National University of Singapore
GEM1518K: Maths in Art & Architecture
War Geometry & Fortresses
GEM1518K Group Assignment
Submitted by Group 15:
Sarinah bte Sahmawi
U024546R
Nurulhuda bte Maamon
U024529J
Deborah Lee Yu Rong
U020032H
Venkatakrishnaprasad Manikandan
U016273R
Ratnam Raguraman
U016300H
Geometry and War: A Brief Introduction
Geometry had played an integral part in the development of warfare: from troop
formation, map-making, fortress building to weaponry. This project will aim to
illuminate how useful geometry was in the art of war. It will start from the earlier
war periods of 1500s to 1800s to modern times, 20th century and beyond. The
main topics we will be covering are fortresses, weaponry and troop formation.
We have learnt much from creating this project and it is our hope that you will
find new and interesting discoveries as you turn through the pages of our work,
just as we have!
Weaponry in Olden Times
In the late 15 and 16th centuries, Mathematicians found an outlet for Geometry
through development in gunnery. When the cannon was in its early conception
period and when a heavy gun in a single metal casting was produced,
mathematician harness this phenomenon into the development of weaponry that
was longer and capable of more accurate fire. In the 16th century, the cannon
came to be used in large numbers and was critical for military victory. At the
same time, instruments were developed in order to harness the capabilities of the
new weaponry. Instruments were made to measure both the inclination of the
barrel and distance to the target. Hence, geometers which related these two
variables were created. Other instruments were also used to harness the
capabilities of the new weaponry. One might be surprised to find out that the
telescope – one of the most symbolic instruments of science was originally
introduced as an instrument of war.
Interesting Stuffs in Store
Table of Contents
-----Introduction----
The Geometry of Fortification
The
of Weaponry
The Mathematics of Troop
Formation
--Conclusion--
The Geometry of Fortification
Fortification in Olden Times
The History of the art and science of fortification stretches for a period of four
centuries, from approximately 1490 to 1890. It has thus seen a lot of changes
through the Renaissance period to the modern twentieth century. The
Renaissance period was the golden
age of fortification. During these 400
years, fortification achieved the stature
of art and science. Fortification’s most
striking
achievement
construction
of
many
was
the
impressive
fortresses found all over the world. In
the twentieth century, technology has
overtaken
the
art
of
military
fortification; while fortification may be
Forts: From the center of everything…
considered irrelevant today, it does not negate the intelligence and sheer effort
that went into its construction.
The 15th Century
During the 15th century, a revolution in the development of arms, in the form of
the canon made it necessary for fortifications and fortresses to be made stronger
and harder to be breached. The original medieval castle walls were high and
constructed to prevent the scaling of the curtain---the castle wall, by means of
ladders. However with the new developments in artillery, the high walls were
easy targets and simply shattered under the accuracy and strength of the
cannon. This necessitated a change in the design of fortifications. Eventually, the
Italian engineers rejected outright the circular walls of medieval times and came
up with the angle bastion---a four-sided projection at the corner of the curtain.
These functioned as flanking points where the defensive could open fire at
attacking forces attempting to breach the curtain. At the same time, the departure
from the circular walls of medieval times also eliminated the problem of dead
ground that could not be covered by flanking fire and which had formerly
provided opportunity for the scaling of castle walls. The picture below shows the
cross-sectional structure of a typical fortress in olden times. Note the square
shape of the buildings.
A key element which influenced the design of the
shape of the fortification in the early 1500s was the
line of defense---the distance from the flank of one
bastion to the tip of the other bastion. For the most
effective flanking fire, most engineers felt that the
Medieval Square Bastioned Forts
line of defense should not exceed the range of
musket fire, which was from 200 to 300 yards by the
eighteenth century. A simple square with bastions
was the first, most basic design at that time. However, the small flanks and sharp
angles characteristic of this design produced cramped interiors and hence limited
the troops and cannon that could be garrisoned there. On the other hand, if one
tried to increase the size of the square bastioned trace, the permissible lines of
defense (200 to 300 yards) were quickly exceeded. Thus, the square bastion
design was quickly replaced by polygonal shaped fortifications. These polygonal
walls offered more sides and were clearly easier to defend. It also allowed for
expansion to achieve even greater interior space---this was carried out by
increasing the number of bastions and the length of the enclosing walls. Although
most theories for a bastioned fortress were based on tidy geometrical design,
nature often called for readjustments in the original design. Many fortifications
had to accommodate terrain with mountains, swamps and rivers and were hence
constructed as irregular polygons, an adaptation from the original conception
(see picture on the flip page).
Venetian architect Michele di Sanmicheli (1484-1559) applied this method in
Verona and Crete and in Venice. Francesco Paciotto da Urbino (1504-1776)
fortified the citadel of Turin with the bastion design.
Irregular forts based on polygonal bastion design
The 16th Century
By the end of the 16th Century, the system of fortification was quite well
developed. Its practitioners were prominent architects, engineers and a few
soldiers. At this time, new elements were added to the bastion design. Outworks,
defenses located near the castle walls but behind the
enclosing ditch were developed. A ravelin, a freestanding triangular outwork equidistant between the
bastions, was situated almost as an island in the moat
in front of the curtain. The ravelin was designed to
protect the curtain as a whole, and to produce crossfire
over the ground in front of the neighboring bastions. If
A Star Fort
an attacker captured the ravelin, he would find himself
isolated in the middle of the ditch, and in the midst of vicious flanking fire. With a
defensive fortification structured in this way, towns within fortress walls were
rebuilt so that their streets radiated out from the town center to each bastion, and
extended along the walls. This facilitated transportation of cannon and
ammunitions from one defensive point to another during period of siege. The final
shape of the new defensive structures resembled a star, and for this reason they
were known as star forts.
