MATH 147 Part II Outline
= 2 sin a/2
Greek astronomers inherited much of
their astronomy from the Babylonians. Beginning with Aristarchus
(4th c.) and Eratosthenes (3rd c.), however, they looked at the cosmos
as being described by geometric patterns (spherical motions) that could
be recorded, measured, and predicted, rather than simply the will of the
Eudoxus (4th c.) developed a geocentric
model that placed the earth at the fixed center of the universe.
This homocentric version posited concentric spheres on which the seven
planets (Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn) traveled around
Apollonius (3rd c.) refined this to
an eccentric model which assumed that the planetary spheres did not share
the same center. This accounted for variability in brightness of
Hipparchus (2nd c.) prepared the first
table of chords to assist with astronomical calculation. For this
he is called "the father of trigonometry".
The chord of an angle is related to
the modern-day sine of the angle by
Claudius Ptolemy (2nd c. AD) wrote
the Almagest, a popular compendium of Greek astronomical knowledge
that became the standard for many centuries (until Copernicus).
In the Almagest, Ptolemy prepared
a much more extensive table of chords than Hipparchus', for angles up to
180°, in steps of 1/2°. For this purpose
he proved a few useful theorems for dealing with chords of supplementary
angles; differences and sums of angles; halves of angles. He also
proved a theorem that bears his name.
Ptolemy's Theorem says that in a cyclic
quadrilateral, the product of the lengths of the diagonals equals the sum
of the products of the lengths of the opposite pairs of sides.
With calculations done in sexagesimal,
suppose the chords of two angles, a
are known. Then the chord of the supplement of a
The chord of the difference of
the arcs is
and the chord of the supplement
of the sum is
The chord of the half of the angle
Know how to...
relate the chord of an angle and its
use the above formulas to calculate
chords of angles as in Ptolemy's Table of Chords.
The development of algebraic techniques
was begun in antiquity with Diophantus of Alexandria, but came to fruition
in the work of Islamic mathematicians like Al-Khwarizmi. Still, none
of these employed a symbolic notation that recognizable as modern.
The evolution of algebraic symbolism
progressed over the fifteenth and sixteenth centuries in Europe; examples
of this development appear in France in the work of Nicholas Chuquet, in
Germany by Christoff Rudolff, and in England by Robert Recorde.
Significantly, this evolution occurs
simultaneously with the introduction of the printing press in Europe.
It is printing that allows notations to become standardized and disseminated.
François Viète, a lawyer
and adviser to the kings of France, writes about the power of "geometric
analysis" which he breaks down into three categories--zetetics, poristics,
and exegetics--for the solution of general classes of algebraic problems.
He is able to this by using letters to stand for numbers and quantities
and applies algebraic manipulations to obtain general solutions.
Beginning in the fifteenth century,
Europeans involved in overseas navigation and its supporting sciences of
cartography and astronomy required their practitioners to perform extensive
arithmetical operations, which were prohibitively tedious. This was
especially true of the multiplications of trigonometric values
which appeared in so many of the investigations of these scientists.
They welcomed methods to speed arithmetical calculation.
The first of these to be successful
was prosthaphaeresis, developed by Clavius and pioneered by Tycho Brahe.
It replaced the multiplication of cosines with the averaging of cosines
of the sum and difference of the given angles: cos a
cos b = [cos(a + b) + cos(a _b)]/2.
John Napier, an amatuer mathematician
and wealthy Scottish landowner, invented logarithms in the 1610s.
He perfected the idea in collaboration with Henry Briggs of Gresham College,
London. The (common) logarithm log x of a number
is the appropriate power of 10 required so that 10log
The basic property of logarithms is
log(ab) = log a + log b. This, together with
the assignments log 1 = 0 and log 10 = 1, imply
that log(a/b) = log a _
log b and log(an) = n
log a. Logarithms turn multiplication into addition
and division into subtraction.
