11institutetext: Haus der Astronomie, Campus MPIA, Königstuhl 17, D-69117
Heidelberg, Germany
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# How the Vikings Navigated With the Sun

M. Nielbock

(Received July 14, 2016; accepted )

The students are introduced to navigation in general, and, in particular, the
skills the medieval people of the Vikings used to navigate on the open sea.
The topics of navigation and the Vikings are introduced by questions and a
short story. A hands-on activity illustrates to the students how the
navigational tool of the shadow board might have helped the Vikings to
determine the cardinal directions and to sail along latitude. A miniature
version of a shadow board simulates how the shadow cast by the Sun was
probably used for this. Finally, a simple math activity for more experience
students demonstrates the geometry that is involved in that technique.

###### Key Words.:

Earth, navigation, astronomy, history, geography, Sun, equator, latitude,
longitude, meridian, celestial navigation, Vikings, seasons, sundial

##  1 Background information

###  1.1 Latitude and longitude

Any location on an area is defined by two coordinates. The surface of a sphere
is a curved area, but using coordinates like up and down does not make much
sense, because the surface of a sphere has neither a beginning nor an ending.
Instead, we can use spherical polar coordinates originating from the centre of
the sphere with the radius being fixed (Fig. 1). Two angular coordinates
remain. Applied to the Earth, they are called the latitude and the longitude.
Its rotation provides the symmetry axis. The North Pole is defined as the
point, where the theoretical axis of rotation meets the surface of the sphere
and the rotation is counter-clockwise when looking at the North Pole from
above. The opposite point is the South Pole. The equator is defined as the
great circle half way between the two poles.

Figure 1: Illustration of how the latitudes and longitudes of the Earth are
defined (Peter Mercator, djexplo, CC0).

The latitudes are circles parallel to the equator. They are counted from
$0\degr$ at the equator to $\pm 90\degr$ at the poles. The longitudes are
great circles connecting the two poles of the Earth. For a given position on
Earth, the longitude going through the zenith, the point directly above, is
called the meridian. This is the line the Sun apparently crosses at local
noon. The origin of this coordinate is defined as the Prime Meridian, and
passes Greenwich, where the Royal Observatory of England is located. From
there, longitudes are counted from $0\degr$ to $+180\degr$ (eastward) and
$-180\degr$ (westward).

Example: Heidelberg in Germany is located at 49$\aas@@fstack{\circ}$4 North
and 8$\aas@@fstack{\circ}$7 East.

###  1.2 Elevation of the pole (pole height)

If we project the terrestrial coordinate system of latitudes and longitudes at
the sky, we get the celestial equatorial coordinate system. The Earth’s
equator becomes the celestial equator and the geographic poles are
extrapolated to build the celestial poles. If we were to make a photograph
with a long exposure of the northern sky, we would see from the trails of the
stars that they all revolve about a common point, the northern celestial pole
(Fig. 2).

Figure 2: Trails of stars at the sky after an exposure time of approximately 2
hours (Ralph Arvesen, Live Oak star trails,
https://www.flickr.com/photos/rarvesen/9494908143,
https://creativecommons.org/licenses/by/2.0/legalcode).

In the northern hemisphere, there is a moderately bright star near the
celestial pole, the North Star or Polaris. At the southern celestial pole,
there is no such star that can be observed with the naked eye. Other
procedures have to be applied to find it. If we stood exactly at the
geographic North Pole, Polaris would always be directly overhead. We can say
that its elevation would be (almost) $90\degr$. This information already
introduces the horizontal coordinate system (Fig. 3).

Figure 3: Illustration of the horizontal coordinate system. The observer is
the origin of the coordinates assigned as azimuth and altitude or elevation
(TWCarlson, https://commons.wikimedia.org/wiki/File:Azimuth-
Altitude_schematic.svg, “Azimuth-Altitude schematic”,
https://creativecommons.org/licenses/by-sa/3.0/legalcode).

It is the natural reference we use every day. We, the observers, are the
origin of that coordinate system located on a flat plane whose edge is the
horizon. The sky is imagined as a hemisphere above. The angle between an
object in the sky and the horizon is the altitude or elevation. The direction
within the plane is given as an angle between $0\degr$ and $360\degr$, the
azimuth, which is usually counted clockwise from north. In navigation, this is
also called the bearing. The meridian is the line that connects North and
South at the horizon and passes the zenith.

Figure 4: When combining the three coordinate systems (terrestrial spherical,
celestial equatorial, local horizontal), it becomes clear that the latitude of
the observer is exactly the elevation of the celestial pole, also known as the
pole height (own work).

For any other position on Earth, the celestial pole or Polaris would appear at
an elevation smaller than $90\degr$. At the equator, it would just graze the
horizon, i.e. be at an elevation of $0\degr$. The correlation between the
latitude (North Pole = $90\degr$, Equator = $0\degr$) and the elevation of
Polaris is no coincidence. Figure 4 combines all three mentioned coordinate
systems. For a given observer at any latitude on Earth, the local horizontal
coordinate system touches the terrestrial spherical polar coordinate system at
a single tangent point. The sketch demonstrates that the elevation of the
celestial North Pole, also called the pole height, is exactly the northern
latitude of the observer on Earth.

###  1.3 Early navigational skills

Early seafaring peoples often navigated along coastlines before sophisticated
navigational skills were developed and tools were invented. Sailing directions
helped to identify coastal landmarks (Hertel 1990). To some extent, their
knowledge about winds and currents helped them to cross short distances, like
e.g. in the Mediterranean.

