Who Was Archimedes?
Sir Isaac Newton (gravity, calculus, optics, etc.) is viewed as the outstanding scientist/mathematician of all time. Einstein (Time Magazine's person of the 20th century) was probably the outstanding scientist/mathematician since Newton; Archimedes (287-212 BC) had no equal in ancient times. So why was Archimedes omitted from Thall's Top 10 Scientists?  Unfortunately, much of our knowledge regarding Archimedes is anecdotal and difficult to verify. Stories about Archimedes are derived from the writings of Plutarch (born 250 years after the death of Archimedes).  Also the ancient Greeks used papyrus rolls (grass-like plant grown in Egypt) to record their works. Since papyrus is fragile, even if left untouched it rots fairly quickly, the only way to preserve manuscripts was to frequently make new copies. Transcriptions made by someone with no technical knowledge would often lead to errors while someone with technical knowledge might change the original text to reflect more modern views.

In mechanics, Archimedes defined the principle of the lever and is credited with inventing the compound pulley. During his stay in Egypt, he invented the hydraulic screw for raising water from a lower to a higher level. As a military consultant, Archimedes constructed a catapult to haul stones and giant mirrors to set ships on fire.

Part of Archimedes' fame comes from solving king Hiero's "crown problem."  It seems Hiero gave the goldsmith a quantity of gold to make a crown.  The finished crown possessed the correct weight but Hiero suspected some silver had been substituted.  The problem was brought to the attention of Archimedes. Upon entering his bathtub, Archimedes noted the overflowed water was proportional to his body submerged. This observation established what has become known as Archimedes' Principle and  proved bad news for the fraudulent goldsmith.  Archimedes later experimented with liquids and
                     discovered density and specific gravity.

However, it was in mathematics that Archimedes excelled; his methods for establishing volumes of geometric figures paved the way for calculus. Let's examine a few of his innovations:

1.  Method of Exhaustion
     To determine area of circle, Archimedes covered circle with a small figure such as a square.  If the
     squares in circle are 0.10 cm², the circle's area can be estimated by counting squares. By making
     squares smaller and smaller, Archimedes devised precise method for finding circle's area.

2.  Method of Summation
     Archimedes represented the first systematic series of any kind with this summation:
3.  Pi Determination
     Archimedes showed that pi was between  3 1/7  and  3 10/71

4.  Square Root of 3
     In calculating pi, Archimedes states that square root of 3 is known to be greater than 265/153 but
     less than 1351/780.  To this day mathematicians do not know how Archimedes arrived at this result.

5.  Large Numbers
     Archimedes represents numbers to powers of 10. This is the basis of our present operations by
     logarithm:  X^mXⁿ = X^m+n
 

In the Sand Reckoner, Archimedes argues that the number of grains of sand fitted into the universe can be expressed(Archimedes developed exponents to represent large numbers). Archimedes has to give the dimensions of the universe and uses a system with the sun at the center with planets (including earth) revolving round it. In quoting dimensions Archimedes gives the results made by his astronomer father Phidias--this is all we know about Archimedes' father.

In 212 BC, the Romans attempt at conquering Syracuse was stalled by huge catapults (designed by Archimedes) hurling 500 pound boulders. Archimedes also constructed large mirrors to set Roman ships on fire.  Syracuse fell eight months later and Archimedes was killed. The story goes that he was drawing figures in the dust and a Roman soldier stepped on the drawings. When Archimedes said, "Don't disturb my circles," the soldier became enraged and killed Archimedes.

Click to see Archimedes Crater

The first really large number to surface in science was related to atoms and molecules. In 1811, Amedeo Avogadro proposed his now famous hypothesis:
Equal volumes of gases at same temperature and pressure contain equal numbers of molecules.
In recognition of this new idea, the number of particles in 22.4 liters of gas at STP (1 atm/0°C) is called Avogadro's number. Although Avogadro had no idea what the actual number of particles might be, his  hypothesis did lead to the eventual determination of this number as 6.02 x 10²³. The unit "mole" was introduced into chemistry around 1900 by Ostwald (originally defined in terms of grams).  Just how big is a mole? To hold one mole grains of sand, freight cars would be needed from the earth to the sun 6 times!
Click to learn how Avogadro's number derived!

October 23rd is National Mole Day
It is a day to:  Celebrate Chemistry
                      Remember the Mole Concept
                      Honor Amadeo Avagadro
Mole Day Jokes
 
 

Georg Cantor (1845-1918) 
For about 2000 years large numbers were ignored--the great mathematician Gauss said infinity should only be used as "a way of speaking" and not as a mathematical value. Most mathematicians followed his advice and stayed away. However, Greg Cantor could not leave it alone. He considered infinite sets not as merely going on forever but as completed entities, that is having an actual though infinite number of members. He called these actual infinite numbers transfinite (from Latin word meaning "beyond limits") numbers. By considering the infinite sets with a transfinite number of members, Cantor made amazing discoveries and was promoted to full professorship in 1879

Cantor established that infinite numbers have different properties than finite numbers and follow different arithmetic rules. For example, consider finite numbers a and b:
if  a > 0  and  b > 0,  it follows that a + b > a    and  a + b > b

Therefore it seemed strrange when Cantor proved that if a were finite and b were infinite, then:
 a + b = b
Einstein showed that if we take the velocity of an object traveling at the speed of light (c) and apply force to increase its velocity by Dv, the object still possesses velocity c or:
c + Dv = c

Cantor's ideas that infinite numbers exist and some are larger than other did not gain immediate acceptance. According to mathematician Henri Poincare, Cantor's theory would be considered by future generations as "a disease from which one has recovered."
 

GOOGOL
Today scientists deal with numbers much larger than the mole. A very large number was concocted in 1938 by Columbia University mathematician Edward Kasner and named after a word  used by his 9 year old nephew. The GOOGOL is 10¹⁰⁰ or 1 followed by 100 zeros:
(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)
The GOOGOL represents a number so vast that it is almost beyond description--the number of particles (protons, neutrons and electrons) in the entire universe is approximately 10⁸⁵.  If the GOOGOL is not large enough for you, then consider the SUPERGOOGOL or 1 followed by a GOOGOL of zeros: