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

就像我第一次接触勾股定理感觉。

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

因为这些结果不能被归结于一个我们需要去证明数学定理，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，学习著名定理案例和故事是相当有用。

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

放弃对地球形状认识？对以太阳为中心宇宙认识？还是对勾股定理认识？对DNA结构认识？

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

他所作出结论真和欧几里定理一样准确。

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

能还记在学校学勾股定理，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，虽然不是什么绝对定理，但是脑外伤发生率在节日会大大升高，甚至像是我们朋友。

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

根据毕达哥拉斯逆定理，这就以形成一个直角三角形，因此，便形成直角。

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在那里存在思维，并将其具体化在扬特定理概念中。

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定理之一，也是实际应用基础，比如建造稳定建筑，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.

从基本计数开始，一直到许多类型无穷大，最终达到康托尔定理深奥程度。

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

利用现有数学定理和逻辑，我们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 年左右一块巴比伦石碑列出了15组满足该定理数。

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

而这正是中心极限定理一个酷炫之处。