1.0 HISTORY OF GEOMETRY

1.1 Ancient Geometry

Geometrys origin approximately 3000 B.C back in ancient Egypt. Ancient geometry can be classified into ancient Egyptian geometry. The development of Egyptian geometry started from the first Egyptian Pyramid, later into area of triangles, and Moscow and Rhind Papyrus.

1.1.1 First Egyptian Pyramid

(Picture taken from wikipidea)

Ancient Egyptians used early stage geometry in several ways, including the surveying of land, construction of pyramids, and astronomy. Around 2900 B.C, ancient Egyptians began using their knowledge to construct pyramids with four triangular faces and a square base. The earliest known Egyptians pyramids are found at Saqqara, northwest Memphis. The earliest among these is the Pyramid of Djoser which said to be constructed during 2630 BC 2611 BC). This Pyramid and its surrounding complex were designed by the architect Imhotep. It is believed the pyramid was built as a tomb for Fourth dynasty Egyptian King Khufu. There are many pyramid constructed after Pyramid of Djoser and the most celebrated pyramid is The Great Pyramid of Giza.The Egyptians used the Pythagoren theorem and the 3-4-5 right triangle was used by rope stretchers or Egyptian engineers. Rope was knotted into 12 sections that stretched out to produce a 3-4-5 triangle.

1.1.2 Egyptian Triangle

Early stage of the 3-4-5 right triangle was used by the ancient Egyptian to measure the area of the land and to make right angles for land boundaries. Ancient Egyptians often used geometry in architecture and agriculture. To measure the right amount of land for each farmer, Egyptians had to have an easy way of measuring area of irregularly shaped fields. Below is the step of how Egyptians got the earliest method of calculating the area of a triangle:

(Image taken from: educ.queensu.ca/.../Activity_3_Measuring_the_Area_of_a_Triangle_)From the step, we already know that the Ancient Egyptians know how to calculate the area of rectangle, that is

From the formula of area of the rectangle, Ancient Egyptians figured out the basic formula of area of triangle.

The earliest known method of calculating the area of a triangle is

a

ch

b

Where the perimeter is a + b + c

1.1.3 Moscow Papyrus

Picture taken from: http://wikipidea

Moscow Papyrus is an ancient Egyptian mathematical papyrus. Based on the paleography and orthography of the hieratic text, the text was most likely written down in the 13th dynasty roughly 1850 B.C. This Moscow Papyrus approximately 18 feet long and varying between 1 and 3 inches wide and Its was divided into 25 problems with less detail solution. This Papyrus is well known of its geometry problems. It is said that seven of 25 problems contained in the papyrus are geometry problems and range from computing areas of triangle, to find the surface area of hemisphere and finding the volume of a frustum.

Simplifying the seven geometry problems, there are two interesting geometry problems that is problem number 10 and number 14. Problem no 10 is basically asking for a calculation of the surface area of a hemisphere or possibly the area of a semi-cylinder. Problem number 14 in the Moscow Papyrus is to calculate the volume of a frustum.

Problem number 10 run like this,

"Example of calculating a basket. You are given a basket with a mouth of 4 1/2. What is its surface? Take 1/9 of 9 (since) the basket is half an egg-shell. You get 1. Calculate the remainder which is 8. Calculate 1/9 of 8. You get 2/3 + 1/6 + 1/18. Find the remainder of this 8 after subtracting 2/3 + 1/6 + 1/18. You get 7 + 1/9. Multiply 7 + 1/9 by 4 + 1/2. You get 32. Behold this is its area. You have found it correctly."

Problem number 14 runs like this,

Problem 14 states that a pyramid has been truncated in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown. The volume is found to be 56 cubic units.

1.1.4 Rhind Papyrus

The Rhind Papyrus, dating from around 1650 BC, is a kind of instruction manual in arithmetic and geometry. A part of the largest surviving mathematical scroll, the Rhind Papyrus (written in hieratic script), asks questions about the geometry of triangles. The surviving parts of the papyrus show how the Egyptian engineers calculated the proportions of pyramids as well as other structures. Originally, this papyrus was five meters long and thirty three centimeters high.

