The parallel shortcut, measured

Picture a line drawn across a map and a town beside it. The sideways method asks where the line crosses the town's own parallel of latitude, then reports the length of the due east or due west trip between the two. In arithmetic it is a subtraction of longitudes and a conversion to kilometres, which is why it is the common implementation.

Two things follow from the geometry, and neither is a matter of taste. The crossing point is one point that sits on the line, so the shortest distance to the whole curve can only be nearer than the trip to that point. And the east-west arc between two points on one parallel is itself never shorter than the great circle joining them. The shortcut therefore never understates. It overstates, or it ties.

When a line runs true north to south, sliding sideways to reach it and walking straight at it are the same trip, and the two answers agree exactly. Tilt the line and they separate, because the sideways trip now travels along the tilt instead of across it. The Midheaven and Imum Coeli lines run north to south by construction. The Ascendant and Descendant lines tilt, and they tilt hardest at high latitude.

There is also a case the shortcut cannot answer at all. A horizon line does not reach every latitude: it curves back on itself, and above its turning point it simply is not there. If the line never crosses your parallel there is no crossing point to measure to, so the sideways method returns nothing, even when the line passes a few kilometres away from you.

The alternative is the one a thumb laid across a line has always meant: the closest the line ever comes to you, in any direction, over the curve of the Earth. It is found by minimising over the line itself rather than by measuring to one chosen point of it, on a single sphere of mean radius, which is the same Earth every other figure on this site is computed against.

It has no special case at high latitude, and no undefined case anywhere: every place on the globe has a nearest point on every line, including the lines that never reach its latitude. That is the measure this engine has used since it was rebuilt, and the methodology page sets out the sources behind it and the orb bands it feeds.

A measurement without a stated corpus is a mood. This one is Albert Einstein, Frida Kahlo, Jimi Hendrix: birth records rated AA, meaning they come from a birth certificate, and the same charts the published example readings on this site are computed from.

Every one of the 40 lines of each chart was measured against every one of the 3,114 settlements in the pinned public-domain gazetteer, which holds places of at least 100,000 people plus every national capital. That is 120 chart-lines and 373,680 place-to-line measurements.

A pair enters the study when its true distance falls inside the outermost published orb of 805 km, because a pair outside orb is not an influence either method would report. That leaves 18,917 in-orb pairs. Both distances are then computed for every one of them. Nothing is sampled and nothing is weighted.

The split is the finding, so the table is arranged around it rather than around a single headline number.

Read the horizon column first. 1,615 of 9,346 horizon pairs, 17.3%, land in a different orb band under the shortcut than under the true distance. A band is not decoration. It is what decides whether a place is described as sitting under a line, inside peak orb, in the background, faint, or not reported at all, and it is the difference between a paragraph and a clause in a reading.

The meridian column is what turns that into evidence. Where the geometry says the two measures should agree they agree, to a median of 0.1 km across 9,571 pairs. If both columns were loud, the honest reading would be that one of the two implementations is broken. Only the tilted lines move.

Across all 18,917 pairs together, the least dramatic true version: 8.6% get the wrong band, and 733 of them, 3.9%, disappear from the answer entirely, either pushed past the outer orb or invisible to the method to begin with.

Aggregates hide their own edges, so here are the two pairs a reader can check by hand. Both are drawn from the table above rather than chosen for effect: one is the largest overstatement in the corpus, the other is the nearest influence the shortcut cannot see at all.

That both exemplars land in the same city is not a coincidence and it is worth saying rather than hiding: high latitude is exactly where horizon lines tilt hardest and where they turn back on themselves, so both failure modes surface in the same places. Near the equator the two methods are close to interchangeable.

If the map in front of you measures sideways, the lines are still drawn where they belong. What moves is the strength attached to the places you care about, and it moves in one direction: a line can be reported a band weaker than it is, or dropped from the answer, but never the reverse. The effect is systematic rather than noisy, and it concentrates on the rising and setting lines at high latitude.

Most tools do not say which distance they compute, so a reader cannot tell which of these numbers they are being shown. That is the part worth taking from this page: not that one method is fashionable and another is not, but that a tool which publishes its method can be checked, and one that does not has to be believed.

Everything above is recomputed from the engine each time this page is built. If the gazetteer grows, if a threshold moves, if the engine changes, these figures change with it and this page cannot quietly keep the old ones.