The Visby-Lenses


Authors

Olaf Schmidt, Dipl.-Ing., Fielmann Akademie Schloss Plön, Plön/Germany

Karl-Heinz Wilms, Ophthalmic Instruments Rodenstock (retired), Emmering

Bernd Lingelbach, Prof. Dr., Aalen University of Applied Science



Key words

rock crystal, vikings, aspheric lens design, aberration, Zernike-Polynomials



Abstract


Purpose

In this study 10 lens-shaped rock crystals, manufactured in the early middle ages have been examined with respect to their image forming qualities.


Methods

The spherical aberration of the lenses served as a scale for comparison. Measurements have been taken with a specially designed optical device granting a non-destructive procedure.

Some of the examined lenses have a silver mounting and have been used as pendants, while others are unmounted and show no signs of use as jewellery.


Results

It turned out that the two largest unmounted lenses have very interesting surfaces rendering a very small spherical aberration of the lens. The top and bottom surface combine to an optical system comparable to the quality of modern aspheric lenses used, e.g., in todays projectors.


Conclusion

According to the results of this study the knowledge and comprehension of optical design has been much further developed in the middle ages than we assume today.



Introduction

In search of an exhibit for the optical department of the German Museum in Munich one of us (K.-H. W.) discovered the image of an aspheric biconvex lens made out of rock crystal in the early middle ages 1 (see Fig.1).

rock crystal lens
Fig. 1: rock crystal lens displayed in [01] and [02]
unfortunately this lens seems to be lost

The aspheric design of the lens is really unusual considering that this lens has been manufactured roughly 1000 years ago. In that era scientists only started to explore the laws of light refraction.

The lens is part of a treasure laid down on Gotland in the end of the viking era, i.e. in the 11th or 12th century. There are hints that they might have been manufactured in Byzanz or in the region around Eastern Europe. There are a few other findings containing lenses of similar shape and size on Gotland. Some of them can be seen at Gotlands Fornsal, the historical museum in Visby. Others are either in different museums or lost.

Of course the lenses discussed in this article have not been examined by the museum in respect of their optical qualities but already O. Ahlström found out that most of them have aspheric surfaces2. It is necessary to check the lenses´ optical properties to find out whether the lens displayed in the above mentioned image is unique or if the other lenses have a comparably refined optical design. Therefore a small expedition went to Gotland in 1997 to measure out the shape of the lenses shown in Visby and find out if the optical qualities of the lenses are really as good as the above mentioned image promises. Furthermore the attempt was made to find out more about the origin of the lenses.



Measurement of the lenses

The examined lenses (see Fig. 2) can be divided into two different groups:

examined lenses
Fig. 2: The lenses examined in Visby
top row: unmounted lenses
bottom row: mounted lenses except the "ball"


1. mounted lenses

These lenses have a silver mounting with different kinds of filligree and have most probably been used as jewellery. The mounting includes a silver plate covering the enterior backside of the lens. Although these mounted lenses have been examined they are not described in this article. Nevertheless, it can be said that, apart from one almost perfect spheric ball, their surfaces are aspheric, as well. Probably the lenses are much older than their silver mountings.


2. unmounted lenses

This category includes both the lenses exposed in Visby and the lens displayed in [01] and [02]. Unfortunately the latter is not part of the exhibition which means that measurements could only be taken on the image.

None of the unmounted lenses show any signs of a usage as jewellery. The surface quality differs between the lenses. While especially the two larger lenses show a very good polishing the surface of one of the others is matt in the circumference of its poles.

All these lenses have one thing in common: they all show a flattening around their poles. This flat part is differently strong pronounced. Furthermore all lenses show a waviness on their surfaces.


