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
26
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