The meridian
In 24 hours, the earth rotates once around the axis which passes through its poles, and which in our times happens to roughly point to the polestar. To an observer on earth, therefore, the sun seemingly rotates, in the opposite direction, around that same axis. Its path is a circle, whose magnitude increases and decreases with the seasons. Generally, part of that orbit is below the horizon, and an observer sees the sun rise, culminate and set. The points of sunrise and sunset are located symmetrically with respect to the culmination point due south. The line connecting sunrise and sunset is parallel to the line East-West, coinciding with it twice a year. (Figure 1.)
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Figure 1. (Polestar deliberately off target w.r.t. the axis of rotation.) |
The sun's rays, hitting a point on earth, e.g. the top S of a vertical rod, create a shadow cone. Any three points on that cone equally far removed from S define a plane 𝛼 parallel to the plane 𝜛 in which the sun seemingly moves. Both planes are orthogonal to the axis of rotation. The intersection of 𝛼 with the horizontal plane of the observer is a straight line parallel to the line East-West. A line orthogonal to it is oriented North-South; it is the local meridian. In the sequel we focus on obtaining a line East-West, which is the crucial part in obtaining the meridian.
The meridian obtained from two shadows of equal length
In the course of a day, the shadow of the top of a vertical rod traces on a horizontal plane a curve which is a conic section, — for most latitudes, a hyperbola. On that curve, any two points equally far from the foot of the rod correspond to positions of the sun that are symmetric with respect to the culmination point at noon. Connecting such points results in a line oriented East-West. This elementary construction yields the meridian from two shadows of equal lengths. In practice, one could extrapolate the hyperbola from any number of observed shadows, then determine where this curve intersects a circle whose centre is the foot of the vertical rod. (Figure 2.)
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| Figure 2. |
The meridian obtained from three shadows of different lengths
In 1614 Muzzio Oddi (relevant pages here) showed how to construct the meridian from any three shadows of unequal lengths. (Figure 3.)
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| Figure 3. Perspective view. |
The rod OS is placed vertically on the horizontal plane of the observer. Let the rays be SA, SB, SC with shadows OB < OA < OC. On the rays SA,SC make SD=SE=SB. We will construct the intersection of the plane BDE and the horizontal plane OABC. As B is in both planes, it suffices to obtain a second point in both planes. Project D,E in F,G respectively. From OB<OC it follows that EG<DF, and the line through D,E will intersect the line through FG in some point J. This point is in both planes: on the line through D,E (points in the plane BDE), and on the line through F,G (points in the plane OABC).
To find J, we will rotate the plan FGED around the axis FG onto the plane OABC; this rotation will not affect the point J, which is on the axis. It is easy to find where D and E will land: they move orthogonally to the axis, and the distances DF and EG are preserved. Hence, if HF=FD and IG=GE, both perpendicular to the axis, the line through HI is the rotated image of the line through D,E. As J is on the line though D,E and on the axis, its rotated image (J again) is on the line through H,I and on the axis.
Figure 4 shows the construction in the horizontal plane. The shadows are OA,OB,OC and the radius of the circle is the length of the rod. The right triangle OBS'' reveals the actual length of SB (red), which in the triangles OAS' and OCS''' leads to D',E', then F,G and finally H,I. Blue segments have the same length, green segments likewise. The line in violet is oriented East-West.
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| Figure 4. Constructions in the horizontal plane. |
For Oddi's construction three separate observations are sufficient. For practical purposes however, the result is very poor. The sun in the sky is not a point, but a disk, and shadows are fuzzy blurs, spoiling the beauty of the geometry.
