A straightedge convicts the rule on five braccia

The rule is exact for a floor that lengthens settled what the taught constant-ratio rule draws. Spacing each receding board a fixed fraction of the one before is an exact perspective of a floor whose boards grow as they recede, seen from the panel’s own horizon, so no test of the transversals alone can convict it: they are a correct picture of something. What says the tiles were meant to be square is the diagonal. On a correct pavement of square tiles the corners where each orthogonal crosses its transversal lie on one straight line; on the rule’s pavement they bow off it by 8.1 pixels on eight braccia, where the fitting test on the transversals finds a fifth of a pixel.

That bow was measured on pavements drawn exactly. A painted pavement is not drawn exactly: every line is laid by a hand, and the corners scatter with it. The essay ended by asking the practical question — whether a bow of three to eight pixels survives a scatter of one or two, how long a pavement has to be before the rule’s bow separates from a hand’s wander 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 lacked, or whether the hand fills in the bow exactly where a pavement is short enough for the rule to have been used.

The answer is that the straightedge works, from five braccia at a pixel of scatter, and fails in a place the question did not anticipate.

Two painted pavements #

Every pavement below is painted, not drawn: the procedure places its transversals, then every line is laid with the hand’s scatter, and every orthogonal’s foot on the ground line is placed with the same scatter before it is ruled to the centric point. The constant-ratio rule has no marks of its own to misplace, so its lines carry the laying scatter alone; the constructed procedures carry their own mark errors as well, at the same precision.

The two pavements look alike, as they must — the rule’s pavement was calibrated so its first and last transversals land where a correct construction puts them, and a pixel of scatter is invisible at this size. The chord through their corners does not look alike. The rule’s corners bow 8.03 pixels off it, which is the bow the earlier essay measured on an exact pavement plus a pixel’s worth of wobble. The constructed pavement’s corners wander 2.47 pixels, which is the hand alone. Over two hundred such pairs of eight-braccio pavements at this scatter, the rule’s diagonal is the more bowed in every one.

The straightedge here is laid the way a reader would lay a ruler across a panel: from the first corner to the last, then measuring how far the worst corner in between falls from it. That choice turns out to matter, and the last figure returns to it.

The page caps the bow, and the hand does not stop #

The earlier essay found the bow growing with the pavement’s length. That was true of pavements drawn with a fixed braccio, and it stops being true once a pavement is drawn to the width of a page.

The rule’s bow, with no hand at all, rises from two pixels on three braccia to 8.15 on eight — and then stops, reaching only 8.24 on twenty-two. Past eight braccia the pavement has filled the page, and a longer pavement drawn to the same width is the same pavement with smaller tiles; the bow is a property of the pavement’s shape against the page, and that shape stops changing. The hand’s wander does not stop. With more corners there are more chances for the worst of them to wander far, and the median wander of a constructed pavement’s corners rises from 1.21 pixels on three braccia to 2.65 on twenty-two at one pixel of scatter, and from 2.43 to 5.30 at two.

That changes the shape of the whole question. The earlier essay’s picture was of a bow growing and a hand’s scatter holding steady, so that a long enough pavement would always convict. On a real panel, whose width is fixed, the bow reaches a ceiling and the hand’s reading climbs toward it. A long pavement is not the best case for the straightedge. A pavement of eight to twelve braccia is.

How cleanly it separates, against the roughness test #

The measure that answers the question directly is the chance that the straightedge ranks a pair of painted pavements correctly — the rule’s reading above the construction’s. A half is chance; one is certainty. The same measure applies to the roughness test, which reads the shape of the residual left after fitting a correct perspective to the transversals and tells a hand that measured every braccio from the ground line from one that stepped them with dividers.

At half a pixel of scatter the straightedge is already reliable on three braccia, ranking the pair correctly 98 times in a hundred. At a pixel it reaches 96 in a hundred on four braccia and certainty from five. At two pixels it climbs more slowly, from 63 in a hundred on three braccia to 90 on six, and levels off near 97 from eight braccia on, the ceiling the page puts on the bow.

