Each system answers its own question

The comparison in the previous essay has a structural weakness and it is worth naming before anything is concluded from it.

The five questions were chosen by somebody who already knew the systems. If the columns were picked so that each system wins one, the resulting table is a restatement of the choice, not a finding — and it would look exactly like a finding.

What a table like this is usually for #

Comparisons of drawing systems are common and they almost all have the same structure: a list of systems, a list of properties, and a conclusion that perspective has the most of them. The conclusion is normally stated as a historical claim — that the other systems are earlier, or partial, or superseded — and the table is the evidence.

The trouble is that such a table is a sampling of properties, and nobody says how the sample was drawn. Given a fixed set of systems it is possible to construct a property list on which any one of them wins, because each system has exact guarantees the others lack. A list on which perspective wins is easy; so is a list on which cavalier wins; so is a list on which the handscroll wins, and this field could have produced one.

So the question is not whether this table is balanced. It is whether the column set was drawn in a way that could have come out unbalanced, and the only evidence for that is the columns doing something the systems’ designs did not predict.

The failure mode has a name here already #

This site has run into this twice and recorded both, and they are the same mistake in different clothes.

A cross-ratio test that measured nothing: four consecutive divisions of a receding row were compared against the value four equally spaced points must have, which gives the equal-steps-by-eye method a perfect score — because four points equally spaced in the picture have the same cross-ratio as four equally spaced in the world. A necessary condition, evaluated at the one input where it cannot fail.

A conformality test that measured nothing: angles were differenced along one tangent basis, which handed the cylinder a perfect score of $5.5 \times 10^{-10}$ degrees, because that basis happens to be the cylinder’s own azimuth and elevation and those two do stay perpendicular. Rotating the right angle through a half-turn and keeping the worst case rejects it by 5.62°.

Both were caught the same way: by asking whether the test could fail on the case it was passing. A comparison table needs the same interrogation, and the question for a table is not about any one cell — it is about the choice of columns.

The test a set of columns has to pass #

A column set chosen to flatter is one where every system’s win is on the property that system was designed around. The diagnostic is therefore:

Does any system win a column it was not designed for?

If none does, the table is a list of design goals with measurements attached and says nothing that reading the systems’ definitions would not. If some do, the columns are measuring something the designers were not aiming at, and the table has independent content.

Run it on the five columns.

Perspective wins a centre and diminution, and it was designed around both. No information.

The parallel systems win true measure, and they were designed around it. No information.

But the parallel systems also win unbounded depth, and nothing about them was aimed at that. Cavalier and cabinet were developed for fortification drawing, where the point is that a length along an axis is a true length; that the resulting picture can be extended indefinitely into depth without accumulating anywhere is a consequence nobody was pursuing, and it is the property that makes the same systems work for a narrative scroll of a palace six hundred years later and a different continent away.

And elevation wins bounded depth — the same column perspective wins — by having no depth axis at all. That is a system winning a column by a mechanism unrelated to the one the column was written to detect, which is the clearest possible sign that the column is measuring rather than describing.

Two wins out of the five that were not designed for. Not a resounding score, and enough to say the table is not purely a construction.

What “designed for” means for a convention nobody designed #

There is a wrinkle in the test and it should be admitted before the test is used, because it is the sort of thing that quietly invalidates an argument.

Cavalier projection was developed for a purpose that can be named. Isometric was standardised, in the nineteenth century, by people who wrote down why. Perspective has a founding literature. For those, “the property it was designed for” is a statement about a documented intention.

A handscroll’s geometry, the removed roof and the aspective figure were not designed in that sense. There is no document, no stated goal and probably no moment at which anybody chose. So the test’s phrase has to be weakened for them to the property the convention is most obviously organised around — which is a judgement rather than a fact, and a judgement made by the same person who chose the columns.

That weakens the test on exactly the systems this phase added, which is unfortunate and is the honest position. The two surprises the test does find — the parallel systems’ unbounded depth, and elevation’s bounded depth by a different mechanism — are both on systems with documented purposes, so they survive the wrinkle. Any surprise found on the scroll or the removed roof would not.

Where the table is weakest #

Honesty requires naming the columns that fail the test outright.

“Does it keep straight lines straight” is won by eight of the nine rows, and lost only by the scroll. A column with one dissenter is barely a column: it is a statement about the scroll wearing a comparison’s clothes. It stays in the table because its absence would be a bigger distortion — a reader comparing systems has straightness in mind and a table that omitted it would be quietly answering it in the affirmative for everything.

“Does size fall with distance” is very nearly the same column as “does it have a centre”, and separates from it on exactly one row. Every system with a centre diminishes, because both are consequences of the same divide; the scroll diminishes without having one, because it divides by depth across its roll while its rays never meet. So the second column carries one row’s worth of information beyond the first — which is more than nothing and much less than a column. They are kept apart because they are separately observable, a reader can see diminution and cannot see a centre, and the single row on which they disagree is the one that says the two are not the same question.

So of five columns: two are informative, two are redundant with each other and with the first, and one is nearly a constant. That is a weak table by the standards this site applies to its own figures, and it is the strongest one available with five properties this site can measure.

How many rows the table actually has #

There is a blunter way to ask how much the five columns are measuring, and it does not depend on judging what anything was designed for: read the columns as a five-letter word for each row and count how many distinct words appear.

Nine rows produce four.

Perspective answers centre, no measure, diminishes, bounded, straight. The handscroll answers no centre, no measure, diminishes, bounded, bent. Elevation answers no centre, measure, no diminution, bounded, straight. And isometric, dimetric, trimetric, cavalier, cabinet and military answer no centre, measure, no diminution, unbounded, straight — all six of them, in every column.

