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Evolution of Modem Battle: An Analysis of Historical Data

A Monograph by Major Charles D. Aden Field Artillery

dDTIC V1TotELECT ISCtAVI$

School of Advanced Military Studies General Staff College United States Army Command andKansas Fort Leavenworth, First Term AY 90-91 Approved for Public Release; Distribution Is Unlimited

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EVOLUTION OF MODERN HISTORICAL DATA

S. FUNDING NUMBERS

BATTLE:. AN ANALYSIS

OF

6. AUTHOR(S)

MAJOR CHARLES D. ALLEN,

USA

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ABSTRACT OF EVOLUTION OF MODERN BATTLE: AN ANALYSIS DATA by MAJ Charles D. Allen, USA, 85 pages.

HISTORICAL

This monograph conducts an analysis of historical data to detect trends in war and the evolution to modern tactical battle. It also investigates the continued use of Lanchester-based attrition models. models Military analysts have used the attrition of of Lanchester in an attempt to capture the dynamics and for tactical modern battle to answer questions involve questions These decisionmakers. strategic weapon systems, force technical evaluations of structure, doctrinal issues, and training. historical data of 1080 The methodology uses study of battles tactical land battles compiled in a The data was partitioned into from the years 1618-1905. two sets, pre- and post-American Civil War, to test the changes around hypothesis that the conduct of warfare form of used a generalized 1865. -he evaluation in the form of a linear Lanchester's equations Lanchester-based model to indicate which regression standard A attrition model best represented the data. test statistical hypothesis test called the Student "t" was applied to determine if the population parameters of and different sets were statistically the data indicated a change in the conduct of battle. that analysis revealed The quantitative not adequately Lanchester's Linear and Square Laws do that This implies describe the battles of this study. by postulated process and battle conditions attrition Logarithmic Peterson's these laws are not supported. that Law is more representative o-F the data and implies the casualty producing power of a force increases as the 4orce size decreases. marked by of modern battle is The evolution to quantitative di'ference as measured by the casualty There is a trend toward decreasing initial force ratio. War that coincides with the post-American Civil ratios innovations of the period. The five technological breechloaders, century: rifled muskets, nineteenth the and smokeless powder were magazines, barbed wire, warfare. catalysts of change in the tactical conduct of increased their The lethality o4 the new weapons with substantial combined with the accuracy and range forced the change in rates of fire increase in battlefield tactics.

SCHOOL OF ADVANCED MILITARY STUDIES MONOGRAPH APPROVAL

Major Charles D. Allen

Title of Monograph:

Evolution of Modern Battle: An Analysis of Historical Data

Approved by:

Monograph Director

. ,_ LTC

Ernest R. Rogers

Director, School of COL W. H. Janes, M

,

Advanced Military

AS

Studies

Director, Graduate Degree Program

Philip J. Brookes, Ph.D.

Accepted this

315t

day of

_

C________"

1990

Table of Contents

I.

Introduction ..................................... 1

II.

Study of Attrition Models ........................ 7

III.

Historical

IV.

Statistical

V.

Evaluation

VI.

Conclusions and

VII.

Appendix

A:

Sample Page from Lex-con ............ 41

VIII.

Appendix

B:

Linear Regression Results ............ 42

Iy.

Appendix

C:

Data Set.

X.

Endnotes.

Xi.

Bibliography .................................... 83

Battles as Data Source ................ 19 Analysis of Historical

Data ..........21

of Results ........................... Implications ..................... -8

............................ 5

........................................ 79

Acces-tton Fo r NTIS GP'A&I I DTIC TAB

By_ Dlstribut ln/ ..... IAatll:iblItv Codos

D-st

!

pGecifl

I.

Introduction

Backaround Military of

analysts in

Lanchester

modern

battle

strategic

an

to

answer

fundamental

in

Training

Center

the

Square

these by

recent

studies

Peterson

casualties

modern the

of

the

simulations Lanchester

and

Army war

equations

historical

data.

of

too

is

that

results. of

the data

Armies

the

in

National

historical

Logarithmic in

Law

data

in

proposed

reflecting

battlefield.

that

the

complex

based

dynamics

to

models.

have

the

failed to validate

currently

games

of

from

appropriate

attrition

the

force

historical

Decay

data

the modern

are

Lanchester

and

on

mixed

to

Ron Johnson

implies that

battlefield

of

analysis

more

conducted

shortcomings

applied

authors suggest

Department

for

The

incurred on

Both

the

analysis

seems

of

involve

systems,

with

Exponential

by Major

Law.

tactical

questions

been

equations

when

"The

A

models

the dynamics

for

weapon

have

discussed

equations

Battle."

attrition

issues, and training.

Lanchester

paper,

These

of

studies

Schneider

his

questions

evaluations

Validation

the

to capture

attempt

structure, doctrinal

Lanchester

used

decisionmakers.

technical

James

have

be

captured

However, using

largely

shown

of

limited

the in the

computer upon

the

validity

Significance of Military

professionals

been concerned best

a

throughout

with developing the best

enemies.

to

nineteenth

Frederick

the

Great

train and

their

force structure.

and

twentieth

armies.-

Napoleon

the use of technology.

Throughout

centuries, military

have used maneuvers to test out

Models are, events

and

representation prediction

and

to varying

new ideas on

tactics and

which

to represent addition

develop

friendly

Spirit.. the

incorporated simulations

The use

of

as

models

and

are

for

an

the

to mimic

"abstract

purpose

understanding

about

of the

U.S. military has used models enemy

forces

in

conflict.

troop maneuvers of

Department

through

interactive

war

the

of

games

REFORGER

Defense

use

of

into

In

has

computer

its

defense

and training. Department

military

engagements such

and

They

used

to the traditional

Team

planning

is

The

combat.

degrees, an attempt

conditions.

to

the

These exercises were models used

real-world process."

and

were

commanders

validate concepts to be employed in actual

real

have

commanders who conducted war games and maneuvers as

tool

to

the ages

equipment, and the best tactics to use in defeating

their both

Military Models

from

FORCEM.

of

models task are

the to

Army

relies

conduct

force

to

employed to

heavily

analysis

corps

on

level.

evaluate

the

on

the

tactical Models, potential

L

effect

of

new

weapon

systems,

the

application

of

doctrine. and the effect of force structure chanaes from task

force

military

through

model

corps-level

is

theater used

commanders

level.

to

The FIRST

train

and

battalion

staffs

in

BATTLE through

the

actions

required to effectively manage resources and fight

their

forces. Table used most These

A

often

models

Center

provides a used by

are

listing of

the computer

TRADOC Analysis Command

currently

in

use

by

the

modela (TRAC).

Combined

Arms

(CAC) Combat Development and Training Development

Directorates. Table A - TRADOC Most Often Used Models

MODEL

USE

ACABUG AMDC APTBASS BABAS BPS CASTFOREM CBS(JESS) COMBAT-SIM CORBAN FIRST BATTLE FORCEM JANUS(T) JANUS(L) JTLS SOTACA

The iroives

UNIT LEVEL

ATTRITION

Analysis Analysis Training Training Decisions Analysis Training Training Training

Task Force Task Force Task Force Battalion Brigade Task Force Corps/Div Task Force Corps/Div

One-on-one One-on-one Lanchester One-on-one Lanchester One-on one Lanchester One-on-one Lanchester

Training Analysis Training Analysis Training Training

Bn/Corps Theater Task Force Task Force Theater Theater

Lanchester Lanchester One-on-one One-on-one Lanchester Lanchester

"real-world" the destruction

process of

of

VALIDATION!

forces

enemy units while

No No No No On-going Yes On-going On-going On-going No No No No No No

in

combat

receiving

friendly

casualties.

For

a

model

to

be

useful

it

must

reflect the destructive process of force attrition.

The

,engine"

of

mathematical A

presents

to

the

methods.

types of

forces,

based

hit,

attrition

on

and

target

detail

two

probabilities

the

models, this

destruction

characteristics

which

tlie of

and

of

is

based

methods

on

will are

force

be

models target

initial

Table

employed

calculate acquisition,

for each

weapon

Lanchester-based discussed

focused

over

a

Lanchester-based

given a hit

engagement.

paper,

of

attrition

attrition

target kill

in

in

model

one-on-one,

One-on-one

attrition

and

simulation

expression of the attrition process.

decrement

target

a

on

time

strength

in

the

based

level

greater rate

on

and

of

force

fighting

effect iveness. in

order

for

representation of process. process

TRAC of

aim

of

confidence

defines

other

the

be

credible

-sLtSt undergo a

model

validatior

the

model's

as

more is

events

valid to

model

by

its

validation "a

ormal is

that

it

outcomes.

increase

inferred

in

behavior

world experiences and

.alidation

that

to

it

that

the real

with

model

model

reality,

ensuring

agreement with comparable

a

the the

is The

degree model

in

of

will

occur. D* neer

the models

considered

listed

in Table A only CASTFOREM

validated

because

it

agreed

has wJ~n

The remaining models either

another established model). not

have

been

The

on-going.

of

use

the

or

validated

that

models

is

process

validation

been

not

have

validated should raise questions on the acceptability of for

results

their

defense

and

planning

training

purposes. In order to properly validate a model,

its outcomes

COL

(Ret) Trevor

should be compared to historical

data.

Dupuy writes that: If we can collect enough reliable data from militarv data from military history we should be able to determine patterns of conduct, performance, and outcomes that will provide insights into the nature of armed basic and that will indicate trends to conflIct, assist military planning for the future.-

purpose

The first

is

to

battle.

tactical

in

monograph

if

the

analysis

war

and

the

determine

trends

reflects

this

of

The

whether

the

continued

models

is

warranted

second use

of

based

historic

of

to

evolution

purpose

The

twofold.

is

to

is

data

modern

investigae

Lanchester-based attrition the

on

results

of

the

analysis. The tactical from

the

methodology battles years

will

compiled

1618-1905.

in

historical

use a

study

of

The data will

be

1500

hypothesis

that

the

conduct

of

of

battles

partitioned

into two sets, pre- and post-American Civil War, the

data

warfare

to test changes

around

1865.

qeneralized equations

The

form

in

evaluation of

the

criteria

of

a

linear

regression

model

will

indicate

attrition

model

best

represents

standard "t"

statistical

test

will

parameters of the

be

are

indicate potential AssUptn1

Por 1 '1 be

applied

which

to

change

attrition

data

called

Student

population

will determine

different

and

if

ca-

in the conduct of warfare.

the purpose of the monograph, the

used of

from

the

only

historical

attrition

models

land

battles

study.

will

time,

and The

tattlefield American Civil

the term

models

modern

conditions War.

thait

as

apply

battle they

to

on vary

homoqeneous

will

occurred

The

focus

deterministic models, those whose parameters do not

forces.

A

s

investigation

Dver

The

sets.

the

compare

The test

a

Lanchester-based

the

statistically

use

regression.

hypothesis test

the data sets.

parameters

combat

Lanchester's

form

will

refer after

to t

Study of Attrition Models

II.

BArkoroind

on Lanch;ster

Frederick

William

Lanchester

(1878-1946)

was

a

brilliant English mathematician, engineer,

and

He

aerodynamics,

was

highly

economics, early

and

the

conduct

number

of

for

military

advocate

envisioned the

respected

for

great

of

his

work

strategy.

the

impact that

war.

Beginning

articles

that

Lanchester

of

use

in

aircraft,

aircraft

in

inventor.

1914

would

attempted

he

to

was

an

and

he

have on

published

quantify

a

that

impact. He

developed

Lanchester

equations, to

engagements. principle

of

His

Modern

"so

many

weapons,

war.'thatJ anything.' analysis

has to

moCnl

of

the men, and is

the

known

firepower

anid

factors

for

the

for

more

ridiculous

to

as

pretend

combat

morale

to

or

demerits

'chances

argued

always begin with

forces available to each side.

forces.

ground

unknown

Lanchester

combat

Lanchester's land

that

such

land

the

air combat.

mocified

attrition

still

would

as

demonstrate

the unattended merits or

However, battle

of

to

the argument

unknown

it

was

adopted

acknowledged

leadership of of

intent

to the conditions of

usage

equations

Lancnester had

it

analysis,

predict the outcome of military

concentration

and then apply

basic

a mathematical

of

calculate that

a counting

any of

Development In

of

his

conditions

Lanchester's Square

original of

ancient

warfare, he contended individuals battle

of

systems are

survival

of

a

where

units

employed

force

man

modern

is

against

fight

man.

against

at

the is

units

systems.

destruction

by The

combat

weapon

the

ancient

forces

warfare,

on

In

the

characterized

equal

against

based

described

warfare.

battles were

combat,

In

activity

modern

approximately

contact.

collective weapon

in

of

Lanchester

and that

engaged

consisted

point

work:

Law

a

and The

of

the

eremv. This

view

of

of

technology

o~f

long-ranged,

battle,

the

immediate

in

warfare

Lanchester's magazine-fed

superiority

under

reflects

time

in

heavier

the

the

fire

tne

progress

introduction

weapons.

superior

a', , .. n)umerically far

with

rifled

"concentration ot

ranka:... [where i-self

modern

in

numbers Ective

infericr ... than

combat

is

return."'

P BLUE B (t

Figure

RED I

1

Mt)

-

Lanrhester'-

Model

R

of

gives an

force it

modern

Eombat

finds

able

to

The

combat dynamics

the

interaction

The

Lanchester

Lanchester's

modeled

between

forces

equations

Square

Law

by Lanchester as

shown

I and

2

below

(equation 3).

The

at any time t for the Blue or Red unit B(t)

and

strengths which The

are

Blue

rate

These rite

R(t) ,

respectively.

represented

destroys at

which

parameters,

Red

coefficients.

Square

Law

Vorce with

would

per

Red p

imply

the greatest

B_

p

p

that

in

force

The

level

force rate

at

is

given

by

1.

Blue

is

given

by

p.

as

the

were

equal,

victory

would

initial

result

initial

R..

known

p

and

1.

time

unit

are

Figure

is represented by

The and

destroys

and If

by

in

reflects

attrition then go

to

the the

strength.

Lanchester's Equations:

EON I EON 2 dB/dt = -pR for B>O dR/dt = -pB for R;O where dB/dt = C) for B=@ where dR/dt = 0 for R=C,

Square Law

EON

A

-

=

p(Roa

-

There are five basic assumption of

the

Square

Law

to

modern

combat.

7

Ra)

for

the application

Those

assumptions

are 1. Two forces are engaged in combat. The forces are homogeneous in composition (i.e. weapons systems are essentially similar, if not identical).

2.

Each

force

is

in

range of

the opponent's

weapons.

Attrition rates are known and do not vary 3. over the course of the battle (this is the condition for a deterministic model). Exact enemy locations are known and battle 4. damage assessment allows shifting to new targets. 5. Weapons surviving targets.

Current simulations opposing scenario

used

forces

constant

is

to

of

the

represent

employ

time

and

to

Square

conditions

"aimed"

acquire

independent

distributed

evenly

application

is

the

fire

of

the

is

number

targets.

The

conditions

where

and occupy

a constant-density defense..

Aimed will

pick

it."

Square the

Law

on

also

opposing

fire assumed targets

is

that the

used

forces

Law

.n

where

fire.

targets

over

Under

the this

assumed

to

be

of

available

to

represent

use

"area"

fire

"each firer on the red side blue

side

and

try

to

kill

This generally applies to the use of direct-fire

weapons--rifle,

tank,

antitank,

and

precision

guided

munitions. Area

fire

is

assumed

where the

individual

not able to acquire a specific target but fires general to

the

location of use or

the targets.

indirect

firer

is

into the

This generally applies

fire weapons,

such

as

artillery.,

along

with

automatic

weapons

and

other

direct

fire

weapons used aaainst a position.

Development of Lanchester's Linear Law Another

condition

characterized where

volley

position.! was

by

more

Lanchester

firing

was

Lanchester common

engagements

under

of

in

modern

as

"firing

employed

combat

conditions

that

than

where

into

brown"

an

enemy

such

in

air

ground

was

the

against

acknowledged

land

warfare

firing

and

sea

forces

are

sometimes unable to acquire and engage specific targets. The Linear Law and

is derived from the following equations 4

5.

FdB/dt n 4 =

L

E

-pBR for BR'::O where dB/dt = 0 for B=O

These

equations

EON 5 for R.BFO dRidt = -AlRB where dR/dt = 0 for R=O

have

included

variable of the targeted unit's strength. process number the

is

now

a function

of

the

number

of targets, and the rate at which

targets.

The solution

of

the above

to the Linear Law of equation 6. Linear Law

EON

--

-

p pRc

-

R)

an

additional The attrition

of

firers,

the

the firers kill equations

leads

Units

evenly distributed

are

4

is only available on the general

4a. Information location. 5a.

assumptions

to

The new assumptions are':

and 5 for the Square Law.

enemy

change

a

Law requires

Linear

The

a general

over

area. Current in

application

representing

area

fire

the

circumstances

and

circumstance

of

a

Linear

is

where opposing

constant-area

appropriate

Law

to

defense.

the

use

the

directed sides A

use

second

Linear

La4

occurs where both forces use aimed fire with the rate of target of

acquisition

targets

longer

(i.e.,

the

individual

inversely proportional the

time

based the

factor

greatly

firepower

coefficients.

number

choose

from,

and

engage

select

the an

Lanchester's Models

simplifying

deciding

to

to

the

:

Lanchester's original overly

targets

reQUired

target)

Shortcomings of

more

to

in

on

the

Each

be attacked based on

conduct

determining

the

that

models have been

force

is of

ratio

reflected the

five

the actual

of

battle.

victory of in

critiqued

the the

basic

in

as -The

battle

opponents attrition

assumptions

is and

rate can

conditions of warfare.

1) Opposing forces are rarely homogeneous. Differing levels of technology can produce weapons which have dissimilar capabilities in the era of modern warf are.

2) Because of the dispersion of the modern battlefield all forces will not be within weapons range of the other. over

Attrition rates are not constant 3) and vary over the duration of the battle.

time

4) The fog of war does not always allow the locations detection of specific and/or general enemy is not Battle damage assessment during battle. available to the extent where multiple hits on a target can be prevented. 5) The friction of war and non-linearity of the battlefield does not accommodate the idea of even distribution of fires or distribution of forces.

There

are

effect of

combat

the

Lanchester

in

characterized

the performance of

models.

soldiers on

the

are

They

the battlefield.

the concept

with

disagreed

Brackney

combat between forces was purely aimed engagements.

combat

not

Lanchester's Equations by Brackney

Adaptation of

forces--

are

that

training and technology on

morale, leadership,

Harold

dynamics

other

an

He

viewed

attactier

represented

combination

of

by

battle as and

a

fire or area

a contest between

defender.

equations

conditions

for

that

7

the

and

His 8

Square

fire two

model

reflects and

Brackney's Equations

I

dB/dt = -pR where dB/dt =

EON 8

-- - - - -

)

for B>O for B=0

for >O dR/dt = -ORB where dR/dt*= 0 for R,B='

---- -

a

Linear

Laws.

EON 7

of

The Blue Red

defender.

fires

in

the

As

a

The

Red

defense,

model

has been applied

Blue

represents the

guerillas

in

government positions. identify the

Mixed Law

the

is given

The

firer

specific aims

and

using

the advantages

place same

and

aimed

fire

concept

from

targets of

the

forces.-

operations

surprised

direction

and

insurgents

soldiers

general

Blue

a

and

to guerilla warfare where the Red

government

specific

acquire

the defender's position

attackers.

ambush

The

the

is able to select

specific

represents

of

to

defender, possessing

upon

f orce

is unable

result,

the direction

area fire. of

attacker

The would

covered

forces but the

Blue

would

would

not

place

attackers.

attack ing fire

and

force

upon

concealed be

able

to

area

fire

in

The

Brackney

by equation 9.

Mixed Law

EON

(/2)

9

The

same

(B

E

-

assumptions

BE)

= p(R.

used

for

-

R)

Lanchester's

Square

and Linear Law are applicable to the Mixed Law.

Adaptation of Lanchester Equations by Peterson Another equations

was

adaptation made

by

to

Richard

the H.

basic

Peterson.

Lanchester He

reasoned

that

the

as

targets are attrition

presented

of

a

force

the attrition

and

this

losses

rate

Peterson

that

is

a

function

inflicted 10

and

are

proportional

to

the

in

its

the

The

10

of

solution

that

incurs

"force size

own

as

early stages

the

Peterson's Equations

dB/dt = -oB where dB/dt

held

Peterson

that

level

and

[are]

of the enemy's numbers."'-

largely independent

SEON

force

of the

Therefore.

the opponent

by

11.

contended

number

cireater

of

process only applied

attrition

battle.

a

increases.

to the opponent.

equations

in

reflected

of

size

unit

for B:O for B=O =

Peterson's

-

EON 11 for R.O dR/dt = -3R where dR/dt = C for R=O-

equations

in

results

the

Logarithmic Law. Logarithmic

[

MON

0 Log

12

The

key

assumption

weapons

are within

weapons

are

through

camouflage of

B=/B = p Log R./R

for

use

the

the range of

intervisible.

the

of

Law

This

cover,

the targets.

