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Journal articles on the topic 'Angle trisection'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 April 2022

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1

De Catalina, Emiliano. "Angle Trisection, Bhaskara’s Proof, and Pythagorean Theorem." Recoletos Multidisciplinary Research Journal 9, no. 1 (June 3, 2021): 1–11. http://dx.doi.org/10.32871/rmrj2109.01.01.

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This paper deals with 1) angle trisection, 2) Bhaskara’s first proof, and 3) Pythagorean theorem. The purpose of this paper is threefold. First, to show a new, direct method of trisecting the 900 angle using unmarked straight edge and compass; secondly, to show Bhaskara’s first proof of the Pythagorean theorem (c2 = a2 + b2) as embedded in this new, direct trisection of the 900 angle; lastly, to show the derivation of the Pythagorean theorem from this trisection of the 900 angle. This paper employs the direct dissection method. It concludes by presenting four points: a) the concept of trisectability as distinct from concept of constructability; b) the trisection of the 900 angle as really a new, different method; c) Bhaskara’s first proof of the Pythagorean theorem as truly embedded in this trisection of the 900 angle and; d) another way of deriving Pythagorean theorem from this trisection of the 900 angle.

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2

Husyan Pour Shad, Mehryar. "Angle Trisection." JOURNAL OF ADVANCES IN MATHEMATICS 17 (September 16, 2019): 165–231. http://dx.doi.org/10.24297/jam.v17i0.8412.

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We seek to increase the development of science, but there are several fundamental questions about what is. Without solving the question is a false reflection of the history of science and the beginning of cognition. We know that their investigation and resolution, with the exception of rooting and knowledge of morphophonemic, do not come. Research on certain natural or pure mathematical phenomena is an example of my fundamental research that will lead to the definition of general principles and scientific theories.

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3

Rediske, Arthur Clair. "The Trisection of an Arbitrary Angle." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 2 (April 18, 2018): 7640–69. http://dx.doi.org/10.24297/jam.v14i2.7402.

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This paper presents an elegant classical geometric solution to the ancient Greek's problem of angle trisection. Its primary objective is to provide a provable construction for resolving the trisection of an arbitrary angle, based on the restrictions governing the problem. The angle trisection problem is believed to be unsolvable for compass-straightedge construction. As stated by Pierre Laurent Wantzel (1837), the solution of the angle trisection problem corresponds to an implicit solution of the cubic equation x cubed minus 3x minus 1 equals 0, which is algebraically irreducible, and so is the geometric solution of the angle trisection problem. The goal of the presented solution is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greek's tools of geometry (a classical compass and straightedge) by changing the problem from the algebraic impossibility classification to a solvable plane geometrical problem. Fundamentally, this novel work is based on the fact that algebraic irrationality is not a geometrical impossibility. The exposed methods of proof have been reduced to the Euclidean postulates of classical geometry.

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4

Muench, Donald L. "An Iterative Angle Trisection." College Mathematics Journal 38, no. 2 (March 2007): 82–84. http://dx.doi.org/10.1080/07468342.2007.11922222.

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5

Bowring, B. R. "TRISECTION OF AN ANGLE." Survey Review 31, no. 244 (April 1992): 354–56. http://dx.doi.org/10.1179/sre.1992.31.244.354.

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6

Leung, S. C. "TRISECTION OF AN ANGLE." Survey Review 31, no. 246 (October 1992): 493. http://dx.doi.org/10.1179/sre.1992.31.246.493.

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7

Ananthapadmanabhan, S. "Trisection of an angle." International Journal of Mathematical Education in Science and Technology 19, no. 6 (November 1988): 907–19. http://dx.doi.org/10.1080/0020739880190618.

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8

Alex, Kimuya M. "The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction)." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 4 (August 4, 2017): 7308–32. http://dx.doi.org/10.24297/jam.v13i4.6175.

