## A New Look At Euclid's Second Proposition (1993)

Venue: | The Mathematical Intelligencer |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Toussaint93anew,

author = {Godfried Toussaint},

title = {A New Look At Euclid's Second Proposition},

journal = {The Mathematical Intelligencer},

year = {1993},

volume = {15},

pages = {12--23}

}

### OpenURL

### Abstract

There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well known results is Mohr's proof in 1672 that all constructions that can be executed with straight-edge and compass can be carried out with compass alone. The earliest such proof of the equivalence of models of computation is due to Euclid in his second proposition of Book I of the Elements in which he establishes that the collapsing compass is equivalent in power to the modern compass. Therefore in the theory of equivalence of models of computation Euclid's second proposition enjoys a singular place. However, like much of Euclid's work and particularly his constructions involving cases, his second proposition has received a great deal of criticism over the centuries. Here it is argued that it is Euclid's early Greek commentators and more recent expositors and translators that are at fault and that Euclid's original...

### Citations

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The Design and Analysis of Computer Algorithms
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(Show Context)
Citation Context ... with each of these operations. For example, one of the preferred conceptually abstract models or ideal machines in which many geometric algorithms are compared is the Real RAM (Random Access Machine =-=[AHU74]-=-) in which each unit of storage space is capable of holding a real number of unlimited precision and in which the primitive operations that can be performed in one unit of time include the arithmetic ... |

1761 |
Computational Geometry An Introduction
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(Show Context)
Citation Context ...roduction In the modern comparative study of geometric algorithms it is imperative to first define the models of computation, i.e., the characteristics of the machine that will execute the algorithms =-=[PS85]-=-. A model of computation specifies the number of processors used, whether they are used sequentially or in parallel, the primitive operations allowed and the cost associated with each of these operati... |

88 |
What is mathematics
- Courant, Robbins, et al.
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(Show Context)
Citation Context ...lapsing compass, 4) the compass, 5) the fixed-aperture compass, 6) the compass with aperture bounded from above, and 7) the compass with aperture bounded from below just to name a few [Sm61], [Ho70], =-=[CR81]-=-, [Ko86]. The collapsing compass deserves elaboration here. With the regular compass one can open it, lock it at a chosen aperture and lift it off the works- space to some other location to draw a cir... |

50 |
The Thirteen Books of Euclid’s Elements
- Heath
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(Show Context)
Citation Context ... Being what it was required to do. End of proof. We remark here that Pedoe's figure, shown in Fig. 2.2, is considerably different from those in other sources on Euclid such as Heiberg [He1883], Heath =-=[He28]-=- and Dijksterhuis [Di55] for example. Much more serious, however, is the fact that Algorithm P given by Pedoe is incorrect! It is clear that for a solution to be obtained by Algorithm P it is crucial ... |

49 | Elementary Mathematics from an Advanced Standpoint: Geometry - Klein - 1940 |

22 |
Journey Through Genius: The Great Theorems of Mathematics
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(Show Context)
Citation Context ...e is drawn, but very many of Euclid's earlier proofs fail before this test... The value of his work as a masterpiece of logic has been very grossly exaggerated." Again, in the words of William Du=-=nham [Du90]: &qu-=-ot;Admittedly, when he allowed himself to be led by the diagram and not the logic behind it, Euclid committed what we might call a sin of omission. Yet nowhere in all 465 propositions did he fall into... |

12 |
Proclus: A commentary on the first book of Euclid’s Elements
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(Show Context)
Citation Context ...from a lack of deep understanding of his original work due in part to the writings of the early Greek commentators of the Elements such as Heron and Theon of Alexandria and others reviewed by Proclus =-=[Mo70]-=- in the 5th century and exacerbated by a 12th century Latin translation of an Arabic manuscript by Adelard of Bath [Bu83] and many English scholars of the 19th century. Furthermore, if we judge the or... |

3 |
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(Show Context)
Citation Context ...wer operations than did Euclid's original constructions. Another active area of research has been to analyze and compare different machine models in terms of their computational power [Ho70], [CR81], =-=[Av87]-=-, [Av90]. For example, in 1672 Georg Mohr [Mo1672] and in 1797 the Italian geometer Lorenzo Mascheroni [Ma1797] independently proved that any construction that can be carried out with a straight edge ... |

