Twenty past four and this table is too hot to lean on — put your elbow on the glass down that end and you'll know about it. Two tubes underneath, both on, and they've been on since one. Right. Nine legs, Kairanga Road corner, council wants the widening line by Tuesday and the two titles don't agree, so what we've got is Bagnall's field book from November 1923 and my own eyes.

What you're looking at is not a drawing of the boundary. It's the arithmetic and nothing else. Each leg is a bearing and a distance; each one resolves into a latitude — that's distance times cosine of the bearing, north positive — and a departure, distance times sine, east positive. Walk them end to end and if the figure genuinely comes back to where it started, the nine latitudes sum to nought and the nine departures sum to nought. So the sums are the error. There's nothing there to interpret. Ours are not nought.

The gap's drawn in red, true scale, and again inside that circle at two hundred times so you can actually see the shape of it. 1.31 m. 968.4 m round. 1 in 739. That's a fence line. That's a good post-and-batten fence. It is not a title boundary, which wants better than 1 in 8,000, and the difference between those two figures is the entire argument between Merrilees and the Coldstream estate.

Now — the thing I want you to look at before anything else. Not the length of the gap. Its bearing. Random error is small and it has no preferred direction; it's a hundred tiny sins sharing themselves out, a bit of chain sag here, a fraction of a minute there. It doesn't point anywhere. A blunder points. One transposed digit in one distance gives you a misclosure whose size is the size of the transposition and whose direction is parallel to that leg. So that red line isn't a puzzle. It's an arrow. Read its bearing off the readout, then go down the column and find the leg that runs the same way, or dead opposite. There. That's your six that's an eight.

(A transposed bearing behaves differently — it swings the whole tail of the figure and the misclosure comes out roughly square to the arm from the offending leg's midpoint. Different animal. I'll know it when I see it.)

Drag any figure in the left column up or down and it flips between the two things that handwriting could honestly be. You don't get to type a number. I don't get to type a number either. His 6 and his 8 close at the top the same way and his 3 and his 5 both start with that same hook, and pretending otherwise is inventing evidence. Eighteen states in that column. One of them closes properly.

Leg re-reads so far this afternoon: 0.

Now. No, Bagnall. No. Leg six, one-seven-one point two. You chained that down the Kairanga bank and booked it as you read it. If you'd chained it on the flat you'd have got seventeen-six and you know it — that's a fall of near enough four metres over forty, that's four centimetres of slope in the leg, and it's the commonest single reason an old traverse comes out long. Every time. I'm not cross about it, it's 1923 and you had a steel band and a plumb-bob and a boy holding the far end.

I'll give you one, though, and I'll say it plainly: your bearings are consistent. All nine of them close as a round of angles to within about forty seconds and that's honest work with a five-inch vernier in November. That's the first point to you.

The second is the reference marks. Bagnall, your RMs are beautiful. Three of them off corner B, all three booked with bearing and distance and a description you could actually use — "iron spike in root of macrocarpa, N side" — and not one of them relying on a fence that could move. I have spent whole days looking for other men's marks. Yours I could find in the dark. Genuinely well done.

The small circle with the cross through it — that's a totara post, in the hedge, sitting 0.94 m off wherever the arithmetic currently puts corner D. He referenced it. Council's man says it's been rammed back in twice since the flood, so it's worth exactly what that's worth, and it isn't in the computation. Back to leg six.

DO NOT ADJUST YET. Take the red gap and drag it and you'll see what I mean. That's a real compass rule — Bowditch — correcting each leg's latitude and departure in proportion to its length over the perimeter, on the assumption that the angles and the chaining are equally reliable. It works. The figure closes exactly. The ratio field goes to PERFECT. And look what it does to the corners: every one of them slides, and the amount is printed beside it in centimetres, and now a 0.9 metre transcription error is smeared across all nine of them so that not one corner is right and there is no evidence anywhere on the sheet that anything was ever wrong. That's burying it. I will not sign that.

Adjust after you've hunted, never before. If I'd been shown a closed plan of this in 1974 I'd have believed it, and I'd have been ten centimetres out at every peg, and the fence would have gone in ten centimetres out, and by now there'd be a hedge on it. Burying it is not a small sin. It's the one that outlives you.

Let go and it springs back to raw. It always springs back to raw.

(The transit rule distributes by latitude and departure instead, on the assumption the angles are better than the chain. Which you choose is a statement about your instrument, not about what you'd prefer to be true. With a five-inch vernier and a band that's been round a fencepost, Bowditch. But hunt first.)

You were going to ask about the north, and I thought of the answer on the stairs coming up, so: yes, his bearings are magnetic. Whole figure has to be swung by the declination of the day he observed plus the change since. That's the knurl on the left margin — turn it and watch. The gap's bearing changes and its length does not move by one millimetre. It can't. Rotating a closed sum of vectors rotates the residual and leaves its modulus alone. Getting that rotation wrong is a very good way to spend an afternoon chasing a blunder round the compass, and I have done it, in 1968, on a Friday.

Temperature field's under the knurl. Steel band, 11.5 parts per million per degree — standardised at 20°, used at 31° on the day, so it reads short by 0.4 mm in forty metres. Put the real correction on all nine and closure improves. It improves by about seven millimetres. Seven. It's honest and you should do it and it will never once save you from a transposed digit. Sag and slope you do at the book, not here.

Half his book's in chains and links and half of it's in metres because somebody converted the back forty pages in 1972 with a slide rule, and I can tell exactly where they stopped because the figures go from four significant places to three mid-column. That's not Bagnall's fault. That one's ours.

They've got the vulcaniser going downstairs, you'll hear it come on. Hot rubber all the way up the stairwell with these windows open, and I'm not shutting them.

Right. Read the gap's bearing. Find the leg it's parallel to. Flip that one figure. If the ratio comes up better than 1 in 8,000 we've got it, and then and only then do we adjust, and then I can plot the widening line and the council can have their answer, and Tuesday is Tuesday.

Leg six, Bagnall. I'm still not happy about leg six.