The Battle of Trafalgar

1
The Battle of Trafalgar

• Historic Battles

2
Fighting the Battle of Trafalgar There is a
theory which says that the rate at which the size
of one group changes is directly proportional to
the size of the opposing group. Consider two
opposing groups of sizes n and m where we
assume that n is greater than m. Then
and
where c1 and c2 are constants.
On multiplying both sides of the first equation
by 2c2n and both sides of the second equation by
2c1m we obtain
which integrates to give c2n2 c1m2 constant
3
This can be rewritten as
where c is a constant.
The ratio will give a measure of the
relative strengths of the two groups for if c1
lt c2 then group n is the superior group in
battle, and if c1gtc2 then group m is the
superior. If c1 c2 then the two groups (or the
sum of the individual units which make up the
groups) could be considered to be of equal
strengths. If at some stage of a battle n N
and m M, say,
then we can calculate the constant c.
We have and so
.
4
Using this equation we can find the number of
survivors S in a battle. Letting n S when
m 0 , the number of survivors S is given by S
int ( N2 M2 1/2) where int is the
integer part of the number if we consider whole
survivors only. We could use for example the
number of guns in each fleet to work out a value
for the ratio and this would act as a measure
of the relative strength of each fleet.
We now wish to apply our theory to a great sea
battle which took place in 1805, namely, the
Battle of Trafalgar. If Admiral Horatio Nelson
had been gracious enough to take our theory on
board, what tactics might he have used in
fighting the battle of Trafalgar? For this
battle, we shall assume that c1 c2 i.e. that
the opposite forces are equally matched in
fighting strength and hence we have the
survivors in a clash being given by
S int ( N2 M2 )1/2
5
By survivor we mean a ship which is still capable
of fighting. A ship considered lost in a
battle does not necessarily mean that the ship
has been sunk. It could mean that the ship has
been badly damaged and has lost its fighting
capabilities. It could have a lot of casualties
in its crew or lost a lot of guns. It could
also have lowered its colours and simply sailed
away from the action or have been scuttled to
prevent it falling into enemy hands. The Battle
of Trafalgar was fought on the 21st of October
1805 off Cape Trafalgar on the Spanish coast,
between the combined fleets of Spain and France
and the Royal Navy. It was the last great sea
action of the period and its significance to the
outcome of the war in Europe is still debated by
historians. The British fleet was led by Admiral
Horatio Nelson and the combined French and
Spanish fleets were led by Admiral Villeneuve.
The British fleet of 33 ships did battle with
the combined Spanish and French fleets of 40
ships.
6
So what does our theory tell us? If Nelson
attacked as a single unit and took on the enemy
in a single battle then the above theory tells us
that
Survivors S int( 402 3321/2 ) 22 A
combined French and Spanish victory with 22
ships surviving.
7
The theory tells us that we will need to employ a
tactic and consider ways in which we can split
the enemy force into two parts and at the same
time split our own fleet, and fight two separate
battles. The survivors of these battles would
then fight it out in a final battle. If we do
this, can we win the battle of Trafalgar? One
possibility would be to try to split the enemy
force into 2 equal groups of 20 ships and at the
same time break your own force into say 8 ships
and 25 ships. The object of this tactic would be
to have a small force of 8 ships acting as a
diversionary force taking on a larger force and
hopefully holding them and giving you time for
your larger force of 25 ships to deal
effectively and swiftly with the other half of
the enemy force. In a sea battle, ships are
spread out and are heavily dependent on the wind
so the tactic of splitting an enemy force is not
a difficult one.
8
This will lead to two separate battles and the
outcomes are Top battle- survivors g int
(202 821/2 ) 18 Bottom battle- survivors
h int (252 2021/2 ) 15
The final battle takes place between the two sets
of survivors Final battle- survivors s int
(182 1521/2 ) 9 and a defeat for Nelson at
Trafalgar.
9
Is there a way in which we can split the force
into two parts and achieve victory? If we look at
other ways which Nelson might have split his
force and consider a p to 33 - p split with
the enemy split in half then the following
result is obtained
The outcomes of these separate battles are Top
battle- Let g int ( 202 p21/2 )
Bottom battle- Let h int ( (33-p)2
2021/2 ) Where g is the number of survivors
on the combined French/Spanish side and h is
the number of survivors in the British fleet. For
victory we require h gt g So we wish to find p