Logarithmic the opponent condition

is

all

but no

two

Law

could

concealment

exist

and/'or

Development of Willard's Linear Regression Model The

first

Lanchester's conducted

Laws

equations

by Willard was

intent

stated

"adequately

battles between generalized 5)

comprehensive

describes

basic

to make them

in

and published

to determine

if

the

To conduct

approximations

and

and

substitution,

or

Linear

outcome

of

the test., Willard

equations

independent of time.

was

Willard's

Square

the

test

data

1962.

course

Lanchester

to

historical

versus

armies."--:

the

attempt

(1

,2

,4 and

Using mathematical he

arrived

at

the

following model.

Willard's Linear Regression Model-

ON3

log B,/R. =

This between force

equation

Biue

levels.

This

is

variable

is

ratio, and

the forces.

R_/B..

Red

respectively.

nodel

reflects

a

rlog

R 0 iB.

relationship

linear

the ratio of casualties and the ratio of

independent force

log E

forces E

a

predictive

the

The

represents

The parameter,

logarithm

casualties

are

model

depicted the

of

the

exchange

B=

the data.

the

initial on and

ratio

ov indicates which

is most representative of

where

inflicted by

initial

the R.,

between

attrition

Table B - Parameter Values

Appropriate Law +1 .5 0 -1

I Results of

Square Law Mixed Law Linear Law Logarithmic

Law

Validation Attempts

As stated earlier, models are designed to represent reality and are used as tools of prediction. of

a

the

model

is

model

to

attempts to is

to

a

the

amount

conflicts

of

available

to

that

the test

claim

for

the

of

the

results

reality

The purpose of

the model

models

comparing

of

collected

degree

number

Lanchester

cases

data

provide

The

process

represent.

predicted by

that

the

Validation

confidence

model that

that

of it

validation the

results

will occur.

of

studies

is

limited.

and

quality

twentieth the

that

have

Researchers of

data

century

models.-.

validation

have

are

not the

the

claimed

collected

Hence,

against

tested

from

readiiV

number

observed

data

of are

very rare. The models

validation have

met

of

with

Schneider

and

Ronald

summary

the

more

of

the

Lanchester-based

mixed Johnson

prominent

success. have

Both

provided

attempts

to

attrition

a

James brief

validate

the

and

Square

Linear

Laws.

attempts and their results.

validation

Validation Studies

Table C - Survey of

I

survey of

a

presents

Table C

Result

Name

Data Source

Busse

Inchon-Seoul Campaign

Unable to validate the Square Law

Engel

Battle of Iwo Jima

Questionable validation of the Square Law

Fain

60 land battles of World War II

Suggestion of Logarithmic Law

Johnson

National Training Center

Suggestion of Logarithmic Law

Schmieman

Bodart's Lex-ican

Square and Linear Laws inadequate

Willard

Bodart's LeAicon

Square and Linear Laws "poor choices"

comprehensive test of

The most was

conducted

validit/ that

of

the

projected

.5.

13

This

.5.

Willard.

Willard

the

assumed

both the Square and Linear Laws and expected

of

that

represented values

Daniel

analysis

combination

equation

by

the attrition models

of

use of

the

performing

would would

result

from

data

area a in

indicate

the data.

ranging

historical

and

aimed

regression y

approx imately

that

the

Mixed

to

-. 87

with

a

using

equal Law

a He

fire.

analysis

His analysis yielded -. 27

reflect

would

to

best

y parameter average

This result refutes Lanchester's basic

near

equaticons

in favor of Peterson's Loaarithmic Law. is

that

the

increases

as

casualty-producing

The implication

power

the force size decreases.

In

of

a

force

other

words,

a smaller unit presents less targets to its opponent and is more efficient This

is

counter

number of to the

in

to

its ability to

Lanchester's basic

casualties

incurred

initial force ratio.

increased

force

inflict

ratio

is

casualties.

premise

that

inversely proportional

Lanchester posited

would

the

reduce

casualties

that for

an the

superior force.

Historical

III.

The the

body of

Battles as a Data Source data

information

1.49. major

the

1618

Wars,

Year's the

to

the

The

i

source

The

Civil

Gaston

study

occurred between begins

through and

on

Bodart's

Bodart's

-

period

War,

is based

the

'ith

the

Napoleonic

concludes

with

1905. accepted of

information

strengths

sustained.

of is

Lexicon

identification

tnitial

in

progresses

American

comprehensive period.

analysis

engagements that

1905.

War,

Russo-Japanese War The

the

Arieqs-Lex icon.

covers the

Thirty

for

conplied

t1itaerhistorisches

years

used

data

as

on

collected

an

accurate

engagements by

Bodart

and for

the

includes

of

combatants,

location

of

of

the

and

CasULalties

forces,

the

battle,

The Appendix

data

is

A

a

is the

categorized classes to

of

organized sample

page

engagements

treffenr

meeting

gefecht

engagements.

belagerLng,

in

einnahme,

chronological

from into

and The

the eight

schlact

text.

Bodart

classes. can

remaining

erstuermung,

order.

be

The

equated

classes

kap i tulIati

of

on,

and

ueberfall are attacks against fortified positions.--The analysis was battles

conducted using

from the period

of

the study.

data on

1080

land

The data set

was

extracted from an annex

of Schmieman's dissertation.

It

consists

initial

and

of

Blue.

the and

(Appendix C)

year, the

casualties

force

strengths

incurred

by

for

each

Red side

IV.

Statistical Analysis of Historical Data The

research

question

is

to

determine

quantitative analysis of historical in the conduct of battle.

evolution

American Civil

modern

in the data that would mark< battle

Schmieman's

categorizing

the

data

yielded

statistically

by

battle

results

that

different.

pre- and post-Civil

War

preliminary

Willard's

linear

The

1

battles

to

arbitrary

analysis

process.

provided

in

was

which

assignment was

and

percent were

analysis

was

not

illitially

conducted

model.

As

l!og

= log E +

the data as

data

size

that

era.

regression

Bodart's

showed

utilizing previously

is:

determine

represented

the

the years of occurrence,

analysis

log B./i'

initial

with

generally

My

regression

presented, the model

EON

beginning

analysis

partitioned the data based upon

The

data reflects trends

War.

William

casualties

of

the

In specific, the analysis was

conducted to identify trends the

if

performed

B

the

10GO

model

best

with

Lanchester-based

indicated by the value of of

done The

Appendix

Ri

Red

to

to

maintain

results B-1.

the

The

of

winning

force

consistency the

value

-.

The in

in

the

regression

-re

-v

=

-. 47'

is

consistent

with

Willard's

ranged

-. 27

to -. 87.11

from

(page 17),

analysis

Referring

the

back

values

to

Table

data better than

B

Law

Peterson's Logarithmic

this suggests that

represents the historical

where

the Square or

Linear Laws. To evaluate the adequacy of descrining

the

required.

A

to

the

null

value

regression. the

data,

regression

point

along

hvpotheses noth

line, the

to

relationship

rt ypothesis confidence freauency

that

conditions.

state is

sample

compared

:onfidence

I- and

Log E

test

is

used

results will

a

95%

to

from

the

level.

a

the

and

the

number

level

"t"

associated computed

the 1 the

actual

asserts

reflect

are

linear

of

the

that

the true statistic

parameters

of

.

level.

computed

population

t-value If

A

null

evaluate

reflect

confidence

100.

a

statement

the hypothesis test will of

that

confidence

the test

intercept The

the data.

tte

slope of

the

indicate

statistical

by

E.

for

variance, to

and Log

a

times out

calculated

mean, is

95

are the

exist

is

Hence,

the results of

1.

av is,

specified

challenge

determined

parameters

would

hypothesis

statistical

this case

This

"t"

level

is

null

the parameters,

does not

a

the

is a

vertical

zero.

at

of

in

parameters

in

that

Student

A

the

regression

posit

equal

test

hypothesis

of

The

a

the regression model

of

the

observations.

It

with "t"

a

speC3fiC

exceeos

the

t-value, the null The tested "t"

null

at

hypothesis is rejected.

hypotheses,

the 95%

Loa

E

=

0

confidence level.

statistics both exceed

and

y

Since

the critical

0,

=

were

the computed

value of

t os

-

1.96. the null hypotheses were rejected. Another

indicator

of

the

adequacy

of

the

the coefficient of determination, R-square. calculated

from

the

degree

independent

and

dependent

The

of

R-square

in

the

value

variabilitv

For

regression. that

the

model

variability very

The when

of

partitioned

provided

in

pre-1661

model

rejected value the results Law mull

at

the

indicates

data

is are

the

R-square

for

15

the

regression.

the

accounted

accounts

amount for

percent

of

by

value

This value does not

into

the

shcws

of

the

indicate a

9-2

suggests y

=

95% that

B-0,

that

the The

confidence

similar.

With

in y

=

null

rejected

the

model. -. 496,

is a better representation of at

data

post-1861

are

Logarithmic

of

the

the

respectively.

level.

percent

for

were

and

and

-. 475.

15

analysis of

pre-1861

accounted

hypotheses

the

is

the regression

Appendix

with

between

is

the data to the model.

results of

appropriate

that

in the data.

good fit

indicates

model,

only

of

is

R-sqUare

correlation

variables

data

this

of

model

Law

hypotheses The

is

were

R-square

variability The

The

in

post-1861

the

Logarithmic

the data.

Again, the

the

q5%

confidence

level.

There

value

to

was a

18

slight

percent

of

improvement the

in

the R-square

variation

in

the

data

explained by the model.

Table D - Summary of Regression Results

Period

Loo

1620-1905

.652

-. 475

15%

Pre-

1861

.700

-. 475

15%

Post-1861

.180

-. 496

18%

regressions

indicate

Although relationship ratio

and

P-square in

the

basic and of

do

Laws

an

of

9_iB

was used

original function .01

logarithm of

not

the casualty

inspire

equations are

attempt

not

to

casualties

as

detect

to

initial

ratio,

"degree

of

manifested

in

through

it

The

transforms

lost.

A

applied.

= -4.605.

trends

initial

factor

data becomes a factor is

a

supported

investigated.

is

the

linear force

the

low

confidence that

the

the

the

Square

analysis

data.

since

magnitude

the

a

The regressions clearly show

Lanchester

historical

R-suare

logarithm of

outcome."

ratio

Log

between

values

In

not

the

the

Linear

E

For

in

the data,

strengths,

logarithmic the

data

and

R /R

function the

of

ten

of

two when the

This characteristic

.I is

=

and was

original

difference

ex ample, Log

the

in

the

logarithm -2.307

and

-eferred to as

"log

compression."

The

use

of

transformed,

compressed

data hinders the analysis of the population parameters.

Table E - Casualty Statistics

[

Red

(Percentages)

Blue

mean

.0858

.1759

st dev

.083

.153

The average casualties sustained by the victor over the

1620-1905

suffered

period

17.6 percent

conducted

at

the

hypothesis that statistic implies

=

t that

This finding are

the

sample of

level

exceeded

are

the

"tto

loser

test was test

the

The computed

t,

=

statistically

1.96

and

different.

losses of the victorious force

less than, the

losing

conducted

by

force

Richard

is

consistent

Helmold

or

a

that

a

ninety-two battles. Schneider

to

qualitatively

He posits that

was

greatly

that percent

in the conduct

style of

while

A paired

confidence

means

change

marked

percent.

casualties.

95%

20.35

analysis

James

[a]

8.6

the t~o means were equal.

the

generally

with

was

the

proposes

an

article

of warfare occurred distinct style of

to give

military

"rise

art.

the American Civil War was the event

transition

military

performed

in

with

art

to

from

the

classical

operational

art.

the ratios oartitioned

"

-

that

Napoleonic A, "t"

into

pre-

test and

Schneider's hypothesis.

data sets to test

post-1861

Table F - Statistics of Partitioned Data

pre-1861

Red post-1861

.0864

.0826

.184Z

.1271

007

.006

.024

.011

mean variance .t.

-. 53

The

-5.82

analysis

strength

ratio

difference 95%

of

implies

between

confidence

the victor's casualties

the

The

level.

analysis

results.

There

confidence between percent

level

the

pre-

is

The

Further and

and

to

no

post-1861

percent

Blue

of

loser

statistical

post-1861 dropped

initial

statistical means

Red

at

the

casualties

yielded

difference the

data

from

percent sets.

18.4

different at

the

95%

casualties The

percent

average of

the

12.7 percent.

analysis was Blue

and

is

to

the two periods.

the

a

there

in the means of

casualties

initial strength

Red

of

that

pre-

remained consistent over

the

Blue pre-1861 post-1861

data

and

conducted by performing

composite data with following results:

a

aggregating both "t"

test

or, the

Table G - Statistics of Aaareaated Data

pre-1861 mean variance

The

hypothesis

post-1861

rejected at the 95% there

is

a

.1354

.1048

.0186

.0087

that

casualty to

post-1861

the

initial

means

the

force ratio are

confidence level.

statistical

of

difference

This in

pre-

and

equal

Was

implies that

the

pre-

ard

set

as

post-1861 means. A

linear

performed

L EON where

regression

the

complete

E

RciRo

is

the

y-intercept

regression

line.

provided

Appendix

in

The

1(BcBo)

+

and

results

B-4.

At

N

is

of

t_,

s

R-square.

=

1.96.

indicates

The that

the 95%

coefficient only

1-7%

of

the

the

the null hypotheses. E = 0 and o= 0, with

data

using the model:

14

E

of

slope

of

ti-e

regression

are

confidence

level.

were both rejected of the

determination. variability

in

data is explained by the model. The results when

i

of

partitioned

the regression analysis of into

pre-1861

and

the

data

post-1861

are

I III I

provided

in

Appendix

analysis of rejected value the

the

at

pre-1861

the

indicates

data

is

B-5 and B-6,

95%

12

accounted

coefficients

Again,

null

confidence

percent

of

model .

in the

the

+orce ratio for

for

post-1861

are

similar.

Post-1861

The data four

periods:

1865 -

1905.

provided

However,

there

R-square

value.

in

the a

percent)

and

is

a

In

data

the

95%

significant this

case

explained

greater

between

at

by

34 the

correlation

the

casualty

Regression

to

I_

Results R-square

.051

.197

173%

.05.

.18

12%

.C) 7

.419

Z4%

was 1620

further -

partitioned

800.

Appendix

1800

-

9-7

into

1861,

through

to note that the analysis

relationship exists Ped

rejected

a

total

1861

-

for

1865'

The results of the regression analysis are

in

interesting

for

were

CE

1861

in

the Red and Blue forces.

Feriod

Pre-

variability

resulting

Table H - Summary ot

i 1620-1905

the

R-square

The

indicates

(approximately 58

The

the model.

variation

This

the

hypotheses were

level. of

In

in

hypotheses

level.

improvement

percent

for

regression the

data, the null

confidence

that

respectively.

Blue.

between That

B-10.

It

is

indicates a linear

the casualty to

relationship

as

force ratios measured

by

R-square

is

generally

and post-Civil Another

improving

through

the

Civil

War

war periods. set

of

aggregated

means

sub-periods.

Table

"t"

tests were

(both I

is

Red

a

performed

and

summary

Blue) of

the

on

the

over

the

descriptive

statistics for the partitioned data. Table I - Summary of

Percent Casualties Ratio

Number of Period

Mean

1620-1800 1800-1961 1861-1965 1865-1905

Variance

.1465 .1175 .1152 .100:-

.0218 .0130 .0074 .0092

Table J provides the results of means

of

the

statistical t

partitioned

1136 704 98 2

the

"t"

data

test between

set.

There

difference at the 95% confidence for

1.96

=

Observations

the

1800-1861 , pre-1800

following and

cases:

1865-1905,

level

the

is

a

where

pre-1800

and

1800-1861

and

1365-1 q05. Table J -

Computed "t"

values

----------- ----------- - -- ---------Ier

-

iod

16201800

1620-1600

..

1800-1861

4.46

......

1861-1865

1.90

.20

1865-1900-

4.4q

2.04

-------------------------------------

--------

1800186118651861 1865 1905 .....

..

1

.. -

I

V.

Results

Evaluation of

Qualitative Analysis of Results The quantitative analysis of the data

is summarized

as follows: 1.

Lanchester's

adequately

describe

the

This

implies

that

laws

are

supported.

not

casualties

to

the

that

independent

of

the

by

Square

a

force

of

the

size

of

the

Law

this

be

study.

by

these

process

that

inversely

The

forces

not

maintains

will

force.

do

Linear

is

Law

largely

engaged

bLtt

is

the attrition rate coefficients fr-

force. .

'Peterson'sLogarithmic

representative of producing

power

decreases. larger

units

-.

the data and of

An

a

force

e-'tension

are

the enemy than

less

The

depicts

linear

casualty

force

ratio.

R-square

this

efficient

generalized

represented

a

implies that

increases of

Law

as

is

the casualty

the

force

assertion

in

their

more

is

size that

attrition

of

are smaller units.

equations

the

Square Laws

battles

attrition

solely a function o+ each

The

size

the

of

1080

and

battle conditions postulated

sustained

proportional predicts

Linear

by

of

Willard's

relationship

ratio

form

regression

between

the

and

the

logarithm

However,

the

relatively

other

factors

indicate

that

Lanchester's

of

logarithm the

low

mav

model of

initial

values

exists

of

that

better explain the variations in the data. 4.

There

conduct of

a quantitative difference

ratio

There

is

for

the

a general

time

pre-

trend

period

Schneider's

assertion

shows

that

and

toward

A

casualty

Willard's

the

there

initial

of

Civil

of

War.

investigation post-1861

conduct

supports

of

warfare

The

analysis

fewer

percent

loser. relationship

ratio

than

model

the

measurable,

conduct

American

periods.

as

exists

is

using

present

measured

with

by

the

reveals

that

of determination.

a

tactical

result

sustains

linear force

analysis

is

the

victor

regression

coefficient The

the

ratios over

with the Civil War.

stronger

to

War

decreasing This

that

casualties than the battle's 5.

post-Civil

studied.

significantly changed also

in

battle as measured by the casualty to initial

force

the

is

of

period

historical

quantifiable

battle

The

next

change

that

occurs

step

is

conditions that

data

that

resulted

in

the

in

the

after

the

qualitative

existed

the

in

the

quantitative

change. Schneider in

his

He

posits

face

of

discusses

article, that

more

lethality

is

"The

Theory

"armies

lethal

the

declining of

the Empty

incur

fewer

weapons."

The

attributable

casualty

to

Battlefield."

casualties weapons'

five

ratios

in

the

increasec

technological

innovations of

powder.' 5

smokeless

a

These

artillery

rifled impact

significant

a

introduction of the machinegun and

direct result are the breech-loading,

as

follow

that

innovations

Other

and

wire,

barbed

magazines,

rifles,

breech-loading

muskets.

rifled

nineteenth century:

the

of

the conduct

on

had

inventions

after

battle

their adoption by the modern armies. Smoothbore by

muzzleloader

prior

armies

the mid-nineteenth

to

of

their

to

development Norton and the

adoption due

rapid The

loading. of

rifle

the

weapons

used

century.

The

1500's did not

with

practical

became

bullet

by

refinement by Captain

The new bullet

problems

the technical

cylindro-conodial

its subsequent

1840's.

in the

rifled musket

early development of lead

the

dominated

the

Captain Minie in the

be dropped down

could

muzzle of the weapon and did not require ramming. When bullet muzzle

the to

musket

spin

velocity

provided

as

was it

fired,

travelled

provided

study

in

troops

armed

with

rifles

the

barrel.

