One point on the sphere for every point of the plane

• the image of 0 lies on the circle ×14
• the ray to 0 passes through the pole ×14
• the image of -2 lies on the circle ×13
• the ray to -2 passes through the pole ×13
• a circle missing the pole projects to a circle, to within a fitted residual ×1
• a circle through the pole projects to a straight line ×1
• and no two of them meet ×1
• and the south chart does have one there ×1
• at least two finite points are carried across the projection ×1
• at the equator the horizon passes through the south pole and projects to a straight line ×1
• between 2 and 24 rays are drawn — one ray is not a correspondence ×1
• between a hundred and four thousand points checked ×1
• between three and seven fibres, each at a latitude strictly inside the sphere ×1
• dim and rays are read by the ray views alone ×1
• each altitude circle has centre cos φ/(sin φ + sin a) and radius cos a/(sin φ + sin a) ×1
• each fibre lies on the unit three-sphere ×1
• each transformation is normalised to determinant one ×1
• every engraved circle of the plate — horizon, altitude, azimuth, ecliptic — is a circle to within rounding ×1
• every patch has the same area on the sphere ×1
• every two fibres are linked exactly once, counted from the drawn curves ×1
• for every star, inside the horizon on the plate is the same as above the horizon by spherical trigonometry ×1
• nearly every sample lay in the overlap ×1
• none of the circles in this list passes through the pole ×1
• only the parabolic map has a repeated fixed point ×1
• the crossing angle is the same on the sphere and in the plane ×1
• the fitted transformation is invertible ×1
• the horizon's circle shrinks as the latitude rises ×1
• the hour angle of sunrise read off the plate equals arccos(−tan φ tan δ) ×1
• the image is on the ray from the pole through the point ×1
• the image lands on the plane ×1
• the north chart has no value at its own pole ×1
• the projected patches are wildly unequal even though the originals are equal ×1
• the projection is drawn from a circle or from a sphere ×1
• the rotation, read in the plane, is exactly the fitted Möbius transformation ×1
• the sixth hour of the night is the lower meridian, a straight line ×1
• the three chosen points determine the transformation ×1
• the trace classifies the transformation as the label claims ×1
• the two charts differ by the reciprocal at every point of the overlap ×1
• the two charts miss different points ×1
• the unequal-hour lines are not circles, and the three-point arc misses each by under two per cent of the plate ×1
• the view is one the family draws ×1
• there are pairs to link ×1

The whole library · What the figures prove