The compass that will not open

A compass has two jobs: it draws a circle, and it carries a length from one place to another. The second is what Euclid’s third postulate grants and what makes the instrument useful, and it is the one a fixed opening removes.

So the question is what remains when the compass may only ever open to one distance. It sounds crippling: no arc of any other radius can be drawn, so the standard constructions — the perpendicular bisector at any scale, the transfer of a segment — all appear to fail as soon as the figure is bigger or smaller than the opening.

The restriction is also not artificial. A compass with a locked opening is a real instrument, more accurate than an adjustable one because nothing moves between marks, and the mediaeval treatises that discuss fixed-opening constructions discuss them for that reason rather than as a puzzle. The mathematical question and the workshop question are the same one.

The answer is that nothing is lost. A straightedge and a compass fixed at any one opening construct exactly what a straightedge and a free compass do.

The two obstructions #

Two things go wrong when the opening is fixed, and they are different.

A segment can be too long. Bisecting $AB$ by two arcs needs each arc to reach past the midpoint, so it needs the opening to exceed half the length. Beyond that the arcs do not meet.

And a segment can be too short, in the sense that a circle of the fixed opening about either end swamps the whole figure and its intersections are far away and badly determined.

The first is the real obstruction and the second is only awkward. The repair for the first is in the hero figure and is entirely elementary: step the opening along the segment until what is left is short enough, then bisect that.

Two details of the walk matter and are easy to skip. Stepping along a line with a fixed opening needs the line, which is why this construction wants the straightedge and the compass-only theorem of the rung below cannot use it. And the walk terminates because a fixed length divides into any other finitely many times — the Archimedean property — so no segment is beyond it, however long. Both are trivially true and both would fail in a setting where they were not, which is the reason to name them. Everything else follows once the midpoint of an arbitrary segment is available, because a midpoint gives parallels — which is the previous rung’s chain — and parallels give the arithmetic.

Carrying a length, which is the real question #

The construction above bisects a piece and leaves the honest question open. Euclid’s third postulate is really given a segment and a point, draw the circle of that radius about that point, and a fixed opening cannot do it directly.

It can do it indirectly, and the construction is a similar-triangle argument.

Suppose the length $\ell$ is to be transferred to a point $P$. Draw any ray from $P$. On it, mark the point at distance $r$ using the fixed opening. Now the problem is to find the point at distance $\ell$ on that ray, given $\ell$ somewhere else and $r$ here — which is a proportion, and proportions are constructed from parallels alone.

So the fixed opening supplies one length and the straightedge supplies every ratio, and between them every length is reachable. That is the whole theorem, and it is why the result is arithmetic rather than geometric: a fixed opening is a unit, and a unit plus the field operations is everything.

It is worth writing the transfer out as a proportion, since “similar triangles” is a name for four lines and a claim.

Put the fixed opening’s length as $r$, and suppose lengths $\ell$ and $u$ are already marked on two rays from a common point $O$ — $\ell$ on the first, $u$ on the second. Joining the two marks and drawing, through the point at distance $r$ on the first ray, the line parallel to that join, gives a point on the second ray at distance $ru/\ell$.

So one parallel computes a quotient. Two applications give a product, and applying it with $u = r$ transfers the length $\ell$ into a multiple of $r$; applying it the other way round recovers $\ell$ anywhere it is wanted. The fixed opening is never asked to change, and the straightedge does the arithmetic.

That is why the theorem is easier than the previous rung’s. There the difficulty was getting a single parallel at all; here parallels are available from the start, since the straightedge is intact and the compass can bisect. The rusty compass theorem is a consequence of the arithmetic once the bisection works, and the bisection works because of a walk.

What the opening’s size does not affect #

The theorem holds for any opening whatever, and it is worth checking that against intuition, since a very small or very large opening seems obviously worse.

A very small opening means many steps to walk along a long segment — the hero figure takes two, and a tenfold smaller opening would take twenty. The construction is longer and reaches the same place. There is no threshold below which it stops working, because the walk is finite for every positive opening.

