Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The insight with exploiting infinitesimals was that objects could still retain certain defined properties, such as angle or slope, even though these objects were quantitatively small. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.
In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size; or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" in the vernacular means "extremely small".
Archimedes used what eventually came to be known as the Method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the Method of Exhaustion. The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus. He exploited an infinitesimal denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{\infty}
Georg Ferdinand Ludwig Philipp Cantor ( /ˈkæntɔr/ KAN-tor; German: [ɡeˈɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
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There's a million, billion, trillion stars but I'm down here low
Fussin' over scars on my soul, on my soul, on my soul, on my soul
On my soul, I am so infinitesimal, oh
They say it started with a big bang
But they say it came out of a small thing
Lately I'm feeling like a big bang
‘Cause I've been making something out of nothing...Like my soul
Millions and billions and trillions of stars but I'm down here low
Fussin' over scars on my soul, on my soul, on my soul, on my soul
On my soul, I am so infinitesimal, oh
They say it started with a big bang
But they say it was really just a small thing
Strangely I'm feeling like a big bang
‘Cause I've been making mountains out of concaves
Do you ever really think about the grains?
Every little one's got a million things
Every little bit's got a billion bits, and that ain't it, no that ain't it
And did you know that when you really get close
Nothing really touches, bro, just kind of floats?
So when you think it might just come to blows
Just so you know, it won't, Because it can't, bro
There's a million, billion, trillion stars but I'm down here low
Fussin' over scars on my soul, on my soul, on my soul, on my soul
On my soul, I am so infinitesimal, oh
They say it started with a big bang
But they say it was really just a small thing
Strangely I'm feeling like a big bang
‘Cause I've been making something out of nothing
[Instrumental]