The rule is exact for a floor that lengthens

The rule is one sentence: each gap between receding boards a fixed fraction of the one before it. The rule that draws another room calibrated it as favourably as possible — ratio chosen so that the first and last transversals land where a correct construction puts them — and found that its pavement, nine and a half pixels from the correct one, is nonetheless a correct perspective to a fifth of a pixel of a room whose horizon is 170 pixels from the panel’s. The natural verdict from there is that the rule is not a projection of anything, and that its error is a disagreement between the transversals and the panel’s own orthogonals.

Both halves of that verdict rest on an assumption that is easy to leave unstated, and stating it changes what the rule is. The fit that found a 170-pixel horizon asked the transversals to be a picture of equal boards. Drop that requirement and ask only what floor the transversals are a picture of, on the horizon the panel actually drew, and the answer is exact — not to a fifth of a pixel but to the last digit of the arithmetic. The rule is a camera. It photographs a particular floor, and it is worth knowing which.

That reframing is not a technicality. It moves the evidence against the rule from a test a reader has to compute to one a reader can draw, and it says why the evidence was so hard to find.

A geometric series is a perspective of something #

The argument is short, and every step of it is a statement about the panel rather than an approximation.

A panel’s transversal sits some distance below the horizon, and on a level ground plane that distance fixes the depth of the ground line it depicts: the further a line is from the picture plane, the nearer its image to the horizon, in a fixed reciprocal relation set by the panel’s viewing distance and eye height. Nothing else enters. So any row of transversals below a horizon is the image of some row of ground lines; the only question is what their spacing is.

The rule puts its k-th transversal at a distance below the horizon that is the first one’s times the ratio to the k-th power. Running the reciprocal relation backwards, the k-th ground line lies at a depth that grows as the inverse ratio to the k-th power. Consecutive depths therefore differ by amounts that are themselves in geometric progression, and each board is exactly the inverse of the ratio times as deep as the one in front of it.

And the horizon this reading needs is the one the panel drew. The rule’s transversals converge on the point where their offsets from the horizon shrink to nothing, which is the horizon itself; it is where the calibration put them. The earlier essay’s 170 pixels is the horizon an equal-board floor would need. The lengthening floor needs none of it.

On the eight-braccio panel, whose ratio comes out at 0.9217, each board is 1.085 times as deep as the last. The front board is 0.74 braccia deep and the back one 1.31. Reprojected, those depths put every transversal back where the rule drew it with no residual at all: the reading is not a best fit, it is an identity.

The total depth is right. The rule was calibrated at both ends, so its last ground line sits at eight braccia exactly as the square pavement’s does. What it gets wrong is the distribution: the boards in front are a quarter too shallow and the boards at the back a third too deep, and the two errors cancel in the sum. That is the whole of the rule’s “bow” in the rule that draws another room, seen from the floor rather than from the page.

Why the orthogonals cannot convict it #

The earlier essay’s strongest instrument was the panel’s orthogonals, which run from the braccia marks on the ground line to the centric point on the horizon. They are drawn with a straightedge, they carry the panel’s actual geometry, and they meet at the horizon the panel drew. The rule’s transversals were said to disagree with them.

Read as a lengthening floor, they do not. Orthogonals are lines on the floor running straight away from the viewer, and a floor’s orthogonals say nothing whatever about how deep its boards are — they are the same lines on a floor of square tiles, oblong tiles or tiles that change from row to row. They fix the horizon and the widths. The rule’s pavement shares that horizon exactly and uses those widths exactly. Every line on the panel, orthogonal and transversal, is consistent with one camera photographing one floor.

The disagreement the earlier essay measured is real, but it is not between two halves of the drawing. It is between the drawing and the belief that its tiles are square. The horizon test computes where an equal-board floor’s horizon would have to be and finds it 170 pixels away; the test is decisive exactly as far as that belief is warranted. For a painted pavement of square tiles, the belief is the whole point of the pavement, and the test is sound. But it is a test of a hypothesis about the floor, run by fitting, and a painter who spaced a floor of lengthening planks by the rule would pass it by failing it.