The

range and

improved

1840 proposed that

German

the

down

additional

which

stability

forced

the rifling

increased

the

the

the

spin

accuracy.

effectiveness and

by three

a

A of

half

. times over those with smoothbore muskets. %

The

adoption

weapons began British

of

in the

replaced

their

rifles

1840-s with traditioral

as

practical

the United Brown

infantry

States. Bess

The

mus :et

with

the

rifle

in

1842.-D and

bureaucratic reluctance

The

with

adopted

use

first

significant

in

the

Crimean

War

smoothbore

muskets

fought

occurred

French

forces

French

overcame

the Tiae

rifle for

equipped

of

where

with

rifles

the

next

breechloader

and

Additionally,

the

the

It

had

was

loaded

rifle

twice the

over

increased

rate of

velocity

which

was

adopted

a

with

by

bolt-action

Prussian

potential

rate

range

the

to

of

army

ir

fire

as the

However,

ball. by

Von

Dreyse.

von

the

Minie

the

the

by

fire was offset

limited

1862 was

rifle.-

1835

in

patented

needlegun

Dreyse's 1840.

innovation

armed

British

against

the Springfield, then the Enfield

combat

forces

Russian

rifles.

in

standard weapon of the American Civil War after

The

their

1857.':

in

its infantry

The

loss of

muzzle

approximately

60)

yards. The on of

above

changing

innovations had a

tactics

smoothbore

required

two

the

of

muskets, use

of

concentrate firepower. narrow

front

battalions firing. due to

with

would

These

the post-1861 with close

range

order

companies

tactics

era.

in

in

line

resulted

lethality of

F'revious

of

200

yards.,

approach

columns and

in

use

formations

Regiments would

d

deploy

the increased

a

effect

substantial

and

employ

highe-

the weapcns.

on

to a

then

volley

Casualties

During that by

"It~he

Civil

War,

exposed

additional

within

value

and

to

the

rifle

had

gained

power

defensive

was

life,

long

to

Mahan

range,

field

of

leaders

the

use

and

both of

give

recognized

Subsequently,

while

must

battle.

that

forces

into

observed

in open assaults,

a

devastate

incorporated

operations

Hart

entrenchments

assaults.

entrenchments

so

military

strengthened the

of

entrenched

Confederate

frontal

Dennis

great destruction

columns

Union

the

"

s

that

defense

conducting

the

use

of

offensive

and

frontal

assaults

was strongly discouraged.-

fire

The

use

and

reload

position f:re.

of

breechloaders

thereby

The

their

weapons

reducing

their

advantage

was

Austro-Frussian War of fire L.-e

at

a

more rapid

AUstria

who

Prussians, with to

achieve

a

1866.

had

in

soldiers the

vulnerability

prone

to

illustrated

to

enemy

in

the

The Prussians were able to

from

to

the

while

clearly

rate

their

six

allowed

prone

ztand

positions

to

against

reload.

The

new weapons and tactics. were able

to

one

firepower

advantage

over

the

Austrians. This

lesson

developed 1870. than

and

The

increased

the

not

fielded

ranqe

twice

was

of

lost the

the

range

of

on the

French

chassepot

chassepot the

was

Prussian

who

quickly

breechloader 1200

in

yards--more

needlegun.

The

lethality of the weapons forced a modification

in

French

pits the

tactics

against

they

the Prussian

Prussians

one-half

where

adopted

adopted

use

attacks. 4 .2

infantry

the

the

American

system of

of

rifle

In

turn,

advancing

of the force in rushes while the remaining half

provided covering fires. The 1849 in

in

magazine America

1879.7

rifles was

the

soldier.

each

firing of

the

Soldiers

and

their

initially patented

vertical

magazine,

increased

each

selection

and

The

innovations,

for

combined

with

potential

did

rate

not

accuracy

targets was

magazine was

not

have

could

to

other

fire

reload

improve

interrupted

patented

the of

by

in

of a'ter

since

the

the need

to

frequently reload. extension

An

development

the

employed

War.

This

the

initrailleuse

machinegun.

which

simultaneously,

represented

magazine

of rapid-fire weapons.

and

barrels

of

a great

and

increase

The French

in

the

fired the in

concept

was

the

developed

Franco-Prussian

thirty-seven

American

rifled

Gatling

iirepower produced

gun by

a

single weapon. The

Russo-Japanese

devastating integrated

impact into

the

of

War

of

rifles

defense

1905 and

of

demonstrated machineguns

fortified

Japanese frontal assaults against Russian

the when

positions.

fortifications

yielded exceptionally heavy casualties. The

introduction of

barbed wire on

the battlefield

restricted

severely breach

the

entrenchments. the

increased

emoloyinq

amount

frontal

the

that

were

uip

-Japanese hUnQ

and

fires.

were

Tactics in

costly

the

ended

assaults

massacred

to

the wire

attackers

very

the

[E) lways

through

the

and machinegUn

assaults

wire

attacking forces

of

time

of

War .

Russo-Japanese

ability

Attempts to get

to the rifle

e :posed

with

Barbed

to greatly enhance the defense.

served

the

on

up

bar-bed

wire... The -nrn

last

imp-Act

the

increase The

accuiracy. the

soldiers

the

introduction, of

Cloud

smol-e

shots

powder

firearms the

provided The

prio,-

to

enemy

being

This Compounded the

experienced The

identified

impAct

of

first

a

impact

and

a

new

concern

gree-ter

Prior created

location Would.

be

deetd

of

specter

of

to

to its

signature

a

the

able 7'h e

psychological

magazine

in

the

charges their

to

The

ot

the

weapon to

fire

s m oL:es

s

invisible

stress already

by the combatants.

demonstrated

nic:powder

them

of

evolution

battlefield.

Presented the additional

enemy.

was

use

and

provided

modern

battlefield.

tne

more

the

irnpact

the

velocity

muzzle

in

second

on

on

powder.

e~~~ssoe~

was

c~n

to

innovation

intense

for

les Wars.

Doer

position. and

rif

British -

accutr:te

and

smokeless

The

initial

artillery It.

in

turn,

cmunter-fire .

powder Use

Of

readily SUbjecteif 5O

the

increased

range

and

accuracy

provided

by

the

new

smokeless powder amplified the effectiveness of the Boer marksmen.

Accounts of

feeling of

"'being

the war

relate the participant's

opposed to an

invisible foe''"'- while

long-range fires.

under the The

introduction

increased

the

battlefield

and

century.

The

of

the

lethality reshaped

need

to

breech-loading

of

the

the

weapons

tactizs

reduce

of

vulnerability

weapons reestablished the superiority led

to

the

adoption

Modern

armies

against

entrenchment

The

of

to

had

tactics evolved

disperse

troop

trenches acCept

would to

for

such

fire

and

soldier the

to

magazine with smokeless

Bloch

and

heavy

powder

accurately

introduction no

larger

of

than

battlefield of

the

and

casualties. order

to

more

on

rate

of

relied

increased

allowed

sector of powder

moral

demands

predicted

long-ranged the

new

assaults

skirmish

the resultinQ

Smokeless

increased

the

fortifications.

the

also

exercise

of

the future would

on

that

firearms,

individual

responsibility on

units to disperse for survivability against enemy

to

defenses.

assume a greater

battlefield.

of

he

nineteenth

frontal

in

assaults

flanking movements against The

that

use

or

of the defense and

and

result

the

the

rifle

forced

the

an

invisible

the

soldiers.

"[blefore

the

battlefields

were

modern

brigades.

The

prove to be much greater

than those of The

conduct

post-1861 above

the past."

era.

were

of

modern

battle

The technological

employed

during

ratios

noted

attributed emerging

in

as

the

the

era

of

and

in

the

presented

shaped

the

The decline in casualty

quantitative

reaction

conditions

innovations

the

tactics by which armies fought.

evolved

of

analysis

the

warfare

armed

in

order

can

be

forces

to

to

protect

themselves from the horrors of the new battlefield.

VI.

Conclusions and

Implications

Summary of Findings The quantitative analysis reveals that Lanchester's Linear

and

Square

Laws

do

not

1060 battles of this study. process

and

are not

supported.

battle

representative of producing

power

decreases. larger the

units

enemy

implies that

conditions postulated

the data and a

force

by

less

are

implies that

increases as

extension

are

than

This

describe

Feterson's Logarithmic

of

An

adequately

of

this

efficient

smaller

the

attrition these

laws

Law is a more the casualty

the

force

assertion

is

size that

in

their

attrition

units.

These

findings

of are

consistent with other studies. The

generalized

represented linear

by

form

Willard's

relationship

of

Lanchester's

regression

between

the

model

equations depicts

logarithm

of

a

the

casualty

ratio

ratio.

However,

indicate

that

Caution derived

conduct

relatively factors

low

or

initial

values

models

of

may

force

R-square

exist

that

be exercised

of

pre-1861

modern

in applying

data

to

battle was

advances

that

modern

shaped

were

the

not

lessons

conditions.

greatly

available

by

the

during

period. The

evolution

quantitative initial

coincides five

of

difference

force

decreasing

ratio.

ratios

with

measured is

the

by

muskets,

change

in the tactical

time

period

of

of

their

marked

by

casualty

to

trend

that

period.

The

War the

nineteenth magazines.,

the catalysts

warfare. The increased

range combined with the substantial

:)

lethality

accuracy

increase

fire forced the change in battlefield

toward

studied

breechloaders.

conduct

with

the

general

smokeless powder were

weapons

is

a

innovations

wire.

new

battle

post-American Civil

barbed

the

and

as

over

the

rifled

modern

There

technological

century:

of

the

the variations in the data.

the

technological that

the

must

from

logarithm of

the

other

better explain

The

and

and

in rates oF

tactics.

Implications The Sauare

Lanchester and

historical

Linear

data.

equations Law

are

as

not

manifested

in

representative

Analysts have persisted

in

these

the of laws

of

because

form

generalized represented

their

and

simulations

by

ease

relative

the

of

of

acceptance.

widespread

Lanchester

the

in

employment

as

the dynamics of battle, then the Logarithmic must

implications

investigate the

should

be

also

accepted.

use of

other

If

equations

is accepted

Willard's model

computer

If more

as

capturing

Law and

not.

the

its

analysts

representative

models. The until

it

acceptance

has

been

decisions

The

result the

impact

training

of

battle, the

are

greatly

underlying model

any

subjected

that

of

of

our

on

model

to

made

the based

force

military

must

be

tentative

validation

process.

the

model's

upon

structure,

doctrine

leadership.

is not representative of

If

and the

the realities

lessons gleaned from the simulations will

be equally fictitious.

APPFENDIX

A

-

Sample Page

from Lex.-ictmr

Deutsch - Fran: rsischer Kricg 1870-1871.

1I7M 27./11.

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Sample Page from Gaston Bodart's, Krejas-Lexicon 16183-1905. Vienna: 562.

Hilitaer-Historisches C. W. Stern, 19038, p.

ALL YEARS MULTIPLE LINEAR REGRESSION Dependent Variable:

LOG(Bc/Rc)

Variable

Mean

intercept LOG E LOG(Ro/Bo)

0. 244

Source .odel Error Tota1

T for HO: Parameter Standard parameter=O) Error Estimate 0.652 -C). 475

DF

Sum o-f Squares

1.v0 1 ,77.00 1 ,078. 00

154.42 887.77 1,042. 1

0.0) .') Mean Square 154.42 (.a-

0.54 0.91 169.36 0. 15 0. 15

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square B-I

5 _2. -13.69

F-Val ue 167

K

en.:

1

~

erE'

i-F-e-

PRE-1861

MULTIPLE LINEAR REGRESSION Dependent Variable:

Variable Intercept LOG E LOG(Ro/Bo)

SGurce Model ErrorTctal

LOG(Bc/Rc)

Mean 0.249

DF 1.00 917.00 918.00

Parameter Standard Estimate Error 0. 700 0.03 -0.475 0.04 Sum of Squares 134.51 775. 10 909.61

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square B-2

L:

Mean Square 134.51 0.85

0.58 0.92 157.99 0. 15 ].15

T for HO: parameter=0 22.05 -12.62

F-Value 159 .14

-

L

me4 e~

~C

re s

i r, r-

!

POST-1861 MULTIPLE LINEAR REGRESSION Dependent Variable:

Variable

LOG(Bc/Rc)

Mean

Intercept LOG E LOG(Ro/Bo) C. 215

Source Model Error Tctal

DF 1.00 158.00 159.00

Parameter Standard T for HO: Estimate Error parameter =0 080 -0.496

C).06 0.08

5.85 -5.85

1um of Mean Squares Square F-Value 21..1 98.26 I.57

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square AdjL sted R-Square

21.31 0. 62

0.27 0.79 289(01 0. 18 0.17

34

.26

ALL YEARS MULTIPLE LINEAR REGRESSION Dependent

Variable:

Variable

Rc/Ro Mean

T for HO: Parameter Standard parameter=O Error Estimate 0.051 0.197

Intercept e 0.176

Source Model Error Total

DF

Sum of Squares 0.98 6.44 7.42

1.00 1 ,078.0 )] I , 079.0C

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square B-4

14.. 2 12.84

0.00 0.()2 Mean Square 0.98 0. 01

0.09

0.08 90.02 0. 13 (. I-

F-Value 164

.86

-

r

L-

R eQre s cwi

-.s.'

PRE- 1861 MULTIPLE LINEAR REGRESSION Dependent

Variable:

Rc/Ro

Variable

Mean

Intercept e Bc/Bc

0.184

Source Model Error Tctal

DF

Parameter Estimate 0. 053 0. 183 Sum of Squares 0.78 5.75 6.53

1 .00 918. 00 919.00

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square

B-5

T for HO: Standard parameter=( Error 0.00 0.02

17. 17 i 14

Mean Square

F-Value

0.78 O. O1

0.09 C0 (8 91.64 0. 12 0. 12

124.01

Ap

.

eer ea's

B;' -

e -

ieres -n Fs-

ItE

POST-1861 MULTIPLE LINEAR REGRESSION Dependent Variable:

Variable Intercept e BC/Bo

Source Model Error Total

Rc/Ro

Mean

Parameter Standard T -for HO: Estimate Error parameter=€) 0.030 0.419

0.127

DF

Sum of Squares

1.00 158.00 159.00

C. 30 0.59 0.89

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square l-6

0.01 0.05 Mean Square 0. 30 0.00

0.08 0.06 7:T.70 0.34 0..4

3.89 9.03

F-Value 81.5e

-:i

71e

-Li

qrear cn

Res

,u t

PRE- 1800 MULTIPLE LINEAR REGRESSION Dependent Variable:

Variable

Rc/Ro

Mean

Parameter Standard T for HO: Estimate Error parameter=0

Intercept e BC/Bo0.20

C.056 0.166 Sum of

Source Model Error Total

DF

Squares

1.00 566. 00 567. 0))

O.46 4.05 4.50

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square B-7

0.01 0. 02

10. 18 7.98

Mean Square '.46 0. 01

0.09

(.08 94.09

0. 10 0. 10

F-Value 63.74

1800-1861 MULTIPLE LINEAR REGRESSION Dependent Variable:

Variable Intercept Bc/Bo

Rc/Ro

Mean

Parameter Standard Estimate Error 0.046 0.)228

e 0.154

Source

DF

Model Error Total

1.-0 .....) 50. 00 .5 .

Sum of Squares , 1.69 20

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square B-8

0.01 0.03 Mean Square 0. 2J. 0. 00

0.08 0.07 86.00 c-. 16 0. 16

T for HO: parameter=z 6.0: 8.15

F-Value 66.45

1861-1865 MULTIPLE LINEAR REGRESSION Dependent

Variable:

Mean

Variable Intercept BC/Bo

Rc/Ro T for HO: Parameter Standard parameter=C) Error Estimate (. 045 C). 402

e o. 17,

Source

DF

Model Error Tota 1

47.00 48. 00

SLIm o-f Squares

t. C00

08 C. 13 0. 21

Dependent Mean Root Mean Square Error Coefficient of Variation R-Square Adjusted R-Square B-9

0.01 C). 08 Mean Square 0.08 U.

00

U. LU

(.05 54.37 1-). 37

3.55 5.21

F-Val ue 27. IQ

re

-

r

isS on-

;Ries

t.

POST-1865 MULTIPLE LINEAR REGRESSION Dependent Variable:

Variable

Rc/Ro

Mean

T for HO: Parameter Standard Estimate Error parameter=O

Intercept e Bc /Bo

0. 125

C.023 o. 421

).01 0.06

Source Model Error Tot a 1

DF 1.00 109. 00 110. C

Sum of Squares 0.22 0.44 0.66

Mean Square -. 0.00

Dependent Mean Root Mean Square Error Coefficient o+ Variation R-Square Adjusted R-Square B- I 0

0.08 U').€U6

e..78 J. 0.37

2.51 7.40

F-Value 54.80

A??ENDIX C - H STOR7CAL DATA SET

YEAR 1620 1621 1622 1622 1623 1626 1626 1628 1630 1631 1631 1632 1632 1632 1633 1634 1634 1634 1634 1635 1635 1635 1635 1635 1636 1636, 1636 1637 1638 1638 1638 1639 1639 1639 1640 1641 1642 1642 1642 1642

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc Rc/Ro 47,500 20,000 26,000 13,000 28,000 26,000 10,000 7,000 10,000 36,000 20,000 60,000 20,000 33,000 17.000 5,000 35,000 30,000 15,000 34,000 7,000 4,000 5.000 3.000 22,000 10,000 5,000 12,000 14,000 5,000 5,000 14.000 18,000 9,000 10.000 10,000 12,000 20.000 25,000 7,500

400 5,000 2,000 2,000 1,000 1,000 800 1,000 500 5,000 400 1,000 4,000 1,000 1,000 100 2,000 8,000 1,000 3,000 400 500 1,000 600 7,000 1,000 500 3,000 1,600 400 500 1,500 600 1,000 2.000 1,000 1,000 800 5.000 400

32,000 20,000 21,000 17,000 22,000 11,000 15,000 6,000 29.000 34,000 8,000 46,000 25,000 27,000 15,000 6,000 25,000 4,000 12.000 16,000 7,000 6,000 7,000 4,000 30,000 12,000 3,000 16,000 17.000 4,000 7,000 20,000 10,000 18,000 18,000 11.000 25,000 18,000 30,000 9,000

6,000 5,000 8,000 2,000 4,000 3.000 2,200 600 800 8,000 1,700 2,000 5,000 3,000 7,000 1,500 6,000 1,400 4,000 5,000 2,400 1,500 1,700 1,500 10,000 2,000 1.000 4,000 2,000 700 4,000 6,000 1.000 2,000 4,000 3,000 2,000 3.000 10,000 4,000

0.0084 0,2500 0,0769 0,1538 0.0357 0.0385 0,0800 0,1429 0.0500 0.1389 0.0200 0 0167 0.2000 0.0303 0-0588 0 0200 0.057! 0.2667 0 0667 0.0882 0.0571 0.1250 0.2000 0.2000 0.3182 0.1000 0.1000 0.2500 0.1143 0.0800 0.1000 0.1071 0.0333 0.1111 0.2000 0.1000 0.0833 0.0400 0.2000 0.0533

Bc/Bo 0.1875 0.2500 0.3810 0.1176 0.1818 0.2727 0.1467 0.1000 0.0276 0.2353 0.2125 0.0435 0.2000 0.1111 0-4667 0.2500 0.2400 0.3500 0.3333 0.3125 0.3429 0 2500 0.2429 0 3750 0.3333 0,1667 0.3333 0.2500 0.1176 0.1750 0.5714 0.3000 0.1000 0.1111 0.2222 0.2727 0.0800 0.1667 0.3333 0.4444

Log RATIO Bc/Rc

Log RATIO Ro/Bo

2.71 0.00 1.39 0.00 1 39 1.10 1.01 -0.51 0.47 0.47 1.45 0.69 0.22 1.10 1.95 2.71 1.10 -1.74 1.39 0.51 1.79 i.10 0 53 0.92 0.36 0.69 0.69 0.29 0.22 0.56 2.08 1.39 0.51 0.69 0.69 1.10 0.69 1.32 0.69 2.30

0.39 0.00 0.21 -0.27 0.24 0.86 -0.41 0.15 -1.06 0.06 0.92 0.27 -0.22 0 20 0.13 -0. 1 0.34 2.01 0.22 0.75 0.00 -0.41 0.34 -0 29 -0 31 -0 18 0.51 -0.29 -0.19 0 22 -0.34 -0.36 0.59 -0.69 -0.59 -0.10 -0.73 0.11 -0.18 -0.18

AFEND:X C - HISTORICAL DATA S"ET

YEAR

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc Rc/Ro

1642 1642 1643 1643 1644 1644 1644 1645 1645 1645 1645 1646 1647 1647 1648 1648 1650 1650 1651 1652 1653 1656 1657 1658 1659 1664 1665 1667 1673 1674 1674 1675 1675 1675 1676 1677 1677 1677 1678 1683

18,000 12,000 22,000 23,000 20.000 20,000 18.000 15,000 16,000 18.000 10,000 17,000 20,000 4,000 14,000 12,000 11,000 11,000 28,000 12,000 15,000 22,000 2,500 15,000 9,000 30,000 14,000 130,000 50,000 4,000 50,000 22,000 6,000 17,000 16.000 8,000 8,000 30,000 40,000 10,000

1,500 10,000 1,000 10,000 1.000 18,000 4,000 25,000 1,500 18,000 8,000 16,000 400 12,000 1,000 18,000 2,000 16,000 6.000 16,000 700 11,000 1,000 12,000 2,000 10.000 2,000 16,000 4,000 18,000 1,000 11,000 1,000 23,000 1,000 9,000 1,000 16,000 4,000 6.000 1,000 25,000 1,500 25,000 200 10,000 2.000 12,000 2.000 6,000 2,000 60,000 1,000 18,000 108,000 20,000 1,000 80,000 300 5.000 6,000 70,000 3,000 14,000 500 6,400 1,100 14,000 4,000 12,000 1,000 7,000 1.300 12,000 4,400 30,000 4.000 40,000 5,000 200,000