A very large opening means the two arcs about the ends of a short segment meet at points far away, so the perpendicular bisector is determined by two widely separated points — which is numerically better rather than worse, though it takes the construction off the page. Again the same place.

And the reachable set does not depend on the opening at all. It is the field generated by the given points’ coordinates, closed under square roots — the opening is the unit, and choosing a different unit rescales everything without changing which ratios are constructible.

That last sentence is worth taking seriously rather than as a formality. Constructibility is always a statement about ratios: the question “is this length constructible” has no meaning until a unit is fixed, and the classical problems all fix one implicitly. A fixed compass opening is a perfectly good unit, so the theorem is in a sense the natural setting rather than a restricted one — and the free compass’s ability to open anywhere is the redundant feature.

Reading those two figures together is the argument for the section that follows. Neither construction resembles the other; one is a chain of arcs and an inversion, the other a walk and a proportion. What they have in common is not a method but a destination, and the destination is a set of numbers rather than a set of drawings.

The three weakenings, together #

This ladder has now asked the question three times, and the answers line up in a way worth stating.

Compass alone, no straightedge: reaches everything. Mohr in 1672, Mascheroni in 1797, and the rung below is its proof by inversion.

Straightedge alone plus one circle with its centre: reaches everything. Poncelet in 1822, Steiner in 1833, and the previous rung.

Straightedge plus a compass of one fixed opening: reaches everything. This rung, and the oldest of the three questions by six hundred years.

Three different amputations, one answer. The pair of instruments is redundant in every direction anybody has tested, and that redundancy is the finding rather than any one of the three theorems.

The explanation is the same in all three cases and it is not geometric. What the instruments reach is a field — the rationals closed under square roots — and a field is generated by very little. Any set of operations that can express addition, multiplication, division and one square root generates the whole of it, and each of the three weakened instrument sets can express those four.

Each theorem is therefore an exercise in expressing four operations, and the ingenuity sits entirely in the hardest one for that particular amputation. For the compass alone it is drawing a line’s worth of points without a line, solved by inversion. For the straightedge with a circle it is the first parallel, solved by the harmonic construction. For the fixed opening it is transferring a length, solved by a proportion. Three different bottlenecks and three different tricks, and once past the bottleneck each proof is routine.

Where a weakening does bite #

Not every restriction is free, and the contrast makes the point.

Restricting to a compass with a fixed opening and no straightedge is a genuine restriction: the constructions above use the straightedge to transfer ratios, and without it the fixed opening cannot express multiplication. What such a compass reaches is a lattice-like set rather than a field.

Restricting the number of steps is a real restriction with a large literature — the geometrography of counting operations, in which every construction is scored by how many lines, circles and point-placements it uses — and it changes what is practical without changing what is reachable. It is also the only one of these questions with genuinely open problems in it, since the cheapest construction for a given target is rarely known to be cheapest.

Restricting to constructions that fit inside a bounded region is not a restriction, which is a small theorem of its own and is worth knowing because it is the one a draughtsman would ask about.

And enlarging the instrument set changes everything, which is the interesting direction. A marked ruler adds cube roots and makes trisection possible; folding does the same. So the boundary is sharp: every weakening tested so far reaches the same field, and the first genuine enlargement jumps to a strictly larger one.

What it costs #

The constructions are long. Transferring a length with a rusty compass takes a dozen steps against the free compass’s one, and each step is a crossing whose accuracy degrades. The theorem is about reach, and a draughtsman would not use it.

The step count grows with the ratio of lengths. Walking a segment of length $L$ with an opening $r$ takes about $L/2r$ steps, so a construction spanning several orders of magnitude in scale is genuinely long. That is a practical statement and it is also the reason the theorem is interesting: the number of steps grows and stays finite, and finiteness is the whole claim.

