The straightedge buys nothing

Two instruments, and the natural assumption is that each does something the other cannot. The straightedge draws lines and the compass draws circles, and a construction is a sequence of both.

The assumption is wrong in one direction and right in the other. The straightedge can be thrown away: any point a construction with both instruments produces can be produced with the compass alone. And the compass cannot be thrown away: the straightedge on its own cannot even bisect a segment.

What a construction produces #

The convention has to be stated before the theorem makes sense, because a compass cannot draw a line and a line is not a point.

A construction starts from a finite set of known points and produces more. A straightedge move takes two known points and draws the line through them; a compass move takes two known points and draws the circle centred on the first through the second. New points arise where two drawn objects cross. The output of a construction is a set of points, and lines and circles are scaffolding.

Once that is agreed, the compass-only claim is not absurd. A line through two known points is never itself the answer; what is wanted is where it meets something, and the theorem says those crossings can be reached another way.

Stepping the radius, which is the whole toolkit #

The elementary move is that a compass opened to a radius can walk around its own circle.

Six chords of length $r$ fit exactly round a circle of radius $r$ — the same fact that makes the hexagon the easiest polygon to draw — because each chord with the two radii to its ends is an equilateral triangle and six sixty-degree angles make a full turn. That is exact rather than approximate, and it hands over the first useful compass-only construction: three steps reach the point diametrically opposite, so a compass can double a segment.

Inversion, done with arcs #

The second ingredient is the one that makes the theorem work, and it is a construction this site has already met from the geometric side.

Inversion in a circle sends a point at distance $d$ from the centre to the point on the same ray at distance $R^2/d$. Its two properties are that it sends circles and lines to circles and lines, and that it is its own inverse.

The point of it here is that inversion can be performed with a compass alone. Given the circle of inversion, centred at $A$ with radius $r$, and a point $C$ outside it, draw the circle centred at $C$ through $A$; it meets the inversion circle at two points $D$ and $E$; the circles centred at $D$ and $E$ through $A$ meet again at the image of $C$. Every step is a circle.

The hero figure is that construction applied to the one place it settles everything. Take a segment $AB$, double it to reach $C$ with $AC = 2r$, and invert $C$ in the circle centred at $A$ of radius $r$: the image sits at $r^2 / 2r = r/2$, which is the midpoint. The figure checks that the point found is at exactly half the distance from each end and exactly on the segment, neither of which was used to place it.

Why the theorem follows #

Mohr’s argument, and Mascheroni’s independently, is that inversion turns every line into a circle.

A construction with both instruments produces points as crossings of three kinds: circle with circle, line with circle, and line with line. The first kind is already available. For the other two, invert the whole picture in some circle. The lines become circles through the centre of inversion, and the crossings become crossings of circles — which the compass can find. Invert back, and the points are the ones wanted.

So the reach of compass alone is the reach of compass and straightedge, and the theorem is a corollary of a change of view rather than a catalogue of clever constructions. It is the same manoeuvre the inversion essay uses to make a tangency problem easy: move to a picture where the awkward objects are of the convenient kind, solve there, and move back. What the constructions supply is the machinery for inverting, which is the figure above.

The reach is therefore the same field of numbers: the tower of quadratic extensions, and everything the degree argument says about it holds unchanged. Dropping an instrument changed the toolkit and not the answer.

Doing without the circle of inversion #

There is a gap in the argument as stated, and closing it is where the compass-only constructions stop being a curiosity and become a technique.

Inverting the picture requires a circle of inversion, and the circle has to be drawn — its centre is a point of the construction and its radius is a distance in the construction. That is available. What is not immediately available is the crossing of two circles that the inverted picture asks for, when one of those circles is the image of a line and passes through the centre of inversion.

The standard treatments handle it by building a short list of compass-only primitives and then composing them:

• double a segment, which is three steps of the radius round a circle;
• invert a point in a circle, which is the four-arc construction the hero figure uses;
• reflect a point in the line through two others, which is one arc about each of the two — the two circles meet at the original point and at its mirror image, and no line is drawn;
• find the centre of a circle through three points, which is two reflections and an inversion.

With those four, every crossing a straightedge could have produced is reachable. The third is worth pausing on because it is the smallest surprise on the list: a line is not drawn and a reflection in that line is performed anyway, using only the fact that both circles are centred on the line and therefore symmetric about it.

That is the pattern the whole theorem runs on. A line is never the object; what a line is for is symmetry, incidence and crossings, and each of those has a compass-only substitute.

The other direction, which fails #

The mirror question is whether the compass can be thrown away instead, and it cannot — not even for the easiest construction there is.

A construction using only a straightedge is preserved by every projective map of the plane, because such a map sends lines to lines and preserves incidence — and incidence is all a straightedge construction ever uses. So if some projective map fixes two given points and moves the point half way between them, no straightedge construction can produce that midpoint from those two points.

The figure exhibits such a map, and it is the least exotic thing in geometry: a central projection from one line onto another. Equally spaced marks go to unequally spaced ones, the ends can be arranged to stay put, and the middle mark lands somewhere else. That is the proof, and it is one picture.

What survives projection is the cross ratio of four points, which the figure computes on both lines and finds equal. So a straightedge can construct anything a cross ratio determines and nothing else — a genuine reach, and much smaller than the compass’s.