This precious manuscript contain of 84 questions which later divided into two books. The second part of the Rhind Papyrus consists of geometry problems involving volumes, areas, and pyramids. Problem 41 46 show how to find the volume of both cylindrical and rectangular based granaries. Problem 48 55 shows to compute an assortment of areas. Five final problems in book 2 related to the slopes of pyramids.

Both Moscow and Rhind Papyrus, the problems contains in the manuscripts are followed by solutions.

1.2 Greek Geometry1.2.1 Thales of Miletus

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Thales contributions to mathematics were mainly to geometry. Among other things, he is reputed to have proposed that vertically opposite angles are equal, the theorem that the angle in a semicircle is a right angle and also studied congruent triangles. He is also accredited with predicting the solar eclipse of 585 BC.

Contribution of Thales to Mathematics

P is any point on the circumference of a circle, centre O and AB is a diameter of the circle. Thales showed that angle APB = 90, or a right angle. It doesnt matter where P lies on the circumference, the result is always true.Thales is accredited with some kind of proof of the theorem although this result was known to the Babylonians.

1.2.2 Pythagoras of Samos

Pythagoras made influential contributions to Greek philosophy and religious teaching in the late 6th century BC. Pythagoras is also known as "the father of numbers". The Pythagoras and his students believed that everything was related to mathematics, and felt that everything could be predicted and measured in rhythmic cycles through mathematics.

Contributions of Thales to Mathematics

Pythagorean TheoremThe theorem states: the sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse. We are most familiar with the form a+b=c. As you can see by solving this puzzle the Pythagoreans ran into a problem when they discovered Pythagorass constant of

Platonic SolidsThe Platonic solids, also called the regular solids or regular polyhedral. An Example of this would be a cube which we know is basically a three dimensional square. In a platonic solid all the faces of the object have the same area and the sides lengths are also equal. There are only five such figures which are the cube, dodecahedron, icosahedrons, octahedron, and tetrahedron and are shown to the right of the text with their open faced pictorials as well as their end plot graphs.

1.2.3 Euclid of Alexandria (c. 325-265 BC)Euclid (c. 325-265 BC), of Alexandria, probably a student of one of Platos students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclids was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt.The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.1. Any two points can be joined by a straight line.2. Any finite straight line can be extended in a straight line.3. A circle can be drawn with any center and any radius.4. All right angles are equal to each other.5. If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate).

1.2.4 Archimedes of Syracuse (287-212 BC)Archimedes (287-212 BC), of Syracuse, Sicily, when it was a Greek city-state, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.1.2.5 After ArchimedesGeometry was connected to the divine for most medieval scholars. The compass in this 13th-century manuscript is a symbol of God's act of Creation.After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.The great Library of Alexandria was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the 4th century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.

1.3 Medieval GeometryArea of an Encircled QuadrilateralJanuary 1, 628 AD - January 2, 628 AD

Brahmagupta's Formula and TheoremBrahmagupta is an Indian mathematician who worked in the 7th century. Among many other discoveries, he had a generalization of Heron's formula:The area S of a cyclic quadrilateral with sides a, b, c, d is given byS = (s - a)(s - b)(s - c)(s - d),where s is the semiperimeter of the quadrilateral: s = (a + b + c + d)/2.

It is interesting to note that Heron's formula is an easy consequence of Brahmagupta's. To see that suffice it to let one of the sides of the quadrilateral vanish. On the other hand, Heron's formula serves an essential ingredient of the proof of Brahmagupta's formula found in the classic text by Roger Johnson.

ProofLet the quadrilateral be ABCD, with AB = a, BC = b, etc. Extend AD and BC to meet at E, outside the circumcircle:

(If AD||BC then consider the other pair of the opposite sides. If those two are also parallel, the quadrlateral is a rectangle, and Brahmagupta's formula reduces to the standard formula for the area of a rectangle.)Denote x = CE and y = DE. Apply Heron's formula to CDE:4Area(CDE) = (x + y + c)(x + y - c)(x - y + c)(-x + y + c). But triangles ABE and CDE are similar, implying:Area(ABE) / Area(CDE) = a / c, from whichS / Area(CDE) = (c - a) / c. We also have the proportionsx / c = (y - d) / a, y / c = (x - b) / a. Adding the two and solving for (x + y) givesx + y = c (b + d) / (c - a). Similarly, subtracting one from the other and solving for x - y we obtainx - y = c (b - d) / (c + a), from which we can find all the terms in Heron's formula. For example,x + y + c = c (b + d) / (c - a) + c = c (b + d + c - a) / (c - a) = 2c (s - a) / (c - a),x + y - c = c (b + d) / (c - a) - c = c (b + d - c + a) / (c - a) = 2c (s - c) / (c - a),x - y + c = c (b - d) / (c + a) + c = c (b - d + c + a) / (c + a) = 2c (s - d) / (c + a), and-x + y + c = - c(b - d) / (c + a) + c = c (-b + d + c + a) / (c + a) = 2c (s - b) / (c + a). A substitution then yieldsArea(CDE) = [c / (c - a)](s - a)(s - b)(s - c)(s - d). And, finally,S = Area(CDE) (c - a) / c = [c - a) / c][c / (c - a)](s - a)(s - b)(s - c)(s - d).

1.4 Modern Age Geometry1.4.1 Rene Descartes1596 AD - 1650 ADDescartes synthesized algebra and geometry by placing points on a coordinate plane. Analytic geometry, branch which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic. Its most common application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax+by+c=0 represents a straight line in the xy-plane, and the linear equation ax+by+cz+d=0 represents a plane in space, where a, b, c, and d are constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection. In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the x-axis. If the line is parallel to the x-axis, its slope is zero. Two or more lines with equal slopes are parallel to one another. In general, the slope of the line through the points (x1, y1) and (x2, y2) is given by m= (y2y1) / (x2x1). The conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equation ax2+bxy+cy2+dx+ey+f=0, where a, b, , f are constants and a, b, and c are not all zero.In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, , , and , the cosines of the angles the line makes with the x-, y-, and z-axes, respectively; these satisfy the relationship 2+2+2= 1. In the same way that the conic sections are studied in two dimensions, the 17 quadric surfaces, e.g., the ellipsoid, paraboloid, and elliptic paraboloid, are studied in solid analytic geometry in terms of the general equation ax2+by2+cz2+dxy+exz+fyz+px+qy+rz+s=0.The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry. Analytic geometry was introduced by Ren Descartes The essence of the method of coordinates consists in the following. Let us consider, as an example, two mutually perpendicular straight lines Ox and Oy lying on plane (Figure 1).

Figure 1These straight lines, together with their indicated directions, the origin of coordinates O, and the chosen scalar unit e, constitute the so-called rectangular Cartesian system of coordinates Oxy in the plane. Straight lines Ox and Oy are called, respectively, the axis of abscissas and the axis of ordinates. The position of any point M in the plane with respect to the system Oxy can be determined in the following manner. Let Mx and My be the projections of M on Ox and Oy and the numbers x and y be the magnitudes of segments OMx and OMv. The magnitude x of segment OMx, for example, is equal to the length of that segment taken with positive sign if the direction from O to Mx coincides with the direction on straight line Ox and with negative sign in the opposite case. The numbers x and y are said to be the rectangular Cartesian coordinates of point M in the system Oxy. Usually they are referred to as the abscissa and the ordinate of point M, respectively. The symbol M (x, y) is used to designate a point M having abscissa x and ordinate y. It is evident that the coordinates of point M define its position with respect to system Oxy.Let L be a certain curve on plane with a given rectangular Cartesian coordinate system Oxy. Using the idea of the point coordinates, it is possible to introduce the concept of an equation of the given curve L with respect to system Oxy in the form of a relationship of type F (x, y) = 0 which is satisfied by the coordinates x and y of any point M located on L and not satisfied by any point which does not lie on L. If, for example, curve L is a circle of radius R with the center at the origin of coordinates O, the equation x2 + y2 R2 = 0 will be the equation of the circle, as is evident from Figure 2. If point M lies on the circle, then from the Pythagorean will be the equation of the circle, as is evident from Figure 2. If point M lies on the circle, then from the Pythagorean theorem we have x2 + y2 R2 = 0 for triangle OMMx. If, however, the point does not lie on the circle, then it is clear that x2 + y2 R2 0. Thus, curve L in the plane can be associated with its equation F (x, y) = 0 with respect to coordinate system Oxy.