The 17th Century
By the seventeenth century, most designs of a permanent fortification
demonstrated essentially the same basic components. While many new designs
were proposed and used, these were mostly variations on the common style that
had taken hold since the 15th century. There were two main ideas that stuck from
the 15th Century onwards: the first goal was to achieve maximum effective fire
power: every sector of a fortress must be swept by converging fields of fire from
both cannon and firearms. But it was also expected that it would be possible to
extend the firepower of cannon within the fortress to the outlying areas, and
attempt to destroy the attackers at a distance from the fortress if possible—this
was the concept of active defense. Acting upon these premises, fortification
engineers were supposed to ensure that every position and outwork was
carefully and mathematically designed to permit the most effective use of the
defenders’ weapons and the maximum degree of mutual support.
Secondly, the fortress had to be able to protect its own garrison and the town
within its walls. To meet these requirements, engineers had already abandoned
the high walls of medieval castles for squat and tremendously thick bastions and
curtain walls, constructed low to the ground in order to counteract enemy
bombardment. Stone-reinforced positions of packed earth, along with deep
ditches, had been added by engineers in order to reduce the damage done by
artillery. As production of cannons increased and more powerful and destructive
cannons were designed, defensive systems expanded into increasingly complex
layers of outworks intended to force the enemy’s batteries farther and farther
away from the fortress itself. Thus, while the concept of effective firepower
carried with it a component of active defense, the development of consecutive
layers of outworks showed that there was a greater understanding of the more
passive defensive possibilities for the fortification.
The basic design of a bastioned fort
The ART of Weaponry
Weaponry in the Olden Times
1. Sector
The arc of this early English sector carries an artillery table drawn up to
record the size and weight of shot and the amount of powder required for
different types of artillery. The names of the pieces were listed in order of
decreasing size, double cannon, double cannon of France, demi-cannon,
demi-cannon of France, culverin,
demi-culverin, saker, minion, falcon
and falconet. The sector is also the
new instrument of the late 16th
century. Equipped with sights for use
by surveyors, it was also provided
with various engraved scales for calculation and measurement. This
unsigned English example is closely related to the form described by
Hood and includes sectoral scales for the graphical calculation of
proportions and for drawing polygons. The reverse of its arc carries a
scale of degrees, subdivided by transversals for greater precision, along
which is an additional series of points for setting out polygons.
2. Protractor and Gunner’s Gauge
This 18th century instrument consisted of a German semi-circular
protractor whose arm carries gunner’s scales for stone, iron and lead.
The small size of the instrument limits the length
of the gauge scales. This means that only small
shot can be measured. These scales are meant
to give direct weights in ounces when a
measured diameter is read against them.
3. Gunnery and Dialing Instrument
This German instrument represents an early attempt to provide a direct
reading of weight simply from the separation of
divider points. The diameter of the shot had to be
taken with a pair of dividers and transferred to the
gauge’s scale in order to read off the weight. This
instrument consists of two main legs whose joint
slides in the slot of a third central leg. There are
smaller link pieces connecting the side legs to the central leg. Alongside
the central leg’s slot are the three standard artillery scales for iron, lead
and stone as well as the inches scale. As the legs are opened or closed,
the joint (which carries a compass box) moves in the slot and the edge of
the compass box acts as an index to the four scales. Thus if the points on
the main legs are set to the diameter of a cannon ball, the weight of the
shot can be read directly from the scales.
4. Gunner’s Calipers
This English instrument clasps a shot between its two
brass
arcs
rather
than
taking
dimensions with points at the end of
curved or straight legs. The arcs are
hinged to pass freely over each other
so that, when clasped around a ball,
the inside edge of each arc intersects a graduated
scale on the other.
5. Surveying and Gunnery Instrument
This instrument is very close to the design of Zubier’s ‘geometrical
gunnery instrument’. It differs principally in having shorter side legs, so
that it cannot be used as a pair of large dividers or calipers. The joint
linking the side and central legs is also different: whereas
Zubier shows all three limbs mounted together on the
same axis and in this instrument, the two side legs are
held in place by blued-steel spring blades. The central
leg carries a double scale of polygons and a scale of
degrees. As in Zubier’s depiction, there is a compass at its end, but in this
case, it is provided with a string-gnomon dial for about 48o latitude.
6. Gunnery and Surveying Instrument
The square format of this instrument is unusual and includes several
different components, several now incomplete. The instrument’s central
slit has a scale of inches and would once have housed a sliding gunner’s
sight. Another missing piece is the plummet, which hung to the side of the
sight. There is a table for converting between feet and paces. The table
correlates with another on the other side of the instrument. A shadow
square and quadrant along with two sighting rules were often used with
this instrument. A pivoted compass and a sundial are both designed to be
used with the square plate upright.
7. Gunner’s sight
An unusual long gunner’s sight – graduated to 9.5 inches – enables a gun
to be set to a high elevation. The instrument’s portability was greatly
enhanced by separating into two parts; the upper part is held in place by a
spring catch. The sight is attached at the side of a sliding cursor. The shot
projection on the underside of the stand enables alignment with the
centerline of the gun’s barrel.