Plotted unmounted lenses


No.

diameter
[mm]

thickness
[mm

]

plotted profiles

surface 1 surface 2

number

1

50.0

32.1

37

37

c.9389:4

2

43.0

22.9

19

16

3

37.0

16.3

10

10

4

27.0

11.8

7

7

5

19.5 x 15.7

10.0

19

1

6

17.0

5.2

1

c.10.720:8

7

19.0

8.0

2

4

c.10.720:8


Measuring of the exhibited lenses

The lenses have been measured by a light section method. Therefore a damaging of the surfaces during the measuring process could be avoided.

The main idea behind this measuring device is to project the profile of the lens and a grid of squares with a defined side lenght together on a screen (compare Fig. 3). After taking a picture of this section, the lens is turned on its z-axis for a defined angle. This method results in a series of images showing the profiles of the lens in different defined angles.


measurement device
Fig. 3: schematic construction of the used measuring device
The images of both the grid and the profile of the lens appear on the screen

Both the crystal and the reference grid are projected through the same lenses L1 and L2. Therefore the images of both undergo the same aberrations and have the same magnification and distortion. The image on the screen serves as a coordinate system with the distortion being exactly the same for the grid and the lens. It can be plotted simply by counting the number of rectangles covered by the profile of the lens on every position on the horizontal axis. For every x-coordinate the corresponding value in z-direction can be found. Because the size of the rectangles is known, the real value can be calculated very easily.

In Figure 4 the result for one profile is displayed.

measured data
Fig. 4: Measured data and best fitted ellipse of one profile

Plotting data

The measured profiles must be described mathematically. A function that describes the course of the measured profiles also describes the surface of the lens.


Three-dimensional fitting of the surfaces

For those lenses which give a convenient amount of plottable profiles it is possible to describe the surface by using a three-dimensional function. Each sectional drawing of a lens‘ datapoints has a shape very closely matching an ellipse. In a first step the best-fitting ellipse for each single profile has been determined (see Fig. 4).

All these profiles then combined to a surface for which, in the next step, the best-fitting three-axis ellipsoid (x-, y- and z-axis) has been determined (see Fig. 5).



3d display
Fig.5: Three-dimensional display of one surface consisting of all sections

This ellipsoid describes the shape of the surface quite precisely. Still it does not show the irregularities of the real lens like the waviness of the surface or the relative flatness of the poles.

In a last step the differences between the ellipsoid and the measured data are described by Zernike-Polynomials. The sum of all terms of the Zernike-Polynomials completely describes the differences between the fitted ellipsoid and the measured data.

In Figure 6 the principle of describing deviations by Zernike-Polynomials is visualized.


fitted ellipsoid
Fig. 6: A best fitted ellipsoid overlayed with the correcting Zernike-Polynomials
In the figure the Zernike-surface's scale is strongly exaggerated. In the same scale
as the ellipsoid the surface would look plane because of the small deviations

The image shows the fitted ellipsoid as a wire frame. On top of this wire frame lies the surface describing the deviations of the measured data from the fitted ellipsoid. If the measured data exactly match the ellipsoid then the Zernike-surface would be a plane. Any deviation from a plane describes a difference between the measured data and the fitted ellipsoid at the exact location on the ellipsoid lying under the surface. In the figure the Zernike-surface's scale is strongly exaggerated. In the same scale as the ellipsoid the surface would look plane because of the small deviations.

Describing the deviations with Zernike-Polynomials has a couple of advantages. Two of them are

  1. The deviation can be discriminated into rotationally symmetric and angle-dependent parts.
  2. There is a one-to-one correspondence between the deviation, its size and the location on the surface of the lens. In this way each lens shows a "finger print" of its surfaces which might give hints where the lenses have been manufactured. The flattened poles of the lenses for example are strongly displayed. If it turns out that at least two of the lenses show similar deviations from the ellipsoid then they might have been manufactured in the same manner and could well have been produced in the same workshop and / or even by the same craftsman.

Optical properties of the lenses

Most imaging aberrations (save chromatic aberrations) are coupled to the spherical aberration of a lens. Therefore, the spherical aberration can serve as a meritfunction for the imaging quality of a lens.

