The slice that has to match

The rings settle a disc by cutting it into pieces whose sizes are already known. The same instinct one dimension up meets an obstacle at once: a ball cut into spherical shells gives shells whose areas are the thing being asked about, so the slicing produces a circular argument rather than an answer. The disc escaped that because a ring’s length was known in advance — it is the circumference, which is where $\pi$ entered the chain — and no corresponding quantity is available for a shell.

The repair is to stop cutting the object and start comparing it, slice by slice, with an object whose volume is known.

The cylinder holds $\pi$ and the cone holds a third of that, so the hemisphere holds $\tfrac23\pi$ and the ball holds $\tfrac43\pi r^3$. The whole derivation is that one area comparison, and the licence to go from equal slices to equal volume is what the rest of this essay is about.

Reading the comparison #

The two cross-sections are the easiest arithmetic in this collection and it is worth doing rather than accepting.

The dome. A hemisphere of radius $1$, sliced at height $h$: the radius of the disc is the other leg of a right triangle with hypotenuse $1$ and one leg $h$, so it is $\sqrt{1-h^2}$ and the disc’s area is $\pi(1-h^2)$.

The cylinder less the cone. The cone has its apex at the centre of the base and opens out to the full radius at the top, so at height $h$ its radius is exactly $h$. The cylinder’s slice is a disc of radius $1$; removing the cone’s leaves a ring between $h$ and $1$, of area $\pi \cdot 1^2 - \pi h^2 = \pi(1-h^2)$.

Identical, at every height, and by an accident that is not one: the Pythagorean relation gives the dome’s radius as $\sqrt{1-h^2}$, and the ring’s area is the difference of two squares — the same relation read as an area instead of as a length.

The principle, and what it actually says #

The step from equal cross-sections to equal volumes is Cavalieri’s principle, stated in 1635:

If two solids lie between the same two parallel planes, and every plane parallel to those cuts them in regions of equal area, the two solids have the same volume.

The left panel is the principle at its simplest and it is worth noticing how strong the statement already is there. A parallelogram and a rectangle of the same base and height have the same area, and the usual proof cuts a triangle off one end and slides it to the other. Cavalieri’s argument does not cut anything: it observes that every horizontal line meets both in a segment of the same length, and concludes.

The modern version of that is an integral, and Cavalieri’s principle is the statement that a volume is the integral of its cross-sectional areas — which is true, is a theorem rather than a postulate, and needs the area function to be integrable. In 1635 it was an axiom, it was controversial, and the controversy was about whether a solid can be regarded as made of infinitely many flat things with no thickness.

Where it lies #

The right-hand panel is the arrangement where the same-sounding statement is false, and it is the reason the word parallel is in the principle rather than decorating it.

Take the unit square and the rays from one corner. Along each ray, mark the point half-way to the boundary. The inner region is bounded by those midpoints; the outer region is what is left. Along every single ray, the inner chord is exactly half the outer one. Equal ratios, at every angle, with no exceptions.

The areas are a quarter and three quarters.

The reason is visible once stated: a fan’s slices are not the same width as each other. A ray at angle $\theta$ sweeps a sector whose area grows like the square of the distance from the corner, so the outer half of a ray — although the same length as the inner half — sits where the sweep is wider and contributes three times as much. The inner region is the square scaled by a half about the corner, and scaling a plane region by a half quarters its area.

So a correspondence between slices is not enough; the slices have to be parallel, which is what makes their widths comparable. That is the content of the hypothesis, and this figure is what happens when it is dropped while everything else is kept.

It is worth saying which of the two regions the usual misuse produces. Somebody reasoning from the ratio would conclude the inner region is half the square, since every chord is half. It is a quarter. The error is a factor of two and it points the obvious way, which is the most dangerous kind: a wrong answer that is wrong by a plausible amount and arrived at by a correct-sounding argument. The same shape of mistake is what makes a scaled map’s area confusing — halving every length quarters every area, and the ray picture is that statement in a setting where the halving is presented as a fact about slices instead of about lengths.

This is also why the argument for the hemisphere was stated slice by slice in areas rather than in radii. The dome’s radius is $\sqrt{1-h^2}$ and the ring’s outer and inner radii are $1$ and $h$; nothing about those three lengths matches. What matches is the areas, and a comparison of lengths would have produced nothing at all.

What the principle is doing in the hemisphere #

With the panel above in mind it is worth going back and checking that the sphere argument satisfies the hypothesis, because it is easy to wave through.

