The hook is the centre, and the eye is not

A perspective drawing is a projection through a centre, and for almost every drawing ever made the centre is a hope. It is where the draughtsman’s eye was supposed to be, more or less, on average, over the hours the drawing took. Nothing records it, nothing enforces it, and the only evidence of where it was is the drawing itself — which is why so much of this collection is about recovering a station point from marks on paper rather than reading it off a label.

There is one family of instruments for which that is not the case, and the difference is worth taking seriously rather than treating as a curiosity of the woodcuts. Dürer’s string frame puts a hook in a wall, runs a taut string from the hook to a point of the object, and records where the string crosses a wooden frame. The centre of projection is then a ring of iron with a position anybody can measure, and the picture plane is a rectangle of timber. The drawing that comes off it is a correct perspective from the hook, whatever the draughtsman’s head was doing.

That single substitution — a physical object where every other method has an assumption — changes what can be said about the resulting sheet. Its viewing distance is not recovered, it is measured with a tape before the drawing begins. Its principal point is not assumed, it is wherever the hook happens to be relative to the frame. And its error under a wandering eye is not small, it is exactly zero.

The viewing distance is an input, not a recovery #

The site’s signature quantity is the distance a picture is correct from: take the focal length in units of the picture’s width, multiply by the width the picture is shown at, and the answer is where a reader has to stand. The essay that established it computes that number from a camera. For a hand drawing there is no camera, so the number has to be got out of the drawing’s own content — three vanishing points, or a rectangle with a known proportion, or a horizon and a pair of perpendicular directions.

For the string frame there is nothing to recover. The hook is 120 centimetres behind the frame and the frame is 56 centimetres wide, so the ratio of the two is the drawing’s focal length in units of its own width, and multiplying by 160 millimetres of print gives 34.3 centimetres. Every step of that is a measurement taken in a room with a tape, before a single mark is made.

That inverts the usual epistemic situation, and the inversion is the reason the instrument is worth an essay. Everywhere else in this collection a picture is interrogated for a station point that its maker did not record. Here the station point is a construction datum, and the picture is a consequence of it. A drawing made this way could be shipped with its correct viewing distance printed on the back, and the number would be a fact about the apparatus rather than an inference from the marks.

It also makes the parameter adjustable in a way that is otherwise only available to a lens. Driving the hook further from the frame lengthens the drawing’s focal length exactly as a longer lens does, and the effect on where a reader must stand is direct.

The range of that adjustment is set by the room, which is a limit worth naming because it is not a limit any of the drawn systems in this collection otherwise has. A lens can be as long as its manufacturer likes. A hook has to be far enough back that a string will still reach the object and near enough that the wall exists, so the achievable focal lengths lie between roughly half a frame width and four of them. That is the same band a working photographer’s ordinary lenses cover, which is not a coincidence: both are bounded at one end by what a subject will fit into and at the other by how far away a person is prepared to work.

The sheet is a projection, and the agreement that says so is weaker than it looks #

The claim that the apparatus produces a correct perspective rather than something that merely resembles one can be checked, and it is worth checking with some care about what the check establishes.

Two routes to the same marks, agreeing to the last bits of the arithmetic, is the shape of evidence this collection uses constantly — and this particular instance is a weak one, so it is better to say so than to bank it. A pinhole camera’s projection is the intersection of a line through the centre with the image plane. Computing the marks as line-meets-plane and then computing them again through a camera object is one theorem evaluated by two pieces of code, not two independent facts converging. The agreement rejects a transposed basis or a sign error in one of the implementations, which is real and is why it is computed; it does not establish that the apparatus is a projection, because that was assumed by whichever route one prefers to call the definition.

What does establish it is the thing the apparatus rules out. A projection through a centre has the property that every mark is determined by the object point and the centre alone — nothing about the recorder enters. That is a claim with a test, because there is a neighbouring instrument in which the recorder does enter, and the two can be given the same perturbation.

The off-centre principal point is worth a sentence on its own before that, because it arrives here for free. The hook is driven in wherever the wall and the subject allow, and that is 122 millimetres off the middle of the frame in this arrangement, so the finished sheet has its optical centre well off its own middle. A trimmed photograph acquires the same defect with a pair of scissors and acquires it silently; the apparatus acquires it with a hammer, in public, and a draughtsman who measured the hook’s position knows the number.

The other instrument, whose centre is on a neck #

Alberti’s veil is the same idea with the centre of projection moved to the wrong side of the picture plane and made of flesh.

The decomposition into a shift and a warp is the useful part of that measurement, and it is worth spelling out why the two behave so differently.

Move the eye and every mark on the pane moves, because every line of sight now starts somewhere else. If the object were at a single depth the whole set would move by one common amount, and a common amount is not a defect: sliding the finished sheet a few millimetres in its mount restores it exactly. What breaks that is depth. A point at depth $b$ behind a pane at depth $a$ has its mark displaced by the wander times $1 - a/b$, so the near parts of the subject move less than the far parts and no single slide of the paper corrects both. The residue after the best common shift — 4.41 millimetres here — is the part that is genuinely a wrong drawing rather than a displaced one.

That factor is also the reason the veil is at its worst on exactly the subjects it was recommended for. A shallow subject close to the pane has $1 - a/b$ near zero and tolerates a wandering head; a deep interior has it near one and does not. The manuals recommend the veil for foreshortened figures and complicated draperies, which are the deep cases.

And the sighting point is not a small piece of apparatus in the drawings that show it. It is a notch, a bead on a stick, sometimes a hole in an upright — a physical attempt to make the eye’s position as definite as the hook’s, and it fails for the reason no amount of joinery can fix. A head can be brought back to a notch to within a few millimetres. A hook does not have to be brought back to anything.