8. Gunnery Instrument
This instrument combines three distinct devices, which are the gunner’s
quadrant, a sight and a gauging rod. The quadrant,
when seen stripped with other components such as
the stand and sight, is of disarming simplicity. Its
plumb bob and line are set against an arc graduated
in points rather than degrees, and its long leg would
have been inserted into a gun’s muzzle in use. Two
small sights attached to the side of this long leg
enable the quadrant to be used for more general
observations. The leg also carries the standard gauge
scales for determining the weight of iron, lead and stone shot.
The quadrant is turned upside down and made to serve as no more than a
frame, to which the stand and sight attachments are screwed to transform
the instrument into a sight that can be placed on the breech of a gun. The
quadrant and sight share no common structural features but their
combination creates an impressive and elaborate spectacle.
9. Surveyor’s Quadrant
This quadrant designed by Lusverg is a model of discreet restraint in
comparison. Nevertheless, its prospects for active
military service were probably no higher. The
upper face of this instrument is marked out for a
surveyor, with folding sights and a quadrant.
However, in the otherwise blank space alongside
the ball and socket joint on the underside, there is
a circular scale of point’s marked ‘Pro Eleuatione Bombardae’. A plumb
bob and line are attached through a hole pierced directly through the plate
of the instrument to measure the elevation of an artillery piece.
10. Gunner’s Level and Gauge
The long leg of this instrument enables it to be placed in the barrel of a
gun to read elevations against a scale of degrees from 45-0-45. The rigid
plummet is intended to give a quicker reading than were possible with a
plumb bob and line, which were liable to continue swinging for longer
period of time. This instrument can also serve as a gunner’s gauge,
carrying scales for stone, iron and lead on one side and scales and
powder on the reverse.
11. Gunner’s Level and Sight, with Sundial and Compass
Taken with its tooled, leather-covered case, the instrument has clearly
been crafted for visual appeal. The main upright plate of this instrument
has its own plummet with a
short leveling arc, supported on
a hinged leaf, which can be
swung round to either side of
the upright. When folded out,
the level reveals a table lettered
in red, which provides data on
shot and powder for various types of artillery. The upright also carries an
accompanying graphical table and has a central slit for a sight, which is
now missing. The whole upright sits in a graduated slot in the arched foot
and can be laterally adjusted, with its position fixed by two screws. The
foot itself has two hinged end-pieces, which raise or lower the instrument.
12. Gunner’s Perpendicular
This instrument has a shaped, fish skin-covered case of wood and
pasteboard. A perpendicular enabled a gunner to establish the centerline
of an artillery piece and thus its direction of fire. The instrument was
placed transversely on the gun’s barrel and once leveled using the spirit
level, the steel plunger was depressed. Marks were made at the muzzle
and breech of the gun and joined by a chalk line.
13. Gunner’s Rule
The rule is not a standard-issue device. Its most distinctive feature is the
diagonal scale labeled ‘Mortar’ which relates
the range of a shot to elevation. Robert
Anderson, a mathematically inclined London
silk-weaver, and bases the scale on a table
of horizontal mortar distances.
In use, the rule provides a graphical solution
to the problem of ranges. Given the range of a particular piece at a given
elevation, a gunner could work by proportion to find the range at any other
elevation and vice versa. The reverse of the instrument makes use of
more familiar materials. One gives the weight of shot and the amount of
powder for artillery pieces. The other indicates the weight of powder
required for mortars of different diameter.
14. Astrolabe
This instrument was made in Paris in 1551. One
side of the astrolabe has the Rojas Universal
projection, while the back has a pivoted alidade
with a shadow square and degree scale, a
zodiacal calendar and a diagram for converting
times between different systems of hours. It is the
alidade and shadow at the back, and possibly the
scale of degrees, that were said to have
applications to gunnery and military surveying.
15. Altazimuth Theodolite
This instrument has a vertical semicircle, which now sits above a
horizontal circle with an inscribed square. Each
quarter of the square has sides divided as a
geometrical
quadrant
and
this
alternative
arrangement has the advantage that a single
alidade, pivoted at the center, can be used with
the circle or with any of the four geometrical
quadrants. The alidade fixed to the vertical semicircle can also be used for
measurements of both coordinates.
16. Circumferentor
Circumferentors were certainly more commonly
used than something so complicated as the
altazimuth theodolite. This instrument could be
used ‘to plant barrels of powder, direct under
Castles, Forts or such like’, according to Hopton.
The instrument too has been generally associated with mining, since the
magnetic needle could be used for orientation underground where no
other sights were possible.
17. Triangulation Instrument
This gilt brass instrument has one fixed and one
sliding pivot, and each of the three arms engraved
with a linear scale. As if to reinforce the gunnery
applications, the reverse side of one arm has scales
relating the size of shot to weight for the three
materials. The sights are in steel and the whole is
very finely made.
18. Military Graphometer and Protractor
Another instrument that links general surveying with fortification is this
graphometer with a scale for polygons. Both the diametric rule and the
pivoted alidade have linear scales and pairs of complex sights, each of
which moves on a short vertical scale. The underneath of the instrument is
flat and the alidade
moves on an open
ring rather than a
central pivot, so that
the center of the
semicircle is always
exposed;
instrument can thus
be
used
the
as
a
protractor.