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This paper is devoted to exposition of a provable classical solution for the ancient Greeks classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837),presented an algebraic proof based on ideas from Galois field showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greeks tools of geometry, and refutethe presented proof of angle trisection impossibility statement. The exposedproof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archimedes proposition of usinga marked straightedge construction [4], [11].

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9

Lamb, John F. "Trisecting an Angle—Almost." Mathematics Teacher 81, no. 3 (March 1988): 220–22. http://dx.doi.org/10.5951/mt.81.3.0220.

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Trisecting an angle with Euclidean tools, a straightedge and compass, has been proved to be impossible. However, many people today are still fascinated by the problem, and some even try to do it in spite of the proof to the contrary. The National Council of Teachers of Mathematics published a book on the subject titled The Trisection Problem (Yates 1971) that contains several methods proposed over the years. One method in particular, given by d'Ocagne, is easy and produces an angle that looks very close to the correct size, so close that only a proof that it fails will convince stubborn believers. The proof presented here uses only algebra and trigonometry and hopefully will show students how the three areas of algebra, trigonometry, and geometry can be interrelated.

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10

Lamb, John F. "Trisecting an Angle—Almost, Part 2." Mathematics Teacher 84, no. 1 (January 1991): 20–23. http://dx.doi.org/10.5951/mt.84.1.0020.

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When students study constructions in high school geometry, they usually make the following conjecture: Since it is easy to bisect a line segment, easy to bisect an angle, and not too difficult to trisect a line segment, there ought to be a way to trisect an angle. The students may even devise ways that they think will accomplish the task. The errors in these methods are sometimes hard to detect. This article highlights one way that has been discovered that comes very close to trisecting an angle with Euclidean tools. Several ways are discussed in one of the books in the Classics in Mathematics Education series published by the National Council of Teachers of Mathematics titled The Trisection Problem (Yates 1971). One of the methods discovered by d'Ocagne was examined in “Trisecting an Angle—Almost” (Lamb 1988). His method was very easy to do, but it was not as accurate as some of the other methods. One of those methods that is also relatively easy to do and gives a much better approximation than the d'Ocagne method for angles between 0 degrees and 90 degrees was discovered by Karajordanoff in 1928 (Yates 1971). The procedure is as follows.

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11

Martins, L. F., and I. W. Rodrigues. "Angle Trisection by Fixed Point Iteration." College Mathematics Journal 26, no. 3 (May 1995): 205. http://dx.doi.org/10.2307/2687344.

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12

Martins, L. F., and I. W. Rodrigues. "Angle Trisection by Fixed Point Iteration." College Mathematics Journal 26, no. 3 (May 1995): 205–8. http://dx.doi.org/10.1080/07468342.1995.11973697.

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13

Porter, A. F. "TRISECTION: THE EUCLIDEAN PROBLEM OF TRISECTING A RANDOM ANGLE AND PROOF OF IMPOSSIBILITY." Survey Review 31, no. 241 (July 1991): 167–74. http://dx.doi.org/10.1179/sre.1991.31.241.167.

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14

Greece, Nikolaos Vrettos. "Angle Trisection by Straightedge and Compass Only." IOSR Journal of Mathematics 10, no. 5 (2014): 48–51. http://dx.doi.org/10.9790/5728-10564851.

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15

Gleason, Andrew M. "Angle Trisection, the Heptagon, and the Triskaidecagon." American Mathematical Monthly 95, no. 3 (March 1988): 185. http://dx.doi.org/10.2307/2323624.

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16

Gleason, Andrew M. "Angle Trisection, the Heptagon, and the Triskaidecagon." American Mathematical Monthly 95, no. 3 (March 1988): 185–94. http://dx.doi.org/10.1080/00029890.1988.11971989.

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17

Koshkin, Sergiy. "100.28 Mixing angle trisection with Pythagorean triples." Mathematical Gazette 100, no. 549 (October 17, 2016): 492–95. http://dx.doi.org/10.1017/mag.2016.115.

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18

Soldatos, Gerasimos T. "One method towards the trisection of the angle." Open Journal of Mathematical Sciences 4, no. 1 (February 22, 2020): 23–26. http://dx.doi.org/10.30538/oms2020.0090.