2,000 4.000 3,000 8,000 6,000 4.000 1,000 3,000 4,000 4,000 1,500 4,000 1,800 8,000 5,000 3,000 3,000 2,000 3,000 2,000 2,000 2,000 3,000 4,000 2,000 8,666 4,000 16,000 20,000 300 8,600 4,000 2,000 2,500 4,000 2,000 3,000 7,000 4,000 48,000

0.0833 0.0833 0.0455 0.1739 0.0750 0.4000 0.0222 0.0667 0.1250 0.3333 0.0700 0.0588 0.1000 0.5000 0.2857 0.0833 0.0909 0.0909 0.0357 0.3333 0.0667 0.0682 0.0800 0.1333 0.2222 0.0667 0.0714 0.8308 0.0200 0.0750 0.1200 0.1364 0.0833 0.0647 0.2500 0.1250 0.1625 0.1467 0.1000 0.5000

Bc/Bo 0.2000 0.4000 0.1667 0.3200 0.3333 0.2500 0.0833 0.1667 0.2500 0.2500 0.1364 0.3333 0.1800 0.5000 0.2778 0.2727 0.1304 0.2222 0.1875 0.3333 0.0800 0.0800 0.3000 0.3333 0.3333 0.1444 0.2222 0.8000 0.2500 0.0600 0.1229 0.2857 0.3125 0.1786 0.3333 0.2857 0.2500 0.2333 0.1000 0.2400

Log RATIO Bc/Rc

Log RATIO Ro/Bo

0.29 1.39 1.10 0.69 1.39 -0.69 0.92 1.10 0.69 -0.41 0.76 1.39 -0.11 1.39 0.22 1.10 1.,10 0.69 1.10 -0.69 0.69 0.29 2.71 0.69 0.00 1.47 1.39 -1.91 3.00 0 00 0.36 0.29 1.39 0.82 0.00 0.69 0.84 0.46 0.00 2.26

0.59 0.18 0.20 -0.08 0.11 0.22 0.41 -. 0.18 0.00 0.12 -0.10 0.35 0.69 -1.39 -0.25 0.09 -0.74 0.20 0.56 0.69 -0.51 -0.13 -1.39 0.22 0.4i -0.69 -0.25 1 87 -0.47 -0.22 -0.34 0.45 -0.06 0.19 0.29 0.13 -0.41 0.00 0.00 -3.00

APPENDIX C - HISTORICAL DATA SET

YEAR

BLUE BLUE RED RED INITIAL CASUALTY INITIAL CASUALTY Rc/Ro Bc Bo Rc Ro

Bc/Bo

Log RATIO Bc/Rc

Log RATIO Ro/Bo

2.20 1.10 3.40 3.00 1.39 1.39 3.22 1.68 3.22 3.00 2.01 1.10 -0.75 0.34 0.69 0.49 0.76 0.92 -0.56 -0.06 0.41 1 10 0.99 -0.81 0.00 2.66 -0.36 1.39 0.22 0.85 1.25 2.89 -0.57 0.26 -0.38 1.10 -0.15 0.15 -0.41 0.69

0.56 -0.34 0.58 0.36 -0.47 -0.18 -1.61 1.85 -0.36 1.39 -0.58 0.42 2.30 -0.41 0.2,7 -0.22 -1.10 -0 69 1.7! -0.10 0.47 0.11 0.26 1.82 0 !8 -0.69 0.92 -1.49 -0.29 -0.92 1.10 -0.55 2.35 -0.18 0.19 0.36 1.32 0.88 0.22 0.08

.......................................................................

1683 1683 1684 1685 1686 1687 1688 1688 1689 1689 1689 1690 1690 1690 1690 1691 1691 1691 1692 1692 1693 1693 1694 1694 1696 1697 1697 1700 170] 1701 1701 1701 1702 1702 1702 1702 1702 1702 1702 1702

28,000 76,000 32,000 60,000 50.000 50,000 3,000 53.000 17,000 20,000 18,000 35,000 50,000 12.000 50,000 20,000 4,000 50,000 46,000 57.000 80.000 40.000 26,000 80.000 60.000 50,000 30,000 9,000 6,000 8.000 15,000 22,000 46,000 25,000 17.000 4.000 15,000 12,000 10,000 25.000

1,000 5,000 100 100 500 2.000 200 1,300 400 100 400 500 9,500 2,000 3,000 2,700 700 8,000 7.000 7,000 8,000 3,000 1.300 18.000 4.000 2,100 10,000 2,000 800 1,200 100 200 3,000 2,700 2,700 200 1.400 1,200 1,200 200

16.000 107,000 18,000 42.000 80,000 60.000 15.000 8.300 40,000 5,000 32,000 23,000 5,000 18,000 38.000 25,000 1-2,000 100.000 8,300 63.000 50,000 36.000 20,000 13.000 50,000 100,000 12,000 40.000 8,000 20.000 5,000 38.000 4.400 30,000 14,000 2,800 4,000 5,000 8,000 23,000

9,000 15,000 3,000 2.000 2,000 8.000 5,000 7,000 10,000 2,000 3,000 1,500 4,500 2.800 6.000 4,400 1.500 20,000 4,000 6,600 12,000 9,000 3.500 8,0 4,0 , 30.000 7,000 8.000 1,000 2.800 350 3,600 1,700 3,500 1,850 600 1,200 1,400 800 400

0.0357 0.0658 0.0031 0.0017 0.0100 0.0400 0.0667 0.0245 0.0235 0.0050 0.0222 0.0143 0.1900 0.1667 0.0600 0.1350 0.1750 0.1600 0.1522 0.1228 0,1000 0.0750 0.0500 0.2250 0 0667 0.0420 0.3333 0.2222 0.1333 0.1500 0.0067 0.0091 0.0652 0.1080 0.1588 0.0500 0.0933 0.1000 0.1200 0.0080

0.5625 0.1402 0.1667 0.0476 0.0250 0.1333 0.3333 0.8434 0.2500 0.4000 0.0938 0.0652 0.9000 0.1556 0.1579 0.1760 0.1250 0.2000 0.4819 0.1048 0.2400 0.2500 0.1750 0.6154 0.0800 0.3000 0.5833 0.2000 0.1250 0.1400 0.0700 0.0947 0.3864 0.1167 0.1321 0.2143 0.3000 0.2800 0.1000 0.0174

7- MsTgE:CAL DATA SET

RED

RED

BLUE

BLUE

INITIAL CASUALTY INITIAL CASUALTY Rc

Bo

Bc

Rc/Ro

Bc/Bo

Log

Log

RATIO

RATIO

Bc/Rc

Ro/Bo

YEAR

Ro

1702 1703 1703 1703 1703 1703 1703 1703 1703 1703 1703

12,000 19,000 26,000 18,000 24,000 12,000 2,000 4,200 1.300 7,000 14.000

2,300 5,000 4,000 900 500 100 500 200 1,300 4,000

22.000 15,000 5,600 22,000 3,500 10,000 12.000 5.000 8.700 2,500 6,000

2,000 2,500 1,800 4,000 300 1,200 1.300 800 2,400 700 1,000

0.0917 0.1211 0.1923 0.2222 0.0375 0.0417 0.0500 0.1190 0.1538 0.1857 0.2857

0.0909 0.1667 0.3214 0.1818 0.0857 0.1200 0.1083 0.1600 0.2759 0.2800 0.1667

0.60 0.08 -1.02 0.00 -1.10 0.88 2.56 0.47 2.48 -0.62 -1.39

-0.61 0.24 1.54 -0.20 1.93 0.18 -1.79 -0.17 -1.90 1.03 0.85

1703 1703 1704 1704 1704 1704 1704 1704 1704 1704 1704 1704 !705 1705 1705 1705 1705 1706 1706 1706 ,706 1706 1706 1706 1706 1706 1706 1707 1707

24,000 23,000 25.000 7.000 3,000 10,000 3,600 6.500 25,000 10,000 50.000 26,000 14,000 2.000 7,000 13.000 22,000 36,000 30,000 30,000 10,500 20,000 5,000 15,000 12.000 60,000 41,000 21,000 3,500

200 1.500 6,000 500 100 400 200 600 3,000 1,500 12,500 1,100 200 50 1,000 600 3,000 1,000 3,000 4,300 3.000 500 200 1.000 2.000 5.000 500 2.000 750

35,000 18,000 11,000 20.000 12,000 2.400 7,400 9.000 4,500 4,000 52.000 7,000 15,000 4.000 20,000 24,000 24,000 12,000 5.500 42,000 36,000 5.000 2,000 10.000 18.000 62,000 19.000 16.000 3,000

3.000 4,500 2.000 2,000 4,000 1,600 800 2,000 2,500 600 14,000 900 2,000 3,500 5,000 6,000 4,000 1,400 1,000 3,800 14,000 300 600 1,500 5,000 2,000 3,000 5,000 750

0.0083 0.0652 0.2400 0.0714 0.0333 0.0400 0.0556 0.0923 0.1200 0.1500 0.2500 0.0423 0.0143 0.0250 0.1429 0.0462 0.1364 0.0278 0.1000 0.1433 0.2857 0.0250 0.0400 0.0667 0.1667 0.0833 0.0122 0.0952 0.2143

0.0857 0.2500 0.1818 0.1000 0.3333 0.6667 0.1081 0.2222 0.5556 0.1500 0.2692 0.1286 0.1333 0.8750 0.2500 0.2500 0.1667 0.1167 0.1818 0.0905 0.3889 0.0600 0.3000 0.1500 0.2778 0.0323 0.1579 0.3125 0.2500

2.71 1.10 -1.10 1.39 3.69 1,39 1.39 1.20 -0.18 -0.92 0.11 -0.20 2.30 4.25 1.61 2.30 0.29 0.34 -1.10 -0.12 1.54 -0.51 1.10 0.41 0.92 -0.92 1.79 0.92 0.00

-0.38 0.25 0.82 -1.05 -1.39 1.43 -0.72 -0.33 1.71 0.92 -0.04 1.31 -0.07 -0.69 -1.05 -0.61 -0.09 1.10 1.70 -0.34 -1.23 1.39 0.92 0.41 -0.41 -0.03 0.77 0.27 0.15

1.100

AffNIIX'

-

HITOP:CAL

:A:A SET

YEAR

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc Rc/Ro

1708 1708 1708 1708 1708 1708 1708 1708 1708 1708 1709 1709 !709 1709 1709 1710 1710 ,710 1710 1710 1710 1710 1710 1710 1711 1711 1711 1712 1712 1712 1712 1712 1712 1712 1713 1713 1714 1715 1715 1715

7,000 35,000 7,000 10,000 3,000 12,000 7,000 17,000 90,000 11,000 40,000 6,000 93,000 16.000 54.000 60,000 21.000 28.000 9,000 12,000 24,000 22,000 31,000 14,000 260,000 30,000 2,500 22.000 25,000 14.000 40,000 30,000 18,000 28,000 40,000 80,000 40,000 18,000 22,000 10,000

1,600 14,000 300 500 200 2,000 1,200 4.000 6,000 1,000 5,400 400 25,000 400 5,000 8,000 4,000 7,000 900 1,500 400 1,600 3.000 3,000 7,000 3.000 800 400 500 1.700 2,200 2,100 3,000 1.000 10,000 10.000 20,000 500 1,200 1,000

33,000 16,000 15,000 16,000 1,200 6,000 3.500 13.000 80,000 22,000 7,000 7,000 90,000 20,000 26.000 8,000 13,600 7,000 3,000 4,000 22,000 20.000 4,000 11,000 40.000 5,000 8.000 7,000 3,200 12,000 18,000 18,000 5,500 2.200 7,000 9,300 16,000 5,000 7,000 3,000

900 7.000 2,000 6,000 200 3.000 3.000 5.000 6,000 2,500 3,200 2,600 11.000 17,000 9,200 3,000 3.000 3,400 1,000 600 1,500 5,000 1,800 4,000 2,100 1,800 2,500 200 1,000 4,000 3,000 2.300 2.000 700 2,000 3,600 6,000 2,500 2,000 700

0.2286 0.4000 0.0429 0.0500 0.0667 0.1667 0.1714 0.2353 0.0667 0.0909 0.1350 0.0667 0.2688 0.0250 0.0926 0.1333 0.1905 0.2500 0.1000 0.1250 0.0167 0.0727 0.0968 0.2143 0.0269 0.1000 0.3200 0.0182 0.0200 0.1214 0.0550 0.0700 0.1667 0.0357 0.2500 0.1250 0.5000 0.0278 0.0545 0.1000

Bc/Bo 0.0273 0.4375 0.1333 0.3750 0.1667 0.5000 0.8571 0.3846 0.0750 0.1136 0.4571 0.3714 0.1222 0.8500 0.3538 0.3750 0.2206 0.4857 0.3333 0.1500 0.0682 0.2500 0.4500 0.3636 0.0525 0.3600 0.3125 0.0286 0.3125 0.3333 0.1667 0.1278 0.3636 0.3182 0.2857 0.3871 0.3750 0.5000 0.2857 0.2333

Log RATIO Bc/Rc

Log RATIO Ro/Bo

-0.58 -0.69 1.90 2.48 0.00 0.41 0.92 0.22 0.00 0.92 -0.52 1.87 -0.82 3.75 0.61 -0.98 -0.29 -0.72 0.11 -0.92 1.32 1.14 -0.51 0.29 -1.20 -0.51 1.14 -0.69 0.69 0.86 0.31 0.09 -0.41 -0.36 -1.61 -1.02 -1.20 1.61 0.51 -0.36

-1.55 0.78 -0.76 -0.47 0.92 0.69 0.69 0.27 0.12 -0.69 1.74 -0.15 0.03 -0.22 0.73 2.01 0.43 1.39 llO 1.10 0.09 0.10 2.05 0.24 1.87 1.79 -1.16 1.15 2.06 0.15 0.80 0.51 1.19 2.54 1.74 2.15 0.92 1.28 1.15 1.20

=

-'

}

TC-i

A'L

:'ATA

SET

YEAR

BLUE RED BLUE RED INITIAL CASUALTY INITIAL CASUALTY Bc Rc/Ro Re Bo Ro

1715 1716 1716 1717 1717 1717 1718 1719 1719 1724 1734 1734 1734 1734 1734 1735 1737 1737 1738 1738 1739 1739 1741 1741 1741 1742 1742 1742 1743 1743 1743 1743 1744 1744 1744 1744 1744 1744 1745 1745

70,000 45.000 63.000 23.000 100,000 50,000 9,300 29.000 18,000 16,600 10,000 53,000 20,000 117,000 40.000 6,000 80,000 6,000 10,000 40,000 80,000 60.000 21,600 10,000 5,000 4,000 18.000 28,000 35,000 5,400 13,000 32,000 80,000 24,000 40,000 24,000 70,000 26,000 70,000 2,500

10,000 12,000 4,500 18,000 4.500 60,000 1,000 2,000 20,000 30,000 5,400 150,000 6.000 1.500 2,000 21,000 5,200 4,000 6,200 800 1,500 6,000 4,000 37,000 800 .40,000 4,500 10,000 5,900 27,000 3.000 2,500 4,000 22,000 2,000 20,000 1,000 20,000 1,300 17,000 8,000 40.000 100 100,000 3,900 15,800 5,000 3.000 1,200 50 7,000 100 8,500 550 4.200 28,000 2,050 26,000 100 6.000 3,200 11,000 9,000 100 150 14,900 6.000 3,000 1,500 10,000 1,400 16,000 16.000 7,000 4,000 25,000 4,800 75,000 100 2,300

57

6,000 3.000 6,000 600 5,000 15,000 1.500 800 900 1,000 1,000 6,000 1,100 1,000 5,800 1,400 17,000 10,000 5,000 2,000 5,200 2,000 3,000 2,500 60 150 250 3,000 2,800 600 1,600 1,000 1,400 1,100 650 700 2,200 3,600 9,600 400

0.1429 0.1000 0.0714 0.0870 0.2000 0.1080 0.1613 0.0690 0.2889 0.0482 0.1500 0.0755 0.0400 0.0855 0.1475 0.4167 0.0500 0.3333 0.1000 0.0325 0.1000 0.0017 0.1806 0.3000 0.0100 0.0250 0.0306 0.1500 0.0586 0.0185 0.2462 0.0031 0.0019 0.1250 0.0375 0.0583 0.2286 0.1538 0.0686 0.0400

Bc/Bo 0.5000 0.1667 0.1000 0.6000 0.1667 0.1000 0.2500 0.0381 0.2250 0.1613 0.1667 0.1622 0.0275 0.2222 0.2148 0.4667 0.7727 0.5000 0.2500 0.1176 0.1300 0.0200 0.1899 0.5000 0.0500 0.0214 0.0294 0.1071 0.1077 0.1000 0.1455 0.1111 0.0940 0.1833 0.0650 0.0438 0.3143 0.1440 0.1280 0.1739

Log RATIO Bc/Rc

Log RATIO Ro/Bo

-0.51 -0.41 0.29 -1.20 -1.39 1.02 0.00 -0.92 -1.75 0.22 -0.41 0.41 0.32 -2.30 -0.02 -0.58 1.45 1.61 1.61 0.43 -0.43 3.00 -0.26 -0.18 0.18 0.41 -0.79 -0.34 0.31 1.79 -0.69 2.30 2.23 -1.00 -0.84 -0.69 -1.98 -0.11 0.69 1.39

1.76 0.92 0.05 3.14 1.20 -1.10 0.44 0.32 1.50 0.98 0.51 0.36 -0.69 3.26 0.39 0.69 1.29 -1.20 -0.69 0.86 0.69 -0.51 0.31 0.69 1.43 -0.56 0.75 0.00 0.30 -0.11 0.17 1.27 1.68 1.39 1.39 0.41 2.30 0.04 -0.07 0.08

-

H:sT3ccAL DATA SET

YEAR

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc Rc/Ro

1745 1745 1745 1745 1745 1745 1745 1746 1746 1746 1746 1746 1746 1746 1747 1747 1747 1748 1749 1753 1755 1756 1756 1757 1757 1757 1757 1757 1757 1757 1757 1757 1757 1757 1758 1758 1758 1758 1758 1758

60.000 17.000 50.000 50,000 35,000 22,000 8.500 10.000 9,000 40,000 60.000 25,000 30.000 110,000 98.000 35,000 7,000 80.000 20,000 11.000 1.000 12,000 30,000 32.000 55.000 15,000 80,000 5.000 50,000 60,000 64,000 54,000 22,000 35,000 10,000 3,000 3,500 11.000 2.600 3.600

5,600 800 1,000 2,000 5,100 3,700 120 300 400 3.000 1,000 800 2,500 4,000 10,000 6,500 300 2,000 700 600 100 2,000 2,600 1,500 6.000 700 5,300 1,000 1,100 2,350 12.600 6.400 600 6,200 300 200 100 600 300 400

50,000 7.000 30,000 4.000 35,000 38,000 5,500 6,000 4,000 44,000 12,000 41.000 25,000 75.000 82,000 26,000 14.000 13.000 9,000 7,000 2.300 2,800 33,000 14,000 25,000 14.000 30.000 3.000 63,000 36,000 61,000 33,000 41.000 65,000 8.000 6,000 2,800 6.700 6,500 15,000

9,000 2,400 1,000 600 3,800 4,500 500 1,000 1,600 7,000 500 300 4,500 7.000 9,000 5.500 5,700 2.000 400 900 1.500 400 2,200 1,500 4,100 400 9.666 1.500 3.000 1,250 9,200 8.600 3.400 10.000 800 400 750 1,100 400 2,000

0.0933 0.0471 0.0200 0.0400 0.1457 0.1682 0.0141 0.0300 0.0444 0.0750 0.0167 0.0320 0.0833 0.0364 0.1020 0.1857 0.0429 0.0250 0.0350 0.0545 0.1000 0.1667 0.0867 0.0469 0.1091 0.0467 0.0663 0.2000 0.0220 0.0392 0.1969 0.1185 0.0273 0.1771 0.0300 0.0667 0.0286 0.0545 0.1154 0.1111

Bc/Bo 0.1800 0.3429 0.0333 0.1500 0.1086 0.1184 0.0909 0.1667 0.4000 0.1591 0.0417 0.0073 0.1800 0.0933 0.1098 0.2115 0.4071 0.1538 0.0444 0.1286 0.6522 0.1429 0.0667 0.1071 0.1640 0.0286 0.3222 0.5000 0.0476 0.0347 0.1508 0.2606 0.0829 0.1538 0.1000 0.0667 0.2679 0.1642 0.0615 0.1333

Log RATIO Bc/Rc

Log RATIO Ro/Bo

0.47 1.10 0.00 -1.20 -0.29 0.20 1.43 1.20 1.39 0.85 -0.69 -0.98 0.59 0.56 -0.11 -0.17 2.94 0.00 -0.56 0.41 2.71 -1.61 -0.17 0.00 -0.38 -0.56 0.60 0.41 1.00 -0.63 -0.31 0.30 1.73 0.48 0.98 0.69 2.01 0.61 0.29 1.61