Figure 2The basic idea behind the method of plane coordinates is that the geometric properties of curve L can be investigated through the study by analytical and algebraic means of the properties of F (x, y) = 0, the equation of the curve. For example, let us apply the method of coordinates to ascertain the number of points of intersection between circle C with radius R and a given straight line B (Figure 3). Let the origin of coordinates of Oxy lie at the center of the circle, and let axis Ox be perpendicular to straight line B. Since straight line B is perpendicular to axis Ox, the abscissa of any point on that line is equal to a certain constant a. Thus, the equation of straight line B will have the form x a = 0. The coordinates (x, y) of the point of intersection of circle C (whose equation has the form x2 + y2 R2 = 0) and straight line B satisfy both equations(1) x2 + y2 R2 = 0, a = 0that is, they constitute a solution of equations (1). Consequently, the geometrical question of the number of points of intersection of a straight line and a circle is reduced to the analytical question of the number of solutions for the algebraic system of equations (1). Solving that system, we arrive at . Thus, the circle and the straight line may intersect at two points (R2 > a2) as shown in Figure 3; may have one common point (R2 = a2), in which case the straight line B is tangent to circle C; or may have no common points (R2 < a2), in which case straight line B lies outside circle C.

Figure 3Analytic geometry studies in detail the geometrical properties of the ellipse, the hyperbola, and the parabola, which are the curves of intersection of a circular cone with planes that do not pass through the apex of the cone. These curves are frequently encountered in many problems in natural science and technology. For example, the motion of a material point under the influence of a central gravity field follows one of these curves; engineering in the construction of projectors, antennas, and telescopes uses the important optical property of the parabola in which light rays proceeding from a certain point, called the focus of the parabola, following reflection from the parabola form a parallel beam.In analytic geometry, a systematic study is made in the plane of the so-called algebraic curves of first- and second-order; these lines in rectangular Cartesian coordinates are defined with first- and second-degree algebraic equations, respectively. First-order curves are straight, and conversely, every straight line is defined by the first-degree equation Ax + By + C = 0. Second-order curves are defined by equations of type, Ax + Bxy + Cy2 + Dx + Ey + F = 0. The basic method of analyzing and classifying these curves consists in selecting a rectangular Cartesian coordinate system in which the equation of the curve will have the simplest possible form and in the subsequent study of this simple equation. It can be demonstrated that by such means the equation of any real second-order curve can be reduced to one of the following simplest forms:

The first of these equations defines an ellipse, the second a hyperbola, the third a parabola, and the last two a pair of straight lines (intersecting, parallel, or coincident).In analytic geometry the method of coordinates is also used in space. Here, the rectangular Cartesian coordinates x, y, and z (abscissa, ordinate, and Z-coordinate) of point M are completely analogous to the plane case (Figure 4). Each surface S in space can be associated with an equation F (x, y, z) = 0 with respect to coordinate system Oxyz.

Figure 4For example, the equation of a sphere of radius R with center at the origin of coordinates has the form x2 + y2 + z2 R2 = 0. Here, the geometrical properties of surface 5 are determined by studying algebraically and analytically the properties of the equation of this surface. The curve L in space is given as the curve of intersection of the two surfaces S1 and S2. If F1(x, y, z) = 0 and F2 (x, y, z) = 0 are the equations of S1and S2, then these two equations taken together constitute the equation of curve L. For example, straight line L in space can be regarded as the line of intersection of two planes. Since a plane in space is defined by an equation of the form Ax + By + Cz + D = 0, a pair of equations of this same type considered together is the equation of straight line L. The method of coordinates, therefore, can be applied to the study of curves in space. Analytic geometry in space systematically studies the so-called algebraic surfaces of the first and second order. Algebraic surfaces of the first order, clearly, can only be planes. Surfaces of the second order are defined by equations of the formAx2 + By2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Mz + N = 0The basic method of studying and classifying these surfaces consists in the selection of a rectangular Cartesian coordinate system in which the equation of the surface has the simplest possible form and in the subsequent study of this simple equation. The most important real second-order surfaces are the ellipsoid, the hyperboloids of one and two sheets, and the elliptic and hyperbolic paraboloids. These surfaces in specially chosen rectangular Cartesian coordinate systems have the following equations:

The most important surfaces of the second order are encountered frequently in various problems in mechanics, solid-state physics, theoretical physics, and engineering. Thus, in the study of stresses arising in solid bodies, use is made of the concept of the so-called stress ellipsoid. Designs in the form of hyperboloids and paraboloids are used in various engineering projects

1.4.2 Carl Friedrich Gauss (1777-1855)

Carl Friedrich Gauss was born on April 30, 1777, in Brunswick, Germany, the son of Gebhard Dietrich Gauss, a bricklayer, and Dorothea Emerenzia Gauss. By 1817Gausshad become convinced that the fifth postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line.

