Book XII of Euclid's Elements
E712323
Book XII of Euclid's Elements is a foundational mathematical text that develops the method of exhaustion to rigorously determine areas and volumes of curved figures such as circles, cones, and spheres.
All labels observed (24)
How this entity was disambiguated
This entity first appeared as the object of triple T8143994 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Book XII of Euclid's Elements Context triple: [Book XIII of Euclid's Elements, isPrecededBy, Book XII of Euclid's Elements]
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A.
Book XIII of Euclid's Elements
Book XIII of Euclid's Elements is the concluding book of Euclid’s mathematical treatise, focusing on the construction and properties of the five regular Platonic solids within the framework of classical Greek geometry.
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B.
Euclid's Elements
Euclid's Elements is an ancient Greek mathematical treatise that systematically presents the foundations of geometry, number theory, and mathematical proof.
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C.
Book X of the Elements
Book X of the Elements is a section of Euclid’s mathematical treatise that systematically develops the theory of irrational magnitudes and incommensurability.
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D.
Book V of the Elements
Book V of the Elements is the section of Euclid’s mathematical treatise that rigorously develops the general theory of proportion, foundational for real number and magnitude theory in geometry.
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E.
Euclides adauctus et methodicus
Euclides adauctus et methodicus is a 17th-century mathematical treatise by Guarino Guarini that expands and systematizes Euclidean geometry for advanced study and architectural application.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Book XII of Euclid's Elements Target entity description: Book XII of Euclid's Elements is a foundational mathematical text that develops the method of exhaustion to rigorously determine areas and volumes of curved figures such as circles, cones, and spheres.
-
A.
Book XIII of Euclid's Elements
Book XIII of Euclid's Elements is the concluding book of Euclid’s mathematical treatise, focusing on the construction and properties of the five regular Platonic solids within the framework of classical Greek geometry.
-
B.
Euclid's Elements
Euclid's Elements is an ancient Greek mathematical treatise that systematically presents the foundations of geometry, number theory, and mathematical proof.
-
C.
Book X of the Elements
Book X of the Elements is a section of Euclid’s mathematical treatise that systematically develops the theory of irrational magnitudes and incommensurability.
-
D.
Book V of the Elements
Book V of the Elements is the section of Euclid’s mathematical treatise that rigorously develops the general theory of proportion, foundational for real number and magnitude theory in geometry.
-
E.
Euclides adauctus et methodicus
Euclides adauctus et methodicus is a 17th-century mathematical treatise by Guarino Guarini that expands and systematizes Euclidean geometry for advanced study and architectural application.
- F. None of above. chosen
Statements (55)
| Predicate | Object |
|---|---|
| instanceOf |
ancient Greek mathematical text
ⓘ
book of Euclid's Elements ⓘ |
| author | Euclid ⓘ |
| contains |
Proposition 1 of Book XII
ⓘ
Proposition 10 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 11 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 12 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 13 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 14 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 15 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 16 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 17 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 18 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 19 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 2 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 20 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 21 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 22 of Book XII ⓘ Proposition 23 of Book XII ⓘ Proposition 24 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 25 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 26 of Book XII ⓘ Proposition 27 of Book XII ⓘ Proposition 28 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 29 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 3 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 30 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 4 of Book XII ⓘ Proposition 5 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 6 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
Proposition 7 of Book XII ⓘ Proposition 8 of Book XII ⓘ Proposition 9 of Book XII ⓘ
linked to:
Book XII of Euclid's Elements
|
| era | Hellenistic period ⓘ |
| focusesOn |
areas of curved figures
ⓘ
volumes of solids ⓘ |
| follows |
Book XI of Euclid's Elements
ⓘ
linked to:
Book XI of the Elements
|
| historicalSignificance | early rigorous treatment of limits ⓘ |
| influenced | development of integral calculus ⓘ |
| language | Ancient Greek ⓘ |
| mainMethod | method of exhaustion ⓘ |
| partOf | Euclid's Elements ⓘ |
| positionInWork | 12 ⓘ |
| precedes | Book XIII of Euclid's Elements ⓘ |
| proves |
areas of circles are proportional to squares of their diameters
ⓘ
volume of a cone is one third the volume of a cylinder on the same base and height ⓘ volume of a pyramid is one third the volume of a prism on the same base and height ⓘ volume relationships of spheres and cylinders ⓘ volumes of cones and cylinders are proportional to bases and heights ⓘ |
| studies |
circles
ⓘ
cones ⓘ cylinders ⓘ spheres ⓘ |
| subject | geometry ⓘ |
| uses | geometric series of inscribed figures ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Book XII of Euclid's Elements Description of subject: Book XII of Euclid's Elements is a foundational mathematical text that develops the method of exhaustion to rigorously determine areas and volumes of curved figures such as circles, cones, and spheres.
Referenced by (24)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements
linked to: Book XII of Euclid's Elements