0.18 0.89 0.51 2.53 0.00 -0.55 0.44 0.51 0.81 -0.10 1.61 -0.49 0.18 0.38 0.18 0.30 -0.69 1.82 0.80 0.45 -0.83 1.46 -0.10 0.83 0.79 0.07 0.98 0.51 -0.23 0.51 0.05 0.49 -0.62 -0.62 0.22 -0.69 0.22 0.50 -0.92 -1.43

AJ'N*

C -

HTOrICAL DATA SET

RED RED BLUE BLUE Log Log INITIAL CASUALTY INITIAL CASUALTY RATIO RATIO YEAR Ro Rc Bo Bc Rc/Ro Bc/Bo Bc/Rc Ro/Bo ------------------------------------------------------------------------------------1758 7.500 2.200 45,000 800 0.2933 0.0178 -1.01 -1.79 1758 30,000 600 18,000 700 0.0200 0.0389 0.15 0.51 1758 33,000 11,000 52,000 18,000 0.3333 0.3462 0.49 -0.45 1758 65.000 5.400 42,000 7,900 0.0831 0.1881 0.38 0.44 1758 31,000 1,800 47,000 5,200 0.0581 0.1106 1.06 -0.42 1758 4,000 1,000 7,300 600 0.2500 0.0822 -0.51 -0.60 1759 38,000 2,700 52,000 4,900 0.0711 0.0942 0.60 -0.31 1759 12,000 50 9,000 100 0.0042 0.0111 0.69 0.29 1759 12,000 200 4,000 300 0.0167 0.0750 0.41 1.10 1759 5.000 200 10,000 350 0.0400 0.0350 0.56 -0.69 1759 70,000 16,000 48,000 18,700 0.2286 0.3896 0.16 0.38 1759 8,000 650 4,500 1,500 0.0813 0.3333 0.84 0.58 1759 72,000 4,800 28.000 7.000 0.0667 0.2500 0.38 0.94 1759 38,000 1,000 13,500 1.300 0.0263 0.0963 0.26 1.03 1759 26.000 900 13.500 500 0.0346 0.0370 -0.59 0.66 1759 36.000 1,800 27.000 2,500 0.0500 0.0926 0.33 0.29 1759 30.000 200 4,000 1,800 0.0067 0.4500 2.20 2.01 1759 20,000 2.700 16,000 1.300 0.1350 0.0813 -0.73 0.22 1760 3,000 300 5,000 300 0.1000 0.0600 0.00 -0.51 1760 13,000 200 8,000 1,200 0.0154 0.1500 1.79 0.49 1760 44.000 10,000 66,000 9.000 0.2273 0.1364 -0.11 -0.41 1760 7,000 800 6,000 1,200 0.1143 0.2000 0.41 0.15 1760 30.000 800 16,000 650 0.0267 0.0406 -0.21 0.63 1760 19.000 1,300 17,000 1,500 0.0684 0.0882 0.14 0.11 1760 22.000 2,000 12.000 1,000 0.0909 0.0833 -0.69 0.61 1760 38.700 3,000 10,800 6.000 0.0775 0.5556 0.69 1.28 1760 30.000 3.300 30,000 3,800 0.1100 0.1267 0.14 0.00 1760 5,400 200 2.800 500 0.0370 0.1786 0.92 0.66 1761 10.000 300 4,000 600 0.0300 0.1500 0.69 0.92 1761 12.500 400 5,500 600 0.0320 0.1091 0.41 0.82 1761 12,000 400 15.000 2,500 0.0333 0.1667 1.83 -0.22 1761 14,000 1,000 7,000 800 0.0714 0.1143 -0.22 0.69 1761 14,000 1.700 4,000 400 0.1214 0.1000 -1.45 1.25 1761 52.000 1,300 100,000 3,000 0.0250 0.0300 0.84 -0.65 1762 12.000 1,100 10.000 700 0.0917 0.0700 -0.45 0.18 1762 24,000 700 16.000 500 0.0292 0.0313 -0.34 0.41 1762 7,000 300 5,000 1,400 0.0429 ^ 2800 1.54 0.34 1762 17,000 750 6,000 700 0.0441 u.1167 -0.07 1.04 1762 50,000 1,000 30,000 1,600 0.0200 0.0533 0.47 0.51 1762 8,000 1,500 10.000 600 0.1875 0.0600 -0.92 -0.22

59

A ?EN: :.

YEAR 1762 1762 1762 1762 1762 1762 1762 1769 1770 1770 1770 1771 1771 1771 1773 1775 1776 1776 1776 1777 1777 1777 1779 1779 1780 1780 1781 1782 1782 1787 1788 1788 1789 1789 1790 1790 1791 1791 1791 1792

7

-

HSTORI:CAL r)ATA SET

BLUE BLUE RED RED INITIAL CASUALTY INITIAL CASUALTY Rc/Ro Bc Bo Rc Ro 17.000 27,000 16.000 23,000 57.000 17,000 16,000 30,000 10.000 40.000 12.000 10,000 15,000 32,000 6.000 2,000 6,000 25,000 2,400 15,000 5,000 18,000 3,200 11,000 2,000 10,000 2,400 20,000 7,000 12,000 8.000 90,000 23.000 27,000 7.000 32.000 36,000 18.000 18.000 45.000

3.000 1,600 1.100 1,600 300 100 450 600 400 1.000 2,500 1,000 500 2,000 800 1,150 430 320 10 500 600 600 20 300 300 300 600 1,100 1,300 1.000 1,600 4,800 300 600 100 10,000 1,000 4,000 500 2,000

12.500 31,000 32,000 20,000 25.000 3,000 13,000 80,000 16.000 15,000 12.000 10.000 40,000 57,000 10.000 1.200 2,000 11,000 1.500 10,000 6,000 8,000 10,000 3,000 4,000 7,000 5,000 10,000 31,000 5,000 8,000 14.000 30,000 50.000 7.000 40,000 64.000 15.000 15.000 13.200

50

3.500 3,000 1,200 1.750 1,500 200 800 3,000 3,000 20,000 6.000 4,000 2,000 4,000 2,000 450 200 2,000 1,300 1,200 600 900 800 400 900 1,000 700 700 6,000 3,000 550 10,000 1,000 5,000 2,000 31,000 3,000 8,000 1.500 1,000

0.1765 0.0593 0.0688 0.0696 0.0053 0.0059 0.0281 0.0200 0.0400 0.0250 0.2083 0.1000 0.0333 0.0625 0.1333 0.5750 0.0717 0.0128 0.0042 0.0333 0.1200 0.0333 0.0063 0.0273 0.1500 0.0300 0.2500 0.0550 0.1857 0.0833 0.2000 0.0533 0.0130 0.0222 0.0143 0.3125 0.0278 0.2222 0.0278 0.0444

Bc/Bo 0.2800 0.0968 0.0375 0.0875 0.0600 0.0667 0.0615 0.0375 0.1875 1.3333 0.5000 0.4000 0.0500 0.0702 0.2000 0.3750 0.1000 0.1818 0.8667 0.1200 0.1000 0.1125 0.0800 0.1333 0.2250 0.1429 0.1400 0.0700 0.1935 0.6000 0.0688 0.7143 0.0333 0.1000 0.2857 0.7750 0.0469 0.5333 0.1000 0.0758

Log RATIO Bc/Rc

Log RATIO Ro/Bo

0.15 0.63 0.09 0.09 1.61 0.69 0.58 1.61 2.01 3.00 0.88 1.39 1.39 0.69 0.92 -0.94 -0.77 1.83 4.87 0.88 0.00 0.41 3.69 0.29 1.10 1.20 0.15 -0.45 1.53 1.10 -1.07 0.73 1.20 2.12 3.00 1.13 1.10 0.69 1.10 -0.69

0.31 -0.14 -0.69 0.14 0.82 1.73 0.21 -0.98 -0.47 0.98 0.00 0.00 -0.98 -0.58 -0.51 0.51 1.10 0.82 0.47 0.41 -0.18 0.81 -1.14 1.30 -0.69 0.36 -0.73 0.69 -1.49 0.88 0.00 1.86 -0.27 -0.62 0.00 -0.22 -0.58 0.18 0.18 1.23

A??EX'::t

YEAR 1792 1792 1792 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 ,793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793 1793

C

DATA )TORICAL SET

-

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Bo Rc Bo Bc Bc/Ro 52,000 13,000 6.000 10,000 30,000 32.000 31,000 43,000 20.000 20.000 45,000 24.000 53,000 25,000 12,000 17,000 38,000 5,000 12,000 10,500 10,000 8,000 8,000 8,000 8,000 20,000 20,000 25,000 11,000 35,000 8,000 2,500 10,000 15,000 7,000 8,000 12,000 8,000 12,000 60,000

300 200 1,000 500 1,500 2,000 2,100 3,000 2,000 2,000 5,000 3,000 1,000 4,000 2,000 2,000 900 200 800 900 100 100 200 200 200 500 500 1,000 300 500 300 100 500 300 400 500 800 600 1.000 600

35,000 1,500 10,000 35,000 13,000 18,000 25,000 22,000 20.000 25.000 30,000 16,000 27,000 40,000 38,000 22,000 22.000 25,000 15.000 6,000 1,200 10,000 34,000 6,000 4,000 8.000 7,000 15,000 6,000 30,000 5,000 8,000 6,000 16,000 18,000 6,000 7,000 16,000 14,000 30,000

200 180 1.000 5,000 3,100 4,000 4.000 4,000 5,000 8,000 2,500 1,600 3,000 8,000 00 4,000 2,000 1,500 200 600 300 2,500 1.200 1,200 1,200 4,000 6.000 1,500 3,000 1,500 4,000 2,000 800 1,200 3,000 1,000 800 2,000 2,000 1.500

0.0058 0.0154 0.1667 0.0500 0.0500 0.0625 0.0677 0.0698 0.1000 0.1000 0.1111 0.1250 0.0189 0.1600 0.1667 0.1176 0.0237 0.0400 0.0667 0.0857 0.0100 0.0125 0.0250 0.0250 0.0250 0.0250 0.0250 0.0400 0.0273 0.0143 0.0375 0.0400 0.0500 0.0200 0.0571 0.0625 0.0667 0.0750 0.0833 0.0100

Bc/Bo 0.0057 0.1200 0.1000 0.1429 0.2385 0.2222 0.1600 0.1818 0.2500 0.3200 0.0833 0,1000 0.1111 0.2000 0.1316 0.1818 0.0909 0.0600 0.0133 0.1000 0.2500 0.2500 0.0353 0.2000 0.3000 0.5000 0.8571 0.1000 0.5000 0.0500 0.8000 0.2500 0.1333 0.0750 0.1667 0.1667 0.1143 0.1250 0.1429 0.0500

Log RATIO Bc/Rc

Log RATIO Ro/Bo

-0.41 -0.11 0.00 2.30 0.73 0.69 0.64 0.29 0.92 1.39 -0.69 -0.63 1.10 0.69 0.92 0.69 0.80 2.01 -1.39 -0.41 1.10 3.22 1.79 1.79 1.79 2.08 2.48 0.41 2.30 1.10 2.59 3.00 0.47 1.39 2.01 0.69 0.00 1.20 0.69 0.92

0.40 2.16 -0.51 -1.25 0.84 0.58 0.22 0.67 0.00 -0.22 0.41 0.41 0.67 -0.47 -1.15 -0.26 0.55 -1.61 -0.22 0.56 2.12 -0.22 -1.45 0.29 0.69 0.92 1.05 0.51 061 0.15 0.47 -1.16 0.51 -0.06 -0.94 0.29 0.54 -0.69 -0.15 0.69

A ?EX:

YEAR

-

H:STOR:CAL DATA SET

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Bo Bc Rc/Ro Ro Re

Bc/Bo

Log RATIO Bc/Rc

Log RATIO Ho/Bo

1793 1793 1793 1793 1793 1793 1793 1793 1794 1794 1794 1794 1794 1794

9,000 26.000 20.000 36.000 43.000 43.000 35,000 24.000 28,000 26.000 28,000 41,000 22,5(0 35,000

1,100 800 1,200 50 2.600 1,800 1.000 1,300 1,500 1,500 400 3,000 2,800 3,000

11,000 32,000 15,000 9.000 41,000 50,000 14.000 8,000 13.000 26.000 60,000 73.000 44,000 50,000

2,000 24.000 700 2.000 3.000 2.000 4,000 1.000 2,000 3,000 3,000 2.000 4.000 10,000

0.1222 0.0308 0.0600 0.0014 0.0605 0.0419 0.0286 0.0542 0.0536 0.0577 0.0143 0.0732 0.1244 0.0857

0.1818 0.7500 0.0467 0.2222 0.0732 0.0400 0.2857 0.1250 0.1538 0 1154 0.0500 0.0274 0.0909 0.2000

0.60 3.40 -0.54 3.69 0.14 0.11 1.39 -0.26 0.29 0.69 2.01 -0.41 0.36 1.20

-0.20 -0.21 0.29 1.39 0.05 -0.15 0.92 1.10 0.77 0.00 -0.76 -0.58 -0.67 -0.36

1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794 1794

22,000 88,000 9,000 20,000 7,000 18,000 81.000 10,000 20,000 20.000 14.000 12.000 12.000 2,000 30,000 60.000 20.000 12.000 10,000 28.000 35.000 9,000 70,000 90,030 8.0uO 6.000

4,000 1,500 50 500 300 1,000 5,000 800 1,000 1,000 500 500 600 100 400 1.000 1.500 1.000 1.000 1.000 1.000 1.000 3.000 1.500 1,000 1,000

28.000 77,000 6,000 30.000 15.000 11.000 46,000 20,000 20,000 20,000 2.000 10.000 12.000 7,000 16,000 60.000 8,000 10.000 7.000 42,000 18,000 6,000 74,000 90.000 12.000 5,000

8,000 3,000 600 900 1,200 800 5,000 1.400 600 2,000 500 3,000 1,300 1,300 1.000 2,500 3,000 6,000 2,000 2,000 1.500 1.500 4.000 7.000 4,000 2,000

0.1818 0.0170 0.0056 0.0250 0.0429 0.0556 0.0617 0.0800 0.0500 0.0500 0.0357 0.0417 0.0500 0.0500 0.0133 0.0167 0.0750 0.0833 0.1000 0.0357 0.0286 0.1111 0.0429 0.0167 0.1250 0.1667

0.2857 0.0390 0.1000 0.0300 0.0800 0.0727 0.1087 0.0700 0.0300 0.1000 0.2500 0.3000 0.1083 0.1857 0.0625 0.0417 0.3750 0.6000 0.2857 0.0476 0.0833 0.2500 0.0541 0.0778 0.3333 0.4000

0.69 0.69 2.48 0.59 1.39 -0.22 0.00 0.56 -0.51 0.69 0.00 1.79 0.77 2.56 0.92 0.92 0.69 1.79 0.69 0.69 0.41 0.41 0.29 1.54 1.39 0.69

-0.24 0.13 0.41 -0.41 -0.76 0.49 0.57 -0.69 0.00 0.00 1.95 0.18 0.00 -1.25 0.63 0.00 0.92 0.18 0.36 -0.41 0.66 0.41 -0.06 0.00 -0.41 0.18

{DATA

YEAR 1794 1794 1794 1794 1794 1794 1794 1795 1795 1795 1795 1795 1795 1795 1795 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796

BLUE BLUE RED RED INITIAL CASUALTY INITIAL CASUALTY Rc/Ro Bc Rc Bo Ro 47,000 60,000 50.000 24,000 25,000 50.000 30,000 35.000 42,000 25.000 43,000 6,000 13,000 27.000 36.000 50,000 28,000 25,000 25,000 15,000 20,000 17,500 4,600 15.000 35,000 36,000 24.000 28.000 26,000 27.000 14.000 9.000 44,000 15,000 35,000 30,000 50,000 14,000 19,500 24,000

500 1,000 600 1.300 1.000 3,000 1,500 600 300 2,500 600 200 500 650 i,400 1,200 5,600 600 1,000 300 750 900 400 300 100 400 600 1,000 650 500 800 700 1,200 1,500 500 600 2,000 2,200 2,100 500

7,000 13,000 5.000 19,000 10,000 45.000 10.000 25.000 60,000 18,000 37,000 12,000 17,000 12,000 33,000 48.000 21,000 9,000 12,000 14.000 10,000 9.500 34.000 6.000 25,000 11,000 12,000 32,000 24,000 10,300 4.000 4.000 30,000 8.000 23,000 24,000 11,000 8,000 10,500 14,000

53

2,000 4.000 1,000 900 1,500 5,500 1,500 2.500 700 3,000 1,100 1,000 1.700 1.500 3,000 1,100 3,000 550 900 400 1,000 400 12.000 1,100 2,000 500 1,600 1,000 1.000 300 2.500 3,000 2,000 3,500 300 2,300 400 800 4,400 1,000

0.0106 0.0167 0.0120 0.0542 0.0400 0.0600 0.0500 0.0171 0.0071 0.1000 0.0140 0.0333 0.0385 0.0241 0.0389 0.0240 0.2000 0.0240 0.0400 0.0200 0.0375 0.0514 0.0870 0.0200 0.0029 0.0111 0.0250 0.0357 0.0250 0.0185 0.0571 0.0778 0.0273 0.1000 0.0143 0.0200 0.0400 0.1571 0.1077 0.0208

Bc/Bo 0.2857 0.3077 0.2000 0.0474 0.1500 0.1222 0.1500 0.1000 0.0117 0.1667 0.0297 0.0833 0.1000 0.1250 0.0909 0.0229 0.1429 0.0611 0.0750 0.0286 0.1000 0.0421 0.3529 0.1833 0.0800 0-0455 0.1333 0.0313 0.0417 0.0291 0.6250 0.7500 0.0667 0.4375 0.0130 0.0958 0.0364 0.1000 0.4190 0.0714

Log RATIO Bc/Rc

Log RATIO Ro/Bo

1.39 1.39 0.51 -0.37 0.41 0.61 0.00 1.43 0.85 0.18 0.61 1.61 1.22 0.84 0.76 -0.09 -0.62 -0.09 -0.11 0.29 0.29 -0.81 3.40 1.30 3.00 0.22 0.98 0.00 0.43 -0.51 1.14 1.46 0.51 0.85 -0.51 1.34 -1.61 -1.01 0.74 0.69

1.90 1.53 2.30 0.23 0.92 0.11 1.10 0.34 -0.36 0.33 0.15 -0.69 -0.27 0.81 0.09 0.04 0.29 1.02 0.73 0.07 0.69 0.61 -2.00 0.92 0.34 1.19 0.69 -0.13 0.08 0.96 1.25 0.81 0.38 0.63 0.42 0.22 1.51 0.56 0.62 0.54

A}E\11x C

YEAR 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1796 1797 1797 1797 1797 1797 1797 1797 1797 1797 1798 1798 1798 1798 1798 1798 1798 1798 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799

-

HI:-TRwICAL

DATA SET

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc Rc/Ro 36,000 22.000 7,000 36,000 20,000 20,000 20,000 40.000 20,000 30.000 20,000 11,000 43,000 40,000 22,000 18,000 9,000 28,000 52,000 45,000 10,000 25,000 20,000 10,000 8,000 3.500 1.000 1,100 25,000 11,000 13,000 8,000 15,000 15,000 12,000 14,000 11,000 8,000 6,000 12,000

2.000 800 1,200 800 400 1,500 1,500 400 2,000 700 3,500 1.200 500 3,800 2.200 200 100 1,200 3,000 2,000 1,500 2.000 300 500 400 200 500 200 400 150 200 200 400 400 400 500 400 400 300 600

45,000 10,000 5,000 34,000 16,000 23,000 14,000 24,000 15,000 35.000 24,000 8,000 24,000 20,000 28,000 17,000 7,000 16,000 34,000 30,000 12.000 25,000 6,000 26,000 4.000 4,500 12,000 5,000 6,000 8,000 4,000 5,000 7,000 15,000 5,000 9,000 12,000 8,000 6,000 10,000

1,300 1,200 2,000 1,200 600 3,000 2,500 600 3,000 1,000 2,200 1.000 700 4,000 4,000 300 800 1,300 2,700 1,000 3,000 15.000 2,000 2,500 500 4,000 1.000 400 750 500 200 300 400 1,200 300 500 1,000 600 500 1,000

0.0556 0.0364 0.1714 0.0222 0.0200 0.0750 0.0750 0.0100 0.1000 0.0233 0.1750 0.1091 0.0116 0.0950 0.1000 0.0111 0.0111 0.0429 0.0577 0.0444 0.1500 0.0800 0.0150 0.0500 0.0500 0.0571 0.1667 0.1818 0.0160 0.0136 0.0154 0.0250 0.0267 0.0267 0.0333 0.0357 0.0364 0.0500 0.0500 0.0500

Bc/Bo 0.0289 0.1200 0.4000 0.0353 0.0375 0.1304 0.1786 0.0250 0.2000 0.0286 0.0917 0.1250 0.0292 0.2000 0.1429 0.0176 0.1143 0.0813 0.0794 0.0333 0.2500 0.6000 0 3333 0.0962 0.1250 0.8889 0.0833 0.0800 0.1250 0.0625 0.0500 0.0600 0.0571 0.0800 0.0600 0.0556 0.0833 0.0750 0.0833 0.1000

Log RATIO Bc/Rc

Log RATIO Ro/Bo

-0.43 0.41 0.51 0.41 0.41 0.69 0.51 0.41 0.41 0.36 -0.46 -0.18 0.34 0.05 0.60 0.41 2.08 0.08 -0.11 -0.69 0.69 2.01 1.90 1.61 0.22 3.00 0.69 0.69 0.63 1.20 0.00 0.41 0.00 1.10 -0.29 0.00 0.92 0.41 0.51 0.51

-0.22 0.79 0.34 0.06 0-22 -0.14 0.36 0.51 0.29 -0.15 -0.18 0.32 0.58 0.69 -0.24 0.06 0.25 0.56 0.42 0.41 -0.18 0.00 1.20 -0.96 0.69 -0.25 -1.39 -1.51 1.43 0.32 1.18 0.47 0.76 0.00 0.88 0.44 -0.09 0.00 0.00 0.18

Ac - EISTOBICAZ

YEAR

DATA SET

BLUE BLUE RED RED INITIAL CASUALTY INITIAL CASUALTY Rc/Ro Be Bo Re Ro

Bc/Bo

Log RATIO Bc/Rc

Log RATIO Ro/Bo

0.34 3.40 1.61 -0.13 2.48 0.74 0.46 1.61 2.30 1.70 0.29 0.05 1.61 0.00 1.79 0.69 -0.90 1.39 0.92 0.00 1.10 1.48 -0.22 0.00 0.69 -0.53 1.05 3.69 0.51 0.34 -0.69 1.20 2.48 1.01 0.39 0.29 1.10 2.39 -1.10 0.69

-0.78 1.04 0.76 0.12 0.34 -0.04 0.66 0.98 -0.37 0.41 1.70 0.62 -0.07 1.27 0.69 0.26 0.19 0.41 0.14 0.36 -0.32 0.60 0.76 0.56 0.41 0.20 0.85 -2.08 0.41 -0.06 0.75 -0.47 -1.87 0.83 0.90 0.13 -1.10 -1.10 0.85 -0.88

.......................................................................