The roughness test does not depend on the hand’s size at all, because it reads the shape of the residual rather than its size, and it rises steadily with length, from 55 in a hundred on five braccia to 98 on twenty-two. So the question the earlier essay posed — at what length does the diagonal separate the rule as cleanly as the roughness separates two hands — has an answer that runs the other way from what it expected. At a pixel of scatter the straightedge is the better test at every length where both can be read. On five braccia it is certain where the roughness test is barely better than a coin. At two pixels the straightedge is still ahead until the longest pavements, where the roughness finally reaches the straightedge’s ceiling.

What a panel says about its maker found that attributing a procedure from the transversals needs many of them. The diagonal needs very few, because it is not asking the transversals to reveal a pattern in their errors. It is asking whether the tiles are square, and a non-square tile announces itself in every corner at once.

Why five braccia #

The threshold falls where it does for a reason that can be read off the second figure. At a pixel of scatter a constructed pavement’s corners wander about one and a quarter to one and a half pixels off the chord on short pavements. The rule’s bow is 1.98 pixels on three braccia, 3.36 on four, and 4.60 on five. The test becomes reliable once the bow is about three times the hand’s typical wander, and that happens between four and five braccia at a pixel — which is the same third-of-the-bow rule the scatter figure below finds from the other side.

The bow grows so quickly on short pavements because it is the difference between a geometric series and the correct reciprocal spacing, measured in the middle of the pavement, and that difference grows faster than the pavement for the first few braccia. The rule that draws another room found the rule already agreeing with a correct construction to a fifth of a pixel on its transversals; the diagonal amplifies the same small disagreement by laying it sideways, across the width of the tiles, where each braccio’s error is multiplied by the tile’s own width.

A threshold a reader can use #

A ranking of pairs is a statistician’s measure. A reader with one panel needs a threshold: a wander beyond which the panel is called the rule’s.

On eight braccia painted with a pixel of scatter, a constructed pavement’s corners wander a median 2.03 pixels off the chord and ninety-five in a hundred stay inside 3.75. The rule’s wander a median 8.64 pixels, and not one of four hundred falls inside 3.75. A reader who lays a straightedge across an eight-braccio pavement, finds a corner four pixels off it, and calls the pavement the rule’s, is wrong about one construction in twenty and right about every rule.

Four pixels, on a drawing 690 pixels across, is about three millimetres on a half-metre panel. The test needs a straightedge and a good reproduction, and nothing else: no fitting, no horizon, no knowledge of how the panel was constructed.

Where the hand does win #

The test gives out when the scatter grows. The figure below finds where, for three lengths of pavement.

On four braccia the straightedge stays above nine in ten until the hand scatters its lines by 1.26 pixels; on eight and on sixteen, until 2.79 and 2.80. The rule’s bow on those pavements is 3.4, 8.1 and 8.2 pixels, so in each case the straightedge gives out once the scatter reaches about a third of the bow it is looking for. The sixteen-braccio pavement tolerates no more scatter than the eight, because its bow is no larger — the page again.

This is where the earlier essay’s worry comes true, in a narrower form than it was put. The hand does fill in the bow, on short pavements drawn by rough hands: three or four braccia at two pixels of scatter, where the bow is between two and three and a half pixels and the test ranks correctly only two or three times in four. Those are exactly the small pavements where a painter might most plausibly have used a rule of thumb rather than a construction. On anything longer, or painted more carefully, the bow survives.

A tiring hand draws a different habit found that a hand’s errors change across a panel in ways a single statistic can hide. The scatter here is a steady hand at a fixed precision; a painter whose precision varies across the pavement would bow the diagonal a little of their own accord, and a test calibrated on a steady hand would then call some constructions the rule’s.

How to lay the straightedge #

There is more than one way to read a diagonal against a straight line, and they are not equally good. The last figure compares three, at the scatter where the choice matters most.