He have discovered the possibility ofnon-Euclidean geometriesbut never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.

An important tool used to measure how much a surface is curved is called the sectional curvature or Gauss curvature. It can be computed precisely if you know Vector Calculus and is related to the second partial derivatives of the function used to describe a surface.

1.4.3 George Friedrich Bernhard Riemann (1826-1866)

Riemann's revolutionary ideas generalized the geometry of surfaces which had been studied earlier by Gauss, Bolyai and Lobachevsky. Later this lead to an exact de_nition of the modern concept of an abstract Riemannian manifold. The development of the 20th century has turned Riemannian geometry into one of the most important parts of modern mathematics. Riemannian geometers also study higher dimensional spaces. The universe can be described as a three dimensional space. Near the earth, the universe looks roughly like three dimensional Euclidean space.The amount that space is curved can be estimated by using theorems from Riemannian geometry and measurements taken by astronomers. For example, there are many different shapes that surfaces can take. They can be cylinders, or spheres or paraboloids or tori, to name a few. A torus is the surface of a bagel and it has a hole in it.

You could also stick together two bagels and get a surface with two holes. How many holes can you get? Certainly, as many as you want. If you string together infinitely many bagels then you will get a surface with infinitely many holes in it. Look at the example given.

1.4.4 Felix Klein (late 1800 and early 1900)

Christian Felix Klein(25 April 1849 22 June 1925) was aGermanmathematician, known for his work ingroup theory,complex analysis,non-Euclidean geometry, and on the connections betweengeometryandgroup theory.Topology is the modern version of geometry, the study of all different sorts of spaces. The thing that distinguishes different kinds of geometry from each other (including topology here as a kind of geometry) is in the kinds of transformations that are allowed before you really consider something changed.

Topology is almost the most basic form of geometry there is. It is used in nearly all branches of mathematics in one form or another. There is any continuous change which can be continuously undone is allowed. So a circle is the same as a triangle or a square, because you just `pull on' parts of the circle to make corners and then straighten the sides, to change a circle into a square.

1.4.5 Fractal Geometry Nature (Benoit Mandelbrot 1924-2010)

He is French and Americanmathematician, noted for developing a "theory of roughness" in nature and the field offractal geometryto help prove it, which included coining the word "fractal". His key discovery is fractal geometry, generates a weird kind of visual beauty.

A graph of the set produces an image of fantastic complexity and strange magnificence. It exhibits the key property of a fractal, which is that it is self-similarit repeats its own pattern on every scale. He says, the fractal geometry of nature is Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

2.0 THE HISTORY OF TRIGONOMETRYThe term trigonometry, although not of native Greek origin, comes from the Greek word trigonon, meaning triangle, and the Greek word -metria, meaning measurement. As the name implies, trigonometry ultimately developed from the studyof right triangles by applying the relationships between the measures of its sides and angles to the study of similar triangles (Gullberg 458). However, the word trigonometry did not exist upon the birth of the subject, but was later introduced by the German mathematician and astronomer, BartholomaeusPitiscus in the title of his work, Trigonometria sive de solutione triangularum tractatus brevis et perspicius, published in 1595. It was then revised in 1600 and published again as Trigonometriasive de dimensione triangulae,(Adamek et al, 2005.Trigonometry is not the work of one man or a nation. In fact, the ancient Egyptians and Babylonians had developed theorems on ratios of the sides of similar triangles (Boyer 158), before trigonometry was ever formalized as a subdivision of mathematics. These two groups had no clear usage of trigonometric functions but were able to use them unknowingly to their advantage.Egyptians used trigonometry to their benefit in land surveying and the building of pyramids. Babylonian astronomers related trigonometric functions to arcs of circles and the lengths of chords subtending their arcs (Gullberg 458).