1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799 1799

3,200 9.600 15,000 46,000 7,000 22,000 29,000 20,000 4,500 9,000 12,000 52.000 6,500 11,000 12,000 4,300 46,000 12,000 11.500 50,000 8,000 8.000 15,000 14,000 6,000 55,000 22.300 1.000 30,000 16,000 18,000 7,500 4.000 23.000 32,000 24,000 3,000 6.000 21.000 5,000

320 100 200 4,000 100 1,000 2.150 400 100 200 300 3.800 200 400 500 200 4,900 600 600 7,000 500 500 1,000 1,000 500 2,200 700 100 1,800 500 2,000 900 500 800 2.100 1,200 500 1.100 6,000 2.000

-li ...... .m,.,.

7.000 3,400 7.000 41.000 5,000 23.000 15.000 7,500 6,500 6,000 2.200 28.000 7,000 3.100 6,000 3,300 38,000 8,000 10,000 35,000 11,000 4,400 7,000 8.000 4.000 45,000 9,500 8,000 20.000 17,000 8.500 12.000 26.000 10.000 13,000 21,000 9,000 18,000 9,000 12,000

450 3,000 1,000 3,500 1,200 2,100 3.400 2,000 1,000 1,100 400 4,000 1,000 400 3,000 400 2,000 2,400 1,500 7,000 1,500 2,200 800 1,000 1,000 1,300 2.000 4,000 3,000 700 1,000 3,000 6,000 2.200 3,100 1,600 1,500 12,000 2.000 4,000

0.1000 0.0104 0.0133 0.0870 0.0143 0.0455 0.0741 0.0200 0.0222 0.0222 0.0250 0.0731 0.0308 0.0364 0.0417 0.0465 0.1065 0.0500 0.0522 0.1400 0.0625 0.0625 0.0667 0.0714 0.0833 0.0400 0.0314 0.1000 0.0600 0.0313 0.1111 0.1200 0.1250 0.0348 0.0656 0.0500 0.1667 0.1833 0.2857 0.4000

0.0643 0.8824 0.1429 0.0854 0.2400 0.0913 0.2267 0.2667 0.1538 0.1833 0.1818 0.1429 0.1429 0.1290 0.5000 0.1212 0.0526 0.3000 0.1500 0.2000 0.1364 0.5000 0.1143 0.1250 0.2500 0.0289 0.2105 0.5000 0.1500 0.0412 0.1176 0.2500 0.2308 0.2200 0.2385 0.0762 0.1667 0.6667 0.2222 0.3333

AFFEM.::X C

YEAR 1799 1799 1799 1799 1799 1799 1799 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1801 1801 1801 1805 1805

-

H:STORCAL DATA SET

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Re Bo Be Rc/Ro 16.400 22,400 33,500 50.000 37,000 17.000 19.000 66.000 17.000 20,000 25.000 5 000 55,000 20.000 52,000 10,000 24,000 14.000 4.000 6,000 6.000 7,000 8.000 9.000 7,000 40.000 12.000 28,000 8.000 5.000 84,000 37.000 11.000 6.000 6.000 15,000 5,500 12.000 11.000 65,000

1,600 1,000 4,000 1,550 5,000 200 100 4,000 700 1.000 2,000 200 2,500 1,000 3,000 600 2,500 1,000 300 500 500 650 1,000 1,000 800 1,000 3,000 6.500 200 200 3,000 2,000 800 500 500 1,300 650 1.500 800 10,000

14,500 8,800 23.000 28.000 33.000 21.000 8.000 50.000 30.000 40.000 20.000 3.000 57.000 7,000 48.000 3,500 12.000 7.000 4,500 4,000 3,000 5.000 10,000 17,000 5.000 10,000 16.000 31.000 5.000 7.000 72.000 35,000 10,000 4.000 3.500 5.000 2.000 10,000 7,000 83.000

6

1,500 2,000 6,000 4.000 9.500 2,000 300 4,100 800 500 1,250 400 5,500 2,000 2,400 1,000 4,000 2,000 500 600 1,450 1,000 950 2,000 1.000 1,000 2,100 7,000 200 250 3,000 1,200 700 400 1,100 700 350 3,000 1,000 16,000

0.0976 0.0446 0.1194 0.0310 0.1351 0.0118 0.0053 0.0606 0-0412 0.0500 0.0800 0.0400 0.0455 0.0500 0.0577 0.0600 0.1042 0.0714 0.0750 0.0833 0.0833 0 0929 0.1250 0.1111 0.1143 0.0250 0.2500 0.2321 0.0250 0.0400 0.0357 0.0541 0.0727 0.0833 0 0833 0.0867 0.1182 0.1250 0.0727 0.1538

Bc/Bo 0.1034 0.2273 0.2609 0.1429 0.2879 0.0952 0.0375 0.0820 0.0267 0.0125 0.0625 0.1333 0.0965 0.2857 0.0500 0.2857 0.3333 0.2857 0.1111 0.1500 0.4833 0.2000 0.0950 0.1176 0.2000 0.1000 0.1313 0.2258 0.0400 0.0357 0.0417 0.0343 0.0700 0.1000 0.3143 0.1400 0.1750 0.3000 0.1429 0.1928

Log RATIO Bc/Rc

Log RATIO Ro/Bo

-0.06 0.69 0.41 0.95 0.64 2.30 1.10 0.02 0.13 -0.69 -0.47 0.69 0.79 0.69 -0.22 0.51 0.47 0.69 0.51 0.18 1.06 0.43 -0.05 0.69 0.22 0.00 -0.36 0.07 0.00 0.22 0.00 -0.51 -0.13 -0.22 0.79 -0.62 -0.62 0.69 0.22 0.47

0.12 0.93 0.38 0.58 0.11 -0.21 0.86 0.28 -0.57 -0.69 0.22 0.51 -0.04 1.05 0.08 1.05 0.69 0.69 -0.12 0.41 0.69 0.34 -0.22 -0.64 0.34 1.39 -0.29 -0.10 0.47 -0.34 0.15 0.06 0.10 0.41 0.54 1.10 1.01 0.18 0.45 -0.24

A??E.\IX C

YEAR

-

IS'TO

CAL D-ATA SET

BLUE BLUE RED RED INITIAL CASUALTY INITIAL CASUALTY Bc Rc/Ro Bo Ro Rc

6,300

Bc/Bo

Log RATIO Bc/Rc

Log RATIO Ro/Bo

1805 1805 1805 1805 1805 1805 1805 1805 1805 1805 1806 1806 1806 1806 1806 1806 1806 1806 1806 1806 1807 1807 1807 1807 1807

49,000 16,000 25,000 20,000 12,000 9,000 7,000 25,000 30,000 13,000 27,000 12,000 27,300 16,000 12,000 96,000 18,000 26,000 4.800 30,000 6,000 45,000 87,000 75,000 12,000

5,700 1,000 4,000 700 600 600 500 1,100 2,000 400 1,000 200 7,100 800 1,000 6.000 850 3,300 350 2,000 3,000 6,000 12,000 23.000 700

46,000 8,000 8,000 15,000 4.000 4,800 8,000 6,000 7,000 3,000 22.000 9,000 50,000 13,700 11,000 54,000 5,500 44,000 4,300 15,000 14.000 16,000 61,000 83.000 9,000

4,000 5,000 800 400 400 400 600 1,200 1,500 600 700 10,000 1,000 1,000 12,000 1,400 3,500 1,700 2,000 5,000 3,000 20,000 23,000 1,100

0.1163 0.0625 0.1600 0.0350 0.0500 0.0667 0.0714 0.0440 0.0667 0.0308 0.0370 0.0167 0.2601 0.0500 0.0833 0.0625 0.0472 0.12bg 0.0729 0.0667 0.5000 0.1333 0.1379 0.3067 0.0583

0.1370 0.5000 0.6250 0.0533 0.1000 0.0833 0.0500 0.1000 0.1714 0.5000 0.0273 0.0778 0.2000 0.0730 0.0909 0.2222 0.2545 0.0795 0.3953 0.1333 0.3571 0.1875 0.3279 0.2771 0.1222

0.10 1.39 0.22 0.13 -0.41 -0.41 -0.22 -0.61 -0.51 1.32 -0.51 1.25 0.34 0.22 0.00 0.69 0.50 0.06 1.58 0.00 0.51 -0.69 0.51 0.00 0.45

0.06 0.69 1.14 0.29 1.10 0.63 -0.13 1.43 1.46 1.47 0.20 0.29 -0.61 0.16 0.09 0.58 1.19 -0.53 0.11 0.69 -0.85 1.03 0.36 -0.10 0.29

1807 1807 1807 1807 1807 1808 1808 1808 1808 1808 1808 1808 1808 1808 1808

20,000 1,000 12,000 65,000 63,000 24,500 32,000 35,000 21,000 13,000 19,000 14,000 50,000 8,000 15,000

1.200 150 2,800 12,500 2,500 200 1.000 700 300 3,000 750 400 6,000 300 1,100

18,000 7,000 6,000 95,000 17,000 11,000 22,000 45,000 19,000 15,500 13,000 15.000 30,000 12.000 22,000

2,500 2,000 1,500 9,000 1,800 2,500 3,000 3,000 400 3,500 1,500 1,000 18,000 200 1,000

0.0600 0.1500 0-2333 0.1923 0.0397 0.0082 0.0313 0.0200 0.0143 0.2308 0.0395 0.0286 0.1200 0.0375 0.0733

0.1389 0,2857 0.2500 0.0947 0.1059 0.2273 0.1364 0.0667 0.0211 0.2258 0.1154 0.0667 0.6000 0.0167 0.0455

0.73 2.59 -0.62 -0.33 -0.33 2.53 1.10 1.46 0.29 0.15 0.69 0.92 1.10 -0.41 -0.10

0.11 -1.95 0.69 -0.38 1.31 0.80 0.37 -0.25 0.10 -0.18 0.38 -0.07 0.51 -0.41 -0.38

-7Y

AEKX

C

-

RED

YEAR 1808 1808 1808 1808 1808 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1609 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809

HTORCAL .}ATA SET

RED

BLUE

BLUE

INITIAL CASUALTY INITIAL CASUALTY Ro Re Bo Bc Rc/Ro 15.000 15,000 25,000 7,000 21.000 5,000 45,000 18,000 54,000 13,000 11.000 15,000 15,000 33,000 10,000 45,000 16,000 35,000 15,000 39,000 12,000 17,500 36,000 60,000 27,000 1.500 7,000 13,300 34,000 66,000 74,000 30,000 12,000 11,300 21,000 5.000 40,000 11,000 21,000 27,000

500 600 1.000 300 1.100 400 3,000 200 6,000 200 200 300 900 2,000 400 4,000 400 4,000 650 3,600 600 1,000 2,000 1,000 150 100 500 1,000 15,000 3,000 4.000 4.600 1,600 800 700 600 300 1,700 2,400 800

4.400 9,000 3,500 3,000 23,000 22,000 25,000 10.000 47,000 12,000 6,300 5 000 20,000 50,000 8,000 40,000 23,000 37.000 3,000 41,000 18,000 23,000 58,000 46,000 13,000 2,300 9.000 11.700 9.400 74,000 60,000 30,000 9,000 20,000 11,000 18,000 39,000 5,000 23,000 18,000

600 1,000 1,500 2,000 900 2,000 1.900 1.700 7,100 2.000 1,100 600 1.500 4,000 1.000 3,000 4,000 3,500 1.000 3.000 2.000 8.000 450 2,800 600 500 1.500 1.500 5,200 6,000 3,200 2.400 700 4,000 13,000 3.000 500 1,000 3,300 1,500

0.0333 0.0400 0.0400 0.0429 0.0524 0.0800 0.0667 0.0111 0.1111 0.0154 0.0182 0.0200 0.0600 0.0606 0.0400 0.0889 0.0250 0.1143 0.0433 0.0923 0.0500 0.0571 0.0556 0.0167 0.0056 0.0667 0.0714 0.0752 0.4412 0.0455 0.0541 0.1533 0.1333 0.0708 0 :'33 G ':'O O.OC" , 0.154) 0.1143 0.0296

B/Bo 0.1364 0.1111 0.4286 0.6667 0.0391 0.0909 0.0760 0.1700 0.1511 0.1667 0.1746 0.1200 0.0750 0.0800 0.1250 0.0750 0.1739 0.0946 0.3333 0.0732 0.1111 0.3478 0.0078 0.0609 0.0462 0.2174 0.1667 0.1282 0.5532 0.0811 0.0533 0.0800 0.0778 0.2000 1.1818 0.1667 0.0128 0.2000 0.1435 0.0833

Log

Log

RATIO Bc/Rc

RATIO Ro/Bo

0.18 0.51 0.41 1.90 -0.20 1.61 -0.46 2.14 0.17 2.30 1.70 0.69 0.51 0.69 0.92 -0.29 2.30 -0.13 0.43 -0.18 1.20 2.08 -1.49 1.03 1.39 1,61 1.10 0.41 -1.06 0.69 -0.22 -0.65 -0.83 1.61 2.92 1.61 0.51 -0.53 0.32 0.63

1.23 0.51 1.97 0.85 -0.09 -1.48 0.59 0.59 0.14 0.08 0.56 1.10 -0.29 -0.42 0-22 0,12 -0.36 -0.06 1.61 -0.05 -0.41 -0.27 -0.48 0.27 0.73 -0.43 -0.25 0.13 1.29 -0.11 0.21 0.00 0.29 -0.57 0.65 -1.28 0.03 0.79 -0.09 0.41

"-

YEAR

TC)RCAL DATA LET

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc Rc/Ro

Bc/Bo

Log RATIO Bc/Rc

Log RATIO Ro/Bo

1.01 -0.46 0.07 1.61 1.61 0.75 0.69 0.85 0.00 1.22 1.39 -0.69 -0.69 0.00 -1.39 1.24 0.92 0.12 1.39 1.61 0.76 0.63 0.36 0.53 0.41 -0.92 0.59 1.39 0.00 0.22 0.74 0.63 1.39 1.54 1.10 0.69 0.13 0.62 0.00 3.00

-0.11 0.21 0.41 0.02 0.32 -0.85 0.41 0.12 -0.08 0.62 -0.63 0.69 0.51 2.54 2.27 -0.59 -0.46 0.41 1.20 -0.20 -0.46 1.53 2.30 0.37 0.09 1.87 -0.25 -0.22 0.76 0.18 0.00 -1.10 -0.84 -0.21 -1.01 -0.69 0.58 0.00 -0.30 0.51

.......................................................................

1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1809 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 18" 1810 1811 1811 1811 1811 1811 1811 1811 1811 1811 1811 1811 1811 1811 1811 1811

52,000 160.000 99,000 12,700 15,500 1,500 18,000 9,000 72,000 26,000 16,000 30,000 25,000 70,000 27,000 32,000 22,000 30,000 6,000 6,000 19,000 23,000 60,000 13,000 12.000 19,500 35,000 8,000 15,000 12,000 6,000 20,000 4,500 17,000 4,000 6,000 32.000 18,000 17,000 5.000

1,000 58,000 30,000 130.000 21,500 66,000 100 12,500 200 11,300 40 3.500 500 12,000 300 8,000 2,000 78,000 400 14,000 2,000 30,000 2,000 15,000 12,000 15,000 1,500 5,500 800 2,800 1,300 58,000 2,000 35,000 1,600 20,000 100 1,800 100 7,300 1,400 30,000 1,600 5,000 700 6,000 1,000 9,000 1.000 11.000 1.000 3,000 1,500 45,000 1.000 10,000 2,000 7,000 2,000 10,000 1.000 6,000 800 60,000 150 10,400 1,500 21,000 300 11,000 500 12,000 7,000 18,000 4,300 18,000 1,000 23.000 100 3,000

2,750 19,000 23,000 500 1,000 85 1,000 700 2,000 1,350 8,000 1,000 6,000 1,500 200 4,500 5,000 1,800 400 500 3,000 3,000 1,000 1,700 1,500 400 2,700 4,000 2,000 2,500 2,100 1,500 600 7,000 900 1,000 8.000 8,000 1,000 2,000

0.0192 0.1875 0.2172 0.0079 0.0129 0.0267 0.0278 0.0333 0.0278 0.0154 0.1250 0.0667 0.4800 0.0214 0.0296 0.0406 0.0909 0.0533 0.0167 0.0167 0.0737 0.0696 0.0117 0.0769 0.0833 0.0513 0.0429 0.1250 0.1333 0.1667 0.1667 0.0400 0.0333 0.0882 0.0750 0.0833 0.2188 0.2389 0.0588 0.0200

0.0474 0.1462 0.3485 0.0400 0.0885 0.0243 0.0833 0.0875 0.0256 0.0964 0.2667 0.0667 0.4000 0.2727 0.0714 0.0776 0.1429 0.0900 0.2222 0.0685 0.1000 0.6000 0.1667 0.1889 0.1364 0.1333 0.0600 0.4000 0.2857 0.2500 0.3500 0.0250 0.0577 0.3333 0.0818 0.0833 0.4444 0.4444 0.0435 0.6667

A??EN:. :x C

YEAR

-

HSTORICAL :ATA ZET

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc c/Ro

1811 1811 1811 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812

33.000 18,000 10,000 30,000 4,000 6,000 3,500 51,000 23,000 28.000 35,000 20 000 32.000 9.000 33.000 90,000 46.000 24,000

1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1812 1813 1813 1813 1813 1813

124.000 36,000 34.000 40,000 180,000 36,000 35,000 27,000 2,000 50,000 33,000 4,500

61600 16.000 8.000 28,000 17,000 80.000 70,000 144.000 15,000 160,000

2,000 600 500 400 200 300 200 1.850 4.300 3.400 1,300 500 10.000 800 10.000 2.000 5,200 6,000

Bc/Bo

Log RATIO Bc/Rc

Log RATIO Ro/Bo

33,000 22,000 12.000 30,000 10,000 1,500 2,000 5,000 20,000 28,000 27,000 7 000 27,000 10,000 87,000 50,000 42.000 24,000

2,000 1,100 1,500 400 3.000 800 250 1,300 3,700 2,700 1.500 700 6.000 1.200 8.000 6,000 10,000 8,000

0.0606 0.0333 0.0500 0.0133 0.0500 0.0500 0.0571 0.0363 0.1870 0.1214 0.0371 0.0250 0.3125 0.0889 0.3030 0.0222 0.1130 0.2500