The chord from the first corner to the last beats a straightedge fitted through all of them at every length. The reason is specific to the rule: its pavement was calibrated to agree with a correct one at its first and last transversals, so its bow is entirely between them, and a chord pinned at the two ends shows all of it. A fitted line splits the bow into a positive half and a negative half and shows only part. The third reading averages every corner’s distance rather than taking the worst, and on long pavements, where there are many corners for the hand’s scatter to average over, it wins: 0.996 on sixteen braccia against the chord’s 0.970. On short pavements the worst corner off the chord is still the better single reading.

So the practical rule is: lay the straightedge from the first corner to the last, and on a long pavement judge by how far the corners fall off it on average rather than by the worst one.

A painter who checked the diagonal was not using the rule #

One historical objection is worth answering before it is raised. Painters of the period knew the diagonal check: drawing a line from one corner of the pavement to the other and making sure it passed through the corners of the tiles was a standard way of verifying, or constructing, a pavement. Could a painter have spaced the boards by the rule and then corrected them against the diagonal, leaving a rule-spaced pavement with a straight diagonal?

No — and the reason is instructive. A painter who places each transversal where an orthogonal crosses the diagonal is not correcting the rule; they are performing the distance-point construction, whose whole content is that the diagonal fixes the spacing. The result has square tiles and correct spacing, and it is not a rule pavement at all. So a straight diagonal is not evidence that the rule was used carefully. It is evidence that a construction was used, and the rule, if it was ever drawn, was overwritten. One hand step each made the parallel point about constructions in general: what survives in the drawing is the step the hand performed, not the recipe the painter had in mind.

What the straightedge is evidence of #

Three procedures, one panel and the slip that leaves no trace found that most of what a hand does leaves the drawing an exact perspective of something, so that the transversals alone cannot say which procedure made them. The constant-ratio rule is the case where that silence is complete: its transversals are an exact picture of a lengthening floor. What the straightedge adds is a second fact the transversals cannot carry — that the floor’s tiles are square — and against that fact the rule is wrong at every corner.

The straightedge convicts a rule pavement of five braccia or more painted to a pixel, and the conviction comes with a stated error rate. It does not convict a short pavement painted roughly, and there the rule may have been used without leaving evidence. The rule that draws another room called the rule a correct picture of the wrong room. A straightedge on the diagonal is how a reader of the panel sees that the room is wrong, provided the painter’s hand was steadier than a third of the bow.

What the measurement assumes #

The hand scatters every line independently, at one precision. Real painters work in passes, and a transversal laid with the same ruler setting as its neighbour shares some of its error; correlated scatter would bow a construction’s diagonal smoothly and make it look more like the rule’s.

The tiles are meant to be square. The diagonal convicts the rule only against a floor of square tiles. A pavement of oblong tiles, deliberately drawn, has a diagonal that meets its distance point at a different place, and its corners lie on a straight line all the same; a reader needs to know the floor was meant to be square, as the earlier essay said.

The pavement is read from a faithful reproduction. Every number is in pixels of the drawings here, 690 across; a reproduction that is itself distorted adds its own bow.

Still open: whether the rule’s bow has a shape the hand does not #

Every reading above measures how far the corners fall from a line: the worst corner, or the average. The rule’s corners do not merely fall off the chord; they fall off it in one direction, with the largest departure near the middle and none at the ends, a smooth arc fixed by the ratio. A hand’s corners fall off it in both directions at random. The two differ in shape, not only in size, and the roughness test’s lesson is that shape survives when size does not — its separation did not depend on the hand’s scatter at all.

That suggests a test on the diagonal that reads the shape of the corners’ departures rather than their size: fit the arc the constant-ratio rule would bend, with its one free parameter, and ask how much of the corners’ departure it explains against how much a hand’s random scatter explains. The measurement that settles it computes that arc-fitting statistic over painted pavements of every length, at scatters up to four pixels, and asks whether it keeps the rule separable where the straightedge failed — on short pavements painted roughly — or whether, at three or four braccia, the arc has too few corners to have a shape at all.