The two solids sit between the same two horizontal planes — the base and the height-one plane. Every horizontal plane cuts both, and the cuts have equal area. Both solids’ cross-sections are measured in the same plane, so the slices are parallel and the widths are comparable. The hypothesis holds exactly.

What the argument does not need is that the two solids resemble each other in any way. One is round and one is a cylinder with a conical hole; nothing is bent into anything; no piece of one is a piece of the other. The only relation between them is an equality of numbers at every height, and that is sufficient.

That is what makes the principle powerful and what makes it feel like cheating. A dissection proof shows where every piece goes. A Cavalieri proof shows nothing going anywhere, and it is complete.

The controversy, which was about something real #

Cavalieri’s principle was not accepted quietly, and the objections were better than they are usually made to sound.

His indivisibles were flat sections with no thickness, and a solid was supposed to be made of them. The obvious objection — infinitely many things of zero volume adding to a positive volume — is one he never answered satisfactorily, and Guldin pressed it hard. Torricelli produced a version of the paradox with teeth: two triangles of different areas can be put in correspondence so that every line of one is matched with a line of the other of the same length, if the correspondence is chosen freely. That is exactly the right-hand panel above, and it is the reason the parallelism requirement is not decoration.

The resolution is that a slice is not a piece of the solid; the volume is the limit of sums of thin slabs, and a slab’s volume is an area times a thickness. Equal areas at every height give equal slabs at every height and therefore equal sums, and the thicknesses are equal because the planes are parallel. The thickness is the thing indivisibles left out and the thing the paradox exploits, and once it is put back the principle is a theorem about integrals rather than a claim about what a solid is made of.

It is worth noticing that this is the same objection the ring dissection has to answer one dimension down: a ring is not a rectangle, it has an inner and an outer edge, and the argument is complete only in the limit. Cavalieri’s version raises the objection more sharply because his slices have no thickness at all, where a ring at least has a width to shrink.

The proof by exhaustion, for comparison #

Archimedes proved the sphere’s volume and he did not have Cavalieri’s principle. It is worth seeing what he did instead, because the contrast is the whole of the exhaustion method.

He brackets. Inscribe in the hemisphere a stack of cylinders of equal height, and circumscribe another; both stacks have computable volume; the stacks differ by one cylinder’s worth, which shrinks as the count rises. Then suppose the hemisphere’s volume is more than two thirds of the cylinder’s and force a contradiction, and suppose it is less and force another. Exactly the method of exhaustion, one dimension up.

That proof is complete, rigorous and about four times as long, and it supplies no comparison object at all. Cavalieri’s argument is two lines and needs a limit theorem that took another two hundred and fifty years to state properly.

The trade is a standing one. A short proof usually assumes a general theorem, and a long one usually assumes nothing, and which is preferable depends entirely on whether the general theorem is available.

What the same comparison gives elsewhere #

The cone. A cone’s volume is a third of the cylinder on the same base, which is itself a Cavalieri argument against a pyramid, and the pyramid’s third comes from three of them assembling into a cube. That factor of three is where the third in the hemisphere’s answer ultimately comes from — and it is the same $n$ in the denominator that integrating a boundary growing like $r^{n-1}$ always produces.

The derivative relation. The ring dissection observes that the derivative of a disc’s area with respect to its radius is its circumference, and that the same holds one dimension up: the derivative of $\tfrac43\pi r^3$ is $4\pi r^2$, the surface. The ball as an accumulation of its own shells is the other slicing — radial rather than horizontal — and it gives the volume from the surface, where this essay’s slicing gives the volume from a comparison and the surface then follows.

The surface, by the same comparison. A sphere’s surface unrolls onto the side of its circumscribing cylinder with no distortion of area at all — Archimedes’ theorem, and the reason the map projection that preserves area on a globe is the one that projects horizontally onto a cylinder. That is a slicing statement again: a horizontal band of the sphere and the matching band of the cylinder have equal area, because the sphere’s band is narrower in one direction by exactly as much as it is longer in the other. A projection that keeps area and a Cavalieri comparison are the same observation, one about a surface and one about a solid.

And the two together settle the sphere completely, which is exactly the pair of results Archimedes asked to have carved on his tomb: the sphere is two thirds of its circumscribing cylinder in both volume and surface. The volume is this essay’s comparison; the surface is the same statement differentiated.

How far the shear may be pushed is worth seeing, because nothing in the principle bounds it. The parallelogram of the left-hand panel can be leaned until it barely overlaps the rectangle it started as, and every horizontal slice is still exactly as wide — so the area is still exactly the same, however unlike the two shapes have become. That indifference to the lean is what makes the principle useful and is also what makes the ray picture beside it so misleading: the eye reads similar shape as similar size, and neither implication holds.

