by Tom M. Apostol One of the
greatest engineering achievements of ancient times is a water tunnel,
1,036 meters (4,000 feet) long, excavated through a mountain on the Greek
island of Samos in the sixth century B.C. It was dug through solid limestone
by two separate teams advancing in a straight line from both ends, using
only picks, hammers, and chisels. This was a prodigious feat of manual
labor. The intellectual feat of determining the direction of tunneling
was equally impressive. How did they do this? No one knows for sure, because
no written records exist. When the tunnel was dug, the Greeks had no magnetic
compass, no surveying instruments, no topographic maps, nor even much
written mathematics at their disposal. Euclids Elements,
the first major compendium of ancient mathematics, was written some 200
years later. There are, however, some convincing explanations, the oldest of which is based on a theoretical method devised by Hero of Alexandria five centuries after the tunnel was completed. It calls for a series of right-angled traverses around the mountain beginning at one entrance of the proposed tunnel and ending at the other, maintaining a constant elevation, as suggested by the diagram below left. By measuring the net distance traveled in each of two perpendicular directions, the lengths of two legs of a right triangle are determined, and the hypotenuse of the triangle is the proposed line of the tunnel. By laying out smaller similar right triangles at each entrance, markers can be used by each crew to determine the direction for tunneling. Later in this article I will apply Heros method to the terrain on Samos. Hero
of Alexandrias theoretical method for working out the line of a
tunnel dug simultaneously from both ends. Heros
plan was widely accepted for nearly 2,000 years as the method used on
Samos until two British historians of science visited the site in 1958,
saw that the terrain would have made this method unfeasible, and suggested
an alternative of their own. In 1993, I visited Samos myself to investigate
the pros and cons of these two methods for a Project MATHEMATICS!
video program, and realized that the engineering problem actually consists
of two parts. First, two entry points have to be determined at the same
elevation above sea level; and second, the direction for tunneling between
these points must be established. I will describe possible solutions for
each part; but first, some historical background. Samos, just
off the coast of Turkey in the Aegean Sea, is the eighth largest Greek
island, with an area of less than 200 square miles. Separated from Asia
Minor by the narrow Strait of Mycale, it is a colorful island with lush
vegetation, beautiful bays and beaches, and an abundance of good spring
water. Samos flourished in the sixth century B.C. during the reign of
the tyrant Polycrates (570522 B.C.), whose court attracted poets,
artists, musicians, philosophers, and mathematicians from all over the
Greek world. His capital city, also named Samos, was situated on the slopes
of a mountain, later called Mount Castro, dominating a natural harbor
and the narrow strip of sea between Samos and Asia Minor. The historian
Herodotus, who lived in Samos in 457 B.C., described it as the most famous
city of its time. Today, the site is partly occupied by the seaside village
of Pythagorion, named in honor of Pythagoras, the mathematician and philosopher
who was born on Samos around 572 B.C. Pythagoras spent little of his adult
life in Samos, and there is no reason to believe that he played a role
in designing the tunnel. Polycrates
had a stranglehold on all coastal trade passing through the Strait of
Mycale, and by 525 B.C., he was master of the eastern Aegean. His city
was made virtually impregnable by a ring of fortifications that rose over
the top of the 900-foot Mount Castro. The massive walls had an overall
length of 3.9 miles and are among the best-preserved in Greece. To protect
his ships from the southeast wind, Polycrates built a huge breakwater
to form an artificial harbor. To honor Hera, queen of the Olympian gods,
he constructed a magnificent temple that was supported by 150 columns,
each more than 20 meters tall. And to provide his city with a secure water
supply, he carved a tunnel, more than one kilometer long, two meters wide,
and two meters high, straight through the heart of Mount Castro. Delivering
fresh water to growing populations has been an ongoing problem since ancient
times. There was a copious spring at a hamlet, now known as Agiades, in
a fertile valley northwest of the city, but access to this was blocked
by Mount Castro. Water could have been brought around the mountain by
an aqueduct, as the Romans were to do centuries later from a different
source, but, aware of the dangers of having a watercourse exposed to an
enemy for even part of its length, Polycrates ordered a delivery system
that was to be completely subterranean. He employed a remarkable Greek
engineer, Eupalinos of Megara, who designed an ingenious system. The water
was brought from its source at Agiades to the northern mouth of the tunnel