0.0606 0.0500 0.1250 0.0133 0.3000 0.5333 0.1250 0.2600 0.1850 0.0964 0.0556 0.1000 0.2222 0.1200 0.0920 0.1200 0.2381 0.3333

0.00 0.61 1.10 0.00 2.71 0.98 0.22 -0.35 -0.15 -0.23 0.14 0.34 -0.51 0.41 -0.22 1.10 0.65 0.29

0.00 -0.20 -0.18 0.00 -0.92 1.39 0.56 2.32 0.14 0.00 0.26 1.05 0.17 -0.11 -0.97 0.59 0.09 0.00

28.000 122,000 4,000 20,000 6,000 22.000 250 1,800 10,000 120,000 800 20,000 8,800 25.000 1,800 25.000 200 32.000 600 700 600 2.600 50 3,500 100 6.000 900 22,000 200 10,000 3,000 25,000 900 16.000 9.000 70,000 4,600 50,000 19,200 93,000 700 12,000 8,000 100,000

43,000 6.000 6,000 800 6.000 2,000 6,000 4.000 550 200 300 500 300 1,500 1,200 2.200 1,900 6,500 5,400 12,000 2.000 16.000

0.2258 0.1111 0.1765 0.0063 0-0556 0.0222 0.2514 0.0667 0.1000 0.0120 0.0182 0.0111 0.0152 0.0563 0.0250 0.1071 0.0529 0.1125 0.0657 0.1333 0.0467 0.0500

0.3525 0.3000 0.2727 0.4444 0.0500 0.1000 0.2400 0.1600 0.0172 0.2857 0.1154 0.1429 0.0500 0.0682 0.1200 0.0880 0.1188 0.0929 0.1080 0.1290 0.1667 0.1600

0.43 0.41 0.00 1.16 -0.51 0.92 -0.38 0.80 1.01 -1.10 -0.69 2.30 1.10 0.51 1.79 -0.31 0.75 -0.33 0.16 -0.47 1.05 0.69

0.02 0.59 0.44 3,10 0.41 0.59 0.34 0.08 -2.77 4.27 2.54 0.25 0.10 -0.32 -0.22 0.11 0.06 0.13 0.34 0.44 0.22 0.47

C - H A}

ORIcAL DATA SE

YEAR

BLUE RED RED BLUE INITIAL CASUALTY INITIAL CASUALTY Rc Bo Bc Ec/Ro Ro

1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1813 1814 1814 1814 1814

16.000 10,000 4,800 20.000 10,000 12.000 32,000 3.000 20.000 40,000 19.000 24.000 24.000 80,000 4.000 21,400 32,000 13,400 12,000 17.000 90,000 60,000 37,000 17,000 42.000 90,000 90.000 60,000 8,000 55.000 100,000 15.400 160.000 187,000 325.000 103.000 15,000 40.000 8.000 36,000

1.900 500 250 1,100 600 1.100 1,500 250 2,000 3.000 2,000 600 1.000 4,000 500 800 1.000 1,800 1,800 700 5,300 6,000 6,000 300 1,600 4,900 5,000 2.500 100 8,000 10,000 250 2,000 21.200 75,000 11.000 300 5,600 450 3,000

15,000 4,000 2.600 8,500 4.000 4.500 18,000 2.500 4.000 20,000 3.500 50.000 18,000 60,000 7,000 18.000 38.000 7,400 11,000 15,000 50,000 40.000 45,000 20.000 32,000 60,000 4.800 50.000 4,000 60,000 200,000 5,000 100,000 97,000 175.000 37,000 4,000 100,000 5.000 30,000

900 1,100 300 2,100 500 600 1,700 500 1,400 6,000 1,350 1,200 1,500 12,000 1,000 1,500 3,600 1,200 2.000 100 3,200 5,000 4,000 2,000 1,100 6,000 1,900 1.500 1,700 11,000 15.000 1.400 3.000 11.000 45.000 9.000 400 5,000 800 3,000

0.1188 0.0500 0.0521 0.0550 0.0600 0.0917 0.0469 0.0833 0.1000 0.0750 0.1053 0.0250 0.0417 0.0500 0.1250 0.0374 0.0313 0.1343 0.1500 0.0412 0.0589 0.1000 0.1622 0.0176 0.0381 0.0544 0.0556 0.0417 0.0125 0.1455 0.1000 0.u162 0.0125 0.1269 0.2308 0.1068 0.0200 0.1400 0.0563 0.0833

Bc/Bo 0.0600 0.2750 0.1154 0.2471 0.1250 0.1333 0.0944 0.2000 0.3500 0.3000 0.3857 0.0240 0.0833 0.2000 0.1429 0.0833 0.0947 0.1622 0.1818 0.0067 0.0640 0.1250 0.0889 0.1000 0.0344 0.1000 0.3958 0.0300 0.4250 0.1833 0.0750 0.2800 0.0300 0.1134 0.2571 0.2432 0.1000 0.0500 0.1600 0.1000

Log RATIO Bc/Rc

Log RATIO Ro/Bo

-0.75 0.79 0.18 0.65 -0.18 -0.61 0.13 0.69 -0.36 0.69 -0.39 0.69 0.41 1.10 0.69 0.63 1.28 -0.41 0.11 -1.95 -0.50 -0.18 -0.41 1.90 -0.37 0.20 -0.97 -0.51 2.83 0.32 0.41 1.72 0.41 -0.66 -0.51 -0.20 0.29 -0.11 0.58 0.00

0.06 0.92 0.61 0.86 0.92 0.98 0.58 0.18 1.61 0.69 1.69 -0.73 0.29 0.29 -0.56 0.17 -0.17 0.59 0.09 0.13 0.59 0.41 -0.20 -0.16 0.27 0.41 2.93 0.18 0.69 -0.09 -0.69 1.12 0.47 0.54 0.62 1.02 1.32 -0.92 0.47 0.18

At'iEND:X C - H:STI'R:CAL DATA SET

YEAR

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Bc Bo Bc Rc/Ro

1814 1814 1814 1814 1814 1814 1814 1814 i814 1814 1814 1814 1814 1814 1814 1814 1814 1814 1814 1814 1814 1814 1814 1814 814 1814 1814 1814 1814 1814 1815 1815 1815 1815 1815 1815 1815 1815 1815 1816

10.000 25,000 7,000 100,000 34,000 12.000 11.000 6,000 15.000 30,000 100.000 5.000 90.000 5.000 50.000 28.000 60.000 123.000 2.700 4.000 25.000 20.000 45.000 30.000 12.000 18.000 14.000 13.000 30.000 100.000 20.000 120.000 10.000 71,000 32.000 40,000 32.000 11.000 6,000 5,000

600 2,100 500 9.000 3,000 500 1.000 650 400 1,900 3,000 600 1.600 500 1,000 2,000 6,700. 6,000 500 770 1.400 500 3.200 800 400 600 1.600 500 2,000 3,500 1,000 19,000 600 12,000 5,100 2.000 2,500 700 20 800

8.800 39,000 4,000 42.000 32.000 10.000 16.000 11,000 10,000 20.000 30.000 4,000 25.000 7,000 20.000 21,000 32.000 41,000 9.000 5.000 13.000 30,000 35,000 13,000 4.000 9.000 25.000 9,000 15,000 50.000 17.000 72.000 5.000 84.000 21.000 20.000 24.000 29.000 12,000 12,000

1,500 3,000 500 7,000 2.800 300 5,000 1.000 800 800 3,400 2,400 2,600 300 1,000 5.000 4,000 3,000 1,900 650 500 3,000 3,000 2,300 600 2,400 1,700 1,000 1.400 4.000 500 25,000 1.600 16,000 4,400 1.000 2,500 1,700 2,000 3,000

0.0600 0.0840 0.0714 0.0903 0.0882 0.0417 0.0909 0.1083 0.0267 0.0633 0.0300 0.1200 0.0178 0.1000 0.0200 0.0714 0.1117 0.0488 0.1852 0.1925 0.0560 0.0250 0.0711 0.0267 0.0333 0.0333 0.1143 0.0385 0.0667 0.0350 0.0500 0.1583 0.0600 0.1690 0.1594 0.0500 0.0781 0.0636 0.0033 0.1600

Bc/Bo 0.1705 0.0769 0.1250 0.1667 0.0875 0.0300 0.3125 0.0909 0.0800 0.0400 0.1133 0.6000 0.1040 0.0429 0.0500 0.2381 0.1250 0.0732 0.2111 0.1300 0.0385 0.1000 0.0857 0.1769 0.1500 0.2667 0.0680 0.1111 0.0933 0.0800 0.0294 0.3472 0.3200 0.1905 0.2095 0.0500 0.1042 0.0586 0.1667 0.2500

Log RATIO Bc/Rc

Log RATIO Ro/Bo

0.92 0.36 0.00 -0.25 -0.07 -0.51 1.61 0.43 0.69 -0.86 0.13 1.39 0.49 -0.51 0.00 0.92 -0.52 -0.69 1.34 -0.17 -1.03 1.79 -0.06 1.06 0.41 1.39 0.06 0.69 -0.36 0.13 -0.69 0.27 0.98 0.29 -0.15 -0.69 0.00 0.89 4.61 1.32

0.13 -0 44 0.56 0.87 0.06 0.18 -0.37 -0.61 0.41 0.41 1.20 0.22 1.28 -0.34 0.92 0.29 0.63 1.10 -1.20 -0.22 0.65 -0.41 0.25 0.84 1.10 0.69 -0.58 0-37 0.69 0.69 0.16 0.51 0.69 -0.17 0.42 0.69 0.29 -0.97 -0.69 -0.88

AE.:,x.. C

YEAR 1828 1828 1828 1828 1828 1828 1828 1828 1828 1829 1829 1829 1829 1831, 1831 1831 1831 1831 1831 1832 1832 1832 1839 1840 1844 1847 1848 1848 1848 1848 1848 1848 1848 1848 1848 1848 1848 1848 1848 1849

-

^ HISTORI CAL DATA SET

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Bc Rc/Ro 24.000 20,000 20,000 7,000 4,500 10.000 7,000 11.000 17,000 19,000 28,000 18,000 37,000 38,000 52.000 50,000 78,000 22,000 80,000 22,000 15,000 14,000 43,000 4,700 10,000 4,400 19,000 55,000 16,000 23,000 19,000 31,000 25,000 16,000 11,000 18,000 42,000 28,000 19,000 28,000

2,000 6,000 1,600 600 1,500 1,800 1,000 500 3,000 100 2,500 100 3,000 4,500 3,800 500 10,600 1,500 10,000 2,000 500 500 500 100 70 720 500 900 500 100 650 900 500 100 100 300 600 350 260 350

30,000 15,000 6,000 11,000 35,000 30,000 20,000 20,000 8,000 35,000 40.000 20,000 15,000 32,000 44,000 26.000 37,000 24,000 50,000 44,000 20,000 17,000 33.000 5,000 30,000 20,000 16,000 75,000 11,000 6.000 4,000 18,000 5,000 6,000 6,000 11,000 10,000 42,000 41,000 38,000

73

4,000 8,000 1,400 3,500 2,500 5,000 1,000 1,500 1,500 1,000 3,000 500 6,000 7,000 2,900 1,000 7,800 1.700 9,000 3,000 2,000 2,500 1,000 2,000 2,000 2,000 700 900 800 300 700 1.400 200 1,000 150 400 300 700 750 1,200

0.0833 0.3000 0.0800 0.0857 0.3333 0.1800 0.1429 0.0455 0.1765 0.0053 0.0893 0.0056 0.0811 0.1184 0.0731 0.0100 0.1359 0.0682 0.1250 0.0909 0.0333 0.0357 0.0116 0.0213 0.0070 0.1636 0.0263 0.0164 0.0313 0.0043 0.0342 0.0290 0.0200 0.0063 0.0091 0.0167 0.0143 0.0125 0.0137 0.0125

Bc/Bo 0.1333 0.5333 0.2333 0.3182 0.0714 0.1667 0.0500 0.0750 0.1875 0.0286 0.0750 0.0250 0.4000 0.2188 0.0659 0.0385 0.2108 0.0708 0.1800 0.0682 0.1000 0.1471 0.0303 0.4000 0.0667 0.1000 0.0438 0.0120 0.0727 0.0500 0.1750 0.0778 0.0400 0.1667 0.0250 0.0364 0.0300 0.0167 0.0183 0.0316

Log RATIO Bc/Rc

Log RATIO Ro/Bo

0.69 0.29 -0.13 1.76 0.51 1.02 0.00 1.10 -0.69 2.30 0.18 1.61 0.69 0.44 -0.27 0.69 -0.31 0.13 -0.11 0.41 1.39 1.61 0.69 3.00 3.35 1.02 0.34 0.00 0.47 1.10 0.07 0.44 -0.92 2.30 0.41 0.29 -0.69 0.69 1.06 1.23

-0.22 0.29 1.20 -0.45 -2.05 -1.10 -1.05 -0.60 0.75 -0.61 -0.36 -0.11 0.90 0.17 0.17 0.65 0.75 -0.09 0.47 -0.69 -0.29 -0.19 0.26 -0.06 -1.10 -1.51 0.17 -0.31 0.37 1.34 1.56 0.54 1.61 0.98 0.61 0.49 1.44 -0.41 -0.77 -0.31

AFPEND:X C --

:STORCAL DATA SET

YEAR

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Be Rc/Ro

1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1849 1850 1850 1850 1853 1854 1854 1854 1854 1854 1854 1855 1855 1855 1856 1859 1859 1859 1859 1859 1860 1860 1860 1861

9.000 16,000 10,000 41.000 37,000 25,000 11,000 30.000 20.000 34,000 25,000 51.000 15,000 12,000 53,000 20,000 4,000 3,300 4,000 39,000 3.000 6,000 20.000 21,000 65.000 30.000 175,000 22.000 50.000 30,000 19.000 48,000 40,500 8.300 14,300 26.000 18,000 3,500 30.000 10,000

1,000 8,000 310 24,000 200 6,000 2,300 59,000 200 45,000 600 15.000 100 8,000 550 20.000 200 8,000 800 40,000 300 38,000 700 50.000 250 8,000 200 7,000 1,200 37.000 1,870 14.000 50 4,000 450 11.500 250 12,000 3,200 26,000 1,000 7,000 2,000 10.000 800 60,000 600 29.000 4,400 35.000 7,000 36.000 54,000 120.000 500 18,000 1,800 74,000 7,000 16,000 1,000 70,000 3,900 62.000 1,000 8.500 800 18,700 1,460 130,000 1,000 18,000 1,500 38,000 700 5,000 200 6,000 1,230 6,000

74

2,000 300 1,000 3,000 500 800 1.100 1,500 2,000 1,200 500 1,300 800 1,300 2.000 1,350 200 700 300 1,750 1.000 3.600 1.600 1.200 5.000 12.000 10,300 1,500 7,300 3.000 3.000 5.700 360 1,100 13,100 1,700 1,700 200 500 1,140

0.1111 0.0194 0.0200 0.0561 0.0054 0.0240 0.0091 0.0183 0.0100 0.0235 0.0120 0.0137 0.0167 0.0167 0.0226 0.0935 0.0125 0.1364 0.0625 0.0821 0.3333 0.3333 0.0400 0.0286 0.0677 0.2333 0.3086 0.0227 0.0360 0.2333 0.0526 0.0813 0.0247 0.0964 0.1021 0.0385 0.0833 0.2000 0.0067 0.1230

Bc/Bo 0.2500 0.0125 0.1667 0.0508 0.0111 0.0533 0.1375 0.0750 0.2500 0.0300 0.0132 0.0260 0.1000 0.1857 0.0541 0.0964 0.0500 0.0609 0.0250 0.0673 0.1429 0.3600 0.0267 0.0414 0.1429 0.3333 0.0858 0.0833 0.0986 0.1875 0.0429 0.0919 0.0424 0.0588 0.1008 0.0944 0.0447 0.0400 0.0833 0.1900

Log RATIO Bc/Rc

Log RATIO Ro/Bo

0.69 -0.03 1.61 0.27 0.92 0.29 2.40 1.00 2.30 0 41 0.51 0.62 1.16 1.87 0.51 -0.33 1.39 0.44 0.18 -0.60 0.00 0.59 0.69 0.69 0.13 0.54 -1.66 1.10 1.40 -0.85 1.10 0.38 -1.02 0.32 2.19 0.53 0.13 -1.25 0.92 -0.08

0.12 -0.41 0.51 -0.36 -0.20 0.51 0.32 0.41 0.92 -0.16 -0.42 0.02 0.63 0.54 0.36 0.36 0.00 -1.25 -1.10 0.41 -0.85 -0.51 -1.10 -0.32 0.62 -0.18 0.38 0.20 -0.39 0.63 -1.30 -0.26 1.56 -0.81 -2.21 0.37 -0.75 -0.36 1.61 0.51

AFEND:X C

RED

YEAR

'ATA SET

-:sTCICCAL

RED

BLUE

BLUE

INITIAL CASUALTY INITIAL CASUALTY Rc/Ro Bc Bo Rc Ro

Bc/Bo

Log

Log

RATIO Bc/Rc

RATIO Bo/Bo

1861

32.000

2,000

35,000

1,600

0.0625

0.0457

-0.22

-0.09

1862 1862 1862 1862 1862 1862 1862 1862 1862 1862 1862 1862 1862 1862 1862 1863 1863 1863 1863 1863 1863 1863 1863 1863 1863 1863 1864 1864 1864 1864 1864 1864 1864 1864 1864 1864 1864 1864 1864

50,000 68,000 94,000 24.000 8,000 12,000 27,000 14,000 60,000 39,000 20,000 25,000 78,000 85,000 45,000 70,000 90.000 18.000 80,000 25,000 65,000 14.000 45,000 30,000 26,000 75,000 70,000 60,000 37.000 55,000 118,000 93,000 120.000 100,000 25,000 52,000 5,000 60,000 43,000

9,400 3,150 19.000 1,800 450 1,200 2.660 1.000 10,200 6,500 1,300 2.200 4,700 11,700 1,900 15,800 17,700 300 10,800 200 5,600 2.000 9,600 2.300 1,100 9,000 4,000 1,700 5.600 3,100 14,300 4,000 39,000 16,200 3.000 4,000 480 700 4,400

70.000 34,000 106,000 15,000 16,000 16,000 17,000 11.000 40,000 51,000 18,000 22.000 117,000 50.000 30.000 57,000 68,000 9,000 130.000 33,000 45.000 16,000 38,000 18,000 5,000 28,000 40,000 100,000 27,000 38.000 62,000 52,000 67,000 50.000 30.000 38,000 5,000 30,000 13,000

10,200 3,700 10,000 1,600 1,050 1,100 2,000 1,000 9,800 4,400 1,800 2,700 11,000 9.400 1.300 11.500 15,300 500 11,400 1,200 2,600 4,000 9,300 3,000 200 7,000 8.500 11,000 1.300 15,000 7.000 1,500 40,000 5,000 3,000 3,000 420 4,600 2,000

0.1880 0.0463 0.2021 0.0750 0.0563 0.1000 0.0985 0.0714 0.1700 0.1667 0.0650 0.0880 0.0603 0.1376 0.0422 0.2257 0.1967 0.0167 0.1350 0.0080 0.0862 0.1429 0.2133 0.0767 0.0423 0.1200 0.0571 0.0283 0.1514 0.0564 0.1212 0.0430 0.3250 0.1620 0.1200 0.0769 0.0960 0.0117 0.1023

0.1457 0.1088 0.0943 0.1067 0.0656 0.0688 0.1176 0.0909 0.2450 0.0863 0.1000 0.1227 0.0940 0.1880 0.0433 0.2018 0.2250 0.0556 0.0877 0.0364 0.0578 0.2500 0.2447 0.1667 0.0400 0.2500 0.2125 0.1100 0.0481 0.3947 0.1129 0.0288 0.5970 0.1000 0.1000 0.0789 0.0840 0.1533 0.1538

0.08 0.16 -0.64 -0.12 0.85 -0.09 -0.29 0.00 -0.04 -0.39 0.33 0.20 0.85 -0.22 -0.38 -0.32 -0.15 0.51 0.05 1.79 -0.77 0.69 -0.03 0.27 -1.70 -0.25 0.75 1.87 -1.46 1.58 -0.71 -0.98 0.03 -1.18 0.00 -0.29 -0.13 1.88 -0.79

-0.34 0.69 -0.12 0.47 -0.69 -0.29 0.46 0.24 0.41 -0.27 0.11 0.13 -0.41 0.53 0.41 0.21 0.28 0.69 -0.49 -0.28 0.37 -0.13 0.17 0.51 1.65 0.99 0.56 -0.51 0.32 0.37 0.64 0.58 0.58 0.69 -0.18 0.31 0.00 0.69 1.20

