The tenth row has neither

A page is bounded by a divide, not a centre introduced a camera for one purpose and then left it off the table it had just been used to correct. The crossed-slits camera draws each point by the ray through a vertical slit at seven metres and a horizontal one at ten and a half; it divides by depth in both directions of the page; and its rays share no point at all, missing any common one by 0.354 m. That was enough to show that what bounds a picture’s page is the divide and not the centre.

It is a system, and the field’s whole method is that a system gets a row and is priced rather than praised — each system answers its own question, and the test that makes the comparison a result is whether any of them wins something it was not designed for. Nine rows have been measured on five questions and then priced in three currencies on four hundred boxes. This one has been measured on a single question and asked to leave.

Putting it on the table answers the sharpest form of the exclusion available. The exclusion says a centre and a true measure do not share a row. A camera with two divides and no centre has neither of them — so the question is whether it has some measure the pinhole lacks, bought with the point it gave up, or whether the point was all it gave up and nothing came back.

The row, measured on the same five questions #

Its cells come out as follows, and each is the same measurement the other nine rows answered.

No centre. Rays fitted from thirteen columns and four rows of the page miss their best common point by 0.354 m, which is a length in the world and not a residual to be tightened. A pinhole’s rays miss by nothing.

No true measure. The midpoint test — a segment from three metres left at four metres out to three metres right at fourteen, with the image of its midpoint compared against the midpoint of its image — puts the drift at 15.597% of the segment’s drawn length. The pinhole’s figure on the same segment is 15.561%. They differ in the fourth decimal place, which is to say that at the precision the table reports they are one number, and that the difference is smaller than the difference between two pinholes of different focal length would be.

Diminution, and bounded depth. A fixed object at six metres is drawn 3.545 times the size of the same object at forty-eight, and the image of the ground at infinity has a limit. Both are what two divides produce, and both are shared with the pinhole. The bounded page is the property this camera was introduced to demonstrate, and it demonstrated it by having it without a centre.

And not straight lines, which is the cell no row with a centre has ever been empty in.

What the point cost, and it is not nothing #

The last of those is the one that does not follow from the first four, and it is worth drawing rather than announcing.

A camera with a centre draws every straight line straight, because its picture is a projective map of space onto a plane and a projective map sends lines to lines. That is the one structural thing a centre delivers, and it is what makes recovering the camera from a photograph possible at all: every recovery of a camera from a photograph reads vanishing points off the images of straight lines, and there are no vanishing points in a picture whose straight lines are curves.

The crossed-slits camera’s two page coordinates divide by two different depths — z + 7 and z + 10.5 — and no single projective map of space does that. So its straight lines bow, by 1.11 px on an ordinary run of ground. The scroll bows them too, and considerably more at 10.33 px, which is the fact a straight line in a scroll is a hyperbola measured.

So the tally on the missing point is: nothing gained, and straightness lost. The row is a pinhole with its centre removed and nothing put back.

Priced, on the same four hundred boxes #

Measured cells are yes-or-no, and a yes in the table is a price established that the interesting quantities are the prices behind them. So the tenth row gets the same treatment: four hundred axis-aligned boxes standing on the ground, every edge read with the single best ruler for that picture, every face with the best area scale, every right angle put to a protractor.

The crossed-slits camera prices within a point of the pinhole on all three, and — the small surprise — slightly better on each. A reader with one ruler misreads an edge by 38.8% under it and 39.5% under the pinhole; a face by 69.9% against 70.8%; a right angle fails the protractor 58.2% of the time against 60.9%.

The near-identity is the result rather than an aside. On every quantity the table measures except the two the exclusion is about, the crossed-slits camera and the pinhole are the same instrument: their length prices differ by 0.7 points, their area prices by 0.9, their angle prices by 2.7, their midpoint drifts by four hundredths of a point, and both bound the page and both diminish. What the point delivers is not a share of any of those. It is two things that no amount of gentler dividing supplies — a place from which the picture is a correct view, and lines that stay lines.

