera vågskålen
a angle of deflexion — ulslagsvinkel
Fig. 3. Sketch showing the lorees working in a ba- lance with equal arms when carrying an overload on the right-hand pan. Re. the symbols, see also the le- gend of Fig. 2. A force is traditionally symbolized by an arrow emerging from lhe point from which the force emanates and poiming in lhe direction thereof.
Forces/arrows can be added, subtracted etc. and may be moved freely along their range. — Skiss över de krafter som påverkar en likarmad våg med en över- vikt på högra vågskålen. Symbolerna är desamma som i Fig. 2. En kraft symboliseras av en pil utgående från kraftens anbringningspunkt och med samma riktning och storlek som kraften. Krafter/pilar kan adderas, subtraheras tn. tn. oeh kan förflyttas fritt ut- med sin riktning.
b u y i n g a n d selling a t h o u s a n d y e a r s a g o . The babnees used for the weighing experiments T a b l e 2 gives t h e m a i n f e a t u r e s of t h e b a l a n c e s a n d f r a g m e n t s i n c l u d e d in t h e p r e s e n t study.
S o m e f u r t h e r facts a r e given below.
Halarue SHM 15115, Värdingc parish, Södermanland.
It is complete with balance beam, pans with chains, lhe pointer and the supporting yoke. It was used by Arne (1918) in his experiment mentioned above. At- tached to the chains are live small picces of sheet bronze, three on one side, two on the other. These piéces greatly puzzled Arne, who believed that they were meant to repair some deficiency, not defmed, attributed to the balance. As a matter of fact, they were almost certainly intended for zero point correc- tion.
Balance SHM 26039, Hållnäs parish. Uppland. Tlie balance beam and ils support were intact. O n e oflhe pans and its chains were heavily corroded. Missing
Förmännen 83(1988)
Viking Age weighing 161
Table 2. Balances studied. — De studerade vågarna.
Balance n u m b e r S H M 15115 Sö
S H M 26039 Hållnäs, U p 6819:536 Ol?
19th century S H M 6104 Visby, G o S H M 1304 S H M 12960 Hellvi, G o Bandlundc Burs, G o
length m m
94
139
82.5
119 100
242 124
122
Beam weighl S
11.8
33
6.3
12.8 14.3
c. 90 23
12
Pointer length m m
31
33
19
33 20
(60?) 31.5
25
Support center vity di 6.5
8.8
4.7
1.8 6
9.5 6.5
6.5 of gra
m m
point distance - suspens
d> m m 9
11.8
6.7
3.0 8.5
14.5 9.5
9
Pans total weighl g
55
54
28
10.3 ( 15?)
(200?) ( 20?)
( 20?)
Figures which could not be measured are in brackets. — Siffror inom parentes är uppskattade.
parts had been replaced by piéces of plastic and stain- less steel wire. As a result thecorroded pan was aboul 5 g too light, net. An appropriate weight was added to reslore the equilibrium.
HalanceSHM 6819 probably slems from Öland. It was bought from a private eollector some 150 years ago. It is a complete balance beam with its pointer, the supporting yoke and the pans. T h e pans have been fitted with new chains in modern times.
The "I9th century halance" was bought in an antique shop together with a few weights. It is assumed to be a coin balance and had gallow ends. According to Kisch (1966) such balances were common in Central fiurope from the 18th century and onwards. T h e weights, if Swedish, probably belonged to one of lhe systems in use before 1855. T h e balance was con- tained in a wooden box marked " M a d e in G e r m a - n y " which casts some doubt on the above dating T h e balance, not being an archaeological arieläct, could be dis- and rcassemblcd al will, which was ni great help.