Six of the nine rows are indistinguishable. That is the sharpest available statement of how coarse this instrument is, and it needs no argument about design goals: two thirds of the table is one row printed six times. A system added to the site’s libraries earns a row automatically, which is the property that makes the table a measurement rather than a description, and it also means most of the rows added so far have carried no information into it.

The six are not the same system. The parallel field separates them sharply — cavalier draws its depth axis at full length and cabinet at half, isometric’s three axes are equal and trimetric’s are all different — and that field can tell any two of them apart in one number. None of those differences is a yes-or-no question, which is why none of them is a column here, and it is the clearest case of the table measuring what it can phrase rather than what distinguishes its subjects.

Two things follow for the claims this field makes.

The absence of a dominance ordering is a statement about four objects, not nine. Checking that no row dominates another across all five columns is four comparisons of pairs rather than thirty-six, and a reader counting nine rows is crediting the check with more than it did.

And the exclusion is unaffected, which is worth saying because everything else in this essay qualifies something. It is a claim about two columns and holds row by row; a table of four distinct rows or of four hundred would show the same thing, because the reason is arithmetic rather than sampling. A coarse instrument can still settle a question that only needs one bit per row, and this is one.

What a stronger comparison would need #

Three columns this site cannot currently fill, named so that the gap is on the record rather than invisible.

Occlusion behaviour. Which surfaces hide which, and whether the hiding is a depth statement. This is the property the removed-roof essay measures for one system and no other, and it is measurable in principle for all of them.

Robustness to the viewer being in the wrong place. A perspective picture is correct from one point; how fast does it degrade as the viewer moves? The viewing field measures this for pinholes, and the parallel systems’ answer is not at all, because it was never correct from anywhere, which is a different kind of answer and would need a column that can express it.

Cost of construction. How much work it takes to make a correct picture in each system. This is not geometry and it is plainly one of the strongest forces shaping which systems traditions actually used; a parallel projection can be drawn with a set square and a scale, and a perspective construction needs a lateral section or a distance point.

That third one is the largest omission in this whole field, and it is the sort of thing a geometric account is structurally unable to see.

Why “each answers its own question” is still worth saying #

Given all that, the slogan in the title is doing less work than it appears to, and it is worth being clear about what remains.

What the table establishes is the exclusion, and the exclusion is not a matter of column choice: no system has both a centre and true measure, and that holds however the other columns are picked, because it is a consequence of the divide. That result is column-independent and it is the field’s one hard finding.

What the table illustrates is that the systems are not ordered. There is no row that dominates another across all five columns — check it — and that is a weaker claim than the exclusion but a useful one, because the account these conventions usually get is precisely an ordering, with perspective at the top and everything else at some distance below it.

And what the table cannot do is tell anybody which system to use, because that depends on what the picture is for, and no column in it measures a purpose.

That last point deserves one more turn, because it is the one a reader is most likely to want overturned. It would be useful to have a rule of the form for this kind of subject, use this system, and the material for one is nearly there: the fields in this phase keep observing that a system’s properties suit the things it was used to draw. Journeys want uniform depth; interiors want comparable rooms; figures want every aspect at full extent.

The reason that does not become a rule is that the fit runs both ways and the measurements cannot tell which direction it ran. A tradition may have kept a system because it suited the subjects, or drawn those subjects because the system suited them, and both stories predict exactly the same correlation. Establishing which would need evidence about the traditions, and this site has none.

So the correlation is reported and the rule is not derived, which is a smaller conclusion than the material seems to offer and is the one the material supports.

The generalisation, and why it is a real one #

The column-choice problem is not peculiar to drawing systems. It is the shape of every comparison in which the things compared were designed by people with different aims, and it has a name outside this site: it is the problem of choosing a benchmark.

Two failure modes, both visible here.

A benchmark drawn from one entrant’s design goals ranks that entrant first, and the ranking is a restatement of the goals. Four of this field’s five columns come from properties perspective’s own theory made interesting — a station point, diminution, straightness — and the fact that perspective wins two of them is not news.

A benchmark that admits every entrant’s goals produces a diagonal in which everybody wins, which is equally uninformative in the other direction. That is what this table would be if the columns had been chosen one per system, and the test above is what distinguishes the two cases.

The test — does anything win a column it was not designed for — is the useful transferable piece, and it is worth stating in the general form: a comparison has independent content exactly to the extent that its results surprise the designs of the things compared. Two surprises out of five here. Not many, and not zero, and the number is on the record rather than implied.

The honest summary #

A comparison of drawing systems is worth making and is worth distrusting, in that order.

The exclusion is real, is a consequence of arithmetic rather than of curation, and survives any reasonable choice of the other columns. The absence of an ordering is real and is weaker. The impression of a neat diagonal, where each system wins the column it was made for, is largely an artefact of choosing columns from properties this site has machinery to measure — which is to say, from properties that perspective made interesting.

That last point is the one worth carrying out of this field. Even a comparison written specifically to avoid treating perspective as the standard ends up measuring the things perspective’s own theory made measurable, because that is where the machinery came from. Being aware of that is not a solution and there may not be one; naming it is what stops the table from being read as more neutral than it is.

What survives all of this #

To state the residue plainly, because an essay of caveats can leave a reader with the impression that nothing was established.

The exclusion in the previous essay is a theorem and survives every objection raised here: it depends on two columns, both of which are forced by the arithmetic of dividing by depth, and no choice of the other three affects it.

The absence of a dominance ordering is an observation and survives too, because it is a statement about the table as it stands rather than about how the table was drawn.

What does not survive is the impression of a clean diagonal — every system winning its own column, fairly, on a neutral set of questions. That impression is partly real and partly a consequence of choosing questions from the properties this site can measure, and the honest accounting is two surprises out of five.