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10Principle · Group 2

Square Law of Strength

Lanchester’s square law in business

“If you concentrate your forces, you win not just a little more but disproportionately more – and if you spread them out, you lose disproportionately.”

Imagine two groups in which every member can reach every opponent. The larger group dishes out more because more people are acting at the same time. And it takes punishment better because the opponent’s hits are spread across more shoulders.

Group 2: Forces and ResourcesT1 Classical Theory of War
Go to the in-depth section ↓
Ten teal dots face five mist-blue dots; above both, the numbers grow into their squares, 100 and 25, and fine lines show how the five fall while only one or two of the 10 fade. Then a long mist-blue row faces a smaller teal group; two golden lines cut the row into parts, the teal dots mass in gold against one part after another, and finally all the dots close into a circle.
In short

If you concentrate your forces, you win not just a little more but disproportionately more – and if you spread them out, you lose disproportionately.

What it’s about: Imagine two groups in which every member can reach every opponent. The larger group dishes out more because more people are acting at the same time. And it takes punishment better because the opponent’s hits are spread across more shoulders. The two advantages multiply. In 1916, Frederick William Lanchester captured this in a formula, known as Lanchester’s square law: fighting strength grows with the square of the numbers. Ten units against five of equal strength then stand not at two to one but at 100 to 25.

In depth

The principle in detail

Three steps through the principle: grasp it, place it, put it to work. Open each topic on its own.

The core messageCore statement and 3 clarifications
In essence

Where forces act at the same time and reinforce one another, concentration pays off disproportionately, and fragmentation costs disproportionately.

It does not say
That the formula makes precise predictions. It is a model; training, leadership, morale and terrain do not appear in it.
It does not say
That many people automatically add up to a concentrated force. If they do not act together, they are fragmented, however numerous they are.
It also says
Concentration has costs. On the way there, you are vulnerable, and other places are left open.
The explanationNever take on a superior force all at once; defeat its parts one after another with local superiority.

Two rules follow from this.

  • Never take on a superior force all at once; defeat its parts one after another with local superiority.
  • And never split your own forces.

A third consequence is uncomfortable: in the model, quality counts only linearly, while numbers count squared. If you had only half as many forces, you would have to be four times as good.

The question

Where are you scattering forces that would achieve far more together?

The signs3 signs when you heed it · 3 when you overlook it
When you heed it
  • You would rather put a whole team on a few projects than one person each on many.
  • You take on a big task in parts instead of tackling everything at once.
  • You present a decision to the outside world together with the others who share responsibility for it.
When you overlook it
  • Every project has exactly one person carrying it alone.
  • Budgets are spread across many fields and show up nowhere.
  • You and your colleagues appear separately and are won over one by one.
The backgroundFrom Frederick William Lanchester’s book “Aircraft in Warfare: The Dawn of the Fourth Arm” (1916). · T1 Classical Theory of War
  1. 1916
    OriginFrederick William Lanchester

    Aircraft in Warfare: The Dawn of the Fourth Arm

  2. Tradition T1
    TraditionClassical Theory of War

    Carl von Clausewitz, Antoine-Henri Jomini, Helmuth von Moltke the Elder, Frederick William Lanchester

    See the tradition on the Origin page
Read more: the full background

From Frederick William Lanchester’s book “Aircraft in Warfare: The Dawn of the Fourth Arm” (1916). Independently of him, M. Osipov arrived at similar equations in Russia in 1915.

The research5 sources: 2 experiments, 1 classic, 1 review, 1 primary study · 1916–2023 · with limitations of the research
5 sources1916–2023 period2 experiments1 classic1 primary study1 review
Strength of evidence
  1. weak
  2. moderate
  3. strong

The model is mathematically clear, and a disproportionate effect of numbers can be seen in battle data, but it is markedly weaker than the square, and animal experiments partly contradict it.

  1. 1916
    Classic
    Frederick William Lanchester

    “Aircraft in Warfare,” Chapters V and VI: the square law for combat in which everyone can hit everyone; Nelson’s plan for Trafalgar as an example.

  2. 2004
    Primary study
    Thomas W. Lucas and Turker Turkes

    Fits Lanchester equations to the battles of Kursk and the Ardennes.

  3. 2005
    Experiment
    Nicola J. R. Plowes and Eldridge S. Adams

    Fights between fire ants do not confirm the square law.

  4. 2016
    Review
    Philip Chan

    Explains the model and its assumptions; historical battles are better fitted by an exponent between 1 and 1.5. The law works as a rule of thumb, not as a precise prediction.

  5. 2023
    Experiment
    Samuel J. Lymbery, Bruce L. Webber and Raphael K. Didham

    Australian meat ants against Argentine ants: the advantage of numbers is weaker than the square law predicts; in complex terrain, individual strength counts for more.