T h e c e n t e r of gravity, t h e o n l y p o i n t n o t easily f o u n d w a s a p p r o x i m a t e d as follows: I m a g i n e t h a t t h e b e a m is a s h o r t c y l i n d e r , 0 c. 5 m m , at- t a c h e d to t w o slightly c o n i c a l i d e n t i c a l a r m s ,
g i v i n g a total l e n g t h of 100 m m . T h e c e n t e r of t h e c y l i n d e r is also its c e n t e r of gravity. S u p p o s e (hal t h e b e a m w e i g h s 10 g a n d t h a t it is fitted w i t h a c o n i c a l p o i n t e r , 30 m m in l e n g t h , w e i g h - i n g 1 g. T h e g r a v i t y c e n t e r will t h e n m o v e a b o u t 1.1 m m t o w a r d s t h e p o i n t e r ' s tip. D u r i n g t h e m c a s u r e m e n t this figure w a s kept in m i n d . N e v e r t h d e s s t h e r a n d o m e r r o r in e s t i m a t i n g t e n d e d to be h i g h . For t h e I9th c e n t u r y b a l a n c e this p r o b l e m d i d n o t exist. T h e b e a m , d e v o i d of all a c c e s s o r i e s , w a s allowed to h ä n g freely. T h e ( c n l c r of gravity, by d e f i n i t i o n , f o u n d its p l a c e o n t h e vertieal line from t h e p o i n t of s u s p e n - sion. P r a c t i c a l p r o b l e m s also a r o s e w h e n t h e w e i g h t s of t h e different p a r t s of t h e b a l a n c e s w e r e to b e d e t e r m i n e d . N o r c o u l d t h e y b e w h o l l y solved. T h e p a r t to b e w e i g h e d w a s p l a c e d o n t h e p a n of a d i g i t a l scale. All o t h e r p a r t s w e r e a s s e m b l e d in t h e p a l m . A n a t t e m p t w a s t h e n m a d e to r e a c h a p o i n t w h e r e t h e p a r t s in t h e p a l m i n f l u e n c e d t h e w e i g h t s h o w n o n t h e scale as little a s possible. T h e task p r o v e d e a s i e r a n d its r e s u l t s less i n a c c u r a t e t h a n h a d b e e n ex- p e c t e d . E r r o r s in excess of ± 1 g w e r e few.
A g a i n , t h e 19th c e n t u r y b a l a n c e w a s very useful w h e n j u d g i n g t h e e r r o r s to b e e x p e c t e d .
H n m m 81(1988)
162 E. Sperber
Table 3. Estimation of the distances from the support point of the beam to the center of gravity of the beam, and to the line joining the two suspension points of the pans. — Bestämning av avståndet från balkens under- stödspunkt till dess tyngdpunkt och till den linje som förenar de två vågskålarnas uphängningspunkter.
M e a s u r i n g mc S H M 15115 V ä r d i n g c , Sö S H M 26039 Hällnäs, U p 6819:536 Ol?
19th century
M e a n — M e d thod
•It al Standard error of mean
— Medeltalets medelfel direct 6.1
6.0
2.4
2.7
T h e center of by we 6.5
8.8
4.7
1.8 ighii
Distance gravity
K diff.
— 0.4
— 2.8
— 2.3
+ 0.9
— 1.2
± 1.0
from support point
di r e d 9.6
10.2
6.0
3.0 to
T h e line Dfsuspe by weigh
9.0
11.8
6.7
3.5
"K ision
diff.
+ 0.6
— 1.6
— 0.7
+ 0.5
— 0.3
± 0 . 4
T h e experiments illustrated in Table 1 to- gether with all data for the functioning balances except d
xand d
2, can be used for a "backward calculation" of d, and d
2. T h e sets of data for zero load and for the highest load tested were used. T h e results are presented in Table 3. T h e values for d, and d
2measured directly or through weighing coincide well within the experimental errors caused by the primitive measuring equipment and the caution neces- sary when handling the ancient artefacts.
T h e figures of Table 2 show very clearly that the weighing properties of a balance can be cal- culated from its main dimensions.
It should be stressed that the sensitivity as measured or calculated here is not of statistical origin, but a constructional parameter, similar to e.g. the swinging time of the balance or the weight of its beam.
T h e dimensions of many of the fragments of balances, kept in the museum still retain most of their dimensions. They can often be used to answer the original question regarding the pre- cision of Viking Age weighing. Four such ba- lance beams, lacking useful pans and with other defects are listed in Table 2. For the calculation the sets of data are complemented by making plausible assumptions, above all regarding the weight of the pans etc. This approach is ge-
nerally regarded with some suspicion by scien- tists. To archaeologists it is often necessary and has frequently yielded important results.