The semicircle scale
is divided to degrees
and the sub-division
of each degree into
minutes is by a steel
index arm, which is
given an epicyclical
motion over a circular
scale divided 0-60 four times and rotates as the alidade is moved around
the center of the semicircle. The semicircle has a second set of divisions
for laying out, in plan or in the field, the angles of regular polygons of
between three and twenty-four sides.
19. Surveying Instrument and Sundial
This instrument is included to show how polygonal scales, originally
justified as useful for laying out fortifications, can be conventionally
included on instruments whose functions are becoming distant from those
required for ordinary military surveying. The general arrangement is
similar to that of the simple theodolite but the central compass is much
larger and, having its own degree scale, can function as a circumferentor.
A prominent equinoctial sundial, whose inclination is adjustable on a
latitude scale, now surmounts it. Shadow- square scales have been added
to the scales on the top surface of the plate and the reverse has a plumb
line moving over a quadrant. The scale of ‘Polygons’ is of the same type
and is in the same position but it is for three to twelve-sided figures.
20. Surveying Instrument for Fortification
The three arms, each with a pair of sights, move on a single pivot and as
they do so a cursor, connected to both outer arms, moves along the
central arm, which is marked with a degree scale, indicating the angle
between the outer arms, and with the number of sides of the regular
polygon formed by repeating this particular angle.
One of the outer arms bears the maker’s
signature,
the
other
is
engraved
‘Pro
declinatione muri’, since these arms indicate
the angle by which the walls of the fortification
decline from each other. If the sights on the
central arm are trained on the center of the
polygonal structure, those along the outer arms
give the directions of the walls. A pin beneath is for mounting the
instrument on a staff or tripod.
21. Military Surveyor’s Protractor
There are basically two scales on the brass plate: an
outer scale of degrees subdivided by diagonals, and
an inner set of scales for drawing polygons of from
three to sixteen sides. To draw any of these regular
polygons, the radial arm is set to the successive
positions marked for the figure in question and the sides drawn with the
tangential arm.
22. Military Architect’s Rule
Both surfaces of the flat rectangle on this instrument are crammed with
tables and scales relevant to military
architecture, recognizing the different
systems of three military engineers:
‘Freytag’, ‘Vavban’ & ‘Klengel’. There
is, for example, an extensive table of
the
sectional
elements
of
a
fortification, with the elements named
and their proportions given for each system, as well as scales for setting
out regular polygons.
23. Military Protractor
This circular silver instrument is used mainly to draw
polygons. The positions of lines from the center to the
corners of all regular polygons with the between three
and twelve sides are marked on ten concentric
circles, each devoted to a single figure.
24. Protractor for Internal and External Angles
The semicircular scale is divided to half-degrees and numbered by 10 in
both directions. For example, 0-180 and 180-0. The set of jointed parallel
rules relate the internal angle, external angle, and angle subtended at the
center for the walls of a regular polygonal fortification. This form of
instrument was known in England as a ‘Parallelogram Protractor’.
25. Military Counters
There are 5 full-length and 4 half-length plates, with the shorter pieces
marked
‘Grenadiers’ and
numbered from 1
to 4. Four of the
longer plates each
represent
a
‘Division’ and are
numbered
in
sequence
from
the first to the
fourth,
each
subdivided
into
three further sections. All the plates have ‘Angle’ marked at one end in
such a way that they are most readily assembled into a square. The final
plate, with a table of numbers, perhaps displays an arrangement of these
or similar plates.
26. Telescope
The invention of the telescope inaugurated a novel
class of scientific instruments. These new optical
instruments, which soon included the microscopes
well as the telescope, were distinct from the
traditional instruments of practical mathematics
through
their use of lenses and mirrors. They
also quickly came to be manufactured by a distinct
group of optical instrument makers. Despite such
differences,
optical
instruments
such
as
the
telescope were, like mathematical instruments,
intimately bound up with military preoccupations.
The sector was an important device for practical
mathematics and several military versions but it
could not match the telescope for spectacle and the
wide-ranging implications of the observations made
with its help. From a military device whose
dissemination states sought to control, the telescope
became a subtler destructive instrument, used to
provide vital new evidence for the Copernican
cosmology.
Modern Weaponry
Modern War (20th century)
Ancient war was mostly depended on geometry. Since geometry is one of the
early developed mathematical subjects, geometrical tactics determined the early
war. Modern war reflects the advancement in science and technology. After the
industrial revolution, the art of modern war began to develop. Mathematics such
as calculus created the ability to define the physical law that initiated the new
technology. Later on, the computer era changed the fashion of war that is now
being in practice.
Artillery
During the ancient time the artilleries fired to the direct target by pointing the
canon to the target. At the beginning of the 20th century it became indirect fire
system (Howitzers).Optical instruments attached to the system and the aiming
range extended far from the visibility. This kind of systems developed and used
during the World WarI period and
its
capabilities
have
been
British Howitzer in 1914
enhanced over the years. The
range
is
changed
by
using
different amount of gun powders
in propelling charge (cartridge).
Many versions of artilleries were
developed during the World Wars
and they were used for special purposes. Artillery became a mobile fire power.
Early day communication helped this to be effective.