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19

Dongho, Joseph, Nguimbous F. L., and Teyou M. R. "On the Existence of Poncelet-Morley Points in Euclidean Geometry." Journal of Mathematics Research 9, no. 3 (May 24, 2017): 68. http://dx.doi.org/10.5539/jmr.v9n3p68.

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20

Rediske, Arthur. "The Trisection of an Arbitrary Angle: A Condensed Classical Geometric Solution." JOURNAL OF ADVANCES IN MATHEMATICS 17 (December 10, 2019): 378–89. http://dx.doi.org/10.24297/jam.v17i0.8487.

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This paper presents a short version of an elegant geometric solution of angle trisection that was published by this author on 2018-04-30 in Volume: 14 Issue: 02 of the Journal of Advances in Mathematics. The style of writing for the above paper was based on how teaching geometry was taught in high schools from 1940 to 1942. Proofs of a problem consisted of a statement that was followed by a valid reason why the statement was made. If the proof was many lines in length, the teacher wanted the students to show each step. The students were not allowed to skip a step or steps to reach the final line of the proof. This short version was generated when a copy of the above paper was reviewed by a retired school teacher, who suggested the proof of the trisection of an arbitrary angle could be shortened. The exposed methods of proof have not changed from the Euclidean postulates of classical geometry.

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21

Aslan, Farhad T., and John F. Lamb. "Error Analysis of the Kopf-Perron Angle Trisection Approximation." School Science and Mathematics 89, no. 1 (January 1989): 3–11. http://dx.doi.org/10.1111/j.1949-8594.1989.tb11883.x.

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22

Farnsworth, Marion B. "Cubic equations and the ideal trisection of the arbitrary angle." Teaching Mathematics and its Applications: An International Journal of the IMA 25, no. 2 (June 1, 2006): 82–89. http://dx.doi.org/10.1093/teamat/hri010.

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23

Aslan, Farhad, John Lamb, and Morris Loe. "An Analysis of the Morris Loe Angle Trisection Approximation Method." School Science and Mathematics 92, no. 4 (April 1992): 220–23. http://dx.doi.org/10.1111/j.1949-8594.1992.tb12176.x.

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24

Tong, Jingcheng. "SOME IMPOSSIBLE EUCLIDEAN CONSTRUCTIONS SIMILAR TO TRISECTION OF AN ANGLE." PRIMUS 8, no. 1 (January 1998): 84–90. http://dx.doi.org/10.1080/10511979808965885.

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25

Rmus, Veselin. "Constructions of squaring the circle, doubling the cube and angle trisection." Vojnotehnicki glasnik 65, no. 3 (2017): 617–40. http://dx.doi.org/10.5937/vojtehg65-13404.

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26

Nigmatullin, R. R. "IS IT POSSIBLE TO SOLVE THE ANGLE TRISECTION PROBLEM WITH COMPASSSTRAIGHTEDGE CONSTRUCTION." ASJ. 1, no. 37 (June 11, 2020): 47–48. http://dx.doi.org/10.31618/asj.2707-9864.2020.1.37.7.

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If you pose the question given in the title of this note you will listen a negative answer. In the Google searcher you will receive about 6 million results. It means to find something new in the problem formulated by Greek mathematician is useless. This problem alongside with the circle squaring is considered as undecidable problem.

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27

A. Willis, Lorna. "Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts." American Journal of Applied Mathematics 3, no. 4 (2015): 169. http://dx.doi.org/10.11648/j.ajam.20150304.11.

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28

Dong, Cong, Ya Ping Chen, Jia Feng Wu, and Rui Bing Cao. "Numerical Investigation on Circumferential Overlap Trisection Helical Baffle Heat Exchanger." Applied Mechanics and Materials 389 (August 2013): 1035–40. http://dx.doi.org/10.4028/www.scientific.net/amm.389.1035.