As was to be expected, the incomplete ba- lances do not diffcr appreciably from those in working order.
The zero adjustment of the Vårdinge babnce
T h e five small piéces of bronze hanging över the pans of the Vårdinge balance greatly puzzled Arne who in fact found no credible explanation for them.
They may simply have been detachable weight piéces that were kept in a convenient way. If so, it remains to be discovered why there were three piéces on one side and two on the other and that, using this arrangement, the needle of the balance happened to point at zero.
By far the most probable explanation for their presence at the balance is that they were intended for zero adjustment. For proper weighing any well defined place for the needle- point may, in fact, be chosen but for a sus- picious customer, constantly afraid of being cheated by a whily merchant, only the suppor- ting yoke would bc the acceptable zero point. A well adjusted zero point would bc an important
"confidence creating measure". O n e might therefore hope that the spacing between the
Form amin, 83 (1988)
Viking Age weighing 163
weight of bronze piéces would give some infor- mation regarding the use of the balance in addi- tion to the evidence reported above.
Consequently, the piéces were measured, their volyme calculated and multiplied with a supposed density of 8.5. T h e weights proved to be:
First side: 0.64, 0.44 and 0.59 g. Sum 1.67 g See. side: 0.32, 0.43 0.75 g T h e differences between the piéces mainly de- pend upon differences in thickness, 0.32 to 0.64 mm. Evidently the piéces were not cut from the same sheet. The weights are crude multiples of 0.15 g. By simple rearrangement of the piéces without omitting any one of them it is possible to obtain the overweights 0.92 (the present figure) 0.90, 0.68, 0.48, 0.38, 0.36, 0.33 and 0.03 g. If we keep to the five piéces — they may have special value for the owner or are partly in- tended to be a decoration of the balance — we find that the above series covers every 0.2 to 0.3 gram interval between 0.0 and 0.9 g or if right and left are reversed —0.9 — 0 — 4-0.9 g.
To summarize: T h e bronze piéces allow a zero adjustment of the Vårdinge balance to the nearest 0.2—0.3 g. This figure is slightly better than the actual sensitivity of the balance esti- mated by weighing or calculation, 0.4 g. T h e two.sets of data for the sensitivity thus largely support each other.
The accuracy of the weight system of Bandlunde, Gotiand
No balance can be better than the set of weights used. Therefore, the weights constitute another important source ofinformation regarding the accuracy of ancient weighing. T h e main ob- stade is, of course, the corrosion which changes the weights to an unpredictable extent. T h e sta- tistically calculated variance will therefore give a blurred picture of the situation a thousand years ago.
Twenty-one weights of the Bandlunde lind belonged to a group which have been intended lo wcigh about 1.4 to 1.5 g each when new, Alter stabilization, they weighed 1.26 to 1.56 g. The group was well separated from adjacent weights and showed normal distribution around its
mean value.
When found, this sample weighed 1.487
±0.087 g; after the stabilization the weight was 1.429 ±0.078 g. T h e weights had löst 0.058 g each without any significant change in the vari- ance. This shows that the dirt and corrosion produets removed did not contribute signifi- cantly to the random deviations present in the samples. Apart from the corrosions removed, large amounts of rather non-reactive corrosion adhere to the weights. In many cases it still covers undamaged parts of the original decora- tion of the weights.
T h e standard error obtained, ±0.08 g, im- plies that deviations from the average weight by more than approximately 0.2 g occur only about once in 25 estimations. In addition it compares well with the insensitivity limits ob- served for the balance found together with these weights, cf. above.
T h e Bandlunde find also contains several globular weights in various states of preserva- tion. These weights have been examined by the present author (Sperber 1986). Measurement ol the weights which had suffered the least from corrosion yielded values up to five times more accurate than those found by direct weighing. A common denominator was found among the 12 weights in question, viz. that they were all mul- tiples of 4.2 g 0 = ±0.08 g i.e.= ±2 %. This fi- gure means that the weights in their present state may sometimes deviate ± 4 % from the average.
