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 T

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Fortification in Olden Times

he History of the art and science of fortification stretches for a period of four

enturies, from approximately 1490 to 1890. It has thus seen a lot of changes

hrough the Renaissance period to the modern twentieth century. The

enaissance period was the golden

ge of fortification. During these 400

ears, fortification achieved the stature

f art and science. Fortification’s most

triking achievement was the

onstruction of many impressive

ortresses found all over the world. In

he twentieth century, technology has

vertaken the art of military

ortification; while fortification may be Forts: From the center of everything…

onsidered irrelevant today, it does not negate the intelligence and sheer effort

hat went into its construction.

The 15th Century

uring the 15th century, a revolution in the development of arms, in the form of

he canon made it necessary for fortifications and fortresses to be made stronger

nd harder to be breached. The original medieval castle walls were high and

onstructed to prevent the scaling of the curtain---the castle wall, by means of

adders. However with the new developments in artillery, the high walls were

asy targets and simply shattered under the accuracy and strength of the

annon. This necessitated a change in the design of fortifications. Eventually, the

talian 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

line of defense should not exceed the range of

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Medieval Square Bastioned Forts musket fire, which was from 200 to 300 yards by the

eighteenth century. A simple square with bastions

as the first, most basic design at that time. However, the small flanks and sharp

ngles characteristic of this design produced cramped interiors and hence limited

he troops and cannon that could be garrisoned there. On the other hand, if one

ried to increase the size of the square bastioned trace, the permissible lines of

efense (200 to 300 yards) were quickly exceeded. Thus, the square bastion

esign was quickly replaced by polygonal shaped fortifications. These polygonal

alls offered more sides and were clearly easier to defend. It also allowed for

xpansion to achieve even greater interior space---this was carried out by

ncreasing the number of bastions and the length of the enclosing walls. Although

ost theories for a bastioned fortress were based on tidy geometrical design,

ature often called for readjustments in the original design. Many fortifications

ad to accommodate terrain with mountains, swamps and rivers and were hence

onstructed 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 free-

standing 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

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

A Star Fort

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; the instrument can thus

be used 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

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.

British Howitzer in 1914

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

achine gun was invented in 1884. It can fire large number of shots in a few

action of time. Usually a machine gun is positioned in a tripod and fired. The

914 machine guns were able to fire 400 to 600 small caliber rounds per minute

nd was the main killer and accounted many deaths and casualties. Three types

f machine guns were used in World War II.

ight machine guns are usually used as offensive weapons against personnel.

hey are mobile and can be carried by a squad during an attack. The come

ipods and were generally magazine fed and are air cooled.

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equipped with b

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

hese massive ships employed in World War II for the first time. Carriers played

major role in World War II. It carried many air planes (fighters) ,had a run way,

nd used like a mobile airport.

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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. Aerial photos of

enemy areas took by the cameras fitted on the

airplanes. Geometrical optics used in the

development of these equipments. Infrared

technology has enabled

noculars to view in night time.

the modern (present)

bi

A powerful binocular used in World War II

World War II vintage training camera. Using 35mm film this is half-frame camera,

Konica Hexar lens.

he above is actually a camera can shoot many photos in quick successions.

Japanese used this to practice for the targeting in machine guns.

The d

World Wars

arting from the

y has gone through an unimaginable

he war has gained a new

ology. 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.

World Wa 1

18 x 24mm format. 75 mm f4.5

T

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World War II

ifference in technology between the

The main features of the modern warfare are aircrafts, submarines, aircraft

carriers, tanks, artilleries, machine guns and communication. St

world war period, the war technolog

development process. During the past two decades t

dimension because of the rapid adva ent in technncem

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 fo

The Mathematics of Troop Formation Ordering of Soldiers

rmations for roman and greek armies are considered apart from the

dian armies in the ancient times.

line or crescent formation also allows individual soldiers to work together and

, and the enemy loses the advantage of their

soldiers break formation and just run from one spot to the

ext. Of course, this means they are very vulnerable while they are traveling, so

in

Typically, formations that spread the men out in a wide, shallow arrangement,

uch as lines and crescents, are best for defending. These formations cover a s

wide area and prevent the enemy from getting through. In the case of spearmen,

a

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

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

n

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

form of slide rule.

ually consists of 600 men. In desperate situations, the size

ithin a legion. A man’s wealth decided

hich century he should fight in. The rich usually served as the cavalry. A

nturies.

a

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 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 us

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 w

w

maniple is a group of two ce

The Roman Quarters

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 city-

states. The integration of the phalanx into tactical warfare became a military

volutionary idea as well as a social evolution. re

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

rmation looked like a hedgehog. The phalanx was composed of a compact unit

lanx was not a permanent formation. Its dimensions and approach

ility to hold its formation.

Troop types of Ancient warfare (in general)

fo

of hoplites (a term used for the Phalanx’s soldiers), often longer in length than in

width. The pha

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 ab

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.

