Stepped, or measured from the zero

Two draughtsmen lay out the same pavement with the same care. One sets each braccio from the edge of the panel with a scale. The other walks a pair of dividers along the ground line, stepping each braccio from the mark before it.

Neither is more careful than the other, and the phrase “the same care” is meant literally: each places a mark to the same precision, and the precision is the only thing about their hands that enters. The drawings they produce are different in a way that is measurable, and the difference has a name and an exponent.

The measurement #

Set both hands to the same precision and lay off pavements of four, six, eight, twelve, sixteen, twenty-four and thirty-two braccia. For each, measure the worst departure of a mark from where it should have been.

The stepped route’s departure grows as the square root of the count — the fitted exponent is 0.42 over this range, converging on a half. The measured route’s grows as the count to the power 0.21, which is not a growth law at all but the slow creep of a maximum over more independent draws.

Both are what the arithmetic predicts, and it is worth doing the arithmetic because the prediction is exact rather than qualitative.

Why a walk grows as a square root #

The stepped draughtsman’s k-th mark carries every slip made before it. If each step is off by a random amount with the same spread, the k-th mark is off by the sum of k such amounts.

A sum of independent errors does not grow like their count; it grows like the square root of it — the same arithmetic how wrong a measurement from one picture can be uses on a chain of readings, because they cancel as often as they accumulate. So the k-th mark is off by roughly the single-step precision times the square root of k, and the worst mark on a pavement of n braccia is off by roughly that times the square root of n. An exponent of one half.

The measured draughtsman’s k-th mark is off by one slip, not k of them, so the worst mark on a pavement of n is the largest of n independent draws. The largest of n draws from a bell-shaped distribution grows like the square root of the logarithm of n, which over a range of four to thirty-two is a factor of about one and a half — an apparent exponent near a fifth, and no growth at all in the sense the other has.

An error with two terms is this collection’s general statement of that distinction: one term responds to effort and one does not, and calling both of them “the error” hides which is which. Here both terms are the same hand, and what separates them is the arrangement of the arithmetic rather than the care.

What it looks like in the drawing #

The exponents are a statement about many drawings. On one drawing the difference appears as the shape of what is left after the best correct perspective has been subtracted.

The statistic is the roughness — the size of the residual’s second difference against the residual’s own size — and for independent errors it is exactly the square root of six. That is not a threshold anybody chose: the second difference of three independent quantities has one plus four plus one times their variance, and the square root of six follows.

At six braccia the two medians are 3.12 and 2.80, a third of a unit apart with distributions that overlap almost entirely. At thirty-two they are 2.51 and 1.14 — the first has arrived at √6 and the second is nowhere near it.

The gap widens, which is the useful part #

A separation that exists but is unusable on any one drawing sounds like no separation at all, and the reason it is not is that the gap is a function of something a reader can look for.

The measured route’s roughness converges on a constant. The stepped route’s falls, because the walk’s own amplitude grows with the count while its step-to-step differences do not — the residual gets larger and smoother at the same time. So the gap opens without bound, and the question for a reader is not “can these be told apart” but “how long a pavement does telling them apart need”.

Measured on this panel at a hand precision of a pixel and a fifth, the gap is a third of a unit at six braccia and one and a half at thirty-two. A confident reading needs a gap several times the spread of either distribution, which on these numbers is somewhere past forty braccia. That is the same shape of answer the distance at which two lamps part gives for a count of lights: the separation exists, and the drawing has to be a certain size before it is reportable.

The gap’s own law is worth writing down, because it has a ceiling. The measured route sits at $\sqrt6$ and the stepped one falls as $n^{-1/2}$, so the gap is $\sqrt6 - c/\sqrt n$ with $c$ fixed by the thirty-two-braccio reading at 6.45. It is 1.43 at forty braccia, 1.64 at sixty-four, and can never exceed 2.449 however long the floor is. Reaching a gap of two takes two hundred braccia — five times the length for a forty per cent improvement, because the approach to the ceiling is a square root.

So the answer to “how long a pavement does telling them apart need” is forty for a reading and nothing at all for certainty. The separation improves without bound in principle and asymptotes in practice, and past a hundred braccia the pavement is no longer the limiting term — the hand’s own consistency is.

Forty braccia is not a painted pavement. It is a marquetry floor, an engraved plate, or a treatise’s demonstration diagram — which is to say the reading works on exactly the drawings made to demonstrate the method and not on the drawings made to be looked at.

The name of the method is not what decides it #

Here is the correction the rung exists for, and it changes what the reading can be said to report.

Alberti’s section is usually executed by measuring each braccio from the panel, and a measuring line is usually executed by walking dividers. But either can be done either way. A draughtsman with a scale and a measuring line measures from the origin every time and produces independent marks; a draughtsman stepping dividers along Alberti’s ground line accumulates and produces a walk.

So the reading does not report which recipe was followed. It reports where the zero was, and that is a fact about how the recipe was executed rather than about which recipe it was.

That is a smaller claim than it looked like at the start of the row and it is a better one, because it is true. One hand step each sets out the four hand steps and this is the sense in which two of them are one step performed two ways.

Whether the distinction is worth anything #

It is, and for a reason that is historical rather than geometric.

Dividers are what a workshop had. A scale marked in braccia, accurate over the width of a panel, is a different instrument and a later one, and the transition from one to the other is a documented thing in the history of drawing practice. A reading that separates stepped from measured marks is a reading that speaks to that transition rather than to the choice between two published recipes.

Whether the reading is good enough to speak to it on any particular panel is the question above, and the answer is usually no. What it can do is say that a claim about a workshop’s instruments made from the spacing of a four-tile floor has nothing behind it, which is the same negative result four marks before anything is said arrives at by counting.