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The circumferential overlap trisection helical baffle shell-and-tube heat exchanger (cothSTHX) is a modification one based on the quadrant helical baffle one to improve the heat transfer performance. The model of cothSTHX has 34 tubes and 3 rods with tube layout of equilateral triangle and a baffle inclined angle of 20°, and the numerical simulation of flow and thermal performances was conducted using the commercial CFD software FLUENT. The distributions of temperature, pressure and velocity, and temperature, pressure or velocity nephograms with superimposed velocity vectors for special slices can provide access to important implicit information in the simulation results. The Dean vortex secondary flow, which is one of the key mechanisms of enhancing heat transfer in heat exchangers, is clearly depicted, and it shows that the fluid in spiral channel flows outward under the centrifugal force, then flows back to the axis under radial differential pressure, thus a single vortex is formed in each baffle pitch cycle. The unfolded concentric hexagonal slices are designed to make up for the defect that the shortcut leakage flow in the V-notch zone of the adjacent baffles could not be depicted along the longitudinal slices. Moreover, the shortcut leakage flow is restricted with the damper effect of the structure that a row of tubes in the circumferential overlap area of adjacent baffles.

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29

Redakcija, "Vojnotehničkog. "Retraction: Rmuš, V.: Constructions of squaring the circle, doubling the cube and angle trisection (65(3), pp.617-640, 2017)." Vojnotehnicki glasnik 68, no. 1 (2020): 131–36. http://dx.doi.org/10.5937/vojtehg68-25019.

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30

Krahe, Jaime Lopez. "«GeOrigaMetry» An Approach to the Accessibility of Geometry for Blind People." Modelling, Measurement and Control C 81, no. 1-4 (December 31, 2020): 67–71. http://dx.doi.org/10.18280/mmc_c.811-412.

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Making geometry accessible for blind people, apart from the formal aspects, can pose some difficulties, especially in terms of accessibility to figures. To deal with this problem this article focuses on paper folding where both Euclidean and origami axiomatic systems are used simultaneously. In the first case, with a ruler and compass, we can solve quadratic problems in a plane. In addition, the axioms of origami allow us to address unanswered questions with classical geometry methods, which involve cubic equations, such as the trisection of an angle. An experiment with INJA (National Institute for Blind Youth, Paris) students and other blind people will take place so that we can see the possibilities offered by this method, which brings a ludic, but rigorous approach to these complex and frequently off-putting issues. We believe that this dynamic pedagogical approach can increase interest and motivation, encourage tactile stimulation and facilitate the development of specific structures of brain plasticity. The article is written in a linear way, accessible to blind people; figures are provided to facilitate understanding for "visually impaired" people, who are not used to following a geometric concept without pictures. Finally, it should be noted that the method is particularly suitable in an inclusive education context.

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31

Lützen, Jesper. "The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle." Centaurus 52, no. 1 (February 2010): 4–37. http://dx.doi.org/10.1111/j.1600-0498.2009.00160.x.

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32

Пшуков, Тагир, Tagir Pshukov, Мурат Мамчуев, and Murat Mamchuev. "Approximate Solution for Squaring the Circle Problem." Geometry & Graphics 6, no. 4 (January 29, 2019): 32–38. http://dx.doi.org/10.12737/article_5c21f593838774.44754853.

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It is known that squaring the circle (the problem consisting in construction of a square with the same area as a given circle), together with duplication of cube and angle trisection, is one of the most famous unsolv able problems of constructive geometry for construction with compass and straightedge. The solution of squaring the circle problem is reduced to the straightening of the circle, that is, to the construction of a segment equal in length to the circle, and its insolvability is connected with the pi character transcendence. In this paper, the limiting case of one of Christian Huygens theorems, which establishes an estimate for length of circumference of a circle through perimeters of regular polygons inscribed in circle and circumscribed about it, is proved. On this basis has been proposed and justified an approximate method for squaring the circle problem solving, which allows consistently construct arbitrarily exact solutions of the problem. We will approximate an arc of a circle whose radius is a multiple of the given circle’s radius, with the help of a segment which is parallel to a shrinking it chord, and then will increase or decrease this segment in the required number of times, so that the resulting segment’s length would be approximately equal to half of the given circle’s circumference. The approximation accuracy will be the higher the smaller arc of the circle we will consider. But possibilities of real tools are limited, and not suitable for both too small and too large drawing scales. In order to cope with this problem, an algorithm for scaled approximation has been proposed, in which it is sufficient to increase (or reduce) the drawing fragment, so that all the time sta y within the sheet of the same size. Perhaps this approach will be useful for other constructions, including the exact ones, where it is necessary to come to very large or vice versa very small drawings’ dimensions.

