Successive steps in the Logic of Geometrical Propositions and Applied Mechanics in Direct Demonstration
Principles of Mechanical Survey. (1)
Successive steps in the Logic of Geometrical Propositions and Applied Mechanics in Direct Demonstration. 

THE FACTS of the observations and experiments presented in the preceding chapters are quite sufficient to demonstrate the earth's non-convexity, and to destroy the premise of the modern system of astronomy. To the mind capable of comprehending a few simple laws of vision, the observations on the water's surface as conclusively demonstrate that the earth is concave, as the more direct processes involved in the Koreshan Geodetic Survey.
Indeed, the fact that we have seen the concave arc, and that it can be seen again under similar circumstances, is a settlement of the question for those who desire ocular demonstrations.
>From these facts, we pass to the consideration of the more direct processes in the demonstration of the earth's contour.
The most accurate measurements of the earth's surface are made by mechanical means. The employment of apparatus for this purpose extends as far back as the history of astronomy.
Starting with the simplest factor of linear measurement, the mind can at once conceive of the necessity of mechanical appliances.
Space exists between any two points on the earth's surface, embracing a number of units of measure-inches, feet, yards, etc.
After a unit of measure has been agreed upon as the standard, its length must be preserved, and the means of preservation reduced to convenience of application or use; wood, iron or other solid material must constitute the embodiment of space units, as the rule, yard-measure, or the surveyor's chain. The process of measuring by all such appliances, is simply that of marking their lengths upon a surface, or by placing them end to end successively throughout the distance to be measured. The measurement of meridian arcs in geodetic survey does not differ in principle from these simple processes; all modern
measurements of the earth involve the same factors.

Ancient and Modern Geodetic Apparatus.
The evolution of geodetic apparatus from the days of Ptolemy to the present time, has resulted from the obvious necessity of obtaining the greatest possible accuracy. Successive improvements have simply increased precision of adjustment. The first apparatus consisted of wooden rods placed end to end upon the
ground; degrees of latitude and longitude have been thus measured, and miles of space determined. In modern times, similar apparatus were used by Picard, the French astronomer.Improvements were made by Osterwald, of Germany; by Mason and Dixon, of London, surveyors of the famous Mason and Dixon line in America.
Beccara mounted the sections of his wooden apparatus upon tripods; Cassini de Thury employed bars of iron with adjusting devices; Bessel, iron and zinc; Helmert, platinum and iridium, resting on iron trusses. U. S. geodetic measurements are made by compound bars of iron and brass, adjusted so as to compensate
for contraction and expansion, -an invention of Prof. A. D. Bache. The rods are enclosed in spar-shaped cases made of wood, and mounted on tripods. The adjustments are made by contact of fine points, determined by a delicate index. Similar apparatus are in use by every government in the world, engaged in geodetic work.

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All such apparatus are employed, not for the purpose of determining a straight line, but to accurately measure distances or meridian arcs; they involve adjustment of single points only. There is no attempt at rectilineation; but the fact remains that their use in modern times by the most skilful geodetic surveyors, proves that accurate adjustments are not only possible, but are actually made by simple contact of metal surfaces. The principles involved in the Koreshan Geodetic Apparatus differ radically from those involved in any other apparatus. It is susceptible of as great precision in linear measurement; and consideration of the principles which distinguish it in its form, purpose, and use, will make apparent its superiority over any other instrument of survey, as a means of determining the actual contour of the earth.
Fundamental Principles of Geometry.
We purpose in this and the succeeding chapters of this work, to make a direct demonstration of the concavity of the earth's surface, involving principles and processes so simple and so absolute as to be both easy of comprehension and conclusive. A direct demonstration proceeds from a premise by regular deduction. First, the premise must be known and absolutely proven, to the utter exclusion of all assumptions; and second, all factors involved in the train of logic must be direct and positive.
In the line of sequences, we must take nothing for granted that would render our conclusion less certain than the premise.
Our every step must be upon the sure footing of fact. We purpose toshow that our premise, as well as resultant conclusion, is so firmly established as to exclude all possibility of doubt or denial.
We begin at the foundation of geometry, and delineate for the mind, step by step, the inevitable conclusions. Geometry is the science of earth-measurement, from (( [Gr. ge], earth, and (((((( [metron], to measure. Modern geometry is a fragment of the true system of the relations and properties of form that existed in the past, from
which its present name is derived.
Every form has its opposite form, and every form has its coordinate form, possessing correlated functions. There is no transmutation possible without segmentation; in the evolution of a circle to its coordinated square, we have the intersection of secant and arc, and relation and bisection of radius and chord, as in the
diagram, Fig. A, on following page. The radius, which is perpendicular to the chord, bisects the chord, and also the arc which it subtends.
The simplest angle to which relations of form can be referred, is the right angle.

There can be no mergence of one circle into another as in figure (diagram) B, without the relations of the cross. If two circumferences intersect each other,the common chord which joins the points of intersection, is bisected at right angles by the straight line which joins their centers.

Our simplest premise, that the angles relating perpendicular and horizontal are equal, is susceptible of being known from a geometrical standpoint.

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The principles upon which depend right angles depend - by which they may be formed, also furnish a conclusive test and demonstration of the perpendicular relations of straight lines. To draw a line at right angles with another, independently of a square as a guide, we may relate the points of intersection of arcs of merging circles to the line connecting their centers, as before illustrated, and as shown in diagrams Fig. D and E.
Having demonstrated the principles of right angles, our premise is proven; and we are ready to take the next step in the line of sequential propositions and arguments.
From the above geometrical facts as a premise, we may construct
a square having four right angles, four sides, two perpendicular and two horizontal parallels, because two straight lines which are
at right angles with a given straight line, are parallel with each other.
We can know as absolutely that the sides of a rectangle are parallel to each other, as we can know that two straight lines are at right angles, and by the same processes.
If we place two rectangles of equal breadth together, we form a new rectangle which is equal in area to the surn of the areas of the rectangles comprising it;
the sides joined being in unity, the extremes are parallel, The other sides are as the extension of straight lines.
If this is the result in the conjunction of two rectangles, the same
results would obtain if ten thousand rect angles were drawn end to end. There could be no possible deviation from a straight rectilinear direction, and we defy any mathematician to show that there could be!

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