There are numerous finds of weights from this period, but complete or nearly complete sets are rare. As a rule only one or two weights are found together. T h e preferred sizes are 11 to
13 g and 31 to 33 g. In the Bandlunde find ap- proximately 1.5 g is also very fréquent. Even if subtractive use of the weights was common, a set consisting of only two or three weights would have been very difficult indeed to work with. It would be necessary to work with provisional weights and, in addition, to try some taring
technique,
Single weights were valuable for another purpose. They could be used to control the weight sets owned by a trading partner. It seems possible that this was the main reason for the many single weights found.
T h e Bandlunde find also contains a frag-
Fmmamun 83(1988)
164 E. Sperber
mentary balance. It has been described in some detail by Koivunen and Derestorp (1987). T h e main dimensions of this balance could be mea- sured or approximated. It is included in Table 2. Its sensitivity could be calculated with a rather high degree of accuracy. It was found to be about 0.2 g/1 mm deflexion at zero load. At a load of 2 x 35 g, the sensitivity was about 2 % per 1 m m deflexion.
T h e similarity between the two figures, the deviation 2 o = ±0.16 g for the small weights and 0.2 g/1 mm deflexion for the balance, are striking. With this balance it could be attested that 95 % of the weights differ no more than 0.2 g from the intended value of 1.49 g. Be- fore corrosion even stricter requirements could probably be encountered.
Regarding the globular weights with Ilat po- lar surfaces it also seems safe to assume that the balance is a little better than the weights in their present state. Here too, it must be inferred that the weights were better a thousand years ago.
T h e balance, on the other hand, has retained its original characteristic dimensions.
To sum up, the weights and the balance from Bandlunde form a well poised system, where the balance and the weights match each other very nicely.
Evidence of weighing accuracy in other parts of the world
T h e collapsible balances constitute a very ho- mogeneous group (Steuer 1984). It is härd to believe that there existed in Sweden a subgroup or modification thereof having much higher ac- curacy than the bulk of the balances. At least, such balances have not yet been found.
O n the other hand, there is considcrable evi- dence showing that many ancient peoples could weigh far more exactly than was customary in Viking Age Swedish commerce.
— T h e Romans had a standard weight, the siliqua with a nominal weight of only 0.189 g.
This weight would have been meaninglcss un- less scales with the proper sensitivity or better were available.
— Several solidus coins from the Migration period have been found in Sweden, some of them of the quality "v. fine" or " f i n e " i.e. not worn to a measurable degree. Most of them
came to Sweden independently by different roads and from different sources. Twenty-onc coins weighed on average 4.426 g, O being
±0.045 g = ±1 %.
— T h e Arabs and other peoples in the Near East likewise had balances with a very high sen- sitivity. Al Chåzini, who lived c. A.D. 1100, de- scribes a balance used for estimating the densi- ty of gold alloys ( I b d 1908). This "Balance of W i s d o m " in the winter at a water temperature around 4°C, gave a density value different from that in the summer, presumably near 30°C.
T h e weight of 0.25 1 of water is 250 g at 0°C but only 249.3 g when the water is warm, i.e.
there is a difference of 0.7 g. T h e weight of 0.25 1 of gold approaches 4.500 g. The sensitivity of the balance, defined in a männer unknown to us, therefore was better than 0.7/4.500, i.e.
about 0.016 %. This rather large balance — its pans measured around 200 mm in diameter — was thus about 100 times more sensitive than those used in Sweden less than a hundred years earlier!
Discussion
T h e most important steps in the development of the two-anncd balance were:
— Recognition of the advantage of using ba- lances with equal arms and development of the theory which allowed work alternativdy with unequal arms.
— Recognition that the sensitivity of a ba- lance depends to a very great extent on the rather short distance between the supporting point of the beam and its center of gravity.
— T h e almost total dimination of the influ- ence of the total load, i n d u d i n g the weight of the panson the sensitivity. Thisisaccomplishcd if the two suspension points of the pans and the supporting point of the beam are all situated on a straight line.
T h e lirst point was largely clarified in Clas- sical times by famous Greek scientists such as Pythagoras, Archimedes, M e n d a o s and others.