That is as far as the projective side is taken here, and the restraint is deliberate. The projective plane as a subject — its invariants, its maps, its incidence theorems and what a camera does with them — belongs to another site in this fleet. What is used above is one projective map, once, as a counterexample: the argument is about an operation set and its reach, and the map is the instrument that measures it.

The refinement is Poncelet’s and Steiner’s: a straightedge together with one circle drawn once, with its centre marked, reaches everything. So the compass is needed, and needed only once.

The centre is not a technicality. A circle without its centre marked is a curve the straightedge can meet but cannot exploit — there is no way to construct the centre of a given circle with a straightedge alone, which is itself a projective argument: a projective map can carry a circle to another conic and will not carry its centre to the new centre.

The reach, priced #

Since the compass alone reaches exactly what both instruments reach, everything already established about that reach carries over without a word of change, and it is worth listing what that is.

The numbers a compass can reach are those lying in a tower of quadratic extensions of the rationals, so every one of them has a degree over the rationals that is a power of two. That closes the three classical problems in one stroke: the cube cannot be doubled, because a cube root has degree three; the general angle cannot be trisected, for the same reason; and the circle cannot be squared, because $\pi$ is not a root of any polynomial with whole-number coefficients at all.

And the regular polygons that can be drawn are exactly those Gauss identified, which is a criterion about the prime factors of the number of sides. The compass-only theorem does not touch any of this; it says the same set of points is reachable and leaves the characterisation alone. That is the useful shape of such a theorem: it changes the toolkit and preserves everything anybody had proved about the toolkit’s reach.

Where it fails, and what it needs #

The compass has to be rigid, or the doubling has to be earned. Some treatments allow only a collapsing compass, which forgets its radius when lifted. The stepping construction above then has to be replaced, and the standard route is to show a collapsing compass can copy a length after all — which it can, in half a dozen arcs.

A compass cannot draw a line, and the theorem does not claim it can. The output of a compass-only construction is a set of points, and the line through two of them exists in the reader’s mind. A picture that draws the segment, as the figure does, is drawing a reading aid and says so.

Inverting the whole picture needs a centre off everything. The argument requires a circle of inversion whose centre is not on any of the lines involved, and such a centre exists because the lines are finitely many. It is the kind of clause that is genuinely necessary and never causes trouble.

The theorem says nothing about accuracy. Mascheroni’s own motive was precision, and the theorem does not deliver it: a compass-only construction has more steps, and every step carries whatever error the instrument has, so the reachable points are the same and the reachable drawings are worse. That is a distinction between an exact question and a physical one, and this field only asks the first.

And the piece count is dreadful. The compass-only version of a construction is far longer than the original — the midpoint alone takes six arcs where a straightedge and compass take three moves. The theorem is about what is reachable, not about what anybody should do.

Where it came from #

Georg Mohr published the result in Amsterdam in 1672, in a small book called Euclides Danicus, and it disappeared completely. Lorenzo Mascheroni proved it again in 1797, in a book dedicated to Napoleon, and the result carried his name alone for a century and a quarter — until a copy of Mohr’s book turned up in a Copenhagen bookshop in 1928.

Mascheroni’s stated motive is worth recording because it is unusually practical: he wanted constructions that would be accurate, and argued that a compass point can be placed more precisely than a straightedge can be laid. Whether or not that is true of instruments, it is a reason from outside mathematics for asking a question that turned out to have a clean answer inside it.

The Poncelet–Steiner theorem, from 1822 and 1833, is the mirror image and has the more interesting statement: the straightedge is almost enough, and what it lacks is exactly one circle. Together the two results say the two instruments are not partners at all. The compass can work alone; the straightedge needs a loan.

What the pictures cannot show #

The compass-only construction drawn here produces one point — a midpoint. The theorem is about every construction there is, and the figure is one instance plus an argument in prose about why the instance generalises.

The arcs are drawn as full circles, because that is what a compass draws, and the picture is consequently busy. A neater version would draw only the arcs near the crossings, and would then be hiding the fact that the whole circle is what the instrument produces.

And the straightedge’s failure is drawn as one projection moving one midpoint. That is a single counterexample and it is exactly what the argument needs — one projective map fixing two points and moving the third is enough to rule out every straightedge construction at once, because the constructions are all preserved and the point is not.

The ladder from here #

Below: what two points can build, where the operation set is introduced and its closure drawn, and every step is a square root, which prices the reach in degrees. Sideways: the map that trades circles for lines, whose inversion is the engine here, and the angle that will not divide by three, where enlarging the operation set is what changes the answer. Above: the Poncelet–Steiner theorem, constructions with a rusty compass of fixed opening, and the reach of a marked ruler.

What is worth carrying away #

Two instruments that look complementary need not be. The right question about an operation set is not what each operation does but what the closure is, and two very different-looking sets can close to the same thing.

The way to find that out is to look for a transformation that respects one set and shuffles the other. Inversion turns lines into circles, which is why the compass can do without lines; projection turns equal spacing into unequal, which is why the straightedge can do without nothing at all. In both directions the answer came from a map rather than from a construction, and the constructions came afterwards to fill in what the map promised.