such that (33-p)2 202 gt 202 p2 This
reduces to 2p2 66p 289 gt 0 which solves
to give p lt 5.2 or p gt 27.8 i.e. a small force
of 4 ships or less is necessary to win the battle.
10
Suppose you are unable to split the enemy into
two equal parts, could we win the battle with an
unequal split in the enemy forces like for
example that shown in the diagram below.
The outcomes of these separate battles are Top
battle- Let g int ( 262 p21/2 )
Bottom battle- Let h int ( (33-p)2
1421/2 ) where g is the number of survivors
on the combined French/Spanish side and h is
the number of survivors in the British fleet. For
victory we require h gt g So we wish to find p
such that (33-p)2 142 gt 262 p2 This
reduces to 2p2 66p 217 gt 0 which solves to
give p lt 3.47 or p gt 62.5 i.e. a very small
force of 2 ships has to be sent to face the 26
ships of the enemy, hardly a practical
proposition. This result suggests that we must
try to split the enemy fleet into two parts and
have these parts as nearly equal as possible.
11
Can we find p in terms of x so that we can
order the correct split in our forces given the
split in the enemy force.
The outcomes of these separate battles are Top
battle- Let g int ( (20x)2 p21/2
) Bottom battle- Let h int ( (33-p)2
(20-x)21/2 ) where g is the number of
survivors on the combined French/Spanish side
and h is the number of survivors in the British
fleet. For victory we require h gt g So we wish to
find p such that (33-p)2 (20-x)2 gt
(20x)2 p2
This reduces to p2 33p 144.5 gt x2 ,
which simplifies to give p lt 16.5 (x2
127.75)1/2 The table in the next slide shows how
to react to the unequal split to ensure victory
12
So we have our plan in place to win the battle of
Trafalgar. If we can split the enemy into 2
equal parts and send 4 or less ships as a
diversion to tackle one half of the enemy while
the rest defeat the other half then we win. If
we cannot split the enemy into 2 equal groups
then select p ships according to x in the
table above. These tactics will ensure a
victory. The big assumption we have made in our
model is that all the fighting units (ships) are
of equal strength as were the strengths of both
fleets in total which was not in fact the case.
While about half of the British fleet was 74
gun ships, some like the Victory, Royal
Sovereign and the Brittania were 100 gun ships
and others like the Phoebe and the Sirius had
only 36 guns. Total strength in guns on the
British fleet was about 2300 guns. The
Spanish/French fleet consisted of a total of
about 2600 guns and they also had more than half
the fleet consisting of 74 gun ships. They also
had the massive Santissima Trinidad which
carried 130 guns and the Santa Ana and Principe
de Asturias both with 112 guns. As well as the
guns, there was also the quality of the crews
themselves to consider. The units making up the
fleets were therefore not all equal.
13
So how did our model do in practice? We will see
that we got some ideas about the battle correct
if we look at what actually did happen at
Trafalgar. We did predict trying to break the
enemy force into two groups and fighting one
battle with a large part of our own fleet. Our
model possibly took the battle a lot further than
the actual battle itself went and it did not
allow at all for the fighting genius of those
involved in the battle who knew not only there
own ships but also those of the enemy and also
to the superior gunnery and ship handling skills
of the British crews. Could we improve our
model at all to achieve a better victory? Is
there a better tactic to fight this battle?
What about splitting the fleet into three
groups? Should we think about grouping our
ships not according to numbers of ships but
according to total firepower and fighting the
battle as firepower groups? And what
actually happened in the battle of Trafalgar?
14
The diagram on the next slide shows the
arrangement of the ships at the start of the
battle. The idea Nelson had was to split the
enemy fleet. Most of his fire power was
concentrated into 2 columns of ships which
sailed right into the heart of the enemy fleet
and this is where most of the action took place.
Column 1 which comprised a total of 11 ships
including Nelsons ship carried a total of 904
guns into the battle while column 2 which
comprised a total of 15 ships carried 1180 guns
into the battle. In the North, the small
British Frigate fleet had a total of 228 guns
and were involved in the action. The outcome? Of
the enemy ships, 17 were taken by the British
fleet, 6 were scuttled, 4 were wrecked in a gale
which followed the battle and the rest sailed
away from the battle. The British lost 449 men
killed and 1242 wounded, some of whom
subsequently died and the French and Spanish
fleets lost 4408 men killed and 2545 wounded.
Admiral Nelson was killed in action.
15
Opposing fleets at Noon 21st October 1805