Soon, the navigators realised that celestial objects, especially stars, can be
used to keep the course of a ship. Such skills have been mentioned in early
literature like Homer’s Odyssey which is believed to date back to the 8th
century BCE. There are accounts of the ancient people of the Phoenicians who
were able to even leave the Mediterranean and ventured on voyages to the
British coast and even several hundred miles south along the African coast
(Johnson & Nurminen 2009). A very notable and well documented long distance
voyage has been passed on by ancient authors and scholars like Strabo, Pliny
and Diodorus of Sicily. It is the voyage of Pytheas, a Greek astronomer,
geographer and explorer from Marseille who around 300 BCE apparently left the
Mediterranean by passing Gibraltar and made it up north until the British
Isles and beyond the Arctic Circle, where he possibly reached Iceland or the
Faroe Islands that he called Thule (Baker & Baker 1997; Cunliffe 2003).
Pytheas already used a gnomon or a sundial, which allowed him to determine his
latitude and measure the time during his voyage (Nansen 1911).

###  1.4 Sailing along latitude

At these times, the technique of sailing along a parallel (of the equator) or
latitude was used by observing circumpolar stars. The concept of latitudes in
the sense of angular distances from the equator was probably not known.
However, it was soon realised that when looking at the night sky, some stars
within a certain radius around the celestial poles never set; they are
circumpolar. When sailing north or south, sailors observe that the position of
the celestial pole change as well, and with it the circumpolar radius.
Therefore, whenever navigators see the same star culminating – transiting the
meridian – at the same elevation, they stay on the “latitude”. For them, it
was sufficient to realise the connection between the elevation of stars and
their course. Navigators had navigational documents that listed seaports
together with the elevation of known stars. To reach the port, they simply
sailed north or south until they reached the corresponding latitude and then
continued west or east.

Nowadays, the easiest way to determine one’s own latitude on Earth is to
measure the elevation of the North Star, Polaris, as a proxy for the true
celestial North Pole. In our era, Polaris is less than a degree off. Due to
the precession of the Earth’s axis, 1000 years ago, it was $8\degr$ away from
the celestial pole.

Figure 5: Vikings probably used the technique of sailing along latitude to
reach destinations west of Scandinavia (red lines). Iceland is on the 64th
northern latitude and 680 nautical miles away from Norway’s coast. The voyage
to Greenland along the 61st northern latitude passes the Shetland and Faroe
Islands. A stopover in Iceland is a viable alternative.

However, using Polaris to determine the north bearing and one’s own latitude
of course only works when it is dark enough to see the 2 mag bright star. At a
clear day, this is only possible during Nautical or Astronomical Twilight,
i.e. when the Sun has set and its centre is more than $6\degr$ below the
horizon. However, at latitudes higher than $61\degr$ north, the Sun can stay
above such low (negative) elevations, especially around summer solstice. This
is the realm north of the Shetland Islands, i.e. certainly the Faroe Islands
and Iceland. Hence, observing Polaris becomes rather difficult during summer,
which is the preferred season for sailing. For latitudes north of the Arctic
Circle, where sea ice can block passages during winter, the sun never sets for
a certain period during summer. Therefore, other techniques were needed for
navigation.

###  1.5 The Vikings

he Vikings were Northern Germanic tribes who were known for their seamanship,
their influential culture and a wide trade network. And they were feared for
their raids and pillages that were executed with roaring brutality. However,
contrary to common urban legends, the Vikings were not the filthy, savage
barbarians that wore horned helmets when going into battle. Instead, they
seemed to be well groomed, and bathed at least once per week (Berg Petersen
2012). Their origins are the coastal regions around western and southern
Scandinavia as well as Denmark. During their explorations, they settled in
Iceland, Greenland, Normandy and the British Isles. However, they ventured as
far as Northern America, all around Europe, the Black Sea and the Caspian Sea
(Fig. 6).

Figure 6: Map of Viking expansion between the 8th and 11th century. Their
origins are the Norwegian coast, Southern Sweden as well as Denmark (Max
Naylor, https://commons.wikimedia.org/wiki/File:Viking_Expansion.svg, public
domain).

The beginning of the Viking Era is commonly dated to 793 CE with the raid of
Christian monastery of Lindisfarne (Graham-Campbell 2001) in Northumbria,
England. However, the Gallo-Roman historian St. Gregory of Tours reports on an
earlier attack by a Danish king named Chlochilaicus on Austrasia, the homeland
of the Merovingian Franks, around 520 CE. It is believed that this Danish king
may be identical to the mythical character of Hygelac in the Beowulf poem
(Susanek 2000). The Viking Era ended with the Battle of Hastings between the
English and the Norman-French, who were descendants of the Vikings, and the
destruction and abandonment of Hedeby, an important Viking settlement and
trading post, both in 1066. As it marked the Norman conquest of Britain, the
Battle of Hastings was such an important turning point in British history that
it was documented with colourful and vivid pictures on the Bayeux Tapestry
(Fig. 7), made in the 1070s which is still in brilliant shape (Hicks 2007).

Figure 7: A segment of the Bayeux Tapestry depicting Odo, Bishop of Bayeux and
half-brother to William the Conqueror, rallying the Norman troops during the
Battle of Hastings in 1066. The Bayeux Tapestry is a 70 metre long embroidered
cloth depicting the Battle of Hastings and the events leading up to the Norman
Conquest of England. It was probably commissioned by Odo himself (Hicks 2007)
(https://commons.wikimedia.org/wiki/File:Odo_bayeux_tapestry.png, public
domain).