This is the same shape of reading the price gave the binary column — a yes or a no replaced by a number that says how much — and it gives the same answer here as there: the boundary in the table is real, and the quantities either side of it vary far more among themselves than across it.

The reason the tenth row prices a shade better is worth a sentence because it is easy to over-read. Its two divides use different depths: the column coordinate divides by z + 7 and the row coordinate by z + 10.5. The second slit is further away, so the vertical direction of the page compresses more slowly than the pinhole’s does, and the picture is very slightly gentler about depth throughout. A gentler divide is not a measure. Being three points less wrong is not a currency anybody was asking about, and the row is not a better instrument for having it.

A measure has a direction in it #

The midpoint test returns one number for a system, and building the tenth row is what showed that the quantity underneath it is not one number.

Ask the question differently. A metre is drawn at some size; does that size depend on how far away the metre is? Ask it separately for a metre running across the scene, a metre standing up, and a metre receding from the eye.

The answers are not what a binary column can carry.

Every parallel system reads 1.00 in all three, to the arithmetic floor, which is what having no divide means.

The pinhole reads 6.43, 6.43 and 38.45. A metre across and a metre standing up are both drawn 6.43 times larger at four metres than at sixty-four, and a metre receding is drawn 38.45 times larger — the square of the first, near enough, because a receding metre is shortened by the divide and by foreshortening at once.

The crossed-slits camera reads 6.45, 5.14 and 25.03. The vertical figure is the one that differs from the pinhole’s, and by exactly the amount its further slit predicts. It keeps nothing.

And the handscroll reads 1.00, 6.45 and 38.73. A metre along the roll is drawn at the same size wherever it lies, exactly; a metre standing up and a metre receding are not. That is the map along and the picture across written as a row of three numbers, and it makes the scroll the only system on the table whose answer to “does it keep a true measure” depends on which measure is asked about.

The elevation splits too, and for a different reason that is worth separating rather than counting alongside: it draws a receding metre as a point. A system that destroys a direction is not a system that measures it badly, and the table keeps the two apart.

What the binary column was hiding, and what it was not #

That gives the measure column a structure, and it is a small structure with one member.

Nine of the ten rows answer the same way in every direction: all three or none. That is why a single column was ever adequate, and why nobody had reason to look — a column that is right about nine rows out of ten looks like a column that is right. The scroll answers differently, and the reason it can is exactly the reason it was interesting in the first place — it is the member of the crossed-slits family with one slit sent to infinity, so one page direction divides by depth and the other does not.

The tenth row is what makes that visible, by being the family’s other end, and the knob is worth turning rather than described.

Read left to right, that figure is the whole family. At 10.5 m the two slits cross: the camera is a pinhole, its rays share a point to 4 × 10⁻¹⁶ m, a straight run of ground is drawn straight to the arithmetic floor, and a metre across the scene is drawn 5.14 times larger at four metres than at sixty-four. Send the slit out and the ratio falls — 4.04 at 16 m, 2.69 at 31 m, 1.55 at 105 m, 1.006 at 10 km — and arrives at 1.000, which is a true scale and is the scroll’s.

It is charged for twice. The rays stop sharing a point almost immediately, missing by 0.07 m when the slit has moved half a metre and by 2.46 m by the time the measure has arrived. And the straight run of ground bows, by 1.4 px at 16 m, 25 px at 105 m, and 99 px at the far end — the bow that makes a handscroll’s straight lines hyperbolas.

The order in which those happen is the point. The centre is gone long before the measure is there. At a slit distance of 21 m the rays already miss by 0.89 m and the ratio is still 3.40 — most of the way to no centre and barely a fifth of the way to a true scale. There is no setting at which the camera has kept anything and lost nothing, and no setting at which it has bought the measure cheaply.

One knob, three positions, and the measure is spent before the centre is bought.

It is worth saying what this does not overturn. The original exclusion — no system with a centre keeps a true measure — is unaffected in every direction, because the pinhole keeps none of the three and the tenth row keeps none of the three. The directional reading makes the measure column finer without moving any row across the boundary, which is what a refinement should do to a result that was right.

Why the near-identity is not a coincidence #

It would be easy to read the closeness of the two rows as a happy accident of the particular slit distances chosen, and it is not.