The above figure illustrates how the artilleries were maneuvered during the WWII
When the target was not in the same level the angle of sight (elevation angle) is
measured and the range, along the line connecting target and the firing position,
is obtained using trigonometry. Angle to fire and the velocity of firing (muzzle
speed) can be calculated for the obtained range. According to the muzzle speed
the cartridge is selected. (Table that had the details about values is used by the
soldiers).In some cases speed of the wind also considered.
Machineguns
Machine gun was invented in 1884. It can fire large number of shots in a few
fraction of time. Usually a machine gun is positioned in a tripod and fired. The
1914 machine guns were able to fire 400 to 600 small caliber rounds per minute
and was the main killer and accounted many deaths and casualties. Three types
pf machine guns were used in World War II.
Light machine guns are usually used as offensive weapons against personnel.
They are mobile and can be carried by a squad during an attack. The come
equipped with bipods and were generally magazine fed and are air cooled.
Medium machine guns are usually water-cooled weapons mounted on large
tripods or mounts. These could fire massive quantities of bullets for a sustained
period but were not easily mobile. Generally used for defense, rather than
offense.
Heavy machine guns were support weapons that had great range and
penetration but were difficult to move and unwieldy. Heavy machine guns are
primarily used for anti-aircraft.
Machine gun 1914
Aircraft carriers and Warships
These massive ships employed in World War II for the first time. Carriers played
a major role in World War II. It carried many air planes (fighters) ,had a run way,
and used like a mobile airport.
British aircraft carrier Triumph 1946
Early modern warships were developed during the world wars. They used
machine guns and different types of cannons (Artilleries). Present days these
ships fires computer guided missiles called cruise missiles.
Optical Instruments
Binoculars and cameras used for reconnaissance purpose.
enemy areas took by the cameras fitted on the
airplanes.
Geometrical optics used in the
development of these equipments. Infrared
technology has enabled the modern (present)
binoculars to view in night time.
A powerful binocular used in World War II
Aerial photos of
World War II vintage training camera. Using 35mm film this is half-frame camera,
18 x 24mm format. 75 mm f4.5 Konica Hexar lens.
The above is actually a camera can shoot many photos in quick successions.
Japanese used this to practice for the targeting in machine guns.
World War 1
World War II
The difference in technology between the
World Wars
The main features of the modern warfare are aircrafts, submarines, aircraft
carriers, tanks, artilleries, machine guns and communication. Starting from the
world war period, the war technology has gone through an unimaginable
development process. During the past two decades the war has gained a new
dimension because of the rapid advancement in technology. War has become
electronically controlled. Nowadays cruise missiles are guided till they find the
target. The new technology has enhanced the traditional art of war. Thus, the
modern warfare is talked more in technology terms rather than in geometrical
terms.
The Mathematics of Troop Formation
Ordering of Soldiers
The ordering of soldiers in regular formations was a frequent topic of
mathematical and military discussion. It was regularly included in expositions of
the use of particular instruments, for example the 'military proteus'. It is
interesting to know that different countries had different kinds of arrangements of
the troops giving rise to an indigenous style of troop formation. To prove this
point formations for roman and greek armies are considered apart from the
indian armies in the ancient times.
Typically, formations that spread the men out in a wide, shallow arrangement,
such as lines and crescents, are best for defending. These formations cover a
wide area and prevent the enemy from getting through. In the case of spearmen,
a line or crescent formation also allows individual soldiers to work together and
support each other.
Formations which take wedge or arrowhead shapes tend to be much better for
attacking than defending. It works just like an actual wedge - a pointed object is
much better for penetrating a surface than a blunt object, because it tends to
spread that surface apart. Thus, the men in a wedge formation will tend to push
the enemy apart and break them up, and the enemy loses the advantage of their
defensive formations.
Certain formations are more mobile than others. Soldiers can turn and re-position
faster without breaking apart. This is important. Any division could be as mobile
as any other if all the soldiers break formation and just run from one spot to the
next. Of course, this means they are very vulnerable while they are traveling, so
instead, a division will try to hold formation as it moves. Because of this, it may
not be as agile as it otherwise could be.
It also came within the bounds of William Oughtred's account of the manifold
calculating uses of his 'circles of proportion'. The instrument consisted of a series
of concentric logarithmic scales operated with rotating indices, and was in effect
a form of slide rule.
Ought red provided some numerical examples on the calculation of troop
formations amongst material that dealt with arithmetic, geometrical problems of
plane and solid measurement, gauging, astronomy and trigonometry. He
pretended to no great military expertise but included his treatment principally as
an exercise in the use of his instrument.
The Roman Quarters
The Roman army was divided equally into 4 quarters. Each quarter was called a
legion, and was usually made up of 4200 men divided into companies known as
cohorts. A cohort usually consists of 600 men. In desperate situations, the size
of the legion could swell to 5000 men. Each legion was designed to be a team
that had its own commanders. Tightly organized and well trained, the Roman
legion had a simplicity that concealed its innovation and true power.
A century is a group of a hundred men within a legion. A man’s wealth decided
which century he should fight in. The rich usually served as the cavalry. A
maniple is a group of two centuries.
Troop Formations in Ancient Greek Army
The Phalanx
In ancient Greek warfare, the Phalanx is the main troop formation. Prior to the
evolution of the phalanx during the seventh-century BC, war was fought by very
limited forces derived exclusively from the social infrastructure of Greek citystates. The integration of the phalanx into tactical warfare became a military
revolutionary idea as well as a social evolution.