###  1.6 Viking navigation

The Vikings were famous for their longships, multi-purpose ships that could be
used on rivers, shallow coastal waters and oceans. They were used for trade,
exploration and warfare. Depending on their size, they could carry from a
dozen up to 80 sailors. Because of their shallow draught, many of them did not
need a harbour to make landfall but could simply be beached. Those ships were
usually decorated with carved ornaments. Propulsion was provided by sail or
oars which could lead to speeds of 15 to 20 knots.

Figure 8: The “Viking”, a replica of the Gokstad Viking ship, at the Chicago
World Fair 1893 (Di Cola & Stone 2012). With a crew of 11, it crossed the
Atlantic and reached Chicago within 2 months (public domain).

Assuming an average speed of 5 knots, crossing the Northern Sea would have
been possible within one or two days. Longer trips, e.g. from Norway to
Iceland would have been achieved within five to seven days.

The Viking sailors were very experienced in interpreting the signs nature
provides. They were able to read the migratory routes of birds (Forte et al.
2005) and whales as well as the smell and sound that the wind carries from
distant shores. The Vikings probably did not have any sea charts, but they
used chants and rhymes that contained sailing information as mentioned in the
medieval Hauksbók chronicle (Sawyer 1997) and were passed on from generation
to generation. For instance, the route from southern Norway to Greenland
passes the Shetland Islands and Iceland. Their sightings could be used to
correct the course, which perfectly coincides with staying on latitude
$61\degr$ north. Therefore, the Vikings must have had skills to follow it.

Figure 9: Illumination of the northern and southern hemisphere of the Earth
during its orbit around the Sun (Tau’olunga,
https://en.wikipedia.org/wiki/File:North_season.jpg, CC0).

As mentioned before, the Sun played an important role for finding a ship’s
course. The difficulty with the Sun compared to the stars is that the Sun
changes its declination, i.e. the elevation above the equator. The reason is
that the Earth with its tilted axis revolves around the Sun. In the northern
summer, the northern hemisphere faces the Sun, while during northern winter,
it is the southern hemisphere. The range, under which the Sun appears in the
zenith, is the latitudes between 23$\aas@@fstack{\circ}$4 north, the Tropic of
Cancer, and 23$\aas@@fstack{\circ}$4 South, the Tropic of Capricorn. For any
given location on Earth, the Sun’s elevation while it transits the meridian –
the line that connects North and South at the horizon through the zenith –
changes by the same amount.

Figure 10: At summer solstice, the Sun is directly above the Tropic of Cancer.
Its apparent position changes during the year (Przemyslaw ”Blueshade”
Idzkiewicz, https://commons.wikimedia.org/wiki/File:Earth-lighting-summer-
solstice_EN.png, “Earth-lighting-summer-solstice EN”,
https://creativecommons.org/licenses/by-sa/2.0/legalcode).

For latitude of $61\degr$ north, the elevation of the Sun above the horizon is
shown in Fig. 11. South is at the centre at an azimuth of $180\degr$.
Throughout the year, the elevation of the Sun at local noon changes by almost
$47\degr$. However, the rate of change is not constant. We can assume a
variation of the solar declination of up to $1\degr$ per voyage to be
acceptable for navigational purposes.

If we allow such a variation during two consecutive days, the Sun could be
used at any time of the year. This means, within two days the declination of
the Sun –- or its elevation at noon -– never changes more than $1\degr$. This
corresponds to a deviation of 8 km after travelling for 240 nautical miles. As
already pointed out, two days are sufficient for voyages through the Northern
Sea. For five to seven day journeys, the allowed period is between end of May
and beginning of July. This is enough to travel from southern Norway to
Iceland. The corresponding drift due to the changing solar elevation amounts
to 25 km or less. For longer travels, e.g. to Greenland, the course can be
adjusted by landmarks on the way when for instance passing the Shetland
Islands, the Faroe Islands and Iceland. Is there evidence for navigational
tools the Vikings used with the Sun?

Figure 11: The diurnal and annual elevation of the Sun above the horizon for a
latitude of $61\degr$ north (Created with the sun chart path program of the
University of Oregon, USA, http://solardat.uoregon.edu/SunChartProgram.html).

###  1.7 The Sun Shadow Board

Sailing along latitude was probably facilitated by a device that was called
solskuggerfjøl (sun shadow board, Fig. 12). 18th century sailors of the Faroe
Islands have been seen using a wooden disk of up to 30 cm in diameter with
engraved concentric rings and a central gnomon whose height could be adjusted
(Tjgaard 2011). It was put inside a bucket with water to cancel out ship
movements. It is quite likely that it was already used during the Viking age.

If we assume a negligible change of the Sun’s declination, the shadow of the
gnomon at noon can be calibrated to any latitude by aligning the tip of the
shadow with a circle. When read during noon of the following days, the shadow
should again touch the same circle. If the shadow is shorter, the position is
too far South; if it is longer, the location is too far north.

Figure 12: Sketch of the Viking sun shadow board (M. Nielbock, own work).

###  1.8 The Sun Compass

The magnetic compass was unknown in Europe during the Viking age. And it would
have been quite useless for them anyway, because the magnetic field of the
Earth is far from homogeneous. The phenomenon that the magnetic poles do not
align well with the geographical ones is called magnetic declination. In
addition, the field lines are strongly curved. And both processes change in
time (Fig. 13). Measuring campaigns like ESA’s SWARM satellites constantly
monitor the magnetic field (ESA 2013).