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33

Aczél, János, and Claudi Alsina. "Trisection of Angles, Classical Curves, and Functional Equations." Mathematics Magazine 71, no. 3 (June 1, 1998): 182. http://dx.doi.org/10.2307/2691201.

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34

Aczél, János, and Claudi Alsina. "Trisection of Angles, Classical Curves, and Functional Equations." Mathematics Magazine 71, no. 3 (June 1998): 182–89. http://dx.doi.org/10.1080/0025570x.1998.11996628.

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35

Silvester, John R. "On cardioids and Morley's theorem." Mathematical Gazette 105, no. 562 (February 17, 2021): 40–51. http://dx.doi.org/10.1017/mag.2021.6.

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Morley’s trisector theorem says that the three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle. See Figure 1. Morley discovered his theorem in 1899, and news of it quickly spread. Over the years many proofs have been published, by trigonometry or by geometry, but a simple angle-chasing argument is elusive. See [1] for a list up to 1978. Perhaps the easiest proof is that of John Conway [2], who assembles a triangle similar to the given triangle by starting with an equilateral triangle and surrounding it by triangles with very carefully chosen angles.

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36

Wen D. Chang and Russell A. Gordon. "Trisecting Angles in Pythagorean Triangles." American Mathematical Monthly 121, no. 7 (2014): 625. http://dx.doi.org/10.4169/amer.math.monthly.121.07.625.

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37

Bhojane, K. A. "71.40 An Apparatus for Trisecting an Angle." Mathematical Gazette 71, no. 458 (December 1987): 299. http://dx.doi.org/10.2307/3617053.

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38

Barton, Lyndon O. "A Procedure for Trisecting an Acute Angle." Advances in Pure Mathematics 12, no. 02 (2022): 63–69. http://dx.doi.org/10.4236/apm.2022.122005.

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39

Coghetto, Roland. "Morley’s Trisector Theorem." Formalized Mathematics 23, no. 2 (June 1, 2015): 75–79. http://dx.doi.org/10.1515/forma-2015-0007.

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Abstract Morley’s trisector theorem states that “The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle” [10]. There are many proofs of Morley’s trisector theorem [12, 16, 9, 13, 8, 20, 3, 18]. We follow the proof given by A. Letac in [15].

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40

McLean, K. Robin. "92.52 Trisecting angles with ruler and compasses." Mathematical Gazette 92, no. 524 (July 2008): 320–23. http://dx.doi.org/10.1017/s0025557200183317.

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41

Radhiah, Radhiah. "Two Celebrated Classical Problems in Geometric Constructions." Jurnal Matematika, Statistika dan Komputasi 17, no. 1 (August 24, 2020): 135–44. http://dx.doi.org/10.20956/jmsk.v17i1.9135.

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The main topic of this paper is about two celebrated classical problems in geometric constructions where the only allowed instruments are compass and ruler with no scale. Two such problems are (1) trisecting an angle, and (2) doubling the cube. In addition, we also study about the construction of 7-gon and 10-gon.

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42

Ma, Xin. "Anatomic Study of Exposed Posterior Malleolus Using the Posterolateral Ankle Approach." Journal of Orthopaedics & Bone Disorders 5, no. 1 (2021): 1–5. http://dx.doi.org/10.23880/jobd-16000203.

