mathematics

Pasch's theorem

Pasch's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pasch's theorem rather than just read about it. In short: In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates. Statement The statement is as follows: [Here, for example, (a, b, c) means that point b lies between points a and c.] Hilbert's use of Pasch's theorem David Hilbert originally included Pasch's theorem as an axiom in his modern treatment of Euclidean ge…

Key takeaways

Pasch's theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pasch's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pasch's theorem from memory before moving on to harder problems.

Reference excerpt

In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates.

Statement The statement is as follows: [Here, for example, (a, b, c) means that point b lies between points a and c.]

Hilbert's use of Pasch's theorem David Hilbert originally included Pasch's theorem as an axiom in his modern treatment of Euclidean geometry in The Foundations of Geometry (1899). However, it was found by E. H. Moore in 1902 that the axiom is redundant, and revised editions now list it as a theorem. Thus Pasch's theorem is also known as Hilbert's discarded axiom. Pasch's axiom, a separate statement, is also included and remains an axiom in Hilbert's treatment.

Worked examples

Example 1 — a first encounter with Pasch's theorem

Start with the simplest possible case. Write down what Pasch's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pasch's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pasch's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pasch's theorem

In research: Pasch's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pasch's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pasch's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary geometry stubs, Euclidean plane geometry, Foundations of geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Pasch's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pasch's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pasch's theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pasch's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pasch's theorem in simple terms?

In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates. Statement The statement is as follows: [Here, for example, (a, b, c) means that point b lies between points a and c.] Hilbert's use…

Why does Pasch's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pasch's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pasch's theorem.