Tables of logarithms quickly supplanted
prosthaphaeresis as the method for speeding calculation. They were
incorporated in the manufacture of slide rules which were a popular tool
of calculation until the 1960s, when they were replaced by hand-held calculators.
Know how to...
use the prosthaphaeretic rule to multiply
use logarithms to multiply, divide
and exponentiate numbers.
operate with the basic logarithmic
Deeply religious, the young Kepler
studied theology and mathematics, and became an advocate for Copernican
cosmology as the imperial mathematician to the Austrian crown at Prague
in the ideologically turbulent 1590s. There he made use of Tycho
Brahe's excellent and voluminous astronomical observations, made at the
Uraniborg observatory off the coast of Copenhagen, to test his mathematical
theories of how the planets moved around the sun. After much trial
and error, he realized what was then formulated in his Astronomia Nova
(A New Astronomy, 1609) as the first two of his laws on planetary
[Elliptical orbits] planets move in
elliptical orbits about the sun, with the sun at one of the focus points.
(The foci are a pair of points F, G, inside the curve, symmetrically
positioned about the center along the major axis, with the property that
as a point P moves around the curve the sum of the lengths PF + PG
[Equal areas in equal times] the areas
of elliptical sectors swept out by a planet as it moves around the sun
are equal for equal times, explaining how planets move more quickly in
their orbits when closer to the sun than when they are farther away.
[Planetary periods are related to distance
from sun (published later in Harmonices Mundi, The Harmony of
the World, 1619)] The length T of a planetary year ( the period
of its orbit) is related to its mean distance a from the sun (the
length of the semimajor axis of the ellipse) by the law T2
= ca3 for some constant c.
Know how to...
use Kepler's third law to deduce the
length of a planetary year.
(distance travelled between
= a and t = b) = .
At the turn of the seventeenth century,
Galileo Galilei challenged the established Aristotelian authority concerning
the natural world, making political
enemies amongst the academics of the universities of Italy and theocrats
in the Church of Rome. He espoused a new
scientific method whereby one left aside explanations of why phenomena
behaved the way they did and instead described them by making experimental
measurments and formulating mathematical models.
After suffering censure and house arrest
under the Church for his views on Copernicanism, he caused a furor with
the 1632 publication in the Italian language of Dialogue Concerning Two
Chief World Systems, in which he creates three characters, Salviati, the
scholar-scientist, Simplicio, the dull Aristotelian, and Sagredo, the layman
and moderator, who discuss the nature of the Ptolemaic and Copernican systems
of the universe. He followed this with the 1638 publication of Discourses
and Mathematical Demonstrations Concerning Two New Sciences, another dialogue
amongst the three characters, in which he explores the motion of objects
in free-fall and of projectiles.
Galileo defined the basic concepts
of mechanics: velocity (speed), which is the rate of change of distance
traveled with respect to time, and acceleration, the rate of change of
velocity with respect to time.
The mean speed law says that the distance
traveled by an object under uniform acceleration between two moments of
time equals the distance traveled by an object moving at constant speed
equal to the speed at the halfway point of the first moveable.
Galileo realized that the distance
traveled by an object corresponds to the area of the region under the velocity
curve; in symbols,
For an object in free-fall, acceleration,
velocity, and distance are given by the relations a = g, v =
gt, and s = (1/2)gt2, where g = gravitational
acceleration = 9.8m/sec2 = 32 ft/sec2.
The science of ballistics arose in
the Reanissance with the introduction of the cannon as a weapon of war.
It provided Galileo with a problem that he was able to resolve: what is
the shape of the path of a projectile?
Resolving the motions of the projectile
into perpendicular directions, at constant speed in the horizontal direction
and uniformly accelerated in the vertical direction, he was able to determine
that the projectile moved along a parabolic arc.
The equations of motion are x =
ct, y = (1/2)gt2, which combine to show that
the path of the projectile is a parabola.
Know how to...
solve problems concerning objects in
free-fall using the ideas outlined above.
determine the position of a projectile
from the Galilean equations of motion.
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last modified 11/20/02