The control that makes the exponents mean something #

Two exponents fitted from a sweep are worth exactly as much as the control beside them, and the control here is that both routes are the same hand.

The stepped and measured pavements in the figure above are generated from the same stream of slips, at the same precision, on the same panel, with the same number of braccia. The only difference in the code is whether the k-th slip is added to the k-th mark or to the running total. Everything that could be a difference in care, in instrument, in scale or in setting is held.

Without that, an exponent of a half and an exponent of a fifth would be consistent with the stepped draughtsman simply being worse. With it, the two exponents are the arrangement of the arithmetic and nothing else — which is the same discipline the residual has a shape applies to a lamp recovery, where two residuals of the same size mean different things because of how they are distributed.

The third arrangement, which is neither #

There is a way of laying off braccia that is neither stepped nor measured, and it is the one a straightedge construction uses.

The bay repeated by a straightedge gets each new transversal from the previous one by drawing a diagonal and finding an intersection. No length is measured and no divider is walked; the hand’s only job is to lay a ruler across two drawn points, and the marks are the crossings.

That arrangement accumulates — the fourth bay is constructed from the third — so it ought to leave a walk. It does, and a slower one: the construction’s intersections are taken at healthy angles, so the leverage between the ruler’s wobble and the crossing’s displacement is close to one, and each step contributes rather less than a divider’s step does. The residual is a walk with a smaller step size, and the roughness sits between the two curves above.

The interesting part is that the construction has no scale to be wrong about. A stepped divider that is set a hair too wide is wrong the same way at every step, which is a systematic error rather than a walk and which the reading would report as a rule; a straightedge construction cannot make that mistake, because it never carries a length. So the same arrangement that makes it accumulate also makes it immune to the one error a walk is usually blamed for.

A divider set wrong, which looks like a rule #

That last point is worth a section, because it is the commonest real failure of a stepped layout and the reading gets it right for an interesting reason.

Dividers are set once and walked many times. If they are set a per cent too wide, every step is a per cent too long — and the k-th mark is off by k times that, not by the square root of k. The departure grows linearly with the count and the residual is a smooth curve rather than a walk.

The reading calls that systematic, and it is systematic: a fixed proportional error in a step length is a rule, in exactly the sense the taught spacing recipe is a rule. What separates the two is where the implied horizon lands. A pavement with a mis-set divider is a correct perspective of a floor whose tiles are a per cent deeper than they should be, so its implied horizon is exactly the drawn one and the panel agrees with itself; the taught rule’s implied horizon is a hundred and seventy pixels away.

So the reading’s two instruments answer two different questions and both are needed. The roughness says whether the departure came from a hand or a rule. The horizon says whether the drawing agrees with its own panel. A mis-set divider fails the first and passes the second; the taught spacing rule fails both.

What the exponents are not #

Two cautions about the fitted numbers, because a power law over a short sweep is the easiest thing in this collection to over-read.

The range is short. Four braccia to thirty-two is less than a decade, and a fitted exponent over less than a decade is a description of the sweep as much as of the law. The half is asserted here because the arithmetic predicts it, not because the fit is convincing on its own; the fit is a check that the arithmetic reaches the drawing.

And one of them is not an exponent at all. The measured route’s 0.21 is a fit to the growth of a maximum, which is logarithmic rather than a power, and a logarithm fitted as a power over a short range gives whatever exponent the range implies. Quoting it as a law would be wrong. It is quoted here as a number to be compared with a half, and its whole content is that it is not one.

What a null result is worth in decades is this collection’s own treatment of the general problem — how much sweep a claimed exponent needs before it is a claim about anything — and this sweep is on the short side of what that essay would accept for a law and comfortably long enough for the comparison being made.

Where the walk goes when it is not visible #

A walk has a second consequence that the roughness does not see, and it is the one that matters most in practice: a walk moves the implied horizon.

The reader’s fit chooses a horizon along with everything else, and a residual that drifts in one direction is partly explained by a horizon a little higher or lower than the one the panel drew. So a stepped pavement’s implied horizon is further from the drawn one than a measured pavement’s, at the same precision, and the difference is measurable.

At a hand precision of a pixel and a fifth on an eight-braccio panel, marks from a zero move the implied horizon by about a pixel and stepped marks by about two. Both are far below the hundred and seventy pixels the taught spacing rule produces, which is why the horizon test separates a rule from a hand cleanly and does not separate the two hands at all.

What a longer pavement does not fix #

There is a temptation, having seen the gap widen, to conclude that the reading simply needs more marks. Two things stop that.

The far transversals contribute almost nothing. A pavement’s tenth braccio is squashed into a fraction of the space its first occupies, so a slip there is a smaller displacement on the page and a smaller contribution to everything measured. Doubling a pavement’s length does not double the evidence; it adds a tail of marks that carry little.

And a long pavement is a different drawing. Nobody lays out thirty-two braccia by walking dividers thirty-two times without checking against a scale somewhere in the middle, which turns the walk into two shorter walks and moves the residual’s shape toward the independent case. Seven is not a power of two is the construction a workshop would use for that check, and it needs no scale at all. The reading’s own subject changes as the pavement grows.

So the honest summary of the sweep is that the two exponents are exact, the separation is real, and the pavements on which it would be decisive are pavements nobody drew that way.

The short version #

Marks stepped from the last accumulate and their worst departure grows as the square root of the count; marks set from a common zero do not accumulate, and their worst grows only as fast as the largest of more draws. The exponents come out at 0.42 and 0.21 over a sweep from four braccia to thirty-two, against the half and the roughly-nothing the arithmetic predicts.

The drawing records which of the two happened, in the roughness of what is left after the best correct perspective has been subtracted — and it records where the zero was, which is not the same question as which recipe was followed.