T h e second point was probably solved by the
Romans but written sources seem meagre pro-
bably because the problem was not properly re-
cognized and expressed. There are many linds
of antique balances. Unfortunatdy, to most
Hnwmm 8 i (1988)
Viking Age weighing 165
a r c h a e o l o g i s t s , t h e i m p o r t a n c e of t h e s h o r t dis- t a m e b e t w e e n t h e s u p p o r t p o i n t of t h e b e a m a n d its c e n t e r of g r a v i t y is far from o b v i o u s . T h i s d i s t a n c e is t h e r e f o r e n o t i n c l u d e d in t h e p u b l i s h e d d e s c r i p t i o n s of b a l a n c e finds.
T h e t h i r d p o i n t w a s solved at t h e latest c.
1800 by t h e S w e d i s h scientist C a h n after t h e p r o b l e m h a d b e e n p r o p e r l y r e c o g n i z e d . It m a y h a v e b e e n e v i d e n t to o t h e r s too, e v e n earlier, b u t t h e final p r a c t i c a l s o l u t i o n w a s p r e s e n t e d by G a h n .
T h e V i k i n g A g e b a l a n c e t y p e a n d s o m e far m o r e sensitive b a l a n c e s w e r e e v i d e n t l y in u s e s i m u l t a n e o u s l y t h o u g h in different c o u n t r i e s a n d p r o b a b l y for different p u r p o s e s . It r e m a i n s to b e e x p l a i n e d w h y t h e y w e r e so p o p u l ä r in S w e d e n o r at least a m o n g foreigners v i s i t i n g S w e d e n . T h e a n s w e r m a y b e t h a t t h e r e w a s n o u r g e n t n e e d for v e r y a c c u r a t e a n d h e n c e m o r e e x p e n s i v e b a l a n c e s in S w e d i s h m a r k e t - p l a c e s . T h e n e e d for h a n d i n e s s w a s m u c h m o r e evi- d e n t . T h e b a l a n c e s h a d a very b c a u t i f u l a p p e a - r a n c e , s h i n i n g from b r a s s o r b r o n z e , g i v i n g a s t a t u s v a l u e t o t h e o w n e r d e s p i t e t h e i r p r o b a b l y m o d e s t p r i c e , m a d e p o s s i b l e by l a r g e - s c a l e p r o - d u c t i o n .
References
Arne, T.J. 1918. Viktsenheterna i Sverige under Vi- kingatiden. Fornvännen.
Berg, Lars och Ottosson, J a n . 1984. Vikingatida hatans- vågar i Sverige. C-uppsats. Institutionen för Arkeo- logi, särskilt Nordeuropeisk. Stockholms Univer- sitet.
Berzelius, J.J. 1818. Lärobok i kemien. Del 3.
I lällström, G G . 1842. Undersökning av ett i Finska Lappmarken gjort fynd . . . Acta Societatis Fenneicae.
Helsingfors.
Kisch, Bruno. 1966. Scales and Weights. Yale University Press. New Haven and London.
Koivunen, S. och Dcrestorp, M. 1986. Vägen från Bandlundeviken. G-uppsats. Institutionen lör Ar- keologi, särskilt Nordeuropeisk. Stockholms Uni- versitet.
Kyhlberg, O. 1980. Vikt och Värde. Stockholm Studies in Archaeology.
Ibel, T h . 1908. Die Waage tm Altertum und Mittelalter.
F>langen.
Sperber, E. 1986. Weight or Volume? A Study of Bronze Covered Iron Weights. Laboraliv Arkeo- logi. Rapport från Stockholms Universitets Arkeologiska Forskningslaboratorium. Nr 1. 1986.
Steuer, H . 1984. Feinwaagen und Gewichte als Quel- len zur Handelsgeschichte des Ostseeraumes. In Archäologische und naturwissenschafiliche Untersuchungen an ländhchen und Fruhslädlichen Siedlungen im deutschen Kustengebiet vom 5. J h . v. Chr. bis zum 11. Jh. n. Chr.
Bd. 2. Handelsplätze des friihen und hohen Mit- lelalters. Weinheim. 273—292.
H u r noggrant kunde vikingarna väga?