3 |
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Citation Context ...sal of the first printed edition of the Latin translation of the Arabic (Ishaq-Thabit) version of Euclid's Elements believed to be made by the monk Gerard of Cremona in Toledo during the 12th century =-=[Bu84]-=- following its discovery in Bagdad, would also appear to be more convincing than examining English texts of the 17th, 18th and 19th centuries. Indeed, apart from the fact that the letters E and F in H... |

3 |
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Citation Context ...compass, 4) the compass, 5) the fixed-aperture compass, 6) the compass with aperture bounded from above, and 7) the compass with aperture bounded from below just to name a few [Sm61], [Ho70], [CR81], =-=[Ko86]-=-. The collapsing compass deserves elaboration here. With the regular compass one can open it, lock it at a chosen aperture and lift it off the works- space to some other location to draw a circle with... |

3 |
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- Mascheroni
(Show Context)
Citation Context ...and compare different machine models in terms of their computational power [Ho70], [CR81], [Av87], [Av90]. For example, in 1672 Georg Mohr [Mo1672] and in 1797 the Italian geometer Lorenzo Mascheroni =-=[Ma1797]-=- independently proved that any construction that can be carried out with a straight edge and a compass can be carried out with a compass alone and Jacob Steiner proved in 1833 that the straight edge i... |

3 |
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(Show Context)
Citation Context ...uctions. Another active area of research has been to analyze and compare different machine models in terms of their computational power [Ho70], [CR81], [Av87], [Av90]. For example, in 1672 Georg Mohr =-=[Mo1672]-=- and in 1797 the Italian geometer Lorenzo Mascheroni [Ma1797] independently proved that any construction that can be carried out with a straight edge and a compass can be carried out with a compass al... |

3 |
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(Show Context)
Citation Context ...concerning the question of cases. For one thing he has been blamed of giving proofs of correctness that depend severely on the figure accompanying the proof. For example according to Bertrand Russell =-=[Ru51]: &qu-=-ot;A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test... The value of his work as a masterpiece of logic has been ver... |

3 |
Jacob Steiner’s geometrical constructions with a ruler given a fixed circle with its center,” (translated from the German 1833 edition
- Stark, Archibald
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(Show Context)
Citation Context ...d a compass can be carried out with a compass alone and Jacob Steiner proved in 1833 that the straight edge is equivalent in power to the compass if the former is afforded the use of the compass once =-=[SA48]-=-. To remind the reader that the straight edge and compass are not yet obsolete computers we should point out that the Mohr-Mascheroni result was strengthened as recently as in 1987 by Arnon Avron [Av8... |

3 |
Methods of geometry
- Smith
- 2000
(Show Context)
Citation Context ...a flurry of activity in England with regards to the writing of text-books on the topic of Euclid's Elements for use in the schools and colleges. A sample of several of these books [La1861], [To1876], =-=[Sm1879]-=-, and [HS1887], yields a common (apart from some trivial notational differences) algorithm and illustrative figure for Euclid's second proposition. However, both the algorithm and figure are quite dif... |

3 |
The Ruler in Geometrical Constructions
- Smogorzhevskii
- 1961
(Show Context)
Citation Context ...uler, 3) the collapsing compass, 4) the compass, 5) the fixed-aperture compass, 6) the compass with aperture bounded from above, and 7) the compass with aperture bounded from below just to name a few =-=[Sm61]-=-, [Ho70], [CR81], [Ko86]. The collapsing compass deserves elaboration here. With the regular compass one can open it, lock it at a chosen aperture and lift it off the works- space to some other locati... |

2 |
The first translation of Euclid's elements into English and its source
- Archibald
- 1950
(Show Context)
Citation Context ...ears to contradict the belief that the first English translation of Euclid's Elements to be printed was translated by H. Billingsley, printed by the famous English printer John Day and issued in 1570 =-=[Ar50]-=-, [Sh28]. What is worth noting about the algorithms in all three of these texts, however, is that 1) they are all identical to each other, 2) like Algorithm 19C, they are ambiguous but, 3) unlike all ... |