Thus, especially at high latitudes, a magnetic compass would have let the
Vikings lose their way more often than it would have aided them in finding the
correct course. But it seems they were able to find the cardinal directions
using the Sun.

Figure 13: Change of the magnetic North Pole and its position relative to
Scandinavia (Created with the NOAA Historical Magnetic Declination Viewer,
http://maps.ngdc.noaa.gov/viewers/historical_declination/).

In 1948, fragments of a small wooden disk were found during historic
excavations of an abandoned monastery of Uunartoq in southwest Greenland (Fig.
14). In the following decades, people started to interpret it as a
navigational tool to determine the cardinal directions using the Sun
(Thirslund 2007). However, even to date, there are doubts that it truly served
that purpose. Nonetheless, there are quite remarkable scientific analyses that
demonstrate that in fact this disk could have been a combination of a sundial,
a compass, and a sun shadow board (Bernáth et al. 2013).

Figure 14: Image of the original wooden disk fragment found in Uunartoq,
Greenland. Annotations denote elements for its possible usage as a sun compass
(Bernáth et al. 2013). When the shadow is aligned with the shadow lines, North
is up. Incisions in that direction permit measuring the shadow length (Lennart
Larsen, Danish National Museum, http://samlinger.natmus.dk/DO/10775,
“Trædisk_Grønland”, background of photograph removed and annotations added by
Markus Nielbock, https://creativecommons.org/licenses/by-sa/2.0/legalcode).

Incised lines have been identified as paths of a shadow cast by a central
gnomon during the days of equinox and summer solstice at a latitude of
$61\degr$ north. These lines hypothetically helped to determine local noon,
i.e. when the Sun attains its highest elevation when crossing the meridian.
This moment indicates the time when the device can be used. At local noon, a
central gnomon produces a shadow that points north. Similar to the sun shadow
board mentioned above, incisions on the wooden board towards the northern
direction can be used to determine possible deviations from the course along a
predefined latitude.

###  1.9 Local time and time zone

The shadow of a gnomon or a sundial is shortest and points north, whenever the
Sun is exactly south (northern hemisphere). This is what defines local noon.
Since the Earth rotates continuously, the apparent position of the Sun changes
as well. This means that at any given point in time local noon is actually
defined for one longitude only. However, clocks show a different time. Among
other effects, this is due to daylight saving time during summer, and the time
zones (Figure 15). Here, noon happens at many longitudes simultaneously.
However, it is obvious that the Sun cannot transit the meridian for all those
places at the same time. Therefore, the times provided by common clocks are
detached from the “natural” local time a sundial shows.

Figure 15: World time zones. Instead of the local time that is based on the
apparent course of the Sun in the sky and valid for single longitudes only,
the common clocks show a time based on time zones which applies to many
longitudes simultaneously (TimeZonesBoy,
https://commons.wikimedia.org/wiki/File:Standard_World_Time_Zones.png,
https://creativecommons.org/licenses/by-sa/4.0/legalcode).

##  2 List of material

The list contains items needed by one student. Some of them can be shared by
two to four individuals.

  * •

Worksheet

  * •

Cardboard (diameter at least 4 cm)

  * •

Toothpick

  * •

Earth globe (+ stable mounting, e.g. inflatable: https://goo.gl/gq1d5N,
http://www.unawe.org/earthball)

  * •

Compass (drawing tool)

  * •

Lamp or spot light

  * •

Scissors

  * •

Cutter or sharp knife

  * •

Glue

  * •

Blu Tack (or similar)

##  3 Goals

With this activity, the students will learn that

  * •

the Vikings were much more than warriors.

  * •

the Vikings were a people that have been very influential for European
history.

  * •

the Vikings were skilled sailors and used to navigate by the Sun.

  * •

the cardinal directions can be determined by the position of the Sun.

  * •

the astronomical (local) noon does not coincide with noon on the clock.

##  4 Learning objectives

At age 14, students should already understand the concepts of latitude and
longitude for defining a position on Earth.

  * •

After reading a story about a Viking voyage, the students will be able to
explain how the Sun can be used to find the cardinal directions.

  * •

They will also be able to explain, how the Sun can be used to maintain a
course along latitude.

  * •

With the same story, the students will be able to describe the region of
origin of the Vikings and some of their typical professions.

  * •

With a hands-on activity, the students will be able to explain the basic
functionality of a shadow board, an old Viking navigational tool.

  * •

More experience students will be able to explain the geometry behind the
shadow board and how trigonometry of a right angle triangle helps to calculate
the length of a shadow.

##  5 Target group details

Suggested age range: 12 – 16 years
Suggested school level: Middle School, Secondary School
Duration: 45 minutes

##  6 Evaluation

The evaluation is done twofold: by checking the results of the activities and
from the answers of the students and their discussions.

  * •

In particular, the teacher may ask the students, to what direction the Sun
attains its highest elevation and how the cardinal directions can be derived
from this.

  * •

The teacher may ask the students, how the elevation of the Sun at noon changes
with latitude, and how this information can be used to navigate along it.

  * •

The teacher can ask the students why the length of the shadow cast by the Sun
is indicative of the local latitude.

  * •

Together with the students, the teacher can develop a sketch that illustrates
the geometry of a stick that produces a shadow when illuminated by the Sun. He
then asks about the trigonometric functions involved in this.

  * •

While showing a map of Europe, the teacher may ask the students, where the
Vikings originated from.

##  7 Full description of the activity

###  7.1 Introduction

It would be beneficial, if the activity be included into a larger context of
seafaring, e.g. in geography, history, literature, etc.

Tip: This activity could be combined with other forms of acquiring knowledge
like giving oral presentations in history, literature or geography
highlighting navigation. This would prepare the field in a much more
interactive way than what a teacher can achieve by summarising the facts.

Tip: There are certainly good documentaries available on Vikings and sea
exploration that could be shown as an introduction. Some suggestions are:

History Channel — The Vikings [New] HD (Duration: 1:28:22)
https://www.youtube.com/watch?v=6mkib3lYA2s

Vikings, The Founders of Europe (Duration: 50:32)
https://www.youtube.com/watch?v=jngPAIGrzJs

Building a Viking Ship (Duration: 1:46)
https://www.youtube.com/watch?v=78kpzwGmBxk

###  7.2 Questions, Answers and Discussion

Ask the students, if they had an idea for how long mankind already uses ships
to cross oceans. One may point out the spread of the Homo sapiens to islands
and isolated continents like Australia.

Possible answers:
We know for sure that ships have been used to cross large distances already
since 3,000 BCE or earlier. However, the early settlers of Australia must have
found a way to cross the Oceans around 50,000 BCE

Ask them, what could have been the benefit to try to explore the seas.
Perhaps, someone knows historic cultures or peoples that were famous sailors.
The teacher can support this with a few examples of ancient seafaring peoples,
e.g. from the Mediterranean and the art of navigation.

Possible answers:
Finding new resources and food, trade, spirit of exploration, curiosity.

Ask the students, how they find the way to school every day. What supports
their orientation to not get lost? As soon as reference points (buildings,
traffic lights, bus stops, etc.) have been mentioned ask the students, how
navigators were able to find their way on the seas. In early times, they used
sailing directions in connection to landmarks that can be recognised. But for
this, the ships would have to stay close to the coast. Lighthouses improved
the situation. But what could be used as reference points at open sea?
Probably the students will soon mention celestial objects like the Sun, the
Moon and stars.

Let the students read the story below (separate document available).

###  7.3 Story: Galmi the Viking

“This is the day”, Galmi thinks by himself as he looks out into the bay of
Vikebygd. He is a Norseman in his early thirties and has spent most of his
life struggling to feed his family on his farm. His wife Sága and his daughter
Oda have endured a lot, but Galmi has made plans that could change things for
the better. He came here 15 years ago with his parents who set out to find
their own little piece of land and live peacefully. However, the last two
winters have struck them hard leaving only little that could be spared for
sowing out next spring.

Last night, Galmi told Sága and Oda that he would go to Avaldsnes and sign on
for the next raid to the land in the West. News had travelled about the riches
his fellow Vikings had brought home from the realm of the Scots and the Picts.
And he definitely wanted a share. “We would be able to buy us food and seeds
to survive until next summer”, he explains. Galmi packs his pouch for the hike
that should take not much more than 9 hours. Sága and Oda are sad and afraid,
but they also know that this would probably be their last chance. And if
everything turns out for the best, Galmi should be back within a week the
latest.

He arrives in Avaldsnes just before sunset. It is a busy little town with
about 15 houses, a market place, the chief’s residency, and, of course, a
harbour with a handful of longships anchoring. Tomorrow will be the last raid
for this season, and Galmi wants to be on board. He enters a tavern and
quickly finds a group of experienced sailors. They share a few drinks and
Galmi befriends them. It turns out that one of them is the nephew of an old
friend of his fathers, Floki. And Ragnar is the leader of this group. They are
about to head for the mead hall where the chief of the island of Karmøy,
Augvald, assigns the raiding parties to their ships. Galmi is lucky. Together
with Ragnar, Floki and 21 other Vikings, he will set sail for Inbhir Nis next
morning, a rich Celtic village at the mouth of the river Ness.

Galmi does not get much sleep during the night. He is torn between the
excitement of the journey and the battle ahead, and the insecure future of his
family. Just as his fatigue starts to win against the racing thoughts in his
head, Ragnar calls him. And Floki is already at his side. It is before sunrise
and the chilling breeze from the North delivers a glimpse of the nearing
autumn. As they reach the shore, three ships have already left the pier and 8
oars at each side of the boats accelerate them quickly north to the mouth of
the fjord. The rest of Galmi’s crew is already here and prepares the ship. It
is a real beauty. “Galmi! Here is your seat”, shouts Rollo. “You’ll be among
the first to row!” Galmi shrugs, “Sure! Why not?” Ragnar and Floki smile and
they jump on board.

One hour later, the ship leaves the fjord and the winds turn out to be
favourable. They set their sail and the ship accelerates smoothly out to the
open sea. Galmi is excited and surprised how steady this ship is despite the
waves and the shallow draught. It is a sunny day and the gods seem to be with
them.

The journey continues without any problem. Galmi uses this opportunity to get
to know the crew. They tell him stories from their last expeditions and how
proud they are being a part of this. “Galmi, bring me a bucket with water”,
Floki suddenly suggests. “What for?”, Galmi returns puzzled. “You’ll see.”
Galmi hands him the bucket while Floki grabs a wooden disk. “It’s noon and the
Sun is shining, let’s see.” He puts it into the bucket where it begins to
float. As Galmi looks closer, he discovers a ring incised about half way
between the edge of the disk and the centre, where a small cone sticks out. A
shadow extends to the edge of the board and slightly beyond the ring. “Okay,”
says Floki, “we have to continue our course to the Southwest for another five
hours. Then we’ll turn west.”