by an underground conduit that followed an 850-meter sinuous course along
the contours of the valley, passing under three creek beds en route. Once
inside the tunnel, whose floor was level, the water flowed in a sloping
rectangular channel excavated along the eastern edge of the floor. The
water channel then left the tunnel a few meters north of the southern
entrance and headed east in an underground conduit leading to the ancient
city. As with the northern conduit, regular inspection shafts trace its
path. The tunnel
of Samos was neither the first nor the last to be excavated from both
ends. Two other famous examples are the much shorter, and very sinuous,
Tunnel of Hezekiah (also known as the Siloam tunnel), excavated below
Jerusalem around 700 B.C., and a much longer tunnel under the English
Channel completed in 1994. The Tunnel of Hezekiah required no mathematics
at all (it probably followed the route of an underground watercourse),
the Tunnel of Samos used very little mathematics, while the Channel Tunnel
used the full power of modern technology. There is no written record naming
the engineers for Hezekiahs tunnel, just as there is none for the
pyramids of Egypt, most cathedrals of Europe, or most dams and bridges
of the modern world. Eupalinos was the first hydraulic engineer whose
name has been preserved. Armed only with intellectual tools, he pulled
off one of the finest engineering achievements of ancient times. No one
knows exactly how he did it. But there are several possible explanations,
and we begin with Heros method. Hero, who lived in Roman Alexandria in the first century A.D., founded the first organized school of engineering and produced a technical encyclopedia describing early inventions, together with clever mathematical shortcuts. One of historys most ingenious engineers and applied mathematicians, Hero devised a theoretical method for aligning a level tunnel to be drilled through a mountain from both ends. The contour map on the right shows how his method could be applied to the terrain on Samos. This map is part of a detailed, comprehensive 250-page report published in 1995 by Hermann Kienast, of the German Archaeological Institute in Athens. The sloping dashed line on the map shows the tunnel direction to be determined. This
schematic diagram shows how Heros method can be applied to the terrain
on Samos. The hypotenuse of the large right triangle is the line of the
proposed tunnel. Markers can be put along the hypotenuses of the small
right triangles at each entrance to help the tunnelers dig in the right
direction. Using Heros
method, start at a convenient point near the northern entrance of the
tunnel, and traverse the western face of the mountain along a piecewise
rectangular path (indicated in red) at a constant elevation above sea
level, until reaching another convenient point near the southern entrance.
Measure the total distance moved west, then subtract it from the total
distance moved east, to determine one leg of a right triangle, shown dashed
on the map, whose hypotenuse is along the proposed line of the tunnel.
Then add the lengths of the north-south segments to calculate the length
of the other leg, also shown dashed. Once the lengths of the two legs
are known, even though they are buried beneath the mountain, one can lay
out smaller horizontal right triangles on the terrain to the north and
to the south (shown in orange) having the same shape as the large triangle,
with all three hypotenuses on the same line. Therefore, workers can always
look back to markers along this line to make sure they are digging in
the right direction. This remarkably
simple and straightforward method has great appeal as a theoretical exercise.
But to apply it in practice, two independent tasks need to be carried
out with great accuracy: (a) maintain a constant elevation while going
around the mountain; and (b) determine a right angle when changing directions.
Hero suggests doing both with a dioptra, a primitive instrument used for
leveling and for measuring right angles. His explanation, including the
use of the dioptra, was widely accepted for almost two millennia as the
method used by Eupalinos. It was publicized by such distinguished science
historians as B. L. van der Waerden and Giorgio de Santillana. But because
there is no evidence to indicate that the dioptra existed as early as
the sixth century B.C., other scholars do not believe this method was
used by Eupalinos. To check the feasibility of Heros method, we
have to separate the problems of right angles and leveling. First, consider
the problem of right angles. The Samians of that era could construct right
angles, as evidenced by the huge rectangular stones in the beautifully
preserved walls that extended nearly four miles around the ancient capital.
Dozens of right angles were also used in building the huge temple of Hera
just a few miles away. We dont know exactly how they determined
right angles, but possibly they constructed a portable rectangular frame
with diagonals of equal length to ensure perpendicularity at the corners.