2 |
A History of Pi, The Golem
- Beckman
- 1971
(Show Context)
Citation Context ...rom limb in the streets of Alexandria by an enraged mob led by Bishop Cyril [La41]. Finally, the Arabs invaded Egypt in 646 A.D. and General Amr ibn-al-As burned the remaining books allegedly because =-=[Be71] "if -=-the books agreed with the Koran they were superfluous; if they disagreed they were pernicious." In short, in all likelihood Euclid's original algos- rithm went up in smoke. In spite of the critic... |

2 | The Elements of Euclid - Todhunter - 1876 |

1 |
Construction problems - why and how
- Avron
- 1989
(Show Context)
Citation Context ... has implications for education. It has already been argued that Euclidean construction problems provide an excellent method of teaching high school students constructive proofs of existence theorems =-=[Av89]-=-. The work presented here suggests that Euclidean constructive geometry can be used as an ideal medium for teaching many of the most modern concepts concerning the design and analysis of algorithms, t... |

1 |
The First Latin Translation of Euclid's Elements Commonly Ascribed to
- Busard
- 1983
(Show Context)
Citation Context ... Elements such as Heron and Theon of Alexandria and others reviewed by Proclus [Mo70] in the 5th century and exacerbated by a 12th century Latin translation of an Arabic manuscript by Adelard of Bath =-=[Bu83]-=- and many English scholars of the 19th century. Furthermore, if we judge the original algorithm and proof of correctness of Euclid's Proposition 2 using today's highest standards in the field of compu... |

1 |
The Medieval Latin Translation of Euclid's Elements
- Busard
- 1987
(Show Context)
Citation Context ...Let the straight lines AE, BF be produced in a straight line with DA, DB." In another 12th century Sicilian Latin translation (of unknown authorship) of Euclid's Elements made directly from the G=-=reek [Bu87] Step 3 is stat-=-ed as follows: "Educantur in directo rectis DA et DB recte AE et BF." This translates to "Lead forth the straight lines AE and BF in a straight line with (in the direction of) the strai... |

1 |
Euclidis Elementorum, Jonae Rosae, Francofurti
- Clavio
(Show Context)
Citation Context ...respectively. It is also much more difficult to find copies of these earlier editions. I have held in my hands only two editions from the 18th century [Wi1703], [Ba1705] and one from the 17th century =-=[Cl1654]-=-, having found all three in the special collection of the library at Queens University in Kingston, Ontario. The 1705 manuscript by Issac Barrow (from Trinity College, Cambridge) has the additional di... |

1 |
The First Book of Euclid's Elementa
- Dijksterhuis
- 1955
(Show Context)
Citation Context ...red to do. End of proof. We remark here that Pedoe's figure, shown in Fig. 2.2, is considerably different from those in other sources on Euclid such as Heiberg [He1883], Heath [He28] and Dijksterhuis =-=[Di55]-=- for example. Much more serious, however, is the fact that Algorithm P given by Pedoe is incorrect! It is clear that for a solution to be obtained by Algorithm P it is crucial that the circle centered... |

1 |
On the Resolution of Doubts in Euclid's Elements and Interpretation of Its Special Meanings, University of Istanbul, 1041 A.D., facsimile edition published by the Institute for the History of Arabic-Islamic Science at the Johan Wolfgang Goethe University,
- al-Haytham
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(Show Context)
Citation Context ...f Doubts in Euclid 's Elements and Interpretation of Its Special Meanings written in 1041 A.D. by Ibn al-Haytham. A copy of this book made in 1084 A.D. was found in the University of Istanbul Library =-=[Ha1041]-=-. As the title suggests this is not a translation of the Elements but a discussion about well known criticisms of Euclid's work. In discussing Euclid's second proposition al-Haytham discusses four bas... |

1 |
Euclide: Les Elements, Editions du Centre National de la Recherche Scientifique
- Kayas
- 1978
(Show Context)
Citation Context ...s Elements a translator can become a traitor. An example will remove any doubt. The most accurate and definitive Greek version of the Elements, the Heiberg edition, was translated in 1978 into French =-=[Ka78]-=-. The book [Ka78] contains, very conveniently, all the propositions in both Greek and French side by side. The introduction contains an interesting discussion on the problems of translation which we p... |