75

A??END:X C

-

H:-ISTO.RCAL

)ATA SET

YEAR

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Be Bo Be Rc/Ro

1864 1864 1864 1864 1864 1864 1864 1865 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1866 1867 1870 1370 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870

72.000 20,000 37,000 40.000 80,000 6,000 5.500 60.000 41,000 220,000 26,000 3.000 24,000 16.000 26,000 25,000 24,000 27,000 75,000 27,000 30,000 25.000 18,000 10.000 14.000 3,000 82,000 38,000 6,000 4,000 35,000 243,000 30.000 200,000 40,000 187.000 35,000 58,000 25,000 14,000

1,700 400 1.200 4,100 2,400 800 430 1,500 350 8,900 1.600 110 400 1,500 250 210 1,200 3,600 5,200 720 1,400 820 950 130 200 400 9,300 3,600 500 400 1,300 10,000 3.500 6.000 1.300 19,700 500 4,800 1,700 500

48.000 11,000 23.000 18.000 40,000 18,000 6,000 30.000 29,000 215,000 44.000 9 200 20.000 9,000 34,000 20,000 31,000 32.000 90.000 22.000 23.000 22,000 14,000 23.000 7.000 7,000 41,000 52,000 8,000 8.000 25,000 400,000 60.000 180,000 36,000 113,000 30,000 84.000 65,000 8,000

76

4,800 700 1,200 1,900 3,000 3,000 900 1,700 920 23,600 2,900 700 1.330 900 600 490 3.700 1,300 3.400 .1,100 3,330 750 750 370 510 1,000 8,000 4,500 1,400 1,800 1,400 16.000 5,000 8,000 1,200 12,800 700 3,200 5,000 800

0.0236 0.0200 0.0324 0.1025 0.0300 0.1333 0.0782 0.0250 0.0085 0.0405 0.0615 o.o367 0.0167 0.0938 0.0096 0.0084 0.0500 0.1333 0.0693 0.0267 0.0467 0.0328 0.0528 0.0130 0.0143 0.1333 0.1134 0.0947 0.0833 0.1000 0.0371 0.0417 0.1167 0.0300 0.0325 0.1053 0.0143 0.0828 0.0680 0.0357

Bc/Bo 0.1000 0.0636 0.0522 0.1056 0.0750 0.1667 0.1500 0.0567 0.0317 0.1098 0.0659 0.8078 0.06 0.1000 0.0176 0.0245 0.1194 0.0406 0.0378 0.0500 0.1448 0.0341 0.0536 0.0161 0.0729 0.1429 0.1951 0.0865 0.1750 0.2250 0.0560 0.0400 0.0833 0.0333 0.0333 0.1133 0.0233 0.0381 0.0769 0.1000

Log RATIO Bc/Rc

Log RATIO Ro/Bo

1.04 0.56 0.00 -0.77 0.22 1.32 0.74 0-13 0.97 0.98 0.59 1.?5 .20 -0.51 0.88 0.85 1.13 -1.02 -0.42 0.42 0.87 -0.09 -0.24 1.05 0.94 0.92 -0.15 0.22 1.03 1.50 0.07 0.47 0.36 0.00 -0.08 -0.43 0.34 -0.41 1.08 0.47

0.41 0.60 0.48 0.80 0.69 -1.10 -0.09 0.69 0.35 0.02 -0.53 -1.12 0.18 0.58 -0.27 0.22 -0.26 -0.17 -0.18 0.20 0.27 0.13 0.25 -0.83 0.69 -0.85 0.69 -0.31 -0.29 -0.69 0.34 -0.51 -0.69 0.11 0.11 0.50 0.15 -0.37 -0.96 0.56

AF-...:x

c - H:STOR:CA!LDATA SET

YEAR

RED RED BLUE BLUE INITIAL CASUALTY INITIAL CASUALTY Ro Rc Bo Ba Rc/Ro

1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1870 1871 1871 1871 1871 1871 1871 1874 1877 1877 1877 1877 1877 1877 1877 1877 1877 1877 1877 1877 1877 1877 1877 1877

40,000 9,000 6,000 200,000 63.000 68.000 26.000 30,000 40,000 78,000 86,000 60,000 16.000 26.000 40,000 35,000 51,000 40.000 45.000 72.000 33.000 23.000 23,000 22,000 25.000 55.000 29.000 20.000 12,000 40.000 12.000 11,000 17.000 15.000 35,000 50.000 24.000 36.000 70.000 18.000

1,000 23,000 500 20,000 500 10,000 8,300 120,000 14,900 113,000 3,400 59,000 1.000 41,000 1,000 20,000 1,000 60,000 2,800 96,000 2.000 64,000 1.500 20.000 1,000 11,000 2,200 18,000 3.500 90,000 4.500 28,000 1,550 6,000 1,700 6,300 1,800 135,000 3.500 88,000 2,500 47.000 1,000 37,000 700 83,000 2,500 32,000 3,500 4,000 2,000 30.000 1,700 13,000 1,300 5,000 2,000 30,000 5,000 60,000 800 5.000 600 16.000 1,000 11,000 2,000 10,000 7.000 95,000 600 10.000 1,600 8,000 3,000 18,000 800 35,000 1,100 19.000

77

2,500 2,000 900 17,000 17,000 5,500 1,100 700 2,200 3,600 3,000 1,000 1,000 4,800 3.500 4,100 1.100 700 4.000 6.000 3.500 1.350 4,000 4,000 1,500 3,000 2.000 1,900 5,000 4.000 1,500 1,000 1,400 1.600 16,000 1.500 2.400 5.000 3.000 1.300

0.0250 0.0556 0.0833 0.0415 0.2365 0.0500 0.0385 0.0333 0.0250 0.0359 0.0233 0.0250 0.0625 0.0846 0.0875 0.1286 0.0304 0.0425 0.0400 0.0486 0.0758 0.0435 0.0304 0.1136 0.1400 0.0364 0.0586 0.0650 0.1667 0.1250 0.0667 0.0545 0.0588 0.1333 0.2000 0.0120 0.0667 0.0833 0.0114 0.0611

Bc/Bo 0.1087 0.1000 0.0900 0.1417 0.1504 0.0932 0.0268 0.0350 0.0367 0.0375 0.0469 0.0500 0.0909 0.2667 0.0389 0.1464 0.1833 0.1111 0.0296 0.0682 0.0745 0.0365 0.0482 0-1250 0.3750 0.1000 0.1538 0.3800 0.1667 0.0667 0.3000 0.0625 0.1273 0.1600 0.1684 0.1500 0.3000 0.2778 0.0857 0.0684

Log RATIO Ba/Rc

Log RATIO Ro/Bo

0.92 1.39 0.59 0.72 0.13 0.48 0.10 -0.36 0.79 0.25 0.41 -0.41 0.00 0.78 0.00 -0.09 -0.34 -0.89 0.80 0.54 0.34 0.30 1.74 0.47 -0.85 0.41 0.16 0.38 0.92 -0.22 0.63 0.51 0.34 -0.22 0.83 0.92 0.41 0.51 1.32 0.17

0.55 -0.80 -0.51 0.51 -0.58 0.14 -0.46 0.41 -0.41 -0.21 0.30 1.10 0.37 0.37 -0.81 0.22 2.14 1.85 -1.10 -0.20 -0.35 -0.48 -1.28 -0.37 1.83 0.61 0.80 1.39 -0.92 -0.41 0.88 -0.37 0.44 0.41 -1.00 1.61 1.10 0.69 0.69 -0.05

"-'-

YEAR

.... '¢

':"CAL :ATA L':

BLUE BLUE RED RED INITIAL CASUALTY INITIAL CASUALTY Be Rc/Ro Bo Re Ro

1877 1877 1877 1877

30,000 14,000 22.000 20.000

1877 1977 1877 1878 1878 1885 1885 1889 1891 1894 1894 1897 1898 1899 1899 1899 1899 1899 1899 1899 1899 1899 1900 1900 1900 1904 1904 1904 1904 1904 1904 1904 1904 1905 1905

10,000 26,000 18.000 18.000 67,0OOO 50,000 32.000 60,000 11.000 12.000 6.000 47.000 12,700 8,700 55.000 10,200 3,400 4,500 25,000 450 4,000 8,5Co 12,000 1.500 5,000 145,000 140.000 135.000 40,000 36,000 24,000 36,000 22,000 40,000 314,000

Bc/o

Log RATIO Bc/Rc

Log RATIO Ro/Bo

36,000 10,000 32,000 33,000

1,000 3,000 1,400 7,500

0.0153 0.1429 0.0455 0.1750

0.0278 0.3000 0.0438 0.2273

0.78 0.41 0.34 0 76

-0.18 0.34 -0.37 -0.50

7.000 200 15,000 1,000 5,400 27.000 300 10,000 6,000 29,000 1,900 32,000 2,500 30,000 8.000 5,000 1,600 14.000 700 15,000 600 5,000 1.800 35,000 1,300 1,850 1,500 330 30 13,500 6.000 500 260 1,200 1,500 230 9,500 180 200 5,000 11,500 120 200 2.500 2,000 430 1,200 20 500 12,000 17,00n 210.000 75,000 40.000 17,500 150.000 4,600 13,000 1,200 38.000 850 12.000 6,000 1,100 1,250 24,000 9,400 58,000 41,000 310,000

1.000 3,000 7.000 2.400 4,500 1,100 5,000 2,200 3,400 3,500 500 500 650 90 910 20 90 110 320 760 900 40 400 160 1.110 46,000 27,500 16,500 900 3,000 1,550 1,800 2.000 13,000 71,000

0.0200 0.0385 0.3000 0.0167 0.0896 0.0380 0.0781 0.0833 0.1455 0.0583 0.1000 0.0383 0.1457 0.0379 0.0005 0.0490 0.0765 0.0511 0.0072 0.4444 0.0300 0.0235 0.0358 0.0133 0.1000 0.1172 0.5357 0.1296 0.1150 0.0333 0.0354 0.0306 0.0568 0.2350 0.1306

0.1429 0.2000 0.2593 0.2400 0.1552 0.0344 0.1667 0.2750 0.2429 0,2333 0.1000 0.0143 0.5000 0.0600 0.0674 0.0033 0.0750 0.0733 0.0337 0.1520 0.0783 0.0160 0.2000 0.1333 0.0925 0.2190 0.6875 0.1100 0.0692 0.0789 0.1292 0.3000 0.0833 0.2241 0.2290

1.61 1.10 0.26 2.08 -0.29 -0.55 0.69 -0.82 0.75 1.61 -0.18 -1.28 -1.05 -1.30 3.41 -3.22 -1.06 -0.74 0.58 1.34 2.01 -1.61 -0.07 2.08 0.80 1.00 -1.00 -0.06 -1.63 0.92 0.60 0.49 0.47 0.32 0.55

0.36 0.55 -0.41 0 59 0.84 0-45 0.06 2.01 -0.24 -0.22 0-18 0.29 2.28 1.76 1.40 0.53 i 04 1.10 0.97 -2.41 -1.06 1.22 1.79 0.22 -0.88 -0.37 1.25 -0.11 1.12 -0.05 0.69 1.79 -0.09 -0.37 0.01

460 2,000 1,000 3,500

Endnotes 1. James Schneider, "The Exponential Decay of Armies in Battle," Theoretical Paper No.1, (Fort Leavenworth, KS: School of Advanced Military Studies, 1985). 2. Ronald L. Johnson, "Lanchester's Square Law in Theory and Practice," Monograph, (Fort Leavenworth, KS: School of Advanced Military Studies, 1990). David 6. Chandler. The Campaigns of Napoleon, (New York:MacMillan Publishing Company, Inc., 1986), p.66-67, 454. 4. James G. Taylor, Modelling, (Arlington, Society of America, 1980),

Force-On-Force Va: Operations p. 5.

Attrition Research

5. This information was extracted from a TRADOC Analysis Command- Fort Leavenworth handout developed by TRAC-FT LVN Training Simulations Directorate, Ft. Leavenworth, KS, 1989. 6. Headquarters, TRADOC Analysis Command, of TRADOC Models, (Fort Leavenworth, KS: Analysis Command, 1988), Appendix A-i.

Inventory TRADOC

7. Trevor N. Dupuy, Numbers. (Fairfax, Va:HERO Books, 1785), p.

and

Prediction 4.

E. "Commentary on Frederick William Lanchester" World of Mathematics, IV , J. Newman, editor, York: Simon and Schuster, 1956), p. 2136. 9. Frederick W. Lanchester, War+are", Engineering 98, (1914), printed in The World of Mathematics, Simon and Schuster, 1956), p. 2144. 10.

Ibid,

p.

11.

Ibid,

p.2139.

War,

in The (New

"Mathematics in p. 422-423 (reIV, (New York:

21Z9

!2. Taylor, p. 31. Note that the equations throughout the paper will use variables R and B to represent the opposing forces in order to maintain consistency between the different models.

Lanchester "Present State of Ladislov Dolansky, 13. 12, Vol Research Operations Combat," of Theory p. 345. 1964), (Mar-Apr p.

23.

14.

Taylor,

15.

Johnson, p.

16.

Lanchester, p.

17.

Taylor,

IS.

Dolansky, p.

.9.

Taylor,

p.

p.

8-9. 2147.

31. 345.

24.

Lanchester-Type Ekchian. An Overview of 20. L. K. NR-041-519. Combat Models for Modern Warfare Scenarios, (Cambridge MA: Massachusetts Institute of Technology, March 1982), p. 7-14. 21. Taylor, p. ,.. The model of insurgency operations was extended by S. J. Dietchman in "A Lanchester Model of Guerilla Warfare," Operations Research 10, (1962), p. 818-827. 2

Schneider,

p.

23.

Taylor, p.

71.

24.

Ibid, p.

77

4.

.

25.5 Daniel A. Willard, Lanchester as Force in History: Analysis of Land Battles of the Years 1618-1905, (McLean, Va: Research Analysis Corporation, 1962), p. 4. 26. Ibid, modified presented

p. 11. The terms of the model have been to be consistent with the previously models use of Blue and Red forces.

27. James G. Taylor. Lanchester-Type Models of Warfare. Vol 2 (Monterey,CA: Naval Postgraduate School, 1980), p. 594. 28.

Willard, p.20.

"The 29. The data was taked from William A. Schmeiman, of Analysis Equations in the Use of Lanchester-Type (Atlanta: Dissertation, Engagments." Past Military Appendix B. Georgia Institute of Technology, 1967), 30.

Willard, p.

9.

31.

Schmeiman, p.

32.

Willard, p.

128.

20.

Historical Helmbold. L. Richard CORG-SP-128, Lanchester's Theory of Combat, Group, Research Operations VA: Combat p. 29.

and Data (Fair+ax, 1961),

and "The Loose Marble-Schneider, James J. 34. March 1989, the Origins of Operational Art," Parameters p. 90. .

Ibid.,

p.

65.

Empty the "The Theory of Schneider. J. James 36. Services United Royal of the Journal Battlefield," Institute for Defence Studies, Sept 1987, p. 37-41. 37.

Ibid.,

p.

38.

the The American Civil War and Edward Hagerman, 36. Indiana (Indianapolis: Warfare, of Modern Origins University Press, 1988) p. 16. In Men Wise, Preston and Sydney F. Richard A. 79. Interrelationships and Its A History of Warfare Arms: and Rinehart Holt, (Chicago: Western Society, with Winston, 1979) p. 244. 40.

Hagerman, p.

17.

41.

Preston and Wise, p.

42.

Schneider, p.

4'.

Hagerman, p.

249.

39. 10.

The Franco-Prussian Michael Howard, 44. York: Rouledge, 1988), p. 4.

War,

(New

1

of Course A Summary of the Hart Mahan, Dennis 45. Defense Attack and the Fortifications and Permanent 1863), p. 229-230 Works. (Richmond, VA: of Permanent Edward, "From Jomini to Dennis Hart quoted in Hagerman, The Evolution of Trench Warfare and the American Mahan: p. September 1967, Civil War History 13, Civil War," 212. 46. Hagerman, Edward, "From Jomini to Dennis Hart Mahan: The Evolution of Trench Warfare and the American Civil War," Civil War History 13, September 1967, p. 218. 47.

Preston and Wise, p.

254.

The American Civil 48. Edward Hagerman, Origins of Modern Warfare, p. 5. 49.

Ibid, p.

50.

Schneider, p.

3?. Waterloo to The Art of War: Indiana University Press, 1974),

The Future of War 52. Jean de Bloch, cal Economic and Political Relations, Peace Foundation, 1914), p. 18. 57.

McElwee, p.

54.

Schneider, Bloch, p.

the

254-254.

51. William McElwee, Mons, (Bloomington, IN: p. 25.

5.

War and

219. p.

39.

41.

in its Techni(Boston: World

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Hiltaer-historisches C. W. Stern, 1908.

Kreiqs

Lexicon

Bloch, Jean de. The Future of War in its Technical Economic and Political Relations. Boston: World Peace Foundation, 1914. K. An Overview of Lanchester-Type Combat Ekchian, L. Models for Modern Warfare Scenarios, NR-041-519. Cambridge, MA: Massachusetts Institute of Technology, March, 1982. DuPuY, Fairfax,

Numbers, Trevor N. Va:HERO Books, 1985.

Prediction

Hagerman, Edward. The American Civil Bloomington, IN: of Modern Warfare. Press, 1988.

and

War.

War and the Origins Indiana University

Headquarters, TRADOC Analysis Command. Inventory of TRADOC Models. Fort Leavenworth, KS: TRADOC Analysis Command, 1988 Helmbold, Robert L. Historical Data and Lanchester's Theory of Combat. CORG-SP-128. Fort Belvior, Va: Combat Oper-ations Research Group, 1 July 1961. Helmbold, Robert L. Lanchester Parameters for Some Battles of the Last Two Hundred Years, CORG-SP-122. Fort Belvior, Va: Combat Operations Research Group, 14 Feb 1961. Hines, William W. and Montgomery, Douglas C. Probability and Statistics in Engineering and Manaaement Science, 2d edition. New York: John Wiley & Sons, 1980. Karr, Alan F. Combat Process and Mathematical Models of Attrition, Paper P-1081. Arlington, Va: Institute for Defense Analyses, 1975.

Stochastic Attrition Models of Lanchester Karr, Alan F. Analyses, Institute for Defense Va, Arlington, Type, 1974. to Waterloo The Art of War: William. McElwee, Bloomington, IN: Indiana University Press, 1974.

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A Men in Arms: Preston, Richard A. and Wise, Sydney F. with Interrelationships its and Warfare of History Western Society, 4th edition. Chicago: Holt, Rinehart and Winston, 1979. Warfare, of Taylor, James G. Lanchester-Type Models School, Naval Postgraduate Monterey, Ca: Volume II. 160 Taylor, James Arlington, Va: 1980

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Tiede, Richard kKS: Manhattan, 1 76.

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Ladislav. Combat."

"Present State of the Lanchester Operations Research 12, (1964):

DuFuy, Trevor N. "The Lanchester Equations: Lanchester's Original Article with a Commentary by Trevor N. DuPuy." History, Numbers, and War. Fall 1977: 142-150. Fain, Janice B. "The Lanchester Equations and Historical Warfare," History, Numbers and War, Spring 1977, p. 74-52.

Hagerman, Edward. "From Jomini to Dennis Hart Mahan: The Evolution of Trench Warfare and the American Civil War," Civil War History 13, September 1967, p. 197-220. Helmbold, Robert L. "Some Observations on the Use of Lanchester's Theory for Prediction," Operations Research 12 (1964): 778-781. Lanchester, Frederick William. "Mathematics in Warfar-e," The World of Mathematics, v4. New York: Simon and Schuster, 1956. p. 2178-2148. Newman, J. The World of Mathematics, Vol Simon and Schuster, 1956. Schneider, James J. of Operational Art,"

IV.

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"The Loose Marble--and the Origins Parameters, March 1989: 85-99.

Schneider, James J. "The Theory of the Empty Battlefield," Journal of the Royal United Services Institute for Defence Studies, Sept 1987: 37-44.

Papers and Unpublished

Works

Bigelman, Paul A. "An Analysis of a Lanchester-Type Combat Model Reflecting Attrition Due to Unit Deterioration and Ineffective Combatants," Thesis, Monterey, Ca: Naval Postgraduate School, March 1978. Farmer, William T. "A Survey of Approaches to the Modeliig of Land Combat," Thesis, Monterey, Ca: Naval Fostgraduate School, 1980. Johnson, Ronald L. "Lanchesters Square Law in and Practice." Monograph, Fort Leavenworth, KS: of Advanced Military Studies, March 1990.

Theory School

Schmieman, William A. "The Use of Lanchester Type Equations in the Analysis of Post Military Engagements." Dissertation, Atlanta, GA: Georgia Institute of Technology. Aug 1967. Schneider, James J. "The Exponential Decay of Armies in Battle," Theoretical Paper No.1, Fort Leavenworth, KS: School of Advanced Military Studies, 1985.