The timeline of trigonometry history

Ancient time : 2000 BC to 500 BCBabylonian (1800 BCE)- a circle = 360 degrees(750 BCE) astronomers had a reasonably accurate means of measuring the elevation (latitude) and lateral direction (longitude) of all objects in the heavens.(500 BCE)- the division of the heavens into twelve regions of 30 degrees each, often referred to as the 12 houses of the zodiac. Hindu Sulbasutras: The earliest written texts we have from this oral tradition date from about 800 BCE. The Sulbasutras are the instructions for constructing various geometrical shapes to make 'fire-altars' using the "Peg and Cord" technique.Chinese: Chinese were the most accurate observers of celestial phenomena before the Arabs. "Oracle Bones" with star names engraved on them dating back to the Chinese Bronze Age (about 2,000 BCE) have been found, and very old star maps have been found on pottery, engraved on stones, and painted on the walls of caves.

Greek Civilisation : 500 BC to 180 CEEudoxus (408-355 BCE)Eudoxus was taught by Archytas, one of the leading Pythagorean philosophers of his time, who maintained that the most perfect shape was a circle.Idea: Planets moved about the Earth in circles on the surfaces of different spheres.Aristarchus (310-230 BCE)Using Eudoxus' theory of proportion, Aristarchus measured the relative sizes and distances to the Moon and Sun and found the Sun to be bigger than Earth! So, he reasoned that the Sun rather than Earth is the centre of the Universe and the Earth is one of the planets.

Applying his logical reasoning to the theory of proportion, he arrived at an estimate for the distance to the Sun of about 19 times the distance to the Moon. This estimate was generally accepted for the next 2,000 years.Hipparchus : Father of trigonometry(190-120 BCE)The first known table of chords was produced by the Greek mathematician Hipparchus in about 140 BC. 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 850 stars (longitude and latitude relative to the ecliptic), and the discovery of the precession of the equinoxes and a measurement of that precession. Menelaos (70 -130 BCE)In about 100 BCE Menelaos compiled a Book of Spherical Proportions Sphaerica, in which he set up the basis for treating spherical triangles by using arcs of great circles instead of arcs of parallel circles on the sphere.

Ptolemy (100 - 178 CE)Ptolemy proved the theorem that gives the sum and difference formulas for chords. Armed with his theorem, Ptolemy could complete his table of chords from 1/2 to 180 in increments of 1/2. Write the book Almagest. The Almagest contains a collection of all the then known astronomical knowledge.

Ptolemy's TheoremPtolemy proved the theorem that gives the sum and difference formulas for chords. Theorem. For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of the diagonals equals the sum of the products of the opposite sides. AC BD = AB CD + AD BCWhen AD is a diameter of the circle, then the theorem says crd AOC crd BOD = crd AOB crd COD + d crd BOC. where O is the center of the circle and d the diameter. If we take a to be angle AOB and b to be angle AOC, then we have crd b crd (180 - a) = crd a crd (180 - b) + d crd (b - a) which gives the difference formula crd (b - a) =crd b crd (180 - a) - crd a crd (180 - b)

d

With a different interpretation of a and b, the sum formula results: crd (b + a) =crd b crd (180 - a) + crd a crd (180 - b)

d

These, of course, correspond to the sum and difference formulas for sines.

Arab civilisation : 180 to 1160 CEFrom China, they learnt how to produce paper, and using this new skill they started a programme of translation of texts on mathematics, astronomy, science and philosophy into Arabic.The quest for knowledge became a lasting and significant part of Arab culture. Al-Mansour had founded a scientific academy that became called 'The House of Wisdom'. This academy attracted scholars from many different countries and religions to Baghdad to work together and establish the traditions of Arabic science that were to continue well into the Middle Ages. Some of this work was later translated into Latin by Mediaeval scholars and passed on into Europe. The Mathematical Treatise of Ptolemy was one of the first to be translated from the Greek into Arabic by Ishaq ben Hunayn (830-910). It was admired for its extensive content and became known in Arabic as Al-Megiste (the Great Book). India: The Sine, Cosine and VersineGreek astronomy began to be known in India during the period 300-400 CE. However, Indian astronomers had long been using planetary data and calculation methods from the Babylonians, and even though it was well after Ptolemy had written the Almagest, 4th century Indian astronomers did not entirely take over Greek planetary theory. Ancient works like the Panca-siddhantica (now lost) that had been transmitted through the version by Vrahamihira and Aryabhata's Aryabhatiya (499 CE) demonstrated that Indian scholars had their own ways of dealing with astronomical problems and that they had great skill in calculation.Trigonometry in the Arab CivilisationThe introduction and development of trigonometry into an independent science in the Arab civilisation took, in all, some 400 years. In the early 770s Indian astronomical works reached the Caliph Al-Mansur in Baghdad, and were translated as the Zij al-Sindhind, and this introduced Indian calculation methods into Islam.

