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Explaining the fundamentals of nature.
The next three books deal with number theory and Euclid's discussion of perfect and prime numbers,
The tenth
book deals with irrational numbers as discussed by Eudoxus, and the last three books deal with the geometry of solids.
In the development of Western culture, the importance of Euclid's geometry in the physical world
can be described as extraordinary and
difficult to estimate.
Euclid's geometry is only susceptible to error for very large quantities and distances,
it is a kind of ordinary sensory world
of mathematics,
and its limitations have only become apparent in the last two centuries.
It was the concept of Euclid that Einstein used as a beginning
to explore
the well-known theory of relativity.
1.2
Archimedes and the Rise of Science
Archimedes
(287
- 212
BC
)
was born and raised in the city of Sinacus, on the island of Sicily
, a city located on the Ionian Sea. is a city on the Ionian coast,
where he spent most of his life.
The walls,
fortifications, ditches, and other remnants of the ancient city can still be seen there today.
Archimedes' father, Phidias
was an astronomer, and Archimedes was a friend of the then king, Hierotheos II
so they were probably related.
Archimedes traveled
through Egypt and studied in the city of Alexandria, the cultural and academic center of Greece at the time
.
Archimedes' achievements include his mathematical research papers and special inventions.
Some of Archimedes' books
on mechanics have been lost, but all the mathematical reasoning on geometry has survived.
The book Equilibrium of the Plane details
Archimedes' proof of the principle of the lever and his study of the center of gravity of an object.
In The Sphere and the Cylinder,
Archimedes discovered how to calculate the surface area of a sphere and the volume of a sphere.
At the same time, Archimedes' work on mathematics was approaching
the theory of calculus,
which would later become the basis for the work of
Newton and Leibniz in the 17th
century.
Archimedes' book
Floating Bodies
contains his most famous Archimedes' Law of Buoyancy,
which states that when a body is submerged in water, the buoyant force on the body is equal to the weight of the body displaced by the water.
The law states that a stone or a rock can be immersed in water only if it has the same weight as the weight of the body in the water.
A stone is heavier than water,
so when a stone is immersed in water
it sinks despite being buoyant,
but a large ship immersed underwater has a weight of water displaced by its hull equal to the weight of the entire ship
so the ship floats.
The law states that when an object is immersed in water, the buoyant force is equal to the weight of the water it is displacing.
This law states that when an object is immersed in water, the buoyant force is equal to the weight of the object displaced by the water.
Archimedes described the principle of buoyancy,
which was also the basis for later hydrostatics.
Another anecdote about Archimedes's law,
but with an unattested allusion,
is the story of the wreath of the King of Hieroglyphics
(commonly
misattributed as a crown)
a wreath of gold that was not entirely pure,
but which contained part of the silver in its composition.
The king of Hiero
wanted to verify the composition of the wreath without destroying it,
because it was sacrilegious.
Archimedes accomplished
this examination.
More than 200
years after Archimedes' death,
the Roman architect Max recorded the following:
While Archimedes was sitting in the bathtub of the
public **** bathhouse,
he noticed that the level of the water in the bathtub rose to a volume equal to the volume of his body submerged in the water,
and therefore he
was not able to see the water in the tub. He
found the answer.
Archimedes immediately jumped out of the tub,
rushed home naked, and shouted with joy.
It is said that ever since
Archimedes' wife forbade him to bathe in the public bathhouse.
Archimedes thus discovered a way to easily calculate the volume of any irregularly
shaped object.
He dipped the king's wreath in water and counted the rise,
then dipped an equal piece of
pure gold in the water and
counted the rise,
and found that the two did not rise at the same rate,
and so it was clear that the wreath was not made of
pure gold.
Archimedes had a number of practical inventions,
the most famous being the Archimedes spiral,
a long, spiral-like tube,
that could draw water from the ground or from a river to the shore.
In addition, he invented a sphere,
constructed like the planets of the solar system, a
model of the workings of the heavens,
powered by water power,
and constructed with great precision.
Another invention was a device to measure the diameter of the sun by refracting its brightness
.
Plutarch describes Archimedes as a man who was preoccupied with mathematics to the neglect of his health:
"
Archimedes used to trace geometrical figures on the ashes of a fire
or to draw on himself with oil, in a state of complete and total concentration,
and he loved science, that is to say, from
science.
He loved science, i.e., he got as much pleasure from it as from religion.
"
Archimedes was killed by Roman soldiers in
212
BC.
According to history,
Archimedes was assisting in the defense
of the city,
when he invented a ballistic device that could hurl stones over long distances and destroy enemy ships,
and a device that could lift enemy boats
out of the water.
Another account of Archimedes' invention of a large mirror that focused sunlight to burn enemy ships is a misnomer,
but the Roman army was indeed repulsed by Archimedes' inventions,
and eventually resorted to a long siege.
Although Roman
general Marcellus hoped that Archimedes would not be killed,
Archimedes was nonetheless killed by the soldiers whose soldiers stormed the city,
and Marcellus
almost went mad with anger as a result.
One of Archimedes' famous mathematical geometric proofs concerned the relationship between a cone, a sphere, and a cylinder,
and he
proved that if all three have the same radius,
and their heights are equal to their diameters, then the ratio of the volumes of a cone, a sphere,
and a cylinder
is is
1:2:3
; moreover, the surface area of the sphere is equal to two thirds of the surface area of the cylinder.
This result so fascinated Archimedes that
it is depicted on Archimedes' tombstone.
About a century later,
Cicero, a Sicilian administrator, found Archimedes' tomb in a pile of overgrown bushes, and Cicero wrote
"
I noticed a column standing in the bushes
and on top of it there were the figures of a sphere and of a cone.
"
Archimedes was not the first to discover the principle of the lever, but he was the first to put a lever and a pulley together
.
Archimedes famously said:
"
Give me a fulcrum and I can lift the earth.
"
2.
Technological development in the post-A.D. period to the Dark Ages
During the Middle Ages (A.D.
476
- A.D.
1453
)
, with the Roman Empire's decline, Western Europe entered the Dark Ages.
The name
is apt,
as much of Roman civilization was destroyed during this period
and replaced by so-called barbarian cultures,
making the next
10
centuries a period of lethargy.
This period was characterized by the absence of a strong ruling regime in Europe,
and the frequent wars brought about by feudalism, which led to the stagnation of science and technology and productivity, and the misery of the people who lived without hope.
The use of this
name is also due to the fact that from this era onwards,
only a few historical documents have been passed down,
allowing people to only have a glimpse of the events of that time by the
shimmering light.
After the collapse of the Western Roman Empire, the barbarians from the north came to dominate the European continent,
and cultural development was interrupted,
with a few scholarly ideas being passed on only in the churches, and the general population living under the manor system,
like serfs, seeking only to
feed themselves.
The whole society was closed and conservative, and the sciences and arts were stagnant.
By some accounts, only
1
000
books were published in Europe during the Dark Ages.
There was very little progress in Western civilization over this long period of time, and the only outstanding
research was in medicine,
because medicine is a practical science,
and the rulers did not generally intervene.
The most famous medical
developments at this time were the second-century Roman physician Galen
(
Galen
), who made new discoveries in the fields of anatomy, physiology, embryology, pathology, medicine, drugs, and so forth, and wrote many books; and tenth-century *** Avicenna
who published Medicine, a book on the subject.