The Phalanx is a dense formation of pike equipped troops. The formation is very
strong at the front but rather vulnerable to flank attacks. From the front, the
formation looked like a hedgehog. The phalanx was composed of a compact unit
of hoplites (a term used for the Phalanx’s soldiers), often longer in length than in
width. The phalanx was not a permanent formation. Its dimensions and approach
to attack depended on the general's tactics and the size of the army. The
Phalanx formation called for each man to trust his neighbouring infantryman,
often a friend or relative. With a shield in his left hand and a spear in his right,
each man depended on his fellow hoplite's shield for full body coverage. Battles
were won and lost depending on the phalanx's ability to hold its formation.
The Phalanx had to meet its enemy with enough momentum to move forward,
but it also had to maintain order within the ranks so as not to leave gaps between
columns. A gap in the chain of infantrymen could be fatal if exploited. As a result,
the best troops were placed at the front and back of the Phalanx. The phalanx
continued its tactical supremacy for many centuries. Later, it was rendered
obsolete by the professional and perfectionist soldiers in the Roman legions.
Troop types of Ancient warfare (in general)
Heavy Cavalry - Mounted soldiers almost always fighting with short spears used
for stabbing but occasionally for throwing (javelins). They typically wore some
kind of armour (protective clothing) and were the equivalent of ‘heavy infantry’
but on a horse.
They fought in a close formation designed to use the momentum of the group top
break up enemy formations. They were unable to ‘charge’ hoplites because the
cavalry did not use saddles and stirrups to hold their seated positions well in a
melee (combat). Persian cavalry sometimes included Skythians and others who
carried a bow and arrows instead of or as well as the spears. Moved quite
quickly (obviously). Well trained. Often ‘nobles’.
Light Cavalry - A version of cavalry which included Skythians and others who
carried a bow and arrows instead of or as well as the spears. They did not wear
armour at all so did not try to come to close combat like heavy cavalry. They
would shoot with javelins or bows and run away from the enemy. When an
enemy was sufficiently weakened by the shooting, then the light cavalry MIGHT
attempt to finish off their victims. Moved very fast as they did not need to keep a
tight formation like heavy cavalry nor carried the same amount of equipment.
Heavy Infantry - Armoured foot soldiers whose main purpose was to fight hand
to hand combat using a close combat weapon such as a spear or sword. Hoplites
were heavy infantry and their long spear gave them reach and weight of thrust vs
their opponents. Heavy infantry almost always carried a shield for further
protection. Immortals had a shorter spear, less protective armour (not so much
metal), bow and arrow and a wicker shield designed to catch arrows shot at them
to shoot back. Heavy infantry was usually the nations best soldier. In Greece, this
meant the well-off classes and higher who could afford their own equipment. In
Persia, it meant the king’s bodyguard and his best troops.
Generally, they moved quite slowly although Hoplites were slowest of all heavy
infantry (except at Marathon!). Generally, they were also well trained although
hoplites were by far better trained than Immortals.
Light Infantry - Infantry who did not wear armour but made up for this by slightly
faster movement. They often fought in tight formations just like the heavy infantry
but not always. Sometimes could move over rougher ground than the heavy
infantry due to looser formation. In the Persian army, they usually fought with
javelins and bows. They were usually the ‘foreign’ contingents from all over the
empire. If they had a shield, it was the wicker type or just light and small. In a
Greek army, this might mean mercenaries who fought like hoplites but could not
afford their own armour. They were for fighting hand to hand. They were trained
at least to stay in basic formations.
Skirmishers - Also known as Psiloi in the Greek army. Psiloi were usually javelin
throwers or bow men who shot and ran. Did not fight in any precise formation
and were not equipped to fight hand to hand. They acted like the light cavalry weaken an enemy by shooting. They were capable of running and operating
over difficult ground such as marshes etc. Persians had plenty of these as well
as the Greeks. In the Persian army, this might have been a very large number as
it required almost no serious training. Generally untrained or not very much!
Peltasts - A special Greek soldier designed as a cross between hand to hand
fighters and Psiloi. They often had the job of chasing and killing Psiloi. They
generally did not wear armour but were equipped for hand to hand combat with
short spears or javelins. They fought in loose formations which, on occasion
especially in the Peloponnesian War, could shoot at an enemy and weaken them
sufficiently before charging their victim and finishing them off. Thracians made
the best Peltasts named after their special shield (a sort of crescent moon shape)
the Pelte. Spartan and Athenian hoplites began their careers as a version of
peltasts before they were old enough to wear armour. These ‘epheboi’ marched
in the rear of the phalanx and ran out between the ranks of the hoplites to chase
away enemy light troops shooting at the Phalanx. Training varied depending on
the uses a polis had for them.
Art of Indian War
Indian history has always been a very hot topic of discussion. One of the most
popular and the oldest such history is the story of Mahabharata tells the story of
two sets of Paternal first cousins--the five sons of the deceased king Pandu
[pronounced PAAN-doo]
(The five Pandavas [said as PAAN-da-va-s]) and the one hundred sons of blind
King Dhritarashtra [Dhri-ta-RAASH-tra] (the 100 hundred Dhartarashtras [Dhaarta-RAASH-tras])--who became bitter rivals, and opposed each other in war for
possession of the ancestral Bharata [BHAR-a-ta] kingdom with its capital in the
"City of the Elephant,"
Hastinapura [HAAS-ti-na-pu-ra], on the Ganga river in north central India. What is
dramatically interesting within this simple opposition is the large number of
individual agendas the many characters pursue, and the Mahabharata – a story
of good wins against the bad. The innermost narrative kernel of the numerous
personal conflicts, ethical puzzles, subplots, and plot twists gave the story a
strikingly powerful development.
