It's like when I first encountered the Pythagorean Theorem.

就像我第一次接触勾股定理的感觉。

Since these results cannot be boiled down to a mathematical theorem that we need to prove, we have to do it some other way.

因为这些结果不结于一个我们需要去证明的数学定理，我们必须换种方式。

It's actually quite useful to have case studies or stories, which are these famous theorems.

事实上，学习著名定理的案例和故事是相当有用的。

The shape of the earth? The heliocentric universe? The Pythagorean theorem? The structure of DNA?

放弃对地球形状的认识？对以太阳为中心的宇宙的认识？还是对勾股定理的认识？对DNA结构的认识？

His conclusions were as infallible as so many propositions of Euclid.

他所作出的结论真和欧几里的定理一样的准确。

You might remember the Pythagorean theorem from school like what the formula is?

你可还记在学校学的勾股定理，比如公式是什么？

Now they didn't have what we call the theorem.

那时他们还没有我们所说的定理。

And it turns out there's a variety of proofs of the Pythagorean Theorem.

其实有很多种证明勾股定理的方法。

And we did that in kind of the intuition behind the binomial theorem videos.

我们是在利用二项式定理的方法来解决问题。

And yet, somehow, he came up with these incredible theorems.

然而，不知何故，他提出了那些令人难以置信的定理。

Well, there's no hard or fast rule, but brain injuries tend to pile up around the holidays, like our friend here.

哦，虽然不是什么绝对的定理，但是脑外伤的发生率在节日会大大升高，甚至像是我们的朋友。

According to the converse of the Pythagorean theorem, that has to make a right triangle, and, therefore, a square corner.

根据毕达哥拉斯的逆定理，这就可以形成一个直角三角形，因此，便可形成直角。

He thought about the mentality that he felt there was there and crystallized it in this concept of Janteloven.

他思考了他在那里所感觉到的在那里存在的思维，并将其具体化在扬特定理的概念中。

Instead, they knew all these particular cases where the theorem held true.

相反，他们知道所有这些定理成立的特殊情况。

This statement is one of the most fundamental rules of geometry, and the basis for practical applications, like constructing stable buildings and triangulating GPS coordinates.

这是几何学中最基本的定理之一，也是实际应用的基础，比如建造稳定的建筑，或对 GPS 点进行三角测量。

Start with basic counting and work your way up to the many types of infinity culminating in the profundity of Cantor's theorem.

从基本的计数开始，一直到许多类型的无穷大，最终达到康托尔定理的深奥程度。

Proofs use existing mathematical rules and logic to demonstrate that a theorem must hold true all the time.

利用现有的数学定理和逻辑，我们可以证明该定理总是成立。

But how do we know that the theorem is true for every right triangle on a flat surface, not just the ones these mathematicians and surveyors knew about?

但是我们怎么知道这个定理对平面上的每个直角三角形都成立，而不是一些数学家和测量人员所推测的呢？

A Babylonian tablet from around 1800 B.C. lists 15 sets of numbers that satisfy the theorem.

公元前 1800 年左右的一块巴比伦石碑列出了15组满足该定理的数字。

And that is a neat thing about the central limit theorem.

而这正是中心极限定理的一个酷炫之处。