In order terms, the Euclidean space time is a closed surface without end, like the surface of the Earth.

按规则来说，欧几里得间-间是一个没有尽头的闭合曲面，就像地球表面。

In the Euclidean approach, the history of the universe in imaginary time is a four-dimensional, curved surface like the surface of the Earth, but with two more dimensions.

在欧几里得方式，虚宇宙历史是一个四维曲面，就像地球表面，但多了两个维度。

In Euclidean space-time there is no difference between the time direction and directions in space.

在欧几里得，间方向和间方向之间没有区别。

The white pyramid in the distance compelled the mind's assent as passionlessly as a Euclidean theorem, and when at last the sun rose into a sky of deep delphinium blue, he felt only a little less than comfortable again.

远处的白色金字塔像欧几里得定理一样冷酷无情地强迫人们同意，当阳终于升入深翠鸟蓝的天，他又觉得不舒服了。

But at the turn of the 19th century, Lobashevsky and Gauss discovered non-Euclidean geometries, and this prompted mathematicians to examine more closely the foundations of their field.

但在 19 世纪初，罗巴舍夫斯基和高斯发现了非欧几里得几何，数学家更仔细地研究该领域的基础。

It helps that Mr. Gaines's prose is itself, poetic and precise The hills and locks of acid for instance, tessellate in shapes like Euclidean art.

大卫·甘格的文笔本身既诗意又精准， 一点很有帮助。例如， 酸液侵蚀的山丘和锁链呈现出像欧几里得艺术般的图案。

In Euclidean geometry, the geometry of Newtonian physics, that is a straight line, but in the geometry of space-time, a geodesic can be curved.

在欧几里得几何， 即牛顿物理学的几何体系，是直线； 但在几何，测地线可以是弯曲的。

Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry.

然而，事实上，除了抛开平行公理并计算出非欧几里德几何外， 别无补救。

It's almost like a Euclidean sort of foundation.

就像是欧几里得式的基础。

Descartes, Hobbes, Spinoza, and Leibniz constructed systems of metaphysics which they deemed to be as valid logically as the Euclidean geometry.

笛卡尔、霍布斯、斯宾诺莎和莱布尼茨构建了形而上学体系，他们认为些体系在逻辑上与欧几里得几何学一样有效。

I'm pretty sure there's Euclidean geometry.

我相当确定存在欧几里得几何。

The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight-as the only remedy for the unfortunate collisions which are occurring.

古典理论家类似于非欧几里得世界的欧几里得几何学家，他们发现在经验看似平行的直线经常相遇，斥责直线不保持直线——是对正在发生的不幸碰撞的唯一补救措施。

A second problem arose from the way space is amazingly flat, meaning that it expands according to normal Euclidean geometry.

第二个问题源于间的惊人平坦性，意味着它按照正常的欧几里得几何学进行膨胀。

To do this, we must perform the sum over histories for all possible Euclidean space-times that have no boundary.

为此，我们必须对所有可能的无边界欧几里得的历史求和。

When someone gets to the point, they're not Euclidean, but there's a word like that, I think, out there.

当某人说到点子上，他们用的不是欧几里得几何学的概念，但我想，外界确实存在一个类似的词。

What we now call Euclidean space-time is very similar except that it has four dimensions instead of two.

我们现在所说的欧几里德非常相似，只是它有四个维度而不是两个维度。