desargues's theorem
Very RareTechnical/Academic
Definition
Meaning
A fundamental statement in projective geometry concerning the perspectivity of two triangles.
In mathematics, specifically in projective geometry, Desargues's theorem states that two triangles are perspective from a point if and only if they are perspective from a line. This theorem is foundational, as projective spaces that satisfy Desargues's theorem can be coordinatized with a division ring.
Linguistics
Semantic Notes
The term is almost exclusively used in mathematical contexts, particularly geometry and algebra. It is a proper noun referring to a specific theorem named after the French mathematician Gérard Desargues. The possessive form ('Desargues's') is standard in mathematical literature, though 'Desargues' theorem' is also seen.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling conventions (e.g., 'theorem' vs. 'theorem') are identical. The possessive form 'Desargues's' is common in both, though some British style guides might prefer 'Desargues''.
Connotations
Identical technical connotations in both dialects.
Frequency
Extremely low frequency in general language, confined to specialised mathematical discourse in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
[Subject] proves/uses/applies Desargues's theorem.Desargues's theorem holds/states that...According to Desargues's theorem, ...Vocabulary
Synonyms
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Exclusively used in advanced mathematics, particularly in geometry courses, research papers, and textbooks on projective geometry or geometric algebra.
Everyday
Not used.
Technical
Core term in projective geometry, computer graphics (for understanding perspective), and some branches of engineering mathematics.
Examples
By Part of Speech
verb
British English
- The configuration desargues.
- We can desargue these two triangles to prove collinearity.
American English
- The triangles desargue from that point.
- You need to desargue the figure to see the perspective line.
adjective
British English
- A Desarguesian plane.
- The configuration is non-Desarguesian.
American English
- A Desarguesian projective space.
- These are Desarguesian properties.
Examples
By CEFR Level
- We learned about an important theorem in geometry today.
- In the lecture on projective geometry, the professor explained Desargues's theorem using a diagram of two triangles.
- The proof relies crucially on Desargues's theorem, which establishes the necessary condition for the triangles to be perspective from both a point and a line.
Learning
Memory Aids
Mnemonic
Think: 'Desargues decides if two triangles agree from a POINT of view and a LINE of sight.' (Point and line are the dual elements in the theorem).
Conceptual Metaphor
GEOMETRIC RELATIONSHIPS ARE PERSPECTIVES (The theorem metaphorically frames a geometric condition as a matter of having a shared viewpoint).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating 'theorem' as 'тезис' (thesis). The correct term is 'теорема'.
- The name 'Desargues' is not transliterated, it remains 'Дезарг' in Cyrillic.
- The possessive 's is not typically reflected in Russian, so it's simply 'теорема Дезарга'.
Common Mistakes
- Mispronouncing 'Desargues' as /ˈdɛsɑːrɡ/ (it is /dəˈzɑːrɡ/).
- Misspelling as 'Desargue's theorem' (missing the final 's').
- Incorrectly applying it to non-projective geometries.
- Confusing it with Pascal's theorem or other projective theorems.
Practice
Quiz
In which field is Desargues's theorem a fundamental result?
FAQ
Frequently Asked Questions
Gérard Desargues (1591–1661) was a French mathematician and engineer, considered one of the founders of projective geometry.
In projective spaces of dimension three or higher, it is always true. However, there exist projective planes (non-Desarguesian planes) where the theorem does not hold.
In projective geometry, the dual of Desargues's theorem is its own converse, making it a self-dual theorem.
It is a foundational result that allows the introduction of coordinates into projective geometry, linking geometric properties to algebraic structures.