Plane and spherical trigonometry

See Article History Trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine sincosine costangent tancotangent cotsecant secand cosecant csc. These six trigonometric functions in relation to a right triangle are displayed in the figure.

Plane and spherical trigonometry

Plane and spherical trigonometry

If disturbed in any way, motion ensues until the flat level is resumed. If dammed up then released, the nature of all liquids is to quickly flood outwards taking the easiest course towards finding its new level. Because if a part of the surface were higher than the rest, those parts of the fluid which were under it would exert a greater pressure upon the surrounding parts than they receive from them, so that motion would take place amongst the particles and continue until there were none at a higher level than the rest, that is, until the upper surface of the whole mass of fluid became a horizontal plane.

If, however, the Earth is a giant sphere tilted on its vertical axis spinning through never-ending space then it follows that truly flat, consistently level surfaces do not exist here!

But this is contrary to the fundamental physical nature of water to always be and remain level! The sense of sight proves this to every unprejudiced and reasonable mind. Can any so-called scientist, who teaches that the earth is a whirling globe, take a heap of liquid water, whirl it round, and so make rotundity?

For example, if the ball-Earth were 25, miles in circumference as NASA and modern astronomers say, then spherical trigonometry dictates the surface of all standing water must curve downwards an easily measureable 8 inches per mile multiplied by the square of the distance.

This means along a 6 mile channel of standing water the Earth would dip 6 feet on either end from the central peak. In Cambridge, England there is a 20 mile canal called the Old Bedford which passes in a straight line through the Fenlands known as the Bedford Level. In the latter part of the 19th century, Dr.

The author, with a good telescope, went into the water; and with the eye about 8 inches above the surface, observed the receding boat during the whole period required to sail to Welney Bridge.

The flag and the boat were distinctly visible throughout the whole distance! There could be no mistake as to the distance passed over, as the man in charge of the boat had instructions to lift one of his oars to the top of the arch the moment he reached the bridge. The experiment commenced about three o'clock in the afternoon of a summer's day, and the sun was shining brightly and nearly behind or against the boat during the whole of its passage.

Every necessary condition had been fulfilled, and the result was to the last degree definite and satisfactory. The conclusion was unavoidable that the surface of the water for a length of six miles did not to any appreciable extent decline or curvate downwards from the line of sight.

But if the earth is a globe, the surface of the six miles length of water would have been 6 feet higher in the centre than at the two extremities. From this experiment it follows that the surface of standing water is not convex, and therefore that the Earth is not a globe!

On the contrary, this simple experiment is all-sufficient to prove that the surface of the water is parallel to the line-of-sight, and is therefore horizontal, and that the Earth cannot be other than a plane!

Rowbotham placed seven flags along the edge of the water each one mile distant from the next with their tops positioned 5 feet above the surface. Near the last one he also positioned a longer, 8 foot staff bearing a 3 foot flag so that its bottom aligned precisely with the tops of the other flags.

He then mounted a telescope at a height of 5 feet behind the first flag and took his observations. If the Earth was a globe of 25, miles, each successive flag would have to decline a definite and determined amount below the last.

The first and second flags simply established the line of sight, the third flag should then fall 8 inches below the second, the fourth flag 32 inches below, the fifth 6 feet, the sixth 10 feet 8 inches, and the seventh flag should be a clear 16 feet 8 inches below the line of sight!

Even if the Earth was a globe of a hundred thousand miles, an amount of easily measurable curvature should and would still be evident in this experiment. But the reality is not a single inch of curvature was detected and the flags all lined up perfectly as consistent with a flat plane.

By positioning them at equal heights aimed at each other successively he proved over and over the Earth to be perfectly flat for miles without a single inch of curvature.

His findings caused quite a stir in the scientific community and thanks to 30 years of his efforts, the shape of the Earth became a hot topic of debate around the turn of the nineteenth century.

Now, when theory does not harmonize with practice, the best thing to do is to drop the theory.planeandspheeical trigonometry, and surveying by attheheels.comrth,a.m.,^ pbofebsorofje^^maticsinphillipsexeteracadejttt. 2ceacl)ers*letiitton* . Course between points.

We obtain the initial course, tc1, (at point 1) from point 1 to point 2 by the following. The formula fails if the initial point is a pole. Problem A g ball at the end of a string is revolving uniformly in a horizontal circle of radius m.

The ball makes 2 revolutions in a second. Spherical geometry and trigonometry used to be important topics in a technical education because they were essential for navigation.

During that time an important element of their presentation was the matter of making accurate computations.

Plane and spherical trigonometry

If the Earth is an extended flat plane, then this fundamental physical property of fluids finding and remaining level is consistent with experience and common sense. mulæ in Plane and Spherical Trigonometry; so as to include an account of the properties in Spherical Trigonometry which are analogous to those of the Nine Points Circle in Plane Geometry.

Introduction to spherical trigonometry