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We compared the effect of the posterolateral ankle approach on the exposed posterior malleolus and vascular nerves in order to reduce the probability of vascular nerve injury during surgical exposure. Five corpses were randomly allocated to incision A and B groups. The tip of the lateral malleolus was used as a starting point, while the lateral line of the Achilles tendon was used as the endpoint to its trisection. Using the two points near the side of the Achilles tendon, we drew two vertical horizontal lines to represent incisions A and B, then measured the horizontal distances between the tip of the lateral malleolus and incision A (a), the tip of the lateral malleolus and incision B (b), and the tip of the lateral malleolus and the midpoint of the sural nerve and small saphenous vein (c). We then exposed the fibula from the posterior portion of the peroneus brevis muscles, dissected the flexor pollicis longus from the posterior edge of the fibula, and used Vernier calipers to measure the maximum length and width of the exposed bone block. There was no statistically significant difference between distances (a) and (c), but there was a significant difference between distances (b) and (c). The length of the exposed posterior malleolus did not differ significantly between incisions A and B, but the width differed significantly. Exposing the posterior malleolus using an approach closer to the lateral Achilles tendon is less likely to injure the sural nerve and small saphenous vein and results in a larger exposed area and easier manipulation. Thus, this could be a better surgical treatment for ankle fractures.

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43

Chen, Y. P., R. B. Cao, C. Dong, J. F. Wu, and M. C. Wang. "Numerical simulation on the performance of trisection helical baffle heat exchangers with small baffle incline angles." Numerical Heat Transfer, Part A: Applications 69, no. 2 (November 30, 2015): 180–94. http://dx.doi.org/10.1080/10407782.2015.1069663.

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44

Dong, Cong, Xin-Fa Zhou, Rui Dong, You-Qu Zheng, Ya-Ping Chen, Gui-Lin Hu, You-Sheng Xu, Zhi-Guo Zhang, and Wen-Wen Guo. "An analysis of performance on trisection helical baffles heat exchangers with diverse inclination angles and baffle structures." Chemical Engineering Research and Design 121 (May 2017): 421–30. http://dx.doi.org/10.1016/j.cherd.2017.03.027.

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45

Lamb, John F., Farhad Aslan, Ramona Chance, and Jerry D. Lowe. "Inscribing an “Approximate” Nonagon in a Circle." Mathematics Teacher 84, no. 5 (May 1991): 396–98. http://dx.doi.org/10.5951/mt.84.5.0396.

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Throughout history even the best mathematicians have been challenged by certain problems. Indeed, some of these problems became famous because they defied solution. Many students of mathematics are familiar with the problems of squaring a circle, doubling the cube, and trisecting an angle. Some may have studied the problem of inscribing regular polygons with a given number of sides in a circle. A few of these polygons, such as those with three sides, four sides, six sides, and eight sides, are easily inscribed. Other such constructions are more involved but are still possible, such as those involving polygons with five sides and with seventeen sides. Still others turn out to be impossible to perform with Euclidean tools, such as those for polygons with seven sides and with nine sides.

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46

Fuchs, Clemens. "Angle trisection with Origami and related topics." Elemente der Mathematik, 2011, 121–31. http://dx.doi.org/10.4171/em/179.

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47

Kang, Kyong-Hun. "A consideration on the history of the angle trisection." Journal of educational Research Institute 6, no. 2 (December 31, 2004). http://dx.doi.org/10.15564/jeri.2004.12.6.2.63.

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48

Kimuya M Alex. "The Possibility of Angle Trisection (A Compass-Straightedge Construction) Kimuya M Alex." Journal of Mathematics and System Science 7, no. 1 (January 28, 2017). http://dx.doi.org/10.17265/2159-5291/2017.01.003.

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49

Blåsjö, Viktor. "Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry." Foundations of Science, April 16, 2021. http://dx.doi.org/10.1007/s10699-021-09791-4.

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AbstractI present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.

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Dreiling, Keith. "Trisecting an Angle Using Mechanical Means." Convergence, 2017. http://dx.doi.org/10.4169/convergence20170103.

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