MATH 147 Part II Outline

Ptolemy

• 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 gods.
• 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 the earth.
• 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 the planets.
• 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 and b, are known.  Then the chord of the supplement of a is

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 a is

• Know how to...
• relate the chord of an angle and its sine.
• use the above formulas to calculate chords of angles as in Ptolemy's Table of Chords.

Viete

 
• 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.

Briggs

• 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 x is the appropriate power of 10 required so that 10^{log x} = x.
• 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 numbers.
• use logarithms to multiply, divide and exponentiate numbers.
• operate with the basic logarithmic properties.

Kepler

• 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 motion:
• [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 is constant.)
• [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.

Galileo

• 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,

(distance travelled between t = a and t = b)   = .

• 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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