Galmi is surprised. Is Floki a seer or a magician? Is he in contact with
Kvasir, the god of wisdom? “Not at all”, laughs Ragnar, “he is just our
navigator. Floki, can you explain what you do?” “Sure. You see, Galmi. The Sun
shines on this board and the shadow points north. You know that the Sun is
south at noon, do you? The length of the shadow tells me, if we are on course.
If the shadow is too long, we are too far north. The Sun is lower there. Is it
too short, we know that we are too far south.” “How do you know all this?”,
asks Galmi. “My grandfather taught me. He was one of the first to scare the
hell out of the Scots”, Floki replies with pride in his eyes.

Four hours after the course correction the steady breeze picks up and dark
clouds appear at the horizon. “Darn! Seems like Thor is angry again!”, Ragnar
shouts. “Get prepared for some rough weather ahead!” The wind evolves into a
storm. Floki strikes the sail. “Okay guys! Take the oars and row!” Ragnar has
to shout in order to be heard through the howling wind. Galmi pulls as hard as
he can. It starts to rain heavily and thunder and lightning seem to tear the
skies apart. “Pitch the tent!”, Ragnar shouts. The crewmen pull up a thick and
long piece of linen and cover the ship almost from bow to stern. The ship
rolls heavily, but the men keep on rowing. Two hours later, the crew at the
oars changes. Galmi tries to get some sleep. He is so tired that neither the
rocking boat nor Thor’s hammer can keep him awake.

Galmi dreams of Sága and Oda. He feels a warm sensation on his cheeks and
opens his eyes. The Sun is rising and the wind has died, but through the
silent hissing of the waves the snoring of the crew sounds like the howling of
the Fenrir wolf. The sail is up and Floki leans at the rudder. “How did it go,
Floki?”, Galmi asks. “Oh, it wasn’t too bad. Nothing we hadn’t managed before.
It is about 12 hours until we reach the shores.” “Come, let me take the rudder
for a while and get some rest. I think I can hold that stick for you.” Floki
smiles. “Thank you, my friend.”

Twelve hours later Ragnar hears the distant calling of seagulls. Everyone on
board knows that this means they are right on course and the shore is not very
far. “Can you smell the smoke, Galmi?”, asks Floki. “The Scots burn peat for
heating their homes.” “Land ahead!” Rollo is the first to discover the
coastline. But it is another four hours until Inbhir Nis. Ragnar discusses the
plan with his crew. They will beach outside the town and camp until the next
morning. After the tiring passage everybody needs the rest. Two days after
they left Avaldsnes, Ragnar, Floki, Rollo and Galmi are prepared as is the
rest of the crew. They ask for Odin’s support and head for Inbhir Nis, whose
clueless inhabitants will be struck by surprise. “This is the day”, Galmi says
to Floki.

####  7.3.1 Questions about the story

Q: What is Galmi’s job?
A: He is a farmer.

Q: Why does he want to go on a raid?
A: Harvests have been poor. He needs resources to buy seeds.

Q: Who is leading the crew of sailors that Galmi joins?
A: Ragnar

Q: What is their destination? Would you know its modern name?
A: Inbhir Nis is the old Gaelic name of Inverness.

Q: What propulsion do longships have?
A: Oars and sail

Q: How do the Vikings protect themselves against heavy rainfall on open seas?
A: A linen tent

Q: How does Floki determine the course?
A: He uses a shadow board to determine their latitude.

Q: How does the crew know that the coast is near?
A: They observe seagulls. They smell smoke.

Q: Do longships need anchors to dock?
A: No, longships are shallow enough to be beached.

###  7.4 Activity: A Game of Shadows

This activity employs the educational tool “Parallel Earth”
(http://www.unawe.org/resources/books/Parallel_Earth/) which demonstrates on
model level the phenomena related to the Sun illuminating the Earth. The
students produce a miniature version of a shadow board and experiment with it.

If there are not many globes available, the teacher may choose to introduce
this activity as a demonstration with only one globe. However, there are
inflatable globes available for only a few Euros that can be distributed among
the students. In that case, let them work in groups of two.

####  7.4.1 Building a miniature shadow board

  1. 1.

Use the compass to draw three concentric circles with radii of 1, 1.5, and 2
cm on a piece of cardboard.

  2. 2.

Cut out the disc along the circle with the radius of 2 cm.

  3. 3.

Cut a 2 cm long piece from the toothpick.

  4. 4.

Run it through the centre of the cardboard disc and glue it. Make sure that it
remains perpendicular to the surface of the disc. The two rings on the disc
must be on the same side like the stick (see Fig. 16).

  5. 5.

Let the glue dry.

Figure 16: Miniature shadow board made of cardboard and a toothpick. Left:
side view, right: top view (own work).

####  7.4.2 Part 1: Shadows

Put the shadow board on the desk. Before doing some experiments, answer the
following questions:

Q: When you illuminate the shadow board, how will the shadow cast by the stick
behave when changing the position of the lamp? What will be the direction of
the shadow relative to the lamp?
A: It always points away from the lamp.

Q: How will the shadow change when you hold the lamp high or low?
A: The higher the lamp, the shorter the shadow.

Take a lamp and illuminate the shadow board. Change the position of the lamp
relative to the board and observe how the shadow changes.

Compare your observations to your predictions. Did the shadow behave as you
have anticipated? Discuss with your classmates:

Q: Imagine the lamp represents the Sun. Where does the shadow point to, when
the Sun is south (northern hemisphere)/north (southern hemisphere) of you?
A: The shadow points north (northern hemisphere)/south (southern hemisphere).