The inevitability of war left both Kauravas and Pandavas to chalk out their
respective strategies and assess the strength and weaknesses of their
opponents. While Arjuna was the best archer on Pandavas side Karna was no
less a warrior on Kauravas side. To the extent some one rated him greater than
even Arjuna! This particular fact caused a great concern in the heart of mother
Kunti.
With this backdrop or the storyline the basic war methodologies now described. It
is really amazing to even think that even in the oldest of days a lot of
mathematics was used fight wars and battles. The war of Kurukshethra
supposedly lasted for 18 days involving lakhs of soldiers fighting. The war
involved usage of bows and arrows with a significant accuracy. But one of the
main surprises was the assembling of the soldiers or the battalion arrangement.
The name given to it was the chakravyuha or the The Circles of
Proportion (also known as circle of death). The chakravyuha consisted of soldiers
forming concentric circles with enemies having to penetrate the circles one by
one to get into the core to kill the main personality and then getting outside again.
In the
Mahabharatha there were only two people in the pandavas who could actually do
this. Arjuna an expert in bows and arrows and his son abimanyu were those. A
very schematic picture of this is shown below.
Abimanyu fighting in the war of kurushetra
In the above picture it is seen that abimanyu (far right) is fighting inside the circle
of death being surrounded by the opponents all around him. Just for interest it is
pointed out that Abimanyu died while getting back from the circle.
How were the troops organized generally!?
Any troop formation would mainly fall under one of the following categories.
The Square Formation
The old notion of fighting in large square battle formations, which remained
relatively unchanged since ancient Greek times, was shown to be outdated and
inefficient by Gustavus Adolphus's brilliant strategy during the war. The large
squares, also known as tercios, were used because a lot of troops can be
concentrated in one large area.
This was not a very efficient way of using available manpower. One of the
biggest drawbacks of the tercios was that it relied on the troops at the front to do
most of the actual fighting while those in the middle and back were left out. In
addition, because of its large size it was difficult to maneuver. Adolphus
organized his troops in linear formation of 6 soldiers wide. This allowed all the
soldiers to be involved in actual fighting and made the formations much easier to
maneuver.
The Column Formation
A formation in which elements are placed one behind the other. This formation
helps to conceal the number of units in a convoy. The enemy can look at the
tracks left by a squad to estimate how many units it is up against. Take 12 trucks
for example, the trucks may travel in columns of 4 or 6. Thus, there will only be 2
or 3 columns. As a result, the enemy may have a hard time tracking the number
of units. It leaves the entire convoy vulnerable to aircraft or mortar fire. For
example, an assault on the convoy can concentrate their attack down the center
of the column and inflict damage to every unit.
The Wedge Formation
A formation in which elements are placed to each side of a central unit, extending
outward and behind refers to the central unit itself. The wedge formation is good
for approaching a battle and offers defense to the convoy, aiding in the
prevention of being flanked to either side. It is best used by infantry or armored
units when traveling between mountains or within wooded areas, where the
threat of being ambushed or flanked is higher. It lacks in the firepower
concentration on specific targets and this formation cannot conceal the number
of units in the convoy.
Invention of telescope and its preliminary
As an astronomical instrument, the telescope is one of the most familiar icons of
science.
Yet
when
invented
it
was
considered more a military device than a
scientific instrument. The first telescopes
were announced in the Netherlands in
1608 and were improved by Galileo in
1609. Galileo's astronomical observations
brought him European fame, but even
before making his discoveries he had
already been rewarded for improving the
instrument's strategic use. While the
telescope, as an optical instrument, is
markedly different from the mathematical
instruments of the period, its origins were equally bound by contemporary
preoccupations with war.
Organization of the Troops (counting & segregation)
When the 18th-century English instrument maker Thomas Wright catalogued the
collection to which this set of brass plates belongs, he was evidently at a loss as
to their function. They clearly did not belong to the contemporary instrumentmaking repertoire and Wright could suggest only that they bore some relation to
gunnery. A more plausible explanation is that they were used to display the
disposition of troops. There are five fulllength and four half-length plates, with the
shorter pieces marked 'Grenadiers' and
numbered from 1 to 4. Four of the longer
plates each represent a 'Division' and are
numbered in sequence from the first to the
fourth, each subdivided into three further sections. All also have 'Angle' marked
at one end of the plate in such a way that they are most readily assembled into a
square. The final plate, with a table of numbers, perhaps displays an
arrangement of these or similar plates. A very sensible conclusion that can be
drawn is that even in the olden days the armies had a count of each and every
soldier who participated in the war.