Heavy Cavalry ort spears used

r stabbing but occasionally for throwing (javelins). They typically wore some

ind of armour (protective clothing) and were the equivalent of ‘heavy infantry’

ut on a horse.

hey fought in a close formation designed to use the momentum of the group top

reak up enemy formations. They were unable to ‘charge’ hoplites because the

avalry did not use saddles and stirrups to hold their seated positions well in a

elee (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’.

y - A version of cavalry which included Skythians and others who

ened by the shooting, then the light cavalry MIGHT

victims. Moved very fast as they did not need to keep a

always carried a shield for further

horter spear, less protective armour (not so much

- Mounted soldiers almost always fighting with sh

fo

k

b

T

b

c

m

Light Cavalrcarried 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 weak

attempt to finish off their

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

protection. Immortals had a s

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

mpire. If they had a shield, it was the wicker type or just light and small. In a

reek army, this might mean mercenaries who fought like hoplites but could not

fford their own armour. They were for fighting hand to hand. They were trained

my. Psiloi were usually javelin

rowers or bow men who shot and ran. Did not fight in any precise formation

ere equipped for hand to hand combat with

hort spears or javelins. They fought in loose formations which, on occasion

e

G

a

at least to stay in basic formations.

Skirmishers - Also known as Psiloi in the Greek ar

th

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 w

s

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

e uses a polis had for them.

pu-ra], on the Ganga river in north central India. What is

ramatically interesting within this simple opposition is the large number of

ment.

ular fact caused a great concern in the heart of mother

unti.

th

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 [Dhaar-

ta-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,"

Art of Indian War

Hastinapura [HAAS-ti-na-

d

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 develop

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 partic

K

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

tion (also known as circle of death). The chakravyuha consisted of soldiers

rming concentric circles with enemies having to penetrate the circles one by

ne to get into the core to kill the main personality and then getting outside again.

the

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

Propor

fo

o

In

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 a ircle

f 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 old notion of fighting in large

latively unchanged since ancient Greek times, was shown to be outdated and

efficient by Gustavus Adolphus's brilliant strategy during the war. The large

quares, also known as tercios, were used because a lot of troops can be

oncentrated in one large area.

ere left out. In

ddition, because of its large size it was difficult to maneuver. Adolphus

ldiers wide. This allowed all the

ier to

aneuver.

l in columns of 4 or 6. Thus, there will only be 2

bove picture it is seen that abimanyu (far right) is fighting inside the c

o

The Square Formation

square battle formations, which remained

re

in

s

c

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 w

a

organized his troops in linear formation of 6 so

soldiers to be involved in actual fighting and made the formations much eas

m

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 trave

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

central unit itself. The wedge formation is good

r approaching a battle and offers defense to the convoy, aiding in the

an astronomical instrument, the telescope is one of the most familiar icons of

d it was

onsidered more a military device than a

use. While the

lescope, as an optical instrument, is

struments of the period, its origins were equally bound by contemporary

preoccupations with war.

outward and behind refers to the

fo

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

science. Yet when invente

c

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

te

markedly different from the mathematical

in

When the 18th-century English instrument m

collection to which this set of brass plates b

to their function. They clearly did not belo

making repertoire and Wright could suggest

gunnery. A more plausible explanation is

disposition of troops. There are five ful

length and four half-length plates, with th

shorter pieces marked 'Grenadiers' an

numbered from 1 to 4. Four of the longe

plates each represent a 'Division' and ar

numbered in sequence from the first to th

fourth, each subdivided into three further sections. All also have 'Angle' marked

at one end of the plate in

Organization of the Troops (counting & segregation) aker Thomas Wright catalogued the

elongs, he was evidently at a loss as

ng to the contemporary instrument-

only that they bore some relation to

that they were used to display the

l-

e

d

r

e

e

such a way that they are most readily assembled into a

quare. The final plate, with a table of numbers, perhaps displays an s

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:

ttp://www.mhs.ox.ac.uk/geometryh www.mahabhrata.com.

.asn.au/teach/ancienthistorydocs/anc_trooptypes.doc http://www.htansw 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-machine-guns.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,

pers) brary Board

New York, 1992

2. The Geometry of War English 355.809 MUS National Library Board

3. Roman Fortresses and their Legions (pa

English q937.06 ROM National Li

T I L L E R Y

little

since the Napoleon times. Although there were a wide

few operated on the same principle of the first artillery pieces centuries before.

A hollow tu

other. A bag of blac

ther loads were designed to be more effective as anti-personnel weapons. The

A R

By the time of the Civil War, technology has advanced the big guns very

variety of designs, all but

a

be was open at one end and closed at the

k 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.

Appendix

O

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

could be deadly, though many of the balls flew straight up i

straight down into the ground, doing nothing, while of the r

forward and sides of the moving ball had any chance

Grapeshot, large iron balls two inches in diameter and arra

sed in the Civil War, but a cousin called canister

rtillery 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.

a line of soldiers, this

nto the air and others

est, only those at the

of killing or injuring.

nged in ‘stands’ of a

u

was the most damaging of all a

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

dozen or more, was not much

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.

y

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

a few were ever used.

temperamental and onl

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 st

thus mowing their way trough infantry. At

was tried, expected to fire each tube imultaneously and send solid shot by

hain, 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 b

Not long, the Confederates experimented with

aper-wrapped cartridge and capped a nipple. Closing the crank closed the block

al design came along too late

the war, and it is just as well for

the men who would have had to face it. The

retch the chain, starting spinning, and

least one double-barreled fieldpiece

s

c

alls broke apart.

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

p

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 practic

for wide use in

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.