Every quantity in the price battery is a statement about how a metre’s drawn size varies over the scene, and both cameras vary it the same way: as one over depth, in both page directions, over a scene whose depths run from two metres to fourteen. The tenth row’s only freedom is that its two coordinates use depths offset by 7 m and 10.5 m rather than by the same number twice. Over a scene fourteen metres deep, a three-and-a-half-metre difference in offset is a modest reweighting of one coordinate against the other, and it moves every price by a point or less.

Push the two slits far apart and the prices do move — that is the sweep above, where the along-scene ratio goes from 5.14 to 1.00. But the moment they are far enough apart to move the prices appreciably, the camera has stopped being a near-pinhole and become a near-scroll, and it has acquired the scroll’s hundred-pixel bow to pay for it. The near-identity holds precisely over the range in which the row is a plausible camera at all, which is why it is the honest summary of what the missing point costs.

Answering the question that was put #

A page is bounded by a divide, not a centre asked whether a row with no centre can have any measure the pinhole lacks, or whether giving up the point buys nothing but the loss of the point. The answer is now definite and it is the second.

It buys nothing. In all three directions the crossed-slits camera’s scale depends on depth, its midpoint drift matches the pinhole’s to the digits reported, and its three prices are within a point of the pinhole’s.

It costs straightness, which is not a price the pinhole pays and not one the question anticipated.

And it identifies where the measure actually lives. The one true measure on the table that a pinhole does not have belongs to the scroll, and the scroll has it because it divides in one direction rather than two. So measure is bought by removing a divide, not by removing the point — and removing the point while keeping both divides removes the one structural thing the point was delivering.

That reading also settles the shape of the family. Two divides bound the page, which is what the tenth row was introduced to prove. One divide keeps one measure. A shared point keeps straight lines. Those are three different properties bought by three different things, and the table’s first two columns had been carrying all three.

What this establishes and what it does not #

It establishes that the exclusion survives its sharpest test. The only system available with two divides and no centre has neither a centre nor a measure, in any direction.

It establishes that a centre buys straight lines, and that this is separable from everything else the table measures. Every row with a centre draws straight lines straight; both rows that divide by two different depths bow them; and no row without a centre draws them straight unless it has no divide at all.

It establishes that the measure column has a direction in it. One row of ten answers differently in different directions, and the reading does not move any row across the exclusion’s boundary.

It does not say the crossed-slits camera is a picture anybody should make. It has no place from which it is a correct view, its lines are curves, and none of the recoveries a photograph permits apply to it. It is on the table as the separating example the exclusion needed, which is what it was introduced as.

It does not settle what a system with one divide about a point would do. The scroll’s one divide is taken about a slit, and a slit is a line rather than a point. Whether the measure belongs to having one divide, or to the divide being about a line, is not decided by any row now on the table.

And it does not price the directions against a scene built for them. The four hundred boxes are axis-aligned, so the three directions are the box edges; a scene of randomly oriented objects would price every system on a subject none of them was built for, and the directional reading would have to be taken over a sphere of directions rather than three. What the three numbers say is what a rectilinear scene costs a reader, and rectilinear is the subject these systems are conventions for.

Still open: whether the exclusion is a theorem or a table #

Ten rows now answer five questions each, and no row is filled in the first two columns at once. That is a fact about ten systems, and every row of it was measured rather than argued.

A table of ten cannot say whether the eleventh would obey. What would say it is a proof, and the directional reading has brought one within reach: a system appears to keep a true measure in a direction exactly when it is affine along that direction, and to have a centre exactly when it divides by depth in both page directions at one and the same depth. If both of those are exactly true rather than nearly so, the exclusion is a two-line consequence of them and the table is ten instances of one statement rather than ten facts.

The measurement that settles it states the two conditions precisely, checks each against all ten rows, and then looks for the case the conditions permit and the table does not contain: a system that divides in one direction only and therefore keeps one measure, with its single divide taken about a point rather than a slit. If that object exists it is a new row; if it cannot exist, the reason it cannot is the theorem.