We chose the spherical aberration for the special case of parallel incident rays. Basing on the elliptical parameters of each surface and each lens the back focus has been calculated for different incidence heights. For comparison the same calculation has been done for a spherical lens resp. surface. For the equivalent radius of these spheres the central radius of the compared ellipse has been applied.



Results


unmounted lenses

Table 2 shows the results of the ellipsoid-fittings. The surfaces of lenses No. 1 to No. 4 were fitted with a three-dimensional function. Because of the different semiaxes a and b on a three-axis ellipsoid the values for the excentricity are not the same in every meridian. Due to the very small deviation average values are displayed.


Nr.

surface

c [mm]

a [mm]

b [mm]

e 2

shape

1

1

32.169

25.961

27.653

0.306

prolate

 

2

15.146

32.069

32.056

3.482

oblate

2

1

59.48

33.645

31.502

0.699

prolate

 

2

48.131

50.585

49.124

0.073

oblate

3

1

10.252

17.473

17.633

1.932

oblate

 

2

32.942

37.391

39.212

0.352

oblate

4

1

8.722

13.811

12.810

1.329

oblate

 

2

7.915

14.120

14.231

2.208

oblate

6

1

15.230

14.945

0.037

prolate

7

1

5.327

9.426

2.132

oblate

 

2

5.221

10.485

3.042

oblate


Both prolate and oblate shapes appear. In respect to an optimized imaging the lenses No. 1 and No. 2 are most interesting. The lenses No. 3 to No. 7 do not show any sign of optical optimization. For this reason the results of the examination of these lenses will not be discussed in this article. They will, however, serve as an example for comparison.

Lens No. 1

The aberration of the fitted ellipsoid is much smaller than the aberration of the comparable sphere. At a height of incidence of 20 mm (which is close to the very rim of the lens) the deviation from the back focus comes out to about 3.5 mm while the deviation of a comparable sphere is about 14mm. This means that the upper surface of this lens is optimized and shows a very small spherical aberration.

The second surface of lens No. 1 degrades the imaging quality of the lens by intensifying its spherical aberration. On the other hand the combination of ellipsoidic surfaces still shows a far lower aberration than the compared spheres. Lens No. 1 is optimized in respect to spherical aberration.


Lens No. 2

On the first glance the surfaces of lens No. 2 seem to be tilted against each other, but during the plotting of the measured data it turned out that they are not. The impression derives from decentrated optical axes of the surfaces. Assuming that this decentration is an error in the manufacturing process it was not taken into account. The displayed results apply for centered surfaces.

The first surface of the lens has a "negative" spherical aberration. The back foci become larger with increasing height of incidence. This "negative" aberration is corrected by the second surface of lens resulting in a spherical aberration displayed in figure 7.

lens 2
Fig. 7: Lens Nr. 2
change of back focus of the lens in comparison to an equivalent sphere

All scales in this and the following figures are in [mm]. The vertical line marks the back focus of the surface without spherical aberration (incidence parallel to the optical axis, height of incidence = 0).

The other two lines represent the course of the spherical aberration with increasing height of incidence for the fitted elliptic and a comparable spherical surface. The larger the deviation from the vertical line the larger the spherical aberration of the surface resp the lens for that height of incidence.

In other words: the smaller the deviation from the vertical line the better the optimization of the surface.

Obviously lens No. 2 has a very small spherical aberration. The course of the aberration with increasing height of incidence is very similar to modern aspheric lenses. Up to a height of about 12mm the deviation from the vertical line is less than 1 mm ! A comparable spherical lens deviates about 5 times more at the same height.

This lens has a combination of surfaces which results in an optimized image.


