Why is it that, in ancient Greece, mathematicians tried to solve geometrical problems using compass and straightedge only and, apparently, only if that failed, they tried to use other tools? Note that this question is not about what they were (and weren't) able to achive using compass and straightedge alone, but about why that was their first choice.
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asked Mar 11, 2018 at 19:49
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1$\begingroup$ The axioms of Euclid allowed only certain very simple constructions, which may be interpreted as straightedge-and-compass constructions. But I do not believe classical Greeks limited themselves to Euclidean constructions only. And the notion that some construction is impossible with certain tools is much more modern. $\endgroup$– Gerald EdgarCommented Mar 11, 2018 at 22:20
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4 Answers
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The real reason is probably that straightedge and especially compass are the simplest, most primitive instruments, and also easy to make. At the same time they are quite accurate. (Straightedge is not so easy to make and is not so accurate, but certainly it stands next to the compass on these criteria). A modern high school student tends to think that to double the cube it is easier to measure the side, then to calculate the cubic root approximately. But this is anachronism. Deprive the student of all modern tools, and prohibit to use the decimal system. How will s/he do the job?
If you think of the tool which is needed to double the cube (a parabola, invented by Menaechmus, exactly for this purpose), it is much more complicated than a straightedge and a compass.
This real reason was disguised: these instruments were probably considered "sacred" at some time. There is a popular theory that in general, geometry has ritualistic origin:
The ritual origin of geometry"
For example, the problem of doubling the cube clearly has some ritualistic origin: the original question was about doubling an Apollo altar. And it is natural to expect that only some sacred tools were permitted.
At a later time the sacred principle became a philosophical principle: the straight line and circle are the only "perfect" figures because "they look the same at all of their parts." (In modern language: isometry group acts transitively on them.) This was explicitly proclaimed by Plato and Aristotle. In particular, Plato declared that the main goal of astronomy is to reduce all celestial motions to uniform circular motions. (Why circular? Why not on ellipses/parabolas?) This philosophy actually slowed down the development of astronomy.
Remark: the ancient Greeks were great theorists in mathematics but not so good in practical applications. They probably thought a straightedge was a great instrument without trying it in practice for precise constructions. That a straightedge is not really useful when high accuracy is required was only discovered by the 18th century by precision instrument makers. If the Greeks had discovered this, they would have limited their geometric constructions to "compass only." In fact, one can do with just a compass everything which one can do with straightedge + compass. That the straightedge is really redundant was proved in 19th century. See Division of the circle and compass constructions for a discussion of this.
answered Mar 11, 2018 at 22:31
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$\begingroup$ I am very grateful for your link. However, this explanation seems incomplete to me, because there is nothing in it that excludes the use of a marked ruler. However, they seem to avoided it. Why? $\endgroup$ Commented Mar 12, 2018 at 16:03
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$\begingroup$ @Jose Carlos Santos: 1. The idea of doing something approximately was foreign to Greek mathematics. 2. Can you explain how exactly you would mark the ruler? $\endgroup$ Commented Mar 12, 2018 at 18:46
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$\begingroup$ I don't understand your first point, since I made no mention whatsoever to doing something approximately. I had this in mind. There is nothing approximated about it. $\endgroup$ Commented Mar 12, 2018 at 18:51
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1$\begingroup$ @Jose Carlos Santos: you did not define a marked ruler exactly, but what I imagine as a "marked ruler" (markings form an arithmetic progression) gives nothing new: in all applications it can be replaced with non-marked ruler + compass. $\endgroup$ Commented Mar 12, 2018 at 19:04
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2$\begingroup$ The idea of approximation was not foreign to Greek mathematics, it was ubiquitous there as Eudoxus's or Archimedes's works attest. "Marked ruler" refers to the ruler with a notch used in neusis, scale marks are not necessary. It is no more "approximate" than straightedge and compass and strictly stronger as a tool. $\endgroup$– ConifoldCommented Mar 16, 2018 at 23:51
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The Greek appear to have been deeply concerned with the problem of change-and- permanance, what we call today 'invariants'. The (straight)line and the circle are the two obvious cases when an object moves and stays at the same place. Actually mechanical physics relies on the two when it admits as basic translations and rotations. Greek geometry is just thinly disguised mechanics and it has already caught this fundamental idea. Every form can be reduced to its permanent constituents which are just straight lines and circles.
Actually the Mohr–Mascheroni theorem showed that straightedge and compass constructions are possible with just a compass.
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The Greek historian Herodotus writes of the origins of geometry in Egypt:
They say that that king [Sesostri, ca. 2000 a. C.] distributed the land among all of the Egyptians, each one having an equal lot in the a square shape, and from these subdivisions obtained tribute, having imposed an annual payment. If the river bore away a part, the owner announced the loss, and officials were sent to observe the extent to which the plot had been diminished for the purpose of adjusting the tribute payment. It is my feeling that this indicates the invention of geometry here, prior to its passing to Greece.
Geometry - literally earth measurement - was the responsibility of the "arpedonapti", those who knot ropes.
It is by tightening ropes that they drew the two simplest and most important lines in geometry: the straight line and the circle. The first, simply tightening a rope between two points, a kind of operation which image is still present in the expressions "to draw a line", "to draw a perpendicular"; the second, making one of the two points turn around the other which is held fixed.
So the suggestion is that the "earth measuring arpedonapti" of ancient Egypt used only lines and circles and this was the art inherited by the Greeks. (I'm not entirely convinced.)
Euclid makes no reference to the edge and compass in his axioms.
Source : The Garden of Archimedes.
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$\begingroup$ One might speculate that drawing figures on wet sand or on a flattened patch of soil is more conveniently done with the 2 tools rather than with a single cord. It is still a miniature version, 'a model', of land measuring. $\endgroup$– sand1Commented Mar 13, 2018 at 9:00
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2$\begingroup$ Euclid may not mention the word "compass", but here is Postulate 3: "Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center." Similarly Postulates 1 and 2 for straightedge. $\endgroup$ Commented Mar 13, 2018 at 11:38
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$\begingroup$ @GeraldEdgar Hi, Gerald. Yes, we can draw circles and straight lines, but we are not restricted by the axioms as to how we obtain such figures. $\endgroup$– nwrCommented Mar 13, 2018 at 15:23
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The basic reason is that the straightline and circle are the most basic of shapes. Aristotle theorises about them in his Physics, for example.
Its still the case today. Except of course the idea has ramified into many new and different ways. For example, when we think topologically, the straight-line is the standard example of a non-compact space and the circle, the standard example of a compact space.
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$\begingroup$ the circle, the standard example of a compact space --- I would have thought the unit interval $[0,1]$ (or $[a,b]$ with $a<b$ more generally) would be standard example, since compactness ideas mostly originated (in the late 1800s) from arguments involving closed and bounded non-degenerate intervals on the real line. $\endgroup$ Commented Apr 27, 2020 at 19:27
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$\begingroup$ Regarding my previous comment, see La genèse du théorème de recouvrement de Borel by Bernard Maurey and Jean-Pierre Tacchi (2005) AND The Borel theorem and its generalizations by Theophil Henry Hildebrandt (1926) AND A pedagogical history of compactness by Manya Raman-Sundström (2015). $\endgroup$ Commented Apr 27, 2020 at 19:38