Historically, has it been considered valid in compass and straightedge constructions to, given two line segments, decide which is longer? It isn't hard to see that this problem is equivalent to the problem of, given 3 co-linear points, choosing the one which lies in the middle. Does assuming this principle actually improve one's power of construction?
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$\begingroup$ You can determine the mid point $m$ between $a$ and $b$. Draw the circle $C$ with center at $m$ and passing though $a$. Then, a third point $p$ on the line is inside $ab$ if the circles with centers $a$ and $b$ and passing through $p$ intersect $C$. When $p$ is outside at least one of them doesn't intersect $c$. Therefore the condition of being in between is already represented in straightedge-compass constructions. $\endgroup$– HellenAug 2, 2017 at 21:49
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$\begingroup$ Therefore, given three co-linear points $x,y,z$, you pick two of them to be $a$ and $b$ do the constructions above. If all the circles intersect, then the third point is in between $a$ and $b$. If not, then change the pair that gets to play the role of $a,b$. $\endgroup$– HellenAug 2, 2017 at 21:55
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1$\begingroup$ @Hellen I agree that you have reduced the problem of determining betweeness to the problem of determining whether or not two given circles intersect. However, it seems to me that the latter principle is just as dubious as the former. I am curious if you know of any geometers that raised objections to that principle, or if compass and straightedge constructions without it have been studied at all. $\endgroup$– user181407Aug 2, 2017 at 22:11
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$\begingroup$ The determination of intersection points in straightedge-compass constructions is assumed to be not dubious. It is an axiom of the construction process. Given two circles you can determine their intersection points. It is a consequence of my comments above that adding the power of determining the point in between doesn't add any extra power to straightedge-compass constructions. $\endgroup$– HellenAug 2, 2017 at 22:15
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2 Answers
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Euclid's algorithm depends on not only deciding one line segment is longer than another, but subtracting the smaller length from the larger repeatedly. As far as deciding betweenness of points on a line, Euclid is silent, assuming it is given implicilty. Axiomatization of betweenness came much later in the 19th century. Also, at that time there was a study of Euclidean constructions under various constraints. This is mentioned briefly in the Wikipedia article Compass-and-straightedge construction.
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Yes, we can use compass-and-straightedge construction to reduce the question of which line segment is longer to an issue of which of three distinct co-linear points is between the other two.
This is an incidence relationship: does point $B$ belong to line segment $AC$? Much like the incidence of a point with a line, the wording of Euclid's Elements seems clearly to assume a geometer can draw that distinction.
Consider the early placement of Book I, Proposition 3:
To cut off from the greater of two given unequal straight lines a straight line equal to the less.
The proof given, a compass-and-straightedge construction, relies on the preceding Proposition 2:
To place a straight line equal to a given straight line with one end at a given point.
The alert Reader will recognize this last as enabling the comparison alluded to in the Question, by transferring the length of one given line segment to a line segment sharing an endpoint with another given line segment. Having accomplished that, the compass is used to draw a circle around the shared endpoint with radius equal to the transferred line segment length.
IF this is the lesser length, THEN the circle intersects the greater length line segment. The geometer recognizes/labels this point of intersection as a cut off.
On the other hand, should the transferred length exceed the other line segment length, then there will not be a point of intersection. In this sense the compass-and-straightedge construction given to prove Book I, Prop. 3 is an effective test to "decide which is longer".
There is of course room for skepticism about exactly what properties of compass-and-straightedge constructions are used in this test. It is fairly explicit in Book I, Postulate 3 that what the compass draws is a circle (as defined in Book I, Def. 15-16 to have all radii of equal lengths).
Missing from these early parts of Euclid's Elements is a direct postulation of (mathematical) trichotomy, i.e. that if two things are not equal, then one is greater and the other is lesser. Lack of such a postulate does not prevent Euclid from appealing to this principle later, cf. Book XII, Prop. 2 inter alia.
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$\begingroup$ Since straighedge-compass construction is more restrictive than Euclid's axioms, that tells you nothing. $\endgroup$– HellenAug 2, 2017 at 22:02
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$\begingroup$ I'm not sure I take your objection. You asked, "Historically,has it been considered valid in compass and straightedge constructions to, given two line segments, decide which is longer?" My answer is yes, and my reference to Euclid is for its applications of compass-and-straightedge constructions as historically accepted. Implicit in those constructions are the ability of the geometer to recognize truth or falsity of various incidence relations: Do two points coincide? Do two lines intersect? What line segments are chords of a circle (because the endpoints lie on the circle)? $\endgroup$– hardmathAug 2, 2017 at 22:51