A carpenters square appears in a mural on a tomb at Thebes, dating
from about 1450 B.C., so its reasonable to assume that the Samians
had tools for constructing right angles, although the accuracy of these
tools is uncertain. In practice, each application of such a tool (the
dioptra included) necessarily introduces an error of at least 0.1 degree
in the process of physically marking the terrain. The schematic diagram
on page 33 shows a level path with 28 right angles that lines up perfectly
on paper, but in practice would produce a total angular error of at least
two degrees. This would put the two crews at least 30 meters apart at
the proposed junction. Even worse, several of these right angles would
have to be supported by pillars 10 meters high to maintain constant elevation,
which is unrealistic. A level path with pillars no more than one meter
high would require hundreds of right angles, and would result in huge
errors in alignment. Therefore, because of unavoidable errors in marking
right angles, Heros method is not accurate enough to properly align
the small right triangles at the two entrances. As for leveling,
one of the architects of the temple of Hera was a Samian named Theodoros,
who invented a primitive but accurate leveling instrument using water
enclosed in a rectangular clay gutter. Beautifully designed round clay
pipes from that era have been found in the underground conduit outside
the tunnel, and open rectangular clay gutters in the water channel inside
the tunnel. So the Samians had the capability to construct clay gutters
for leveling, and they could have used clay L-shaped pieces for joining
the gutters at right angles, as suggested in the illustration on the left.
With an ample supply of limestone slabs available, and a few skilled stonemasons,
Eupalinos could have marked the path with a series of layered stone pillars
capped by leveling gutters that maintained a constant elevation while
going around the mountain, thereby verifying constant elevation with considerable
accuracy. In 1958,
and again in 1961, two British historians of science, June Goodfield and
Stephen Toulmin, visited the tunnel to check the practicability of Heros
theory. They studied the layout of the surrounding countryside and concluded
that it would have been extremely laboriousif not actually impossibleto
carry out Heros method along the 55-meter contour line that joins
the two entrances, because of ravines and overhangs. They also noticed
that the tunnel was built under the only part of the mountain that could
be climbed easily from the south, even though this placed it further from
the city center, and they suggested that an alternative method had been
used that went over the top, as shown in the photo above. Armed with
Goodfield and Toulmins analysis, I checked the feasibility of Heros
method. Its true that the terrain following the 55-meter contour
is quite rough, especially at the western face of the mountain. But just
a few meters below, near the 45-meter contour, the ground is fairly smooth,
and it is easy to follow a goat trail through the brambles in a westerly
direction around the mountain. Eupalinos could have cleared a suitable
path along this terrain and marked it with stone pillars, keeping them
at a constant elevation with clay leveling gutters, as described earlier,
or with some other leveling instrument. At the western end of the south
face, the terrain gradually slopes down into a stream-bed that is usually
dry. An easy walk along this streambed leads to the northern face and
a gentle slope up toward the northern entrance of the tunnel. Not only
is the path around the mountain almost level near the 45-meter contour,
it is also quite shortfrom one tunnel entrance to the other is at
most only 2,200 meters. Goodfield
and Toulmin suggested that the most natural way to establish a line of
constant direction would be along the top of the mountain, driving a line
of posts into the ground up one face of the hill, across the top, and
down the other, aligning the posts by eye. To compare elevations on the
two sides of the hill, they suggested measuring the base of each post
against that immediately below it, using a level. This approach presents
new problems. First, it is difficult to drill holes in the rocky surface
to install a large number of wooden posts. Second, aligning several hundred
posts by eye on a hill-side is less accurate than Heros use of right
angles. Keeping track of differences in elevation is also very difficult.
There is a greater chance of error in measuring many changes in elevation
along the face of a hill than there would be in sighting horizontally
going around on a path of constant elevation. Because errors can accumulate
when making a large number of measurements, Eupalinos must have realized
that going over the top with a line of posts would not give a reliable
comparison of altitude. To ensure
success, he knew it was essential for the two crews to dig along a nearly
level line joining the two entrances. The completed tunnel shows that
he did indeed establish such a line, with a difference of only 60 centimeters
in elevation at the junction of the north and south tunnels, so he must
have used a leveling method with little margin for error. Water leveling
with clay gutters, as described earlier, provides one accurate method,
but there are other methods that are easier to carry out in practice. Recently,
my colleague Mamikon Mnatsakanian conceived an idea for another simple
leveling tool that could have been used, a long wooden bow suspended by
a rope from a central balancing point so that its ends are at the same
horizontal level. The bow need not have uniform thickness and could be
assembled by binding together two lengthy shoots from, say, an olive tree.
