Chapter 2

Question 1

*1. What is meant by irrational numbers? Give some examples.

Answer 1

a. Irrational numbers are numbers that can’t be expressed as a ratio of two integers. Examples: pi, e, square root of 2

Question 2

*2. What were the connections perceived by the Pythagoreans between irrational numbers, geometry, and the concept of atoms?

Answer 2

a. Pythagoreans believed that all things are composed of individual blocks called atoms. Since they are distinct units, they are countable and pythagoreans supposed geometry as a branch of arithmetic. But as the existence of irrational numbers was realized, it meant that many triangles (subject of geometry) cannot be built upon atoms since isosceles right triangles, for example, have a hypotenuse equal to length * square root of 2 which is an irrational number and therefore cannot represent a whole number of atoms.

Question 3

*3. Why did the ancient Greeks consider a circle to be a perfect figure?

Answer 3

a. Because no matter how much you rotate the circle about its center, it maintains its original appearance or that it has constant appearance. For greeks, perfection is identified with constancy, something that is perfect cannot be improved, so it must remain constant.

Question 4

4. What does Plato’s Allegory of the Cave have to do with science?

Answer 4

a. Plato’s allegory of the cave illustrates the essential objective of science which is to explain what is observed by the senses through reason

Question 5

*5. Why did the ancient Greeks consider that heavenly objects had to move in circular paths?

Answer 5

a. Because circle is the perfect geometrical figure

Question 6

*6. What is the stellar or celestial sphere? Describe its appearance.

Answer 6

a. The celestial sphere is like a giant spherical canopy or dome surrounding the earth where the stars (which appear to be little pinpoints of light), sun, moon and other planets are mounted

Question 7

8. What is the ecliptic?

Answer 7

a. The path of the sun (and roughly by the planets) tilted an angle of 23.5 degrees from the celestial equator.

Question 8

9. What is diurnal rotation?

Answer 8

a. Daily rotation of the celestial sphere

Question 9

11. What is retrograde motion?

Answer 9

a. Retrograde motion is when a planet varies its direction of motion (from east to west relative to fixed stars) rather than following its overall west to east path. It wanders slightly off the ecliptic as well.

Question 10

*What is alternate motion?

Answer 10

b. Alternate motion is illustrated when a planet is just ahead of the Sun on the ecliptic and at other times it is just behind the sun. If the planet is just west of the sun, the daily rotation will be seen as morning star and if it just east of the sun, it will be seen as an evening star.

Question 11

*13. What is meant by a homocentric model of the universe?

Answer 11

a. The earth is a motionless sphere surrounded by eight other rotating “concentric” spheres which carry the sun, the moon and the planets. (Spheres have the same center –> homocentric)

Question 12

14. Name the prime substances described by Aristotle.

Answer 12

a. In Sublunar world, the prime substances are: earth, water, fire and air
b. In astronomical world, it is ether

Question 13

15. Name and describe the devices used by Ptolemy in his calculations.

Answer 13

a. Eccentric – center of sphere carrying the planet
b. Epicycle – smaller sphere where planets orbit whose center is carried by a larger sphere called the deferent
c. Equant – point inside the sphere with an equal distance on the other side of the eccentric.

Question 14

• 16. What particular aspects of Ptolemy’s approach did Copernicus not like?

Answer 14

a. Ptolemaic system was too complicated; Ockham’s razor: Simple explanations are preferable to complicated explanations
b. Inconsistent ways in which Ptolemy applied his various devices
c. Use of equant is contrary to the very idea of perfect circular motion

Question 15

17. How does the heliocentric theory account for retrograde motion?

Answer 15

a. Retrograde motion and all the slowing down and speeding up of planets is simply an optical illusion which happens when an inner laps an outer planet. It will then appear that the outer planet changes its direction into west-to-east fashion and then eventually continue its east-to-west path

Question 16

18. Which of Ptolemy’s devices did Copernicus have to use and why?

Answer 16

a. Copernicus used the eccentric, deferent and epicycle in an attempt to fit astronomical data but failed in doing so.

Question 17

*What were the two objections to Copernican theory

Answer 17

philosophical and religious objections

Question 18

*21. What are the common characteristics of Tychonic theories of the universe?

Answer 18

a. Earth was at the center of the sphere and the sun and moon orbits around the earth which was deemed to be stationary. The other five planets travel in orbits centered about the sun
i. The outer sphere of the stars rotates once in 24 hours, giving rise to diurnal motion
ii. The earth rotates daily on its axis, while remaining at the center of the sphere

Question 19

22. What was Tycho Brahe’s real contribution to the science of astronomy?

Answer 19

a. His accurate measurements of the positions of heavenly bodies (Sun, moon, planets) for 20 years

Question 20

*23. What were the new discoveries and arguments that tipped the balance toward the heliocentric theory?

Answer 20

a. The occurrence of nova + comet; the comet was observed in the astronomical/heavenly world. The heaven changed so it was not perfect; then there is no reason why the motion of the Earth should be any different than the motion of a planet
b. the invention of telescope

Question 21

Galileo’s works made important discoveries like:

Answer 21

i. Surface of the moon is marred with mountains and craters; heavenly bodies are not perfect
ii. There were other centers of rotation in the universe than the earth; the planet Jupiter has moons and the sun spins on its own axis
iii. Venus went through phase changes; it could be fully illuminated like full moon or partially illuminated like crescent moon. Venus is smaller (further away from the earth) when fully illuminated compared when it appeared as a crescent. This can only be explained when Venus orbits the sun
iv. There were many more stars; Even when viewed from telescope, stars were seen as mere pinpoints of light, indicating that they were indeed at least as far away as Copernicus claimed
v. Galileo began the development of physics as opposed to Aritostelian philosophy

Question 22

24. State Kepler’s laws of planetary motion.

Answer 22

a. The orbit of each planet about the sun is an ellipse with the sun at one focus of the ellipse
b. The line joining the sun and planet (called the radius vector) sweeps over equal areas in equal times as the planet travels around the orbit
c. The square of the period of revolution (time for one complete orbit) of a planet about the Sun is proportional to the cube of the average distance of the planet from the sun.