That matters for what a reader can do. The horizon test needs the transversals’ implied horizon, which is a least-squares computation over the whole row. It is easy to think of it as a straightedge test, since the orthogonals are found with one, and it needs rather more than that. The straightedge test the belief actually calls for is a different line.

The diagonal that knows the tiles are square #

A floor of square tiles has one property no other floor has, and it can be checked on the page with a straightedge alone: the corners of its tiles along a diagonal lie on one straight line. That line runs to the distance point, which the distance point is the viewing distance identifies as the one place the panel’s viewing distance is drawn as a length. On oblong tiles the corners along a diagonal still exist — each is where the k-th orthogonal crosses the k-th transversal — but they lie on a straight line only if the tiles’ proportions are the same in every row.

So the test is: find the corners where orthogonal k meets transversal k, and lay a straightedge from the first to the last.

On the rule’s eight-braccio pavement the corners bow 8.1 pixels off the chord at their worst. On the correct pavement the same corners lie on the chord to the arithmetic floor. And the bow is on one side, toward the horizon, because the rule’s front tiles are too shallow — their corners too near the ground line — and its back tiles too deep.

Eight pixels on a 420-pixel panel is small. It is not small against the other test.

Forty-seven times the evidence, from a straightedge #

The reason the rule survived five centuries is that the reader’s own projective test — fit a correct perspective to the transversals with the horizon free and measure the worst miss — returns a fifth of a pixel on a pavement of ordinary length. Four marks before anything is said counts how little a short row can testify, and the rule’s pavement exploits exactly that.

The diagonal reads the same pavement differently, and the comparison is direct because both are in pixels on one drawing.

At eight braccia the transversals miss a perspective by 0.17 pixels and the diagonal bends by 8.1: 47 times more evidence, from an instrument that needs no computation. At four braccia, where the transversals testify to almost nothing, the diagonal already bends by 3.4 pixels. Past eight braccia, on a page of fixed width, both readings level off together, because a longer pavement on the same sheet is drawn with narrower braccia and the rule’s floor approaches a single shape.

There is a reason the diagonal is so much more sensitive, and it is the same reason the transversals are so insensitive. A perspective fitted to one row of marks has three free numbers, and a geometric series of a few terms is almost exactly absorbed by them — which is the fitting test’s weakness. The diagonal has no free numbers. It asks a question the drawing has already answered — whether row k’s tile has row 1’s proportions — and the only way to absorb a wrong answer is to move the corners, which a painter would have to do on purpose.

Every tile names its own viewing distance #

The lengthening floor has a second consequence a reader meets without looking for it, and it is the one that would matter to anyone using the pavement as an instrument.

The classical way to recover a panel’s viewing distance is to take one square tile, extend its diagonal to the horizon, and measure the distance from the centric point: that is the distance point, and it is the viewing distance on the page. It works from any tile of a square pavement and gives the same answer from every one. On the rule’s floor the tiles are not square and are not the same shape as each other, so each tile’s diagonal runs to a different point.

The front tile names 704 pixels; the back tile names 398; the panel was drawn from 520. A reader who took the front tile — the largest and easiest to measure, and the one a demonstration always uses — would conclude the panel is meant to be seen from a third further away than it was drawn; one who took the last tile, a quarter nearer. A picture’s correct viewpoint depends on the size it is shown at, which turns that viewing distance into centimetres in front of a reproduction, and a reproduction measured this way puts the reader somewhere between those answers depending on which tile was trusted.

This is the same fact as the diagonal’s bend, stated as a number a reader would actually compute. The bend is the straightedge noticing that the tiles do not share a distance point. The spread of distance points is what that disagreement costs a reader who does not notice.

The rule as a camera #

So the rule has a clean description that is neither “wrong” nor “approximately right”. It is an exact camera pointed at the wrong floor.

Its camera is the panel’s: same horizon, same centric point, same viewing distance, which the orthogonals and the calibrated end transversals fix. Its floor has the panel’s braccia across and a depth that grows geometrically from front to back, with the growth factor set by the ratio the painter used. Given the ends, the ratio is not free — the rule that draws another room shows it is forced by where the first and last transversals go — so the painter did not choose the lengthening either. It is what the rule does to any pavement it is calibrated on.

