Outline of the History of Sicily B.

History of Trigonometry Outline Trigonometry is, of course, a branch of geometry, but it differs from the synthetic geometry of Euclid and the ancient Greeks by being computational in nature. For instance, Proposition I. That is, if you want to know the remaining angle and the remaining two sides, all you have to do is lay out the given side and the two angles at its ends, extend the other two sides until they meet, and you've got the triangle.

No numerical computations involved. But the trigonometrical version is different. If you have the measurements of the two angles and the length of the side between them, then the problem is to compute the remaining angle which is easy, just subtract the sum of the two angles from two right angles and the remaining two sides which is difficult.

The modern solution to the last computation is by means of the law of sines.

All trigonometrical computations require measurement of angles and computation of some trigonometrical function. The modern trigonometrical functions are sine, cosine, tangent, and their reciprocals, but in ancient Greek trigonometry, the chord, a more intuitive function, was used.

Trigonometry, of course, depends on geometry.

The law of cosines, for instance, follows from a proposition of synthetic geometry, namely propositions II. And so, problems in trigonometry have required new developments in synthetic geometry.

An example is Ptolemy's theorem which gives rules for the chords of the sum and difference of angles, which correspond to the sum and difference formulas for sines and cosines. The prime application of trigonometry in past cultures, not just ancient Greek, is to astronomy.

Computation of angles in the celestial sphere requires a different kind of geometry and trigonometry than that in the plane. The geometry of the sphere was called "spherics" and formed one part of the quadrivium of study.

Various authors, including Euclid, wrote books on spherics. The current name for the subject is "elliptic geometry. Thus, spherical trigonometry is as old as plane trigonometry. The Babylonians and angle measurement The Babylonians, sometime before B. Degree measurement was later adopted by Hipparchus.

The Babylonians were the first to give coordinates for stars. They used the ecliptic as their base circle in the celestial sphere, that is, the crystal sphere of stars. The celestial sphere rotates around the axis through the north and south poles.

The Babylonians measured the longitude in degrees counterclockwise from the vernal point as seen from the north pole, and they measured the latitude in degrees north or south from the ecliptic.

Hipparchus of Nicaea ca. Hipparchus was primarily an astronomer, but the beginnings of trigonometry apparently began with him.

Certainly the Babylonians, Egyptians, and earlier Greeks knew much astronomy before Hipparchus, and they also determined the positions of many stars on the celestial sphere before him, but it is Hipparchus to whom the first table of chords is attributed.

It has been hypothesized that Apollonius and even Archimedes constructed tables of chords before him, but there is no reference to any such earlier table.

Some of Hipparchus' advances in astronomy include the calculation of the mean lunar month, estimates of the sized and distances of the sun and moon, variants on the epicyclic and eccentric models of planetary motion, a catalog of stars longitude and latitude relative to the eclipticand the discovery of the precession of the equinoxes and a measurement of that precession.

According to Theon, Hipparchus wrote a book work on chords in a circle, since lost. That would be the first known work of trigonometry. Since the work no longer exists, most everything about it is speculation. But a few things are known from various mentions of it in other sources including another of his own.

It included some lengths of chords corresponding to various arcs of circles, perhaps a table of chords. Besides these few scraps of information, others can be inferred from knowledge that was taken as well-known by his successors.

Chords as a basis of trigonometry In a modern presentation of trigonometry, the sine and cosine of an angle a are the y- and x-coordinates of a point on the unit circle, the point being the intersection of the unit circle and one side of the angle a; the other side of the angle is the positive x-axis.

The Greek, Indian, Arabic, and early Europeans used a circle of some other convenient radius. For this description of trigonometry, we'll leave the radius unspecified as r and it's double, the diameter, we'll denote d. Some properties of chords could not have escaped Hipparchus' notice, especially in a book work on the subject.

Hipparchus probably constructed his table of chords using a half-angle formula and the supplementary angle formula. What other relations among the chords of various angles that Hippocrates would have known remains speculation.

The earliest work on spherical trigonometry was Menelaus' Spherica.Chapter Outline. Chapter Civilizations of the Americas, ca − They also made advances in astronomy and wrote books of history. Maya civilization reached its peak between and C.E.

A combination of factors led to the Maya’s decline. Civilization arose 5, years ago in the Near East (Mesopotamia and Egypt) The emergence of civilization was characterized by the emergence of: Cities that were larger, more populous, and more complex than Neolithic villages.

Outline of Ancient Mesopotamian Civilizations Posted on July 6, by jamesmarlandson This post is part of a series of outlines of civilizations that influenced the Western world. An Outline of the Theory of Civilization. Chapters , 10, Appendix.

pp. , October (Discussion): Emergence of the Japanese Empire Andrew Gordon. The Outline of History, subtitled either "The Whole Story of Man" or "Being a Plain History of Life and Mankind", is a work by H. G. Wells that first appeared in an illustrated version of 24 fortnightly instalments beginning on 22 November and was published as a single volume in It sold more than two million copies, was translated into many languages, and had a considerable impact.

Outline of the history of western civilization wikipedia, the following outline is provided as an overview of and topical guide to the history of western civilization, late middle ages:

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A Brief History Of Chinese Civilization - barnweddingvt.com