Bibliography
Websites:
http://www.mhs.ox.ac.uk/geometry
www.mahabhrata.com.
http://www.htansw.asn.au/teach/ancienthistorydocs/anc_trooptypes.doc
http://www.members.cts.com/funtv/j/jjartist/Armydesc.html
http://www.pvv.ntnu.no/~madsb/home/war/romanarmy/romanarmy00.php3
http://members.tripod.com/~nigelef/index.htm
http://www.spartacus.schoolnet.co.uk/
http://www.jodavidsmeyer.com/combat/military/weapons-machineguns.html
http://www.firstworldwar.com/
www.cmp.ucr.edu/cameras/ Machine_Gun_Camera.html
http://www.mhs.ox.ac.uk/geometry/essay.htm
Book References:
1. Ramparts: Fortification from the Renaissance to West Point
Marguerita Z. Herman, Avery Publishing Group Inc. Garden City Park,
New York, 1992
2. The Geometry of War
English 355.809 MUS National Library Board
3. Roman Fortresses and their Legions (papers)
English q937.06 ROM National Library Board
Appendix
A
R
T
I
L
L
E
R
Y
By the time of the Civil War, technology has advanced the big guns very little
since the Napoleon times. Although there were a wide variety of designs, all but
a few operated on the same principle of the first artillery pieces centuries before.
A hollow tube was open at one end and closed at the
other. A bag of black powder was rammed into the
muzzle, the open end, and shoved to the back of the
tube. The projectile was pushed in after it. The piece was
simply detonated either by the old-fashioned method of
applying a flame or lit fuse to a touchhole at the
breech, or, more often, a copper priming fuse
was inserted into the vent, and its spark set off
by jerking a friction primer. The solid shot,
literally a round ball of iron, and of little effect
except when it hit an opposing artillery piece.
Other loads were designed to be more effective as anti-personnel weapons. The
shell, either round or occasionally, cylindroconoidal, was hollow inside and
contained a powder charge. The spherical case shot was more effective. Again,
a hollow round ball containing up to 78 lead musket balls and an exploding
charge made up the shell. When it went off in the midst of a line of soldiers, this
could be deadly, though many of the balls flew straight up into the air and others
straight down into the ground, doing nothing, while of the rest, only those at the
forward and sides of the moving ball had any chance of killing or injuring.
Grapeshot, large iron balls two inches in diameter and arranged in ‘stands’ of a
dozen or more, was not much used in the Civil War, but a cousin called canister
was the most damaging of all artillery loads. Gunners would ram down a tin can
filled with 27 cast iron balls used
against attacking infantry when within
300 yards or less. On being fired, it
turned
the
cannon
into
a
huge
shotgun. The artillery of both sides in
the war was dominated by a basic
fieldpiece design little changed from
the time of Napoleon. Mush larger smoothbores, monsters with bores up to 20
inches and more in diameter, and capable of firing projectiles weighing more than
half a ton, were built for seacoast defense in the North and to protect large
stationary fortifications.
Smoothbore was capable of hurling a ball nearly five miles out to sea! However,
the mortars were more often used instead of the smoothbores. These mortars
had specific purposes and they were designed to sit low to the ground, and to fire
a heavy exploding ball high up into the air in an
arching trajectory that could take it over and behind
earthworks or masonry fortifications, to explode in
their rear. Very few were actually being used with field
armies for they were of no use in conventional battles.
The 3-inch ordnance rifle came to be the favored
piece. Its 3-inch bore, with deep rifling groves, imparted a spin to its elongated
shells that gave them greatly increased range and accuracy. Maximum efficiency
was achieved from the powder charge. The presence of a wrought iron band
around the breech help reinforced the ten-pounder cast iron tube for large loads.
Armstrong designed a powerful hollow
screw for the breech of his gun. It
allowed a solid breechblock to be
removed, the projectile and charge
shoved
in,
and
the
breechblock
replaced. Unfortunately, the breechloaders proved to be temperamental and only
a few were ever used.
INTERNAL MACHINES
Many new kinds of exploding shells designed by artillerists were aimed to set
ablaze fortifications. Others tried joining two solid shot with a length of chain,
expecting that upon firing the balls would stretch the chain, starting spinning, and
thus mowing their way trough infantry. At least one double-barreled fieldpiece
was tried, expected to fire each tube simultaneously and send solid shot by
chain, though the foe. All such efforts failed, one observer noting that when the
double-barreled cannon was test fired, it plowed up a field, knocked over a
couple of saplings, and the balls broke apart.
Not long, the Confederates experimented with
a rapid-fire, large-bore cannon called the
Williams Machine Gun, which theoretically
could fire more than sixty 1.57 caliber balls per
minute. A gunner operated the crank that
opened the breechblock, and cocked a hammer, while another man inserted the
paper-wrapped cartridge and capped a nipple. Closing the crank closed the block
and tripped the hammer. Few were actually being used, and they proved to be
temperamental. Much more efficient was the Agar machine Gun, which looked
for the entire world like a crank operated coffee
mill. It could shoot 120.58 bullets per minute.
The most practical design came along too late
for wide use in the war, and it is just as well for
the men who would have had to face it. The
Gatling Gun was first patented in 1862. It had 6 barrels mounted to make a
hollow cylinder. Turning a
crank rotated the barrels, and as each one came in line, the crank fed a cartridge
from a hopper into the breech of the barrel and fired it. The government failed to
adopt it back then. Not until 1865, and a new model, were all the imperfections
worked out, at which time the Gatling became a truly devastating killing machine.
Fortunately, the war was over then.
Confederate engineers also experimented with land mines, called ‘torpedoes’.
The use of such weapons was controversial, but then in a war in which
technology was just as much a combatant as the armies themselves, almost
anything could be deemed legitimate. Even exploding bullets were attempted,
designed to go off after entering a man’s body.