Uses of construction in problems and theorems in Euclid’s Elements I–VI

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.

Original languageEnglish
Pages (from-to)403-452
Number of pages50
JournalArchive for History of Exact Sciences
Volume72
Issue number4
DOIs
Publication statusPublished - 2018 Jul 1

Fingerprint

Euclid
Theorem
Postulate
Straightedge
Geometric object
Euclid's Elements

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • History and Philosophy of Science

Cite this

Uses of construction in problems and theorems in Euclid’s Elements I–VI. / Sidoli, Nathan Camillo.

In: Archive for History of Exact Sciences, Vol. 72, No. 4, 01.07.2018, p. 403-452.

Research output: Contribution to journalArticle

@article{16a6f996b17c48249bfaf57e30078ffd,
title = "Uses of construction in problems and theorems in Euclid’s Elements I–VI",
abstract = "In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.",
author = "Sidoli, {Nathan Camillo}",
year = "2018",
month = "7",
day = "1",
doi = "10.1007/s00407-018-0212-4",
language = "English",
volume = "72",
pages = "403--452",
journal = "Archive for History of Exact Sciences",
issn = "0003-9519",
publisher = "Springer Verlag",
number = "4",

}

TY - JOUR

T1 - Uses of construction in problems and theorems in Euclid’s Elements I–VI

AU - Sidoli, Nathan Camillo

PY - 2018/7/1

Y1 - 2018/7/1

N2 - In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.

AB - In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.

UR - http://www.scopus.com/inward/record.url?scp=85049055039&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049055039&partnerID=8YFLogxK

U2 - 10.1007/s00407-018-0212-4

DO - 10.1007/s00407-018-0212-4

M3 - Article

AN - SCOPUS:85049055039

VL - 72

SP - 403

EP - 452

JO - Archive for History of Exact Sciences

JF - Archive for History of Exact Sciences

SN - 0003-9519

IS - 4

ER -