The sighting point is also where the veil’s arithmetic and its ergonomics pull against each other, which the string frame is spared. Bringing the pane closer to the eye widens the angle it covers and makes the marks larger and easier to place, and it also increases $a/b$ for everything in the subject, which reduces the wander term. Moving it away does the reverse. So the veil rewards working with the pane close, and working with the pane close is exactly the arrangement in which a small movement of the head is a large change of viewpoint relative to the pane’s own size. The instrument’s two errors are traded against one another by the same adjustment, and the manuals give no rule for setting it.

There is a further asymmetry between the two centres that has nothing to do with steadiness. The hook’s position can be measured after the drawing is finished, by anybody who walks into the room with a tape. The eye’s position cannot be measured at all, before or after, because the thing to be measured no longer exists — and a sighting point is a record of where the eye was asked to be rather than of where it was. That distinction is the reason the veil’s marks carry no evidence of their own reliability while the frame’s carry a measurement of theirs.

The control, and its zero is exact rather than small #

Put the same perturbation to both instruments and the comparison stops being an argument about which is better made.

A control that reads exactly zero is a different kind of evidence from a control that reads small, and the distinction is the one this collection keeps returning to. A small reading says the effect is weaker in the control case, which invites the thought that the difference is one of degree and that a careful enough draughtsman closes it. An exact zero says the mechanism is absent. The head can move as far as the room allows and the string still runs from the hook, so there is no quantity to be careful about.

It is also a control that could have failed. Had the string frame been modelled with the draughtsman sighting along the string — which is one plausible reading of how a person actually operates the thing — the head would have entered and the line would have risen with the veil’s. It does not, because the operation is to pull the string taut and read where it crosses the frame, and that reading is a geometric fact about two objects in a room.

What that buys, stated exactly, is one thing and not several. The apparatus fixes the centre of projection; it does not fix the recording. A draughtsman still has to mark the crossing point on the frame, transfer it to a sheet, and do it hundreds of times, and every one of those steps has an error that has nothing to do with projection. The instrument removes one source of error completely and leaves the others untouched, which is worth having and is not the same as being accurate.

What the station point is worth, when it is not known #

The value of a measured centre is easiest to see against the case where there is none, and that case is drawn elsewhere in the collection.

That arc is the ordinary condition of a hand-made perspective drawing. The marks are consistent with a whole family of cameras, one for each position along the horizon that the principal point might occupy, and each of them reconstructs a differently-proportioned rectangle from the same four corners. The essay that draws the arc makes the family explicit, and the one that follows it shows that the choice usually made — assume the middle — is a choice about the subject’s proportions wearing the costume of a choice about the picture.

The string frame removes the arc. Its principal point is where the hook is, its focal length is how far back the hook is, and both were written in a workshop notebook before the drawing existed. A reconstruction from such a drawing has no free parameter, and its answer is checkable against a room rather than against a plausibility judgement.

The size of that indeterminacy is not academic. Standing in the wrong place measures what a reader loses by occupying a station the picture did not draw, and the arc above says that for an ordinary hand-made drawing nobody knows which station that is to begin with. A drawing whose centre is a hook is the only case where the two questions come apart cleanly — the correct station is known exactly, so any discrepancy that remains is entirely the reader’s position and not partly an unknown of the drawing.

That is the same move Brunelleschi made with a hole through his panel, a century earlier and from the other side. A hole enforces the reader’s eye; a hook enforces the maker’s. Both replace an assumption about where a person was with an object that a person has to put their face against or drive into a wall, and both are the only cases in this collection where the centre of projection is not being inferred.

What the apparatus does not settle #

Three limits, and they are all about the difference between a geometry and a practice.

The measurements above are of an idealised instrument. A real string sags, a real frame is not perfectly planar, a real hook is a finite ring rather than a point, and the marks are made by a hand. None of those is modelled here, and each of them is a genuine source of error in a drawing actually made this way. The claim is about which errors the geometry of the instrument admits, and the answer is that a wandering eye is not among them.

The comparison with the veil is a comparison of geometries and not of the drawings the two produced. The veil is quicker, needs no wall, and works on a subject that will not hold still for a string; the frame is slower and needs a room built round it. Which produced better drawings is a question about workshops rather than about projection, and this collection has no instrument pointed at it.

And nothing here says a drawing made from a hook looks right, or that a reader standing at 34.3 centimetres sees what the draughtsman saw. That is a claim about perception. The geometric statement is narrower and is the whole of what is offered — the sheet is the exact projection of the subject through a specified point, and the specified point is one anybody in the room could have measured.

An instrument that states its own premise #

The general shape here is not about Dürer, and it is worth extracting because it applies to instruments this collection has not met yet.

Every drawing system in the collection makes a claim about a centre of projection. Most of them make it silently: the construction assumes a station point, the draughtsman supplies one implicitly by standing somewhere, and the finished sheet carries the assumption without recording it. The recoveries elsewhere in the collection exist because of that silence — reading the camera back out of the picture it drew is only necessary where the camera was never written down.

An apparatus that puts the centre of projection into hardware is doing something a construction cannot. It is turning a premise into a component. The premise is then checkable by someone who was not there, adjustable by moving one object, and immune to the operator’s care in a way no instruction ever is. That is a stronger form of the same idea as the grid method’s exactly-placed corners, where a construction supplies the points a judgement would otherwise have to supply — except that the grid leaves the judgement inside each cell, and the hook leaves none at all.

The corresponding lesson for reading pictures is a question to ask of any drawing whose method is known. Which of its parameters were measured, and which were assumed by the act of standing somewhere? For a photograph the answer is usually all of them and none. For a construction on a board the answer is usually none and all. For the string frame it is the first, and that is rare enough in five centuries of drawing to be worth the hardware.