Such a bow, about eight meters long, would weigh about four pounds and
would be easy to carry. And it has the advantage that no prior calibration
is needed. To use the
bow as a leveling tool, place it on the ground and slowly lift it with
the rope. If the two ends leave the ground simultaneously, they are at
the same elevation. If one end leaves the ground first, place enough soil
or flat rock beneath that end until the other end becomes airborne. At
that instant, the bow will oscillate slightly, which can be detected visually,
and the two ends will be at the same elevation. Three people are needed,
one at the center and one at each end to fix the level points. To check
the accuracy, turn the bow end-to-end and make sure both ends touch at
the same two marks. In this way, the device permits self-checking and
fine tuning. There will be an error in leveling due to a tiny gap between
the endpoint of the bow and the point on which it is supposed to rest
when level; this can be checked by the human eye at close range, and would
be one or two millimeters per bow length. By sighting horizontally along the tops of two pillars set eight meters apart, a target point can be located up to 100 meters away; a possible error of two millimeters in the elevations of the two pillars expands to an error of 25 millimeters in the elevation of the target point. To employ
the leveling bow to traverse Mount Castro at constant elevation, one can
proceed as follows. Construct two stone pillars, eight meters apart, and
use the leveling bow to ensure that the tops of the pillars are at the
same elevation. At the top of each pillar, a horizontal straightedge can
be used for visual sighting to locate other points at the same elevation
and on the same linethe same principle as used for aiming along
the barrel of a rifle. Points approximately at the same elevation can be sighted visually at great distances. Choose such a target point and construct a new pillar there with its top at the same elevation, and then construct another pillar eight meters away to sight in a new direction, using the leveling bow to maintain constant elevation. Continuing in this manner will produce a polygonal path of constant elevation. The contour map on the right shows such a path in green, with only six changes in direction leading from point B near the northern entrance to point D near the southern entrance. The distance around Mount Castro along this path is less than 2,200 meters (275 bow lengths), so an error of two millimeters per bow length could give a total leveling error of about 55 centimeters. This is close to the actual 60-centimeter difference in floor elevation measured at the junction inside the tunnel. In contrast to Heros method, no angular or linear measurements are needed for leveling with Mamikons bow. Using
pillars and the leveling bow, a polygonal path, in green, can be traced
around the mountain at a constant elevation above sea level to join point
B near the northern entrance to point D near the southern. Without Heros
method, how could Eupalinos locate the entry points and fix the direction
for tunneling? Kienasts report offers no convincing theory about
this, but using information in the report, Mamikon and I propose another
method that could have determined both entries, and the direction of the
line between them, with considerable accuraccy. Kienasts contour
map (right) shows that the north entrance to the tunnel, point A on the
map, lies on the 55-meter contour line, and the source at Agiades is near
the same contour line. His profile map in the report, which I have adapted
for the diagram above, reveals that persons standing at the crest of the
mountain cannot see Agiades in the north and the seashore in the south
at the same timebut they could if they were on top of a seven-meter-high
tower at point T. The Samians could easily have constructed such a tower
from wood or stone. Mamikon has conceived a sighting tool, based on the same principle as the Roman groma, that could have been used to align T with two points: one (N) near the source in the northern valley and one (S) on the seashore in the south within the city walls. This sighting tool consists of a pair of two-meter-high fishing poles about five meters apart, with a thin string hanging vertically from the top of each pole to which a weight or plumb bob has been attached. Each pole would be mounted on the tower so that it could turn around a vertical axis, allowing the hanging strings to be aligned by eye while aiming toward a person or marker at point N. Sighting along the same device in the opposite direction locates S. From
a point seven meters above the top of the profile, there is a direct line
of sight to Agiades in the north and the seashore to the south. It would
have been easy for Eupalinos to build a surveying tower (T ) there. Two
sighting poles such as the ones shown below could have been installed
on the tower to align with markers N and S in Agiades and on the shore,
respectively. Using the
same type of sighting instrument from point N, and sighting back toward
T, the tunnels northern entrance (point A) can be chosen so that
S, T, N, and A are in the same vertical plane. Then the leveling bow can
be used to ensure that A is also at the same elevation as N. For the southern
entrance, we need a point C inside the city walls at exactly the same
elevation as A. Starting from a nearby point B at a lower elevation easier
to traverse and in the vertical plane through N and A, a series of sighting
pillars can be constructed around the western side of the mountainusing
any of the leveling methods described earlieruntil point D, at the
intersection of the sighting plane through T and S, is reached. The two
terminal points B and D will be at the same elevation above sea level,
and the vertical plane through points B, T, and D fixes the line for tunneling.