U n d e r v i k i n g a t i d e n v a r h o p l a l l b a r a , l i k a r m a d e v å g a r m e d d ä r t i l l h ö r a n d e v i k t e r v a n l i g a i Sve- rige. D e a n v ä n d e s i n o m v i k t s o m r å d e t 1 till 100 g r a m . M a n h a r u t g å t t från att g u l d och silver v a r d e e n d a v a r o r i d å t i d e n s h a n d e l tillräckligt d y r b a r a för att p å k a l l a n o g g r a n n v ä g n i n g i n o m d e t t a o m r å d e och a t t v å g a r n a d ä r f ö r a n v ä n d e s vid k ö p av d e s s a ä m n e n . Ä r 1984 f a n n s 115 vå- g a r eller f r a g m e n t i s v e n s k a m u s e e r ( B e r g och O t t o s s o n 1984) t i l l s a m m a n s m e d flera h u n d r a vikter.
K ä n s l i g h e t e n h o s e n b a l a n s v å g b e r o r pä dess k o n s t r u k t i o n . Särskilt viktigt ä r det lilla a v s t å n - d e t , v a n l i g e n u n d e r 10 m m , m e l l a n v å g b a l k c n s s t ö d p u n k t ( e g g e n ) och d e s s t y n g d p u n k t . D e n n a r e l a t i o n h a r föga eller i n t e alls u p p m ä r k s a m - m a t s av a r k e o l o g e r n a och a v s t å n d e t i fråga a n -
ges d ä r f ö r i n t e i p u b l i c e r a d e r a p p o r t e r o m fyn- d e n .
D ä r e m o t tycks r e l a t i o n e n h a varit m e r eller m i n d r e väl k ä n d b l a n d r o m e r s k a o c h a r a b i s k a v å g f ä b r i k a n t e r , s o m av å t e r f u n n a m y n t och v i k t s a t s e r att d ö m a k u n d e tillverka v å g a r m e d k ä n s l i g h e t n e r i m i l l i g r a m o m r å d e t . S v e n s k e n C a h n s o m l e v d e 1745—1818, gavs av B e r z e l i u s ä r a n för att t e o r e t i s k t h a u t r e t t och p r a k t i s k t h a löst p r o b l e m e t s a m t att h a s p r i t t sitt v e t a n d e till d e n v e t e n s k a p l i g a v ä r l d e n .
F ö r f a t t a r e n h a r g e n o m t i l l ä m p n i n g av enkel v å g t e o r i , g e n o m p r a k t i s k a v ä g n i n g s f ö r s ö k och g e n o m u p p m ä t n i n g av d i m e n s i o n e r o c h ö v r i g a d a t a hos sju v i k i n g a t i d a v å g a r och v å g f r a g m e n t k u n n a t visa att d e n u r s p r u n g l i g a k ä n s l i g h e t e n k a n b e r ä k n a s för t. o. m illa m e d f a r n a v å g f y n d .
Förmannen 83(1988)
166 E. Sperber
Härför måste först ej tillgängliga data ersättas med sannolika siffror (exempelvis vikten hos felande vågskålar o. dyl.).
Känsligheten hos samtliga sju vågar visade sig ligga mellan 0,2 och 0,4 g för 1 m m utslag på visaren vid nära noll grams total belastning.
Vid hög belastning sjunker känsligheten. Vid vägning av ca 50 g vara ligger noggrannheten av vägningen omkring 2 %, vilket bekräftas av ett vägningsförsök från 1842, utfört av den ut- märkte finske fysikern G. G. Hällström år 1842.
H a n s våg kunde inte skilja 100,3 g från 102,1 g.
Aven om de gamla vågarna inte var särskilt känsliga var de av en enkel och praktisk kon-
struktion. Säkerligen gjorde den stora produk- tionen priset överkomligt och de var vackra att se på. De fyllde därför en stor upgift i handeln med silver och guld. Omfattningen av denna handel visas bland annat av de många fynden av silverskatter från denna tid.
Noggrannheten hos de vikter som användes till vågarna var väl anpassad till vågarnas käns- lighet. Hos ett fynd från Gotland, Burs sn, Häffinds var spridningen mellan ett tjugutal kubooktaedriska vikter som vägde omkring 1,5 g, ±0,08, vilket innebär att avvikelser från me- delvärdet med ±0,2 g ( = vågens känslighet) var ovanliga.
F.omanmn 83(1988)