Q: Imagine the lamp represents the Sun. Are the shadows longer in winter or
summer? Explain!
A: As the Sun attains lower elevations above the horizon during winter, this
is the season when the shadows are longer.

####  7.4.3 Part 2: Navigate

The Vikings had travelled between Scandinavia and Greenland. Fig. 5
demonstrates that when embarking from southern Norway and travelling along the
61st northern latitude, they would end up at the southern tip of Greenland. It
is quite likely that at least during parts of such a voyage they used the
navigational tool of the shadow board.

Figure 17: The lamp that represents the Sun illuminates the globe. Northern
summer is achieved, when the axis of the globe and the line connecting it with
the lamp subtends an angle smaller than $90\degr$ (own work).

To simulate the point of embarkment, attach the miniature shadow board with a
Blu Tack on the globe at the southern tip of Norway. Since the Vikings
predominantly sailed during summer, the angle between the axis of the globe
and the connecting line to the lamp should be less than a right angle
($90\degr$, see Fig.17) – although this is not very important for the purpose
of this exercise. The easiest way to do this is to keep the North Pole up.

Important: Always try to keep the lamp, the shadow board and the North Pole
within the same vertical plane.

In the beginning, put the globe in a position so that the shadow on the shadow
board points up, i.e. to the North Pole. Adjust the distance between the globe
and the lamp until the shadow touches one of the rings (inner ring in Fig.
18).

Figure 18: Starting position of the shadow board with the gnomon located at
southern Norway (own work).

Q: What is the local time of the day, when the lamp (the Sun), the shadow
board and the axis of the globe (Earth) are within the same plane? From the
perspective of the shadow board on the Earth, where would be the Sun in the
sky?
A: The Sun just passes the meridian, i.e. it is due south. Therefore, this
situation is local noon.

While keeping the orientation of the globe, rotate it around its axis and
watch how the shadow changes (Fig. 19).

Figure 19: Change of shadow for situations before (left) and after local noon
(right). The orientation of the shadow indicates time (own work, not presented
to the students).

Q: Which situation corresponds to the configuration with the shadow board east
of the previous orientation of the globe?
A: This is the morning of the day.

Q: In which way does the shadow change?
A: It turns away from the previous direction and becomes longer.

Q: When is the shadow shortest?
A: At local noon.

Now, return to the initial configuration. Then change the latitude of the
starting position by moving the shadow board north and south (Fig. 20).

Tip: If an inflatable globe is used, its orientation can be fixed by mounting
it on a bowl.

Figure 20: Change of shadow during local noon for situations more northern
(left) and more southern (right) than the initial position. The length of the
shadow at noon indicates latitude (own work, not presented to the students).

Q: In which way does the shadow change?
A: The length changes. It is shorter for more southern positions, and longer
for more northern positions.

You will now simulate a voyage to the southern tip of Greenland. During each
measurement with the shadow board adjust the globe such that it indicates
local noon for the board.

Now put the shadow board on a position west along the current latitude. Rotate
the globe until the shadow points north. Repeat this procedure a few times
until you reach the southern tip of Greenland. Make sure, the tilt of the axis
is kept constant.

Figure 21: Functionality of the shadow board when travelling west along
latitude. When measured at local noon, the orientation and the length remains
the same (own work, not presented to the students).

Q: Explain why the length of the shadow during local noon is always the same
anywhere on the same latitude.
A: The configuration between the lamp (the Sun) and any position on a given
latitude is always the same. Rotating the globe (Earth) does not change the
angle under which the shadow board is illuminated.

###  7.5 The Math of the Shadow Board (optional, for higher terms,
trigonometric functions needed)

This topic is an interesting application of simple trigonometry. Therefore,
the example of the shadow board can be used to motivate reinforcing
trigonometry.

For simplicity, the calculations assume:

  * •

The Sun is above the equator (equinox).

  * •

The radius of the Earth is negligible in comparison to Sun’s distance.

  * •

Local noon at the location of the shadow board.

  * •

The local horizon at the position of the shadow board is flat.

Figure 22: Sketch that represents the illumination of a gnomon at a certain
latitude on Earth by the Sun. For simplicity, the Sun is located above the
equator (equinox). Local noon is assumed (own work).

The basic configuration of the Sun and Earth is shown in Fig. 22. There is a
gnomon (i.e. the shadow board) extending from the surface of the Earth that,
illuminated by the Sun, casts a shadow. Since the Earth is small in comparison
to the Sun’s distance, the rays of light hitting the equator and the gnomon
are considered to be parallel to each other (Fig. 23).

Figure 23: Sketch that demonstrates the geometric relations of a gnomon
(stick) of height $h$ at latitude $\varphi$ that, illuminated by the Sun,
casts a shadow of the length $\ell$. For simplicity, the Sun is located above
the equator (equinox). The local horizon is assumed to be flat (own work).

Since the local horizon at the position of the shadow board is a tangent at
the surface of the Earth, the gnomon and the horizon subtend a right angle.
The height of the gnomon is $h$ and the length of the shadow is $\ell$. The
geometry of Fig. 23 demonstrates that the latitude $\varphi$ is also the angle
opposite of $\ell$.

As a task, develop Fig. 23 with the entire class or group.

Our local coordinate system that extends to the horizon is a tangent area that
touches the surface of the Earth. When looked at from the side, this area
appears like a line that touches the circumference of the Earth – a circle.

Q: What is the angle between a tangent and a line connecting it to the centre
of the circle?
A: It is a right angle.

Let them identify the right angled triangle.