It is then a simple matter to sight a level line from the top of a pillar
at D to determine the southern entrance point, C, that has the same elevation
as A. Markers could be placed along a horizontal line of sight at each
entrance to guide the direction of excavation. How accurate
is this method of alignment? In sighting from one fishing pole to its
neighbor five meters away, there is an alignment error of one millimeter,
which expands to an overall error of 22 centimeters in sighting from point
T at the top of the mountain to point N, 1,060 meters away to the north,
plus another error of 13 centimeters from T to S, 644 meters away to the
south. So the total error in sighting from N to S by way of T is of the
order of 35 centimeters. Adjusting this to account for the alignment error
from N to A by way of B, and from S to C by way of D, gives a total error
of the order of less than 50 centimeters between the two directions of
tunneling. This is well within the accuracy required to dig two shafts,
each two meters wide, and have them meet, and is much more accurate than
the other proposed methods of alignment. Moreover, as with the alternative
method of leveling, no angular or linear measurements are needed. No matter how it was planned, the tunnel excavation itself was a remarkable accomplishment. Did the two crews meet as planned? Not quite. If the diggers had kept faith in geometry and continued along the straight-line paths on which they started, they would have made a nearly perfect juncture. The path from the north, however, deviates from a straight line. When the northern crew was nearly halfway to the junction point, they started to zigzag as shown in the diagram on the left, changing directions several times before finally making a sharp left turn to the junction point. Why did they change course? No one knows for sure. Perhaps to avoid the possibility of digging two parallel shafts. If one shaft zigzagged while the other continued in a straight line, intersection would be more likely. Or they may have detoured around places where water seeped in, or around pockets of soft material that would not support the ceiling. In the final stretch, when the two crews were near enough to begin hearing each other, both crews changed direction as needed and came together. The sharp turns and the difference in floor levels at the junction prove conclusively that the tunnel was excavated from both ends. At the junction itself, the floor level drops 60 centimeters from north to south, a discrepancy of less than one-eighth of a percent of the distance excavated. This represents an engineering achievement of the first magnitude. This
picture was taken inside the tunnel looking toward the southern entrance.
Although the floor is more or less level along the whole length, the roof
slopes in line with the rock strata. Metal grillwork now covers the water
channel on the left to prevent hapless tourists from falling in; the photo
on the next page shows how treacherous it was before. No one knows
exactly how long it took to complete the project, but estimates range
from 8 to 15 years. Working conditions during excavation must have been
extremely unpleasant. Dust from rubble, and smoke from oil lamps used
for illumination, would have presented serious ventilation problems, especially
when the workers had advanced a considerable distance from each entrance.