It is worth ending with the other instrument beside this one, because between them they are the whole of how a curved quantity is reached here. One brackets and one compares; one produces a number and refutes every other value; one produces an equality between two objects and never bounds anything at all. They share no step, and a reader who has both has the two moves available whenever a shape resists being cut up.

Two outlines, and the principle between them #

Both solids are drawn in profile and neither is a solid. What is on the page is two outlines and two line segments, and the areas in the caption are computed from the geometry rather than measured off the drawing. A reader looking at the figure sees a chord and a pair of chords, not a disc and a ring.

The equality is checked at four hundred heights and claimed at all of them. Since both areas are the same polynomial in $h$, the claim is an identity and the checking is a check on the arithmetic rather than evidence for the identity — but the figure cannot say that; what it can say is that four hundred samples agree.

The principle itself has no picture. The step from equal slices to equal volumes is the thing being relied on, and the left panel of the Cavalieri figure illustrates it in a case where it is obvious. That it holds in general is a theorem about integration, and the right panel shows only that a different statement is false.

And the failure panel’s regions are computed, not drawn accurately at every ray. Thirteen rays are drawn out of infinitely many; the areas of a quarter and three quarters come from integrating in polar coordinates, and what the drawing conveys is the arrangement rather than the measurement.

The same trick in four dimensions, and where it stops #

The comparison has an obvious continuation and it is worth following one step, because what it reveals is that the pattern does not settle.

A ball in four dimensions, sliced at height $h$, gives a three-dimensional ball of radius $\sqrt{1-h^2}$ and therefore volume $\tfrac43\pi(1-h^2)^{3/2}$. That is not the slice of any cylinder-minus-cone, because the exponent is no longer an integer — so the comparison object that worked in three dimensions has no analogue, and the slicing gives an integral rather than a subtraction.

Doing the integral gives $\pi^2/2$ for the unit four-ball, and continuing gives a sequence of volumes $2$, $\pi$, $\tfrac43\pi$, $\tfrac12\pi^2$, $\tfrac{8}{15}\pi^2$, $\tfrac16\pi^3$, … which rises to a maximum at five dimensions and then falls to nothing. The unit ball in twenty dimensions has volume about $0.026$; in a hundred, about $10^{-40}$.

That is worth knowing because it is entirely invisible from the three-dimensional argument, and because it explains a fact about high-dimensional geometry that sounds absurd on first hearing: almost all of a high-dimensional cube is outside its inscribed ball. The comparison in this essay is exact and delightful and it is also a two-and-three-dimensional accident, and the honest reading of it is that a slicing argument gives whatever the slices give.

Still open here: which dissections exist at all #

Cavalieri’s principle gives the sphere’s volume without cutting anything up, and that raises the question of whether a genuine dissection exists — a finite cutting of one solid into pieces that reassemble into another of the same volume.

In the plane the answer is yes for every pair of polygons of equal area, which is the Wallace–Bolyai–Gerwien theorem and is the setting every dissection proof here belongs to. In space it is no, and that is Hilbert’s third problem, settled by Dehn in 1900: a cube and a regular tetrahedron of equal volume cannot be cut into finitely many polyhedral pieces that reassemble, because a quantity built from the dihedral angles — the Dehn invariant — is unchanged by cutting and differs between them. That is an invariant in the sense a count that survives every way of drawing a solid is one: the arbitrary part is the cutting, and what is left over is the object.

So the licence Cavalieri’s principle gives is not one a dissection could have given. A limit is genuinely required in three dimensions, and the slicing argument is the shape it takes. Curved solids are further out still: no finite dissection turns a hemisphere into anything polyhedral at all, so the comparison in this essay’s hero figure is not a shortcut past an available dissection — it is the argument.

Comparing rather than cutting #

The habit is worth naming because it is the opposite of the one the dissection starts with.

The rings argument is constructive: take the object apart, and the pieces’ sizes add up. It works exactly when the pieces’ sizes are already known, and it fails for a ball because a shell’s area is the question.

The slicing argument is comparative: find a second object whose volume is known, and show that the two agree slice by slice. It needs no relation between the objects beyond a numerical one, and it produces no account of where anything went.

When the pieces of a decomposition are not measurable in advance, the move is to compare rather than to cut — and the price is that the resulting proof, however short, explains nothing about the shape. Archimedes’ sphere result is two lines and gives no feeling whatever for why a ball is two-thirds of its cylinder.