Famous for his algebra book, Abu Ja'far Muhammad ibn Musa al-Khwarizmi had also written a book on Indian methods of calculation (al-hisab al-hindi) and he produced an improved version of the Zij al-Sindhind. Al-Khwarizmi's version of Zij used Sines and Versines, and developed procedures for tangents and cotangents to solve astronomical problems. Al-Khwarizmi's Zij was copied many times and versions of it were used for a long time.

Many works in Greek, Sanskrit, and Syriac were brought by scholars to Al-Mansur's House of Wisdom and translated. Among these were the works of Euclid, Archimedes Apollonius and of course, Ptolemy. The Arabs now had two competing versions of astronomy, and soon the Almagest prevailed.

The Indian use of the sine and its related functions were much easier to apply in calculations, and the sexagesimal system from the Babylonians continued to be used, so apart from these two changes, the early Arabic versions of the Almagest remained faithful to Ptolemy. Abu al-Wafa al-Buzjani (Abul Wafa 940-998) made important contributions to both geometry and arithmetic and was the first to study trigonometric identities systematically. The study of identities was important because by establishing relationships between sums and differences, and fractions and multiples of angles, more efficient astronomical calculations could be conducted and more accurate tables could be established.

The sine, versine and cosine had been developed in the context of astronomical problems, whereas the tangent and cotangent were developed from the study of shadows of the gnomon. In his Almagest, Abul Wafa brought them together and established the relations between the six fundamental trigonometric functions for the first time. Al-Biruni's (973-1048) treatise entitled Maqalid 'ilm al-hay'a (Keys to the Science of Astronomy) ran to over one thousand pages and contained extensive developments in on trigonometry. Among many theorems, he produced a demonstration of the tangent formula. Abu Muhammad Jabir ibn Aflah (Jabir ibn Aflah, 1100 - 1160) probably worked in Seville during the first part of the 12th century. His work is seen as significant in passing on knowledge to Europe. Jabir ibn Aflah was considered a vigorous critic of Ptolemy's astronomy. His treatise helped to spread trigonometry in Europe in the 13th century, and his theorems were used by the astronomers who compiled the influential Libro del Cuadrante Sennero (Book of the Sine Quadrant) under the patronage of King Alfonso X the Wise of Castille (1221-1284).

A result of this project was the creation of much more accurate astronomical tables for calculating the position of the Sun, Moon and Planets, relative to the fixed stars, called the Alfonsine Tables made in Toledo somewhere between 1252 and 1270. These were the tables Columbus used to sail to the New World, and they remained the most accurate tables until the 16th century.

By the end of the 10th century trigonometry occupied an important place in astronomy texts with chapters on sines and chords, shadows (tangents and cotangents) and the formulae for spherical calculations. There was also considerable interest in the resolution of plane triangles. But a completely new type of work by Nasir al-Din al-Tusi (Al-Tusi 1201-1274) entitled Kashf al-qina 'an asrar shakl al-qatta (Treatise on the Secrets of the Sector Figure), was the first treatment of trigonometry in its own right, as a complete subject apart from Astronomy. The work contained a systematic discussion on the application of proportional reasoning to solving plane and spherical triangles, and a thorough treatment of the formulae for solving triangles and trigonometric identities. Al-Tusi originally wrote in Persian, but later wrote an Arabic version. The only surviving Persian version of his work is in the Bodleian Library in Oxford.Al-Tusi invented a new geometrical technique now called the 'Al-Tusi couple' that generated linear motion from the sum of two circular motions. He used this technique to replace the equant used by Ptolemy, and this device was later used by Copernicus in his heliocentric model of the universe. Arab Science and Technology Reaches EuropeThe Arab astronomers had learnt much from India, and there was contact with the Chinese along the Silk Road and through the sea routes, so that Arab trading posts were established in India and in China. Through these contacts Indian Buddhism spread into China and was well established by the 3rd century BCE, probably later carrying with it some of the calculation techniques of Indian astronomy. However few, if any, technological innovations seemed to have passed from China to India or Arabia.

