Hellenistic Science: Euclid, Aristarchus, & Archimedes

HPS 340

Euclid: Biographical

• Dates (probably) 365-275 BCE

• Active at the Library in Alexandria

The Elements

• His most famous text; this system of geometry has remained essentially constant since.

• A systematic approach: begins with definitions, postulates, and axioms. Each proof follows a given format.

Elements: Definitions

1. A point is that which has no part.2. A line is breadthless length.3. The ends of a line are points.4. A straight line is a line which lies evenly with the points on itself.5. A surface is that which has length and breadth only.6. The edges of a surface are lines.7. A plane surface is a surface which lies evenly with the straight

lines on itself.8. A plane angle is the inclination to one another of two lines in a

plane which meet one another and do not lie in a straight line.9. And when the lines containing the angle are straight, the angle

is called rectilinear.

Elements: a Proposition

To find the center of a given circle.

Let ABC be the given circle. It is required to find the center of the circle ABC.

Draw a straight line AB through it at random, and bisect it at the point D.

Draw DC from D at right angles to AB, and draw it through to E. Bisect CE at F.

I say that F is the center of the circle ABC.

For suppose it is not, but, if possible, let G be the center. Join GA, GD, and GB.Then, since AD equals DB, and DG is common, the two sides AD and DG equal the two sides BD and DG respectively.

And the base GA equals the base GB, for they are radii, therefore the angle ADG equals the angle GDB.

But, when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, therefore the angle GDB is right.

But the angle FDB is also right, therefore the angle FDB equals the angle GDB, the greater equals the less, which is impossible. Therefore G is not the center of the circle ABC.Similarly we can prove that neither is any other point except F.

Therefore the point F is the center of the circle ABC.

Method of Exhaustion

Optics

• Euclid believed that vision was the result of “visual rays” traveling from the eye along straight lines.

• Objects are seen when the visual rays strike them.

• The more rays striking an object, the more accurately it is seen.

Sample proof from Optics

Of equal intervals that are on the same straight line, those that are seen from further away appear smaller.

For let BG, GD, DZ be equal, and let K be an eye, from which let visual rays KB, KG, KD, KZ fall on them. And let KB be at right angles to BZ.

Then since in right-angled triangle KBZ (segments) BG, GD, DZ are equal, angle E is greater than angle H, and angle H is greater than angle T.

Therefore BG appears greater than GD, and GD than DZ.

Aristarchus: Biographical

• Lived circa 310-230 BCE

• From Samos, Greece

Aristarchus’ cosmology

• Aristarchus is the first to propose a heliocentric cosmology.

• The sun is at the centre of the universe, the Earth revolves about it, and the sphere of fixed stars is much farther out: far enough that parallax is not apparent.

Geometrical Methods

• Aristarchus’ program was to apply geometric methods to astronomy

• He was able to determine ratios of distances and sizes for the Earth, moon, and sun.

Method of Lunar Dichotomy

• Earth-Moon-Sun is a right angle at the half-moon

• Can measure Moon-Earth-Sun

Lunar Dichotomy

• Aristarchus measured the angle Moon-Earth-Sun to be 87°

• This put the ratio of distances between 18:1 and 20:1

• The problem with this method is that a very small error in angle leads to a large error in ratios. A more correct angle measurement is 89°50’, and the actual distance ratio is about 390:1

Eclipse Diagram

• During a lunar eclipse, the moon is fully eclipsed by the Earth’s shadow

• Aristarchus’ observations put the moon at half the diameter of the shadow

Archimedes: Biographical

• Lived in Syracuse (Sicily)

• Dates are probably 287-212 BCE

• Famous for aiding the defense of Syracuse from the Romans

Archimedes

• His task is to show that the properties of the universe can be expressed numerically: Sand Reckoner

• Improved Euclid’s Method of Exhaustion

• Calculated an improved value for π

Law of the Lever

• Problem is to determine at what point two unequal weights will balance

• Know that two equal weights will balance at their midpoint

Centre of Gravity

• Archimedes introduces the principle of centre of gravity in order to solve this problem.

• Centre of gravity is a physical property: “the point at which the object can be suspended and remain motionless.”

Law of the Lever

• Thus, weights on the lever balance at their centre of gravity.

• We need only to be able to determine the centre of gravity common to two unequal weights.

Commensurable Weights

• Divide two weights into units and determine centre of gravity

Floating Bodies

• Problem is to determine what makes floating bodies stable or unstable

• This analysis also depends on the idea of centre of gravity

Four Basic Principles

1. If an object is cut into two pieces, its centre of gravity lies on the line segment joining the centers of gravity of the pieces.

2. Any solid less dense than water will, if placed in water, be so far immersed that the weight of the solid will equal the weight of the displaced water.

Four Basic Principles

3. A solid immersed in water is driven up by a force equal to the difference between its weight and the weight of the displaced water

4. A body forced upward in water is forced upward along the perpendicular to the surface of the water that passes through the centre of gravity of the submerged part.

Forces on a Floating Body

Forces on a Floating Body