Lens No. 3

This lens is an example of a lens which is not optimized in respect of spherical aberration. In Figure 8 the strong deviation is clearly visible. The elliptic shape of the lens causes an even larger aberration than a comparable sphere would have.

aberration
Fig. 8: spherical aberration of lens Nr. 03 in comparison to an equivalent sphere


Presentation of the differences between ellipsoid-fitting and measured data by Zernike-Polynomials

The size of the differences between measured data and fitted ellipsoides varies considerably. All lenses show an increasing deviation towards the periphery. Consequently, the deviation is larger on lenses with a larger diameter. Figure 9 displays a few examples of Zernike-Polynomials describing the lenses Nr. 2 and Nr. 4.

zernike
Fig. 9: Zernike-Polynomials describing the differences between measured data and fitted ellipsoid.
(Top row: Lens Nr. 2; Bottom row: Lens Nr.4; left side: first surface; Right side: second surface)


Analysis of the lens described by Schmitz1 and Ahlström2

The plotting of the lens displayed in Fig. 1 is discussed separately because the conditions of the analysis differ from the other lenses. It is not known to which extent the objective used for the picture distorted the shape of the lens. Furthermore a scale to judge the exact diameter and thickness is missing. We only have one picture of this lens so that rotational symmetry is not guaranteed.

For the analysis we assume a diameter of 50 mm (as described by Schmitz1) and a centre thickness of 28.5 mm. A best-fitted ellipse has been determined iteratively.

Because of the above mentioned reasons the displayed results depend on the assumption that the image corresponds exactly to the real lens and that the lens is rotationally symmetric.

Results of the examination


The aberration curve is very similar to modern aspheric lenses used e.g. for spectacles. Up to a height of incidence of about 17 mm the departures from the ideal focus remain smaller than 1 mm! From that height on, the aberration increases very swiftly. Considering the appearance of total reflection at heights of incident higher than about 21.5 mm only a small periphere area with visible aberration remains. This lens seems to have a better optimization than any other lens examined in this study.

The surfaces of the lens have an almost perfect elliptical shape. The combination of its upper and lower side combine to a lens with very low spherical aberration.

Discussion


Age and origin of the lenses

The examined rock crystals were found in different Gotlandic treasures. It is estimated that they were laid down in the second half of the 11th century. The processing of rock crystal has been common everywhere in the known world at that time. Therefore it is difficult to say where the lenses originally came from. Furthermore they were found in different treasures so that a common origin cannot be taken for granted4. The analysis of the lenses delivered no evidence in respect of a common origin.

The mounted lenses could have been imported readily grinded but without their mounting which might have been added by a Gotlandic goldsmith. The mounting of the "ball" shows typical Gotlandic details of the Viking era4. As for the other crystals it is at least questionable whether the mountings were manufactured on Gotland. Most probably all mounted lenses except for the ball have been readily imported 4.

How did the crystal become the property of a Gotlandic? Unfortunately, it is unknown who laid down the treasures. This would give an idea on this question. It could, for instance, have been an adventurer or a trader. There are a couple of ways how the lenses reached Gotland. One of them is that they have simply been bought somewhere. The trading connections of the Vikings have certainly been wide enough. They held contact to the whole known world. During the time in question the Swedish Vikings (and Gotland as well) concentrated on the trade in the East and South-East. Therefore it seems logical to search for the origin of the lenses in these places. M. Stenberger reports that the crystals were brought from the Orient or Persia to West- or South-West-Russia, where the silver mountings were added to the lenses. From there the lenses might have been traded to Gotland.

But there are alternatives to this version. A member of the Varangian guard could have taken lenses like these from Byzanz to Gotland. After retiring they most probably returned home to Gotland. The number of Scandinavian members in the Varangian Guard detectably decreased in the second half of the 11th century5, the men were sent home and took their precious goods (and booty) with them.

Apart from these two possibilities there is a third one. Both mounted and unmounted lenses appeared on Gotland quite suddenly in the end of the 11th century and disappeared just as quick as they came6. Can it be that all rock crystal lenses of that kind have reached Gotland at one occasion ? This sounds plausible as well but does not give any answer to the question: how, and in which manner have the lenses been brought to Gotland ? Perhaps it was a tradesman who bought a lot of these lenses somewhere and sold them to a Gotlandic goldsmith.

Or was the gaining of the lenses a little less peaceful ? They could well be part of a Viking plundering in Russia or the Orient. The Vikings have not been only tradesmen, but warriors, too, as can be seen on the amount of quarrels with e.g. Byzanz.


How have the lenses been produced ?

The shape of the lenses` surface show only minor departures from rotational symmetry. This leads to the conclusion that they were manufactured on some kind of turning-lathe. The surfaces are almost perfectly elliptic. The almost perfectly spherical ball shows that craftsmen were well able to produce spheres. This suggests that the elliptic shape was intended. The flattening of the poles and some irregularities show that the lenses have been reworked. The reason for this rework is unclear. The rim of the lenses has possibly been reworked to form an even profile which makes the mounting easier. The flattening of the poles cannot be explained that easily. Most probably it is a result of the processing method. Since this flattening of the poles can be found on all of the examined lenses the conclusion can be drawn that all lenses have been manufactured using the same method.

Apart from the mentioned irregularities at the poles and the rims of the lenses all lenses show a waviness on their surfaces. These "waves" perhaps resulted from the reworking process and could have different origins. Smaller dips could have been caused during the polishing of the lenses. On the other hand both smaller and larger irregularities might have been caused by "repair-works", for example if the lens had slight splits on the surface or a minor damage occured while a "customer" used it.


Has the magnification of the lenses been utilized deliberately ?

The lenses examined in Visby, especially the larger ones, show an obvious magnification. Furthermore the imaging quality is very high. The aberration of the largest unmounted lens is very small up to the rim. It is hard to imagine that these properties have not been observed by the manufacturer or the user of the lens. The mounted lenses which have a silver plate on their back side strongly suggest that the optical properties have not only been discovered but also used deliberately. The silver plate reflects the incoming light and produces distinct images of objects in front of the piece of jewellery. This means that both the backside of the lens and the "front" side of the silver must have a polished surface. In the middle ages polishing caused considerable extra work. Why should a gold- or silversmith expend time if not to achieve a certain aim: to produce a better light reflection behind the back side of the stone and a more interesting effect of the jewellery.

Jay M. Enoch described the rather complicated manufacturing process of an antique greek ring7. The ring is covered with a piece of polished glass. Underneath the glass lies a piece of gold foil with the effect that "the gold seems to float under the glass". The goldsmith who produced this ring definitely knew which effect he wanted to achieve. The same applies to the manufacturer of the described mounted lenses with the silver plate.

This does not imply that the craftsmen of the middle ages had detailed knowledge about the laws of refraction and reflection. But they most certainly had an empiric knowledge about how a piece of rock crystal must be shaped to achieve a certain effect.

We find a couple of hints that the imaging or magnifying properties of lenses have at least been used in the design of jewellery. In this context it is an interesting question whether reading stones have been used before the time of Roger Bacon. The opinions about this question vary from author to author. One central question is: How did older (presbyopic) people read and how did older craftsmen like goldsmiths perform their filigree work. Gorelick and Gwinnet were of the opinion that finer work was done by young or myopic people. This argument is plausible up to a certain degree. On the other hand: the filligree work of some pieces manufactured in the early middle ages suggests that the craftsman must have had a lot of experience so that a young man simply could not do it7. Perhaps both variations existed beneath each other.

It stands to reason if the unmounted lenses have been designed to be used as reading stones or magnifiers. At least it is not out of the question. We know that the magnification of a water filled ball has been used as a reading aid by Lucius Annaeus Seneca, who lived around 55 BC- 41 AC, long before the time in question1,7.

The imaging quality of the lenses examined in Visby differs considerably. Especially the largest of the lenses could definitely be used as a loupe. Its imaging quality is much better than that of the later produced spheric reading stones of e.g. Roger Bacon.

