Horopter — the series
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The depth a pair calls zero
Two eyes verged on a point agree — the same coordinate in both pictures — not on a plane at the fixation distance but on a circle through both eyes and that point. Found by bisection along 121 azimuths and fitted rather than assumed, it is a circle to 0.0000 cm; at 26° off centre it lies 23 cm nearer than a flat wall does.
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Both coordinates agree on a circle and a line
Two eyes fixating a point straight ahead see their horizontal image coordinates agree on a whole vertical cylinder over the Vieth–Müller circle, the same radius at every height to the last bit. Their vertical coordinates agree on almost none of it — 7.35 px apart at 26° aside and 30 cm up, 29.26 px when the fixation is brought to 60 cm. The points where both agree are the circle and one vertical line, and the line is the axis of the motion that carries one eye onto the other.
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Raise the gaze, and the line is gone
Turn two eyes 20° aside in the plane they share and the horopter keeps its vertical line — but the line stays in the median plane, 1.277 m ahead, not at the point being looked at. Raise the gaze as well and the rule by which each eye rolls decides the rest: Helmholtz's rule keeps a line; Listing's law and Fick's rule make the eyes' relative motion slide 3.93 mm and 7.09 mm along its axis, no point stays put, and the horopter becomes one curve.
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A sliding pair keeps its line only near the middle
Two eyes that roll by Listing's law lose the horopter's straight line as soon as they look up and aside, and the question worth a number is whether they lose it by much. Measured along the line they nearly keep, the disparity is exactly the slide seen by one eye — 3.20 px at 20° aside and 20° up, a metre and a bit away — and the slide does not shrink with distance. At arm's length the line survives to within a pixel only in a narrow cross through the middle of the field.
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A quarter of a degree of roll keeps the line
A Listing pair's horopter loses its straight line because the motion between the eyes slides along its own axis. Helmholtz's rule cures that by turning each eye several degrees. A roll a tenth that size, in opposite senses, cures it just as exactly, and its size is a short law: a quarter of the elevation times the vergence, with hardly any dependence on the azimuth.