1 |
A Text-Book of Euclid's Elements
- Hall, Stevens
- 1887
(Show Context)
Citation Context ...tivity in England with regards to the writing of text-books on the topic of Euclid's Elements for use in the schools and colleges. A sample of several of these books [La1861], [To1876], [Sm1879], and =-=[HS1887]-=-, yields a common (apart from some trivial notational differences) algorithm and illustrative figure for Euclid's second proposition. However, both the algorithm and figure are quite different from Pe... |

1 |
Alexandria - Shrine of mathematics
- Langer
- 1941
(Show Context)
Citation Context ...ning books. The last mathematician alive in Alexandria, a woman by the name of Hypatia and daughter of Theon was torn limb from limb in the streets of Alexandria by an enraged mob led by Bishop Cyril =-=[La41]. Fin-=-ally, the Arabs invaded Egypt in 646 A.D. and General Amr ibn-al-As burned the remaining books allegedly because [Be71] "if the books agreed with the Koran they were superfluous; if they disagree... |

1 |
The First Six Books of the Elements
- Lardner
- 1861
(Show Context)
Citation Context ...shed) there existed a flurry of activity in England with regards to the writing of text-books on the topic of Euclid's Elements for use in the schools and colleges. A sample of several of these books =-=[La1861]-=-, [To1876], [Sm1879], and [HS1887], yields a common (apart from some trivial notational differences) algorithm and illustrative figure for Euclid's second proposition. However, both the algorithm and ... |

1 |
Geometry and the Liberal
- Pedoe
- 1976
(Show Context)
Citation Context ...l algorithm and proof are beyond reproach. Since this proposition uses Proposition 1 to obtain a solution we begin by outlining the latter. 2. Euclid's First Two Propositions According to Pedoe Pedoe =-=[Pe76]-=- contains a lively discussion of Euclid's elements of geometry applied to painting, sculpture and architecture throughout recent history and to illustrate Euclid's method he presents the first two pro... |

1 | Oeuvres D'Euclide - Peyrard, Les - 1819 |

1 |
Elements of Geometry; Containing the First Six
- Playfair
- 1831
(Show Context)
Citation Context ...the extremity of BC connected to A in Step 1, thus avoiding one of Taylor's problems. Another variation occurs in a much earlier Scottish book on Euclidean geometry published in 1831 by John Playfair =-=[Pl1831]-=- which has a variation of Algorithm 19C. In this book we are asked to extend DA and DB to E and F respectively and thus the ambiguity of Algorithm 19C is also present here. However, unlike Algorithm 1... |

1 |
The first English Euclid
- Shenton
- 1928
(Show Context)
Citation Context ...contradict the belief that the first English translation of Euclid's Elements to be printed was translated by H. Billingsley, printed by the famous English printer John Day and issued in 1570 [Ar50], =-=[Sh28]-=-. What is worth noting about the algorithms in all three of these texts, however, is that 1) they are all identical to each other, 2) like Algorithm 19C, they are ambiguous but, 3) unlike all other al... |

1 |
Euclid's Elements of Geometry
- Taylor
(Show Context)
Citation Context ...A similar problem exists with Step 6. The ambiguities observed in the algorithms described in [Sm1879], and [HS1887] which are exemplified here as Algorithm 19C are absent in the exposition by Taylor =-=[Ta1895]-=-, if not in the body of the algorithm at least in the subsequent discussion where it is indicated that we are free to choose one or the other alternative as is Step 1. It is therefore instructive to e... |

1 |
Computational geometric thinking as kinesthetic thinking
- Toussaint
- 1984
(Show Context)
Citation Context ...tely afterwards. Even today too many writers still publish geometric algorithms without including a proof of correctness in spite of the many geometric algorithms that have been found to be incorrect =-=[To84]-=-. These authors could certainly take a lesson here from Euclid. Sometimes, as we shall see below, the algorithms in the Elements include unnecessary steps for obtaining the solution but these steps ha... |

1 |
Elements of Euclid Explained and Demonstrated in a New and Most Easy Method
- Williams
- 1703
(Show Context)
Citation Context ...bout 325 and 280 in the 18th and 17th centuries, respectively. It is also much more difficult to find copies of these earlier editions. I have held in my hands only two editions from the 18th century =-=[Wi1703]-=-, [Ba1705] and one from the 17th century [Cl1654], having found all three in the special collection of the library at Queens University in Kingston, Ontario. The 1705 manuscript by Issac Barrow (from ... |