Seen that way, a rule-spaced panel is not an incoherent drawing and not a nearly correct drawing of a square floor. It is a coherent drawing of a floor nobody built: planks laid by a carpenter who made each one about eight and a half per cent longer than the last. Everything a reader can test about the drawing’s internal consistency comes back clean, because the drawing is consistent. What cannot come back clean is the diagonal, because it is the one test that encodes what the floor was supposed to be.

A long pavement on the same page #

The drag on the pavement figure runs from four braccia to twenty-two, always drawn to the same width, and one more placement shows where that leads.

On sixteen braccia the boards run from 0.72 to 1.33 braccia and the diagonal bends 8.2 pixels, nearly the same as on eight. The reason is the calibration. With the ends pinned and the page width fixed, adding braccia divides the same depth of floor into more, narrower strips, and the ratio creeps toward one as the count grows; the geometric floor it describes converges on a single shape whose front-to-back growth is about 1.85 overall. So the rule’s error, measured as a shape, is a property of the pavement’s depth rather than of its tile count, and a painter cannot reduce it by adding tiles. Only a shallower floor — a pavement that stops nearer the viewer — makes the lengthening smaller.

What the lengthening floor does not settle #

The reading is on a level floor. The reciprocal relation between a transversal’s height and its ground line’s depth assumes the pavement is horizontal. On a floor that tilts toward or away from the viewer the same transversals depict another row of depths, and a rule-spaced pavement on a stage raked toward the audience is a different floor again. Nothing here measures that.

It assumes the orthogonals are right. The corners come from orthogonals drawn to the centric point. A panel whose orthogonals miss the centric point has corners in the wrong places independent of the transversals, and the diagonal then bends for two reasons at once. Separating them needs the orthogonals checked first, which both vanishing points on the paper treats as its own problem.

The bend is measured without a hand. Every corner here is exact. A painter’s hand scatters the corners as it scatters the transversals, and a scatter of a pixel or two added to an eight-pixel bow is still a bow; at four braccia, where the bow is 3.4 pixels, the question of whether a hand hides it is live and is not asked here. One hand step each measures what a hand leaves in the transversals and would be the model for doing the same in the corners.

And a floor that really lengthens would pass. A pavement of genuinely lengthening planks, drawn correctly, has a bent diagonal too. The diagonal test convicts the rule only on a floor that was meant to be square. That is the same limitation the horizon test has, made visible instead of hidden inside a fit — which is the improvement.

The floor, the horizon, and the diagonal #

The constant-ratio rule, calibrated at both ends, is an exact perspective on the panel’s own horizon of a floor whose boards grow by the inverse of the ratio — 0.74 braccia at the front of an eight-braccio pavement and 1.31 at the back. The orthogonals agree with that floor, so they cannot convict the rule; the earlier horizon test convicts it only by assuming square tiles, and does so by fitting.

The test that states the assumption on the page is the diagonal. The corners where orthogonal k meets transversal k bend 8.1 pixels off their chord, where the transversals miss a fitted perspective by 0.17 — 47 times the evidence, from a straightedge. And every tile of the rule’s floor names a different viewing distance, from 704 pixels at the front to 398 at the back, against the 520 the panel was drawn from.

Still open: whether a painted floor’s diagonal survives the painter’s hand #

The bend is large against the fitting test and small against a brush. On a real panel the corners are where painted lines cross, each placed with the painter’s own scatter, and the test’s value depends on whether a bow of three to eight pixels can be seen through a scatter of one or two.

The measurement that settles it takes each of the procedures three procedures, one panel draws — the distance point, Alberti’s section, the stepped measuring point, and the rule — adds a hand’s scatter to both the transversals and the orthogonals, and measures the diagonal’s bend across many drawings at each pavement length. It then asks for the length at which the rule’s bow separates from a hand’s scatter as cleanly as the roughness statistic separates two hands — and whether a straightedge on the diagonal is, in practice, the attribution test a small painted pavement has been lacking, or whether the hand fills its bow in exactly where the pavement is short enough for the rule to have been used.