''Spherical Excess''

Zetetic Astronomy, by 'Parallax' (pseud. Samuel Birley Rowbotham), [1881], at sacred-texts.com

"SPHERICAL EXCESS."

As a proof of the earth's rotundity, many place great reliance upon what is called the "spherical excess," which has been observed on making trigonometrical observations on a large scale.

"The angles taken between any three points on the surface of the earth by the theodolite are, strictly speaking, spherical angles, and their sum must exceed 180 degrees; and the lines bounding them are not the chords as they should be, but the tangents to the earth. This excess is inappreciable in common cases, but in the larger triangles it becomes necessary to allow for it, and to diminish each of the angles of the observed triangle by one-third of the spherical excess. To calculate this excess, divide the area of the triangle in feet by

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the radius of the earth in seconds, and the quotient is the excess:" ¹

"The theodolite used to measure the angles (in the English survey) surpassed in its dimensions and elaborate workmanship, every instrument of the kind that had been seen in Europe; it measured angles with such precision, that it became necessary, in the calculation of the triangles, to take into consideration the excess of three spherical angles above two right angles, a quantity that had hitherto been too minute to be ascertained by any instrument, and was only known by theory to have any existence. The amount of the total error in the sum of the three angles never exceeded three seconds, so that the angles. generally must have been measured to the nearest second." ²

In this so-called argument for rotundity we have another instance of the manner in which the most practical men of science are led astray. Just as the differences observed in the reading of chronometers as compared with those of the logs and dead reckonings when sailing in the southern regions, navigators, having had an education which involved the doctrine of rotundity, could not possibly see the real explanation which demonstrable truth afforded, but were forced to adopt the idea that ocean counter currents existed, overlooking altogether, and not daring to face the obvious fact that the differences were observed whether sailing east or west, and therefore that they were parties to the contradictory notion that the currents of the sea were moving in contrary directions carrying ships right and left, or backward and forward, at the same time; so the most skilful observers connected

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with the ordnance survey of Great Britain and Ireland, could not see that the angles which were too large for agreement with their general operations were the result of slight divergence in the rays of light passing through the lenses of their telescope; but, contrary to every principle of reasoning, assumed that the tops of the high places on and to which observations were made, were divergent from the common centre of a globular earth, and hence the so-called "spherical excess," for which they made such allowances as were necessary to make their observations agree with the theory of rotundity. Had they known that such a theory was contrary to fact, and that the earth was a plane, they would have sought an explanation of discrepancies in the proper quarter. They would have recognised the influence of refraction or "collimation" in their instruments; for they could not be ignorant of the optical peculiarities which necessitate so many observations upon the same point before they could decide upon the "average of errors" as their proper reading. The rule that the greater the number of observations made "averaging the errors," the more correct the deductions, ought to have led them to seek the "spherical excess" only in the optical character of the telescopes employed. In the operations connected with the Mont Cenis Tunnel the leading observations were many times repeated before the proper angles were ascertained. Mr. Francis Kossuth, one of the Royal Commissioners of Italian Railways, in his report on the tunnel, after describing the processes adopted in surveying over the mountain, says:--

"The whole system consisted of 28 triangles; and 86 was

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the number of measured angles. All of these were repeated never less than 10 times, the greater part 20, and the most important as many as 60 times." 1

In many of the triangulations connected with the British ordnance survey, the observations were repeated upwards of a hundred times, in order to diminish the personal and instrumental errors to which all such operations are liable. In page 41 of this work it is shown that a levelled theodolite pointed towards the sea represents the horizon as below the horizontal cross-hair, on account of what is technically called "collimation," or "a slight divergence of the rays of light from the axis of the eye on passing through the several glasses of the theodolite." The same "collimation" exists in connection with the vertical cross-hair; and hence the slight excess of the three angles over 180 degrees so often observed when taking very long sights--such, for instance, as those between Kippure and Donard, in Ireland, and Precelly, in Wales.

Footnotes

262:1 "Treatise on Levelling." By Castle.

262:2 Dr. Rees's "Cyclopœdia," article "Degree."