And a physical rusty compass is not the object in the theorem. The theorem’s compass has one exact opening; a real one has an opening that drifts, and a construction that assumes exactness is not robust to that. The whole subject is about idealised instruments, and the idealisation is doing more work here than in the free-compass case, where a drifting opening matters only once.

Who asked, and the century it took #

The question is old and the answer is comparatively recent, which is unusual for a subject this elementary.

Constructions with a fixed opening appear in Abu al-Wafa’s tenth-century treatise on the constructions a craftsman needs, where the motivation is practical: a compass that can be locked is more accurate than one that has to be reset, and a craftsman resetting a compass between marks introduces error every time. The Renaissance geometers — Ferrari, Cardano, Tartaglia — treated fixed-opening constructions as a recreational challenge and produced many individual results.

Nobody proved the general theorem until much later. Steiner’s 1833 work contains the essential ideas, and the clean statement — a straightedge and a compass of any single fixed opening reach exactly what the free pair reaches — is generally credited to him and completed afterwards.

The delay is worth an explanation, and it is the usual one. The theorem is easy once the constructible numbers are understood as a field, because it reduces to “these operations generate that field”, and the field-theoretic view of construction dates from the nineteenth century. Before it, every such question had to be answered by exhibiting constructions one at a time, and exhibiting infinitely many is not a method.

That is the same reason the impossibility results took so long. Trisection and the duplication of the cube resisted for two thousand years and fell within a decade of the algebra becoming available, and the possibility results fell in the same decade for the same reason.

What the pictures cannot show #

The walk is drawn on one segment. The claim is that any segment can be walked along in finitely many steps, which is the Archimedean property of the reals — that no length is infinitely many times another — and that is an axiom rather than a picture.

The length transfer is described and not drawn. It needs a ray, a unit mark, a given length elsewhere, and two parallels, which is four constructions and a page. The figure of the parallel stands in for the whole of it.

The proportion is drawn as a parallel and is a length transfer. The figure showing the harmonic construction is standing in for a different construction that uses it, and the substitution is signposted in the caption and is still a substitution. What is actually needed is two rays, three marks and one parallel, and the figure shows a segment, an apex and two diagonals.

And “reaches everything” is a statement about all constructions. Every figure here performs one, and the theorem is a claim about a set closed under a chain of operations. What makes it a theorem is the arithmetic of the middle sections; the figures verify instances.

Where the ladder goes next #

Named here as debts. The number of steps — geometrography, and the question of the cheapest construction for a given target, which is where the practical interest of these instruments actually lies. And the rusty compass without a straightedge, mentioned above as a genuine restriction and not characterised here.

Sideways, the field of constructible lengths is priced in square roots, the enlargement that does change the answer is the marked ruler, and the two other weakenings are the rung below and the rung between.

That figure is the object all three theorems are about, and it is worth ending on because it is the reason the theorems feel less surprising once they are understood. The closure is defined by what it is closed under, not by how one gets there. A set closed under four field operations and square roots is the same set whichever route reaches it, and a theorem saying that a restricted route reaches it is a theorem about the route rather than about the set.

Put that way the three results stop being three coincidences. Each one exhibits a route; the routes are ingenious; and the destination was fixed before any of them was found.

What is worth carrying away #

When several different restrictions of a tool all leave its reach unchanged, the reach was never about the tool.

Compass alone, straightedge with one circle, straightedge with a stuck compass — three amputations, three theorems, one answer. What that says is that the classical constructions were never characterised by the instruments; they were characterised by a field, and any instrument set expressing that field’s operations does the same work.

The habit worth taking is to look for the algebraic invariant behind a set of operations. Once it is found, the instruments become interchangeable and the impossibility results become statements about numbers — which is where they are provable.

The same reading explains why the classical impossibilities are so robust. Doubling the cube is out of reach for compass and straightedge, for compass alone, and for straightedge with a rusty compass, and the three statements are one statement: the field these tools generate is closed under square roots and the answer needs a cube root. No rearrangement of the instruments changes a field.