Q: What is the trigonometric function that describes the relation between
$\ell$ and $h$?
A: The tangent.

Q: If $h=10\,{\rm cm}$, what is the length of the shadow $\ell$ at latitudes
of $\varphi=40\degr$ and $60\degr$?
A: $\tan\varphi=\frac{\ell}{h}\leftrightarrow\ell=h\cdot\tan\varphi=8.4\,{\rm
cm}$ and $17.3\,{\rm cm}$.

###  7.6 Conclusions

Q: Why does the shortest shadow during the day point north (northern
hemisphere)?
A: The shadow is shortest, when the Sun is highest during the day. In the
northern hemisphere, this is when the Sun transits the meridian, i.e. it is
exactly south. The shadow hast to point north.

Q: What time was it, when the Sun was in the South (northern hemisphere)?
A: Most probably not 12 h, noon.

Q: Why does the clock show something else than 12 h at local noon, when the
Sun attains its highest elevation?
A: Time zones, daylight saving time, equation of time (bonus knowledge)

Q: The length of the shadow at local noon changes throughout the year. Why?
A: The declination of the Sun changes. This is because the Earth with its
titled axis revolves around the Sun. From the surface of the Earth, the Sun
appears to be alternating between the Tropic of Cancer and the Tropic of
Capricorn. This modifies the elevation of the Sun seen from any point on
Earth.

##  8 Conclusion

The activity shows how the cardinal directions can be determined with the
shadow of a gnomon caused by the Sun during the day. At the same time, the
length of the shadow is a measure of the latitude on Earth. This knowledge
represents a navigational skill that Vikings during the 8th until the 11th
century used to find their course to distant destinations across the open
seas. A short story is included that tells the students something about what
the life of a Viking may have looked like and generates interest to study this
important and influential era of European history.

###### Acknowledgements.

This resource was developed in the framework of Space Awareness. Space
Awareness is funded by the European Commission’s Horizon 2020 Programme under
grant agreement no. 638653. The hands-on activity is based on the “Parallel
Earth” tool developed by Carme Alemany and Rosa Maria Ros within the
educational programme EU-UNAWE.

## References

  * Baker & Baker (1997) Baker, R. F. & Baker, C. F. 1997, Ancient Greeks: Creating the Classical Tradition (New York, USA: Oxford University Press)
  * Berg Petersen (2012) Berg Petersen, I. 2012, What Vikings really looked like, Science Nordic, http://sciencenordic.com/what-vikings-really-looked
  * Bernáth et al. (2013) Bernáth, B., Blahó, M., Egri, A., Barta, A., & Horváth, G. 2013, Proceedings of the Royal Society A, 469, 20130021
  * Cunliffe (2003) Cunliffe, B. 2003, The Extraordinary Voyage of Pytheas the Greek, paperback edn. (Harmondsworth, UK: Penguin Books)
  * Di Cola & Stone (2012) Di Cola, J. M. & Stone, D. 2012, Images of America: Chicago’s1893 World’s Fair (Charleston, South Carolina, USA: Arcadia Publishing)
  * ESA (2013) ESA. 2013, ESA’s magnetic field mission Swarm, European Space Agency, http://www.esa.int/Our_Activities/Observing_the_Earth/The_Living_Planet_Programme/Earth_Explorers/Swarm/ESA_s_magnetic_field_mission_Swarm
  * Forte et al. (2005) Forte, A., Oram, R., & Pedersen, F. 2005, Viking Empires (Cambridge, UK: Cambridge University Press)
  * Graham-Campbell (2001) Graham-Campbell, J. 2001, The Viking World, 3rd edn. (London, UK: Frances Lincoln)
  * Hertel (1990) Hertel, P. 1990, Das Geheimnis der alten Seefahrer, Geographische Bausteine No. 38 (Gotha, Germany: Hermann Haack Verlagsgesellschaft mbH)
  * Hicks (2007) Hicks, C. 2007, The Bayeux Tapestry: The Life Story of a Masterpiece (London, UK: Vintage Books)
  * Johnson & Nurminen (2009) Johnson, D. S. & Nurminen, J. 2009, The History of Seafaring, 2nd edn. (National Geographic), authorised German Edition
  * Nansen (1911) Nansen, F. 1911, In Northern Mists: Arctic Exploration in Early Times, Vol. 1 (New York: Frederick A. Stokes company)
  * Sawyer (1997) Sawyer, P. 1997, The Oxford Illustrated History of the Vikings, ed. P. Sawyer (Oxford, New York: Oxford University Press)
  * Susanek (2000) Susanek, C. 2000, in Reallexikon der Germanischen Altertumskunde, 2nd edn., ed. J. Hoops, Vol. 15 (Berlin, Germany: Walter de Gruyter GmbH & Co. KG), 298–300
  * Thirslund (2007) Thirslund, S. 2007, Viking Navigation (Roskilde, Denmark: Viking Ship Museum)
  * Tjgaard (2011) Tjgaard, G. 2011, Windjamming to China (Durham, UK: Strategic Book Group)

## Glossary

##  Appendix A Supplemental material

This unit is part of a larger educational package called “Navigation Through
the Ages” that introduces several historical and modern techniques used for
navigation. An overview is provided via:

Navigation_through_the_Ages.pdf

The supplemental material is available on-line via the Space Awareness project
website at http://www.space-awareness.org. The direct download links are
listed as follows:

  * •

Worksheets: astroedu1648-Vikings-WS.pdf

  * •

Story of the Viking Galmi: astroedu1648-Vikings-Story.pdf