Transcript Hellenistic Science: Euclid, Aristarchus, & Archimedes

Hellenistic Science: Euclid,
Aristarchus, & Archimedes
HPS 340
Euclid: Biographical
• Dates (probably) 365275 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.
2.
3.
4.
5.
6.
7.
8.
9.
A point is that which has no part.
A line is breadthless length.
The ends of a line are points.
A straight line is a line which lies evenly with the points on
itself.
A surface is that which has length and breadth only.
The edges of a surface are lines.
A plane surface is a surface which lies evenly with the straight
lines on itself.
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.
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-EarthSun 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