Another aspect is the question whether an "optimization" of lenses has been intended or not. Some of the examined lenses show a spherical aberration so small that a purpose seems very likely. The mathematical background needed for an optimization has not been explored in that time so it could only be done by trial and error. A skilful and very experienced craftsman might well have been able to produce optimized lenses. On the other hand it must be stated that most of the examined lenses do not have optimized surfaces. Considering this it is just as likely that the shape of the "better" lenses is only the result of a "nice" design and the imaging quality is a by-product.

The lens described by Schmitz has a stunning similarity to the largest lens examined in Visby in respect to shape and size. Both do have an optimized imaging. It seems possible that both lenses come out of the same hands. These two lenses could well be "twins". If it turns out that both lenses have really been produced in the same workshop then the thesis of an intended optimization is strongly supported.



Conclusion

The examination of the lenses exhibited in Visby shows that some of them do have much better optical properties than later produced spherical reading stones do. The imaging quality is almost as good as aspherical lenses produced nowadays. The conclusion that the aspheric shape resulted coincidentially cannot be drawn because the "ball" proofs that craftsmen were very well able to produce almost perfect spheric surfaces. Therefore the elliptic shape of the lenses must be assumed as intended.

Although it is well known that the foci of aspheric mirrors were calculated much earlier1 we found no hint in literature that elliptic surface optimization was performed on lenses in the middle ages. This does not mean that it has not been tried to improve the imaging qualities of lenses. The impressing imaging quality of the lenses suggests that the craftsmen knew better than the scientists in that time. Obviously the optimization has been done by craftsmen long before mathematicians were able to describe the shape and properties of aberration-corrected lenses. It seems as if a small number of stone grinders (perhaps only one) has been able to manufacture lenses refined in the described way.

Unfortunately it is not clear where the lenses come from. It would be interesting to know whether they come out of one workshop or whether they are a collection from different places. In this context the question if the lens described by Schmitz is a copy of the largest Visby-lens is interesting as well.

The only way to answer this question is to examine this lens and to compare the results to this study. Unfortunately this lens seems to be lost. Neither the National Museum in Stockholm nor Gotlands Fornsal in Visby can tell where this lens is at the moment.

Facing the results of this study we have to think about the impression we have of the optical knowledge in the middle ages. Lenses showing imaging properties of that quality are not produced out of lack of knowledge. It seems that this knowledge got lost for at least some 500 years until Descartes calculated the ideal focusing lens shape but, lacking the necessary technical equipment, could not produce it.



Acknowledgments

We thank the "Förderverein Deutscher Augenoptiker" as well as the Aalen Institut für Augenoptik for their generous support and Dr. Ralf Blendowske for his useful comments during the preparation of this article. Also, we thank Gotlands Fornsal, especially Mrs. Malin Lundquist for her uncomplicated help.



References

  1. Schmitz E-H. Handbuch zur Geschichte der Optik,
    Band 1: von der Antike bis Newton.
    Verlag J.P. Wayenborgh, Bonn 1981

  2. Ahlström O Swedish Vikings used optical lenses.
    The Optician1950; ,459-469

  3. Born M, Wolf E. Principles of Optics. 6th ed. Pergamon Press, 1986

  4. Stenberger M. Die Schatzfunde Gotlands der Wikingerzeit.
    Kungl. Vitterhets Historie och Antikvitets Akademien
    Stockholm 1947, 1958

  5. Oxenstierna E G. Die Wikinger.
    Kohlhammer Verlag Stuttgart 1979

  6. Gotlands Fornsal, Gotländisches Landesmuseum
    personal info. 1989-1991

  7. Enoch J M. Early lens use: lenses found in context with their original objects.
    Optom Vis Sci 1996;73:707-15

Principal Author: Olaf Schmidt, Aalen University of Applied Science Gartenstrasse 135, D-73430 Aalen, GERMANY

e-mail: olaf.schmidt@o2online.de