Where the research reaches its limits
  • The law applies only when all units fight at the same time and can reach every opponent.
  • It does not apply to one-on-one fights, to armies only part of which is engaged, or to area fire.
  • Morale, leadership and terrain do not appear in the formula.
  • In real battles, numbers have a disproportionate effect, but less than the square.
  • Applications to markets, elections or organizations are analogies, not laws of nature.
Read more: what research shows

In 1916, Frederick William Lanchester distinguished between two kinds of combat. If each fighter engages only one opponent, strength grows with the numbers. If everyone can hit every opponent, it grows with the square of the numbers. As evidence, he interpreted Horatio Nelson’s plan for Trafalgar. The equations were taken up widely in operations research. Comparisons with real battles, however, point more to an exponent between 1 and 1.5, as Philip Chan summarizes; Thomas W. Lucas and Turker Turkes studied Kursk and the Ardennes in this light. Experiments with animals do not paint a uniform picture either: in fire ants, Nicola J. R. Plowes and Eldridge S. Adams found no square law, and in fights between Australian meat ants and Argentine ants, the advantage of numbers was weaker than predicted.

The related principles07 Concentration of Force · 08 Relative Superiority · 09 Economy of Force · 77 Network Effects
Concentration of ForceRelative Superiority

To go deeper, see 07 Concentration of Force (the square law explains why concentration pays off so strongly) and 08 Relative Superiority (even local superiority in numbers leads to a clear victory)

Economy of Force

09 Economy of Force shows the costs of concentration elsewhere

Network Effects

77 Network Effects shows the same self-reinforcement through size in the marketplace

Read more: the full text

To go deeper, see 07 Concentration of Force (the square law explains why concentration pays off so strongly) and 08 Relative Superiority (even local superiority in numbers leads to a clear victory); 09 Economy of Force shows the costs of concentration elsewhere, and 77 Network Effects shows the same self-reinforcement through size in the marketplace.

The action steps5 things you can actually do

  1. Count how many projects in your area are each carried by a single person, and merge some of them.

  2. Concentrate budget and attention on a few fields where you can become visible.

  3. Break an overwhelming task into parts and take them on one after another with full strength.

  4. On important decisions, present a united front instead of making separate attempts.

  5. Use the formula as a thinking aid, not as a calculation: check whether your forces really act at the same time and reinforce one another.
The reflection questions3 questions for you or for a conversation
  1. Which projects in your area are each carried by one person alone, and which could you carry together?
  2. Where are you facing a big task all at once instead of tackling it in parts?
  3. On which decision would a united front be stronger than two separate attempts?

Take your time with one question. Write down your answer – or talk it over with someone you trust.

To finish
The picture to rememberThe animation once more, with an explanation
  1. Ten teal dots face five mist-blue dots
  2. Above both, the numbers grow into their squares, 100 and 25, and fine lines show how the five fall while only one or two of the 10 fade
  3. Then a long mist-blue row faces a smaller teal group
  4. Two golden lines cut the row into parts, the teal dots mass in gold against one part after another, and finally all the dots close into a circle
Strategy Check

Where are you spreading yourself thin?

Get your profile across all 12 groups in about 10 to 15 minutes.

Start the Strategy Check
In context

Group 2: Forces and Resources

Guiding question: Where do you commit your forces, and how long will they last?

Stage 3 of 5: Choose the path

Where the 92 principles come from: eight traditions of strategy, from classical war theory to dealing with uncertainty.

More about the origin

Background: WEGVISOR’s own selection from the strategy literature · More about the origin

Examples

The principle in practice

From the different spheres of society.

Government

Horatio Nelson, Trafalgar and the Square Law of Strength

At Trafalgar in 1805, the British fleet is smaller than the Franco-Spanish one. Horatio Nelson wins anyway because he cuts the enemy line and strikes only one part of the enemy at a time with concentrated force.

Square Law of Strength
Government

The SNP, the 2015 General Election and the Square Law of Strength

In the 2015 UK general election, the Scottish National Party wins 56 seats with 4.7% of the vote, while UKIP wins just one with 12.6%. What decides the outcome is not the number of votes but how concentrated they are.

Square Law of Strength
BusinessMedia

VHS versus Betamax and the Square Law of Strength

Sony launches Betamax in 1975; JVC follows with VHS in 1976. JVC licenses its format to many manufacturers, while Sony keeps control. The combined strength of the VHS alliance grows faster than any single company, and Betamax loses despite its good technology.

Square Law of Strength
GovernmentCulture & Sports

Movie: Gladiator

In Ridley Scott’s “Gladiator” (2000), Maximus and his fellow fighters are meant to die in the Colosseum as the losing side of a reenacted battle. He gets the individual fighters to fight together, and they defeat the chariots.

Square Law of Strength
FAQ

Questions about the Square Law of Strength

What does the Square Law of Strength say?

When two sides act against each other at the same time, their fighting strength in the model grows with the square of their numbers. Twice as many forces are then not twice as strong but four times as strong. Concentration pays off disproportionately, though less so in reality than in the model.

Where does the Square Law of Strength come from?

From the British engineer Frederick William Lanchester, who published it in 1916 in his book “Aircraft in Warfare.” Independently, M. Osipov developed similar equations in Russia in 1915.

What are some examples?

The classic case is Horatio Nelson’s victory at Trafalgar in 1805. The basic idea also shows up in the 2015 UK general election, when the concentrated SNP won far more seats than the scattered UKIP, and in the victory of VHS over Betamax.

How well is the Square Law of Strength supported by research?

Evidence: moderate – the model is mathematically clear, and a disproportionate effect of numbers can be seen in battle data, but it is markedly weaker than the square, and animal experiments partly contradict it.

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