Extracted text
Editorial transcription, lightly normalized from the scan. Line-break hyphenation, spacing, capitalization, and unmistakable OCR errors have been repaired; the claims remain those of The Flaming Sword.
The Open Court of Inquiry.
THE EDITOR.
Problem of Astronomical Triangulation.
Editor Flaming Sword:— In a recent letter I asked you to show why astronomical triangulations have heretofore been incorrect. You explained in the Court of Inquiry, that no triangulation could be made with any degree of accuracy until it was decided whether the earth's surface is convex or concave. This I am satisfied is correct, following the usual method of deflecting the observed angle to the heavenly bodies from the zenith; but in this connection I desire to submit a theory accompanied by a sketch [see diagram below], of right line triangulation, using for one leg the chord of the Peruvian arc. This arc has been accurately measured and the length of the chord calculated; and if my theory proves correct, it will either definitely prove or disprove the Koreshan Cosmogony. My theory is as follows: Measurement of distance to the moon.—Let observers be stationed at the ends of the arc BC, having their instruments level, elevating the angles A′CM and ABM. Now add to those angles respectively, the angles A′CB and ABC. The last two angles may be found by subtracting 90° (that being the angle A′CD—or ABF—the angle between the level and perpendicular) from the angle DCB or FBC. These additions will give the angles MCB and MBC; and the remainder of the process can be carried out in the usual manner—no account being taken as to whether the surface of the earth is concave or convex. I should be pleased to see this answered in the Open Court of Inquiry.—M. P. H., Marietta, O.
Let us observe in the first place, that neither the Peruvian arc nor any other meridian arc so far accurately measured upon the earth's surface, is long enough for accurate observations of the lunar parallax, and hence could not be used for purposes of astronomical triangulation. Points of simultaneous observation are usually several thousand miles apart, as, for instance, Greenwich and Cape Town, or Chicago and Valparaiso, Chili.
Our correspondent has given one of several methods of computing the value of observed angles as applied to the convex arc. The method is correct as far as it goes, but it is incomplete, not only in the fact that the proposed arc is too short, but because it ignores the angle of the two radii extending from the center of the earth. Before the distance to the moon can be determined, from the basis of convexity of the earth, the four angles of a quadrilateral, bounded by two radii of the earth, and the lines extending from the points of observation to the moon, must be taken into consideration. The chord constitutes the one base of two triangles, whose apexes point in opposite directions. The angle of the two radii extending from the center of the earth must first be known, else it would be impossible to relate the other angles of the quadrilateral.
Our correspondent imagines that in his method the question as to whether the arc is convex or concave is ignored; but in this he is mistaken. All that he has done is to lay out a diagram with the lines of sight extending from the points of observation outward from the convex side of the arc. So far as the diagram is concerned, the concave arc drawn does not enter into the matter as a factor, but the convex arc does constitute the very basis of the relations laid out. Of course, after the diagram is drawn and the angles related, the calculation may proceed on the basis of dealing with straight lines and angles and not with arcs; but the one factor of relating these angles is the size and curvature of the earth—and in this and other instances of astronomical triangulation, it is the convex arc.
No method of calculation (and different methods from the same premise must give the same results) can possibly change the fact that the angles of observation from opposite sides of the arc are not the same; for in one case the lines extend outward into imaginary space between ever-diverging perpendiculars, while in the other, from the concave side, the perpendiculars converge above us at the center of the earth, so that the two lines of sight are drawn inward and downward toward the arc itself, as shown in our former treatment of the subject. If we take the points of observation say, 40° north and south latitude, the lines of sight will nearly constitute a chord of the arc 80° in length. Such relation is apparent by merely drawing diagrams in contrast from the two sides of the arc, without making any calculation at all. If we proceed from the basis of the concave arc, there could be no external angles to be considered. The usual angles which give enormous distances to the heavenly bodies are entirely fictitious, because projected from a hypothetical premise.
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The source scans remain the record. This page preserves the periodical's text as an editorial transcription; its claims remain those of The Flaming Sword.
The Flaming Sword · Vol. 17, No. 15 · Feb 27, 1903 · PDF p. 10