A visit to
the tunnel today reveals its full magnificence. Except for some minor
irregularities, the southern half is remarkably straight. The craftsmanship
is truly impressive, both for its precision and its high quality. The
tunnels two-meter height and width allowed workers carrying rubble
to pass those returning for more. The ceiling and walls are naked rock
that gives the appearance of having been peeled off in layers. Water drips
through the ceiling in many places and trickles down the walls, leaving
a glossy, translucent coating of calcium carbonate, but some of the original
chisel marks are still visible. The floor is remarkably level, which indicates
that Eupalinos took great care to make sure the two entrances were at
the same elevation above sea level. Of course,
a level tunnel cannot be used to deliver a useful supply of water. The
water itself was carried in a sloping, rectangular channel excavated adjacent
to the tunnel floor along its eastern edge. Carving this inner channel
with hand labor in solid rock was another incredible achievement, considering
the fact that the walls of the channel are barely wide enough for one
person to stand in. Yet they are carved with great care, maintaining a
constant width throughout. At the northern end, the bottom of the channel
is about three meters lower than the tunnel floor, and it gradually slopes
down to more than nine meters lower at the southern end. The bottom of
the channel was lined with open-topped rectangular clay gutters like those
in the drawing on page 34. When the
tunnel was rediscovered in the latter part of the 19th century and partially
restored, portions of the water channel were found to be covered with
stone slabs, located two to three meters above the bottom of the channel,
onto which excess rubble from the excavation had been placed up to the
tunnel floor. The generous space below the slabs along the bottom of the
channel provided a conduit large enough for ample water to flow through,
and also permitted a person to enter for inspection or repairs. In several
places the channel was completely exposed all the way up to the tunnel
floor, permitting inspection without entering the channel. Today, most
of the rubble has been cleared from the southern half of the channel,
and metal grillwork installed to prevent visitors from falling in. To visit
the tunnel today, you first enter a small stone building erected in 1882
at the southern entrance. A narrow, rectangular opening in the floor contains
a steep flight of wooden steps leading down into a walled passageway about
12 meters long, slightly curved, and barely wide enough for one person
to walk through. The sides of the passageway are built of stone blocks
joined without mortar, and capped by a gabled roof formed by pairs of
huge, flat stones leaning against each other in a manner characteristic
of ancient Greek construction. This passageway was constructed after the
tunnel had been completely excavated. Excess rubble from the excavation,
which must have been considerable, was used to cover the roof of this
passageway, and a similar one at the north entrance, to camouflage them
for protection against enemies or unwelcome intruders. Electric lights
now illuminate the southern half, up to a point about 100 meters north
of the junction with the northern tunnel, where there is a barrier and
landfall that prohibits visitors from going further. For centuries,
the tunnel kept its secret. Its existence was known, but for a long time
the exact whereabouts remained undiscovered. The earliest direct reference
appears in the works of Herodotus, written a full century after construction
was completed. Artifacts found in the tunnel indicate that the Romans
had entered it, and there is a small shrine near the center from the Byzantine
era (ca. A.D. 500900). In 1853, French archaeologist Victor Guérin
excavated part of the northern end of the subterranean conduit, but stopped
before reaching the tunnel itself. An abbot from a nearby monastery later
discovered the northern entrance and persuaded the ruler of the island
to excavate it. In 1882, 50 men restored the southern half of the tunnel,
the entire northern underground conduit, and a portion of the southern
conduit. On the foundation of an ancient structure, they built a small
stone house that today marks the southern entrance. In 1883, Ernst Fabricius
of the German Archaeological Institute in Athens surveyed the tunnel and
published an excellent description, including the topographic sketch shown
on page 30. After that, the tunnel was again neglected for nearly a century,
until the Greek government cleared the southern half, covered the water
channel with protective grillwork, and installed lighting so tourists
could visit it safely. Today, the
tunnel is indeed a popular tourist attraction, and a paved road from Pythagorion
leads to the stone building at the southern entrance. Some visitors who
enter this building go no further because they are intimidated by the
steep, narrow, and unlit wooden staircase leading to the lower depths.
At the north entrance, which can be reached by hiking around Mount Castro
from the south, modern stone walls lead to a staircase and passageway.
A portion of the northern tunnel has been cleared, but not lighted. The tunnel, the seawall protecting the harbor, and the nearby temple of Herathree outstanding engineering achievementswere constructed more than 2,500 years ago. Today, the tunnel and seawall survive, and although the temple lies in ruins, a single column still stands as a silent tribute to the spirit and ingenuity of the ancient Greeks. The videotape on the Tunnel of Samos has been a favorite of the educational Project MATHEMATICS! series (www.projectmathematics.com) created, directed, and produced by Professor of Mathematics, Emeritus, Tom Apostol. He is, however, better known to students both here and abroad as the author of textbooks in calculus, analysis, and number theory, books that have had a strong influence on an entire generation of mathematicians. After taking a BS (1944) and MS (1946) at the University of Washington, he moved to UC Berkeley for his PhD (1948), then spent a year each at Berkeley and MIT before joining Caltech in 1950. Although he became emeritus in 1992, his mathematical productivity has not slowed downhe has published 40 papers since 1990, 16 of them jointly with Mamikon Mnatsakanian.
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