At this time many religions and races coexisted in Iberia, each contributing to the culture. The Muslim religion was generally very tolerant towards others, and literacy in Islamic Iberia was more widespread than any other country in Western Europe. By the 10th century Cordoba was said to have equally good libraries and educational establishments as Baghdad, and the cities of Cordoba and Toledo became centres of a flourishing translation business.

Between 1095 and 1291 a series of religiously inspired military Crusades were waged by the Christians of Europe against the Arab Empire. The principal reason was the restoration of Christian control over the Holy Land, but there were also many other political and economic reasons

By 790 CE, the Arab empire had reached its furthest expansion in Europe, conquering most of the Iberian peninsula, an area called Al-Andalus by the Arabs.During the twelfth and thirteenth century hundreds of works from Arabic, Greek and Hebrew sources were translated into Latin and the new knowledge was gradually disseminated across Christian Europe.

Geometrical knowledge in early Mediaeval Europe was a very practical subject. It dealt with areas, heights, volumes and calculations with fractions for measuring fields and the building of large manors, churches, castles and cathedrals.

Europe Trigonometry : 1160 onwardsRichard of Wallingford (1292-1336)he wrote an important work, the Quadripartitum, on the fundamentals of trigonometry needed for the solution of problems of spherical astronomy. The first part of this work is a theory of trigonometrical identities, and was regarded as a basis for the calculation of sines, cosines, chords and versed sines.Georg von Peuerbach (1423-1461)Peuerbach's work helped to pave the way for the Copernican conception of the world system; he created a new theory of the planets, made better calculations for eclipses and movements of the planets and introduced the use of the sine into his trigonometry.Johannes Muller von Konigsberg or Regiomontanus (1436-1476)Regiomontanus had become a pupil of Peuerbach at the University of Vienna in 1450. Later, he undertook with Peuerbach to correct the errors found in the Alfonsine Tables. He had a printing press where he produced tables of sines and tangents and continued Puerbach's innovation of using Hindu-Arabic numerals.Nicolaus Copernicus (1473 - 1543)Copernicus wrote a brief outline of his proposed system called the Commentariolus that he circulated to friends somewhere between 1510 and 1514. By this time he had used observations of the planet Mercury and the Alfonsine Tables to convince himself that he could explain the motion of the Earth as one of the planets. Georg Joachim von Lauchen called Rheticus (1514-1574)In 1551, with the help of six assistants, Rheticus recalculated and produced the Opus Palatinum de Triangulis (Canon of the Science of Triangles) which became the first publication of tables of all six trigonometric functions. This was intended to be an introduction to his greatest work, The Science of Triangles.

Bartholomaeus Pitiscus (1561 - 1613)The term trigonometry is due to Pitiscus and as first appeared in his Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus, published in 1595. A revised version in1600 was the Canon triangularum sive tabulae sinuum, tangentium et secantium ad partes radii 100000. The book shows how to construct sine and other tables, and presents a number of theorems on plane and spherical trigonometry with their proofs.However, soon after Rheticus' Opus Palatinum was published, serious inaccuracies were found in the tangent and secant tables at the ends near 1 and 90. Pitiscus was commissioned to correct these errors and obtained a manuscript copy of Rheticus' work. Many of the results were recalculated and new pages were printed incorporating the corrections. Eventually, Pitiscus published a new work in 1613 incorporating that of Rheticus with a table of sines calculated to fifteen decimal places entitled the Thesaurus Mathematicus.

References:1. History of Geometry Activity 3 (http://www.educ.queensu.ca)2. www.wikipidea.org3. www.touregypt.net4. Jim Loy (1999). The 3-4-5 right triangle in ancient Egypt (www.jimloy.com)5. www.slideshare.com6. http://encyclopedia2.thefreedictionary.com/Analytic+Geometry7. http://www.cut-the-knot.org/Generalization/Brahmagupta.shtml