How could I have discovered the normal distribution? What was the first derivation of the normal distribution, can you reproduce that derivation and also explain it within its historical context? 
I mean, if humanity forgot about the normal distribution, what is the most likely way I would  rediscover it and what would be the most likely derivation? I would guess the first derivations must have come as a byproduct from trying to find fast ways to compute basic discrete probability distributions, such as binomials. Is that correct?
 A: Stahl ("The Evolution of the Normal Distribution", Mathematics Magazine, 2006) argues that the first historical traces of the normal came from gambling, approximations to the binomial distributions (for demographics) and error analysis in astronomy.
A: I would guess the first derivations must have come as a byproduct from trying to find fast ways to compute basic discrete probability distributions, such as binomials. Is that correct?
Yes.

The normal curve was developed mathematically in 1733 by DeMoivre as
  an approximation to the binomial distribution. His paper was not
  discovered until 1924 by Karl Pearson. Laplace used the normal curve
  in 1783 to describe the distribution of errors. Subsequently, Gauss
  used the normal curve to analyze astronomical data in 1809.

Source : NORMAL DISTRIBUTION
Other sources with historical context:


*

*The Evolution of the Normal Distribution

*Wikipedia history section
Nowadays the fact that the Normal distribution is an approximation for Binomials for large $n$ is considered as a special case of the Central Limit Theorem. It can be found in most text books and is considered as elementary. You can find a proof on Wikipedia. The exponential just shows up as $e^x=\lim(1+\frac{x}{n})^n$ after some Taylor expansion of the characteristic function that yield $-\frac{t^2}{2}$. Sometimes you still find special proofs for Binomials in textbooks and this is known as DeMoivre-Laplace theorem.
A: The question's historical part was answered already, possibly, multiple times on this forum, e.g. see the accepted answer to a similar question. No, it was not discovered as an approximation to discrete distributions. I doubt there was even a notion of probability distribution at the time. It was discovered by guys who's be called physicists or mathematicians these days, I guess nature philosophers at the time.
How would another civilization discover the normal distribution is an interesting question. Anyone who studies errors and disturbances of any kind would have found it. It happened so that our civilization found it while studying celestial bodies. I doubt that it is likely that other humans would develop statistics before physics or mathematics.
A: What' special about the normal distribution is the Central Limit Theory. For details and derivation/proof see:
https://en.wikipedia.org/wiki/Central_limit_theorem
A: I also asked myself that question and this youtube video is the best answer I have found
https://www.youtube.com/watch?v=cTyPuZ9-JZ0
I don`t think that it is the original derivation but the description of the video says "This argument is adapted from the work of the astronomer John Herschel in 1850 and the physicist James Clerk Maxwell in 1860. "
A: It's hard to parse this question. Who is the "I" in this question? And when is the time in question? An almost trivial answer is finding a location/scale family that is $\propto \exp(-x^2)$. The OP then goes on to ask "If humanity forgot about the normal distribution, in what manner would it be rediscovered"? This is an altogether different question. I think a relevant answer here is one that 1) borrows the perspective of modern science 2) provides an answer that is different from the most frequently encountered historical answer, aka the Central Limit Theorym. 
In quantum mechanics, information theory, and thermodynamics, the entropy quantifies the state of a system. In these fields, the quantum state is in fact, wholly random or stochastic. Contrast this with classical mechanics. In classical mechanics, states are fixed but our observation is imperfect due to the contribution of hundreds or millions of unobserved influencing factors: this kind of result gives rise to the CLT.
In quantum mechanics, we use Bayesian probability to quantify our belief about the state of the system. Along those lines, proofs have been presented, and tweaked, that the Gaussian or normal random variable has maximum entropy among all random variables with finite mean or standard deviation.
https://www.dsprelated.com/freebooks/sasp/Maximum_Entropy_Property_Gaussian.html
https://en.wikipedia.org/wiki/Differential_entropy
http://bayes.wustl.edu/etj/articles/brandeis.pdf
