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

按规则来说，欧几里得空间-时间是一个没有尽头的闭合曲面，就像地球表面。

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

在欧几里得时空中，时间方向和空间方向之间没有。

It's almost like a Euclidean sort of foundation.

就像是欧几里得式的基础。

I'm pretty sure there's Euclidean geometry.

相当确定存在欧几里得几何。

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

们现在所说的欧几里德时空常相似，只是它有四个维度而不是两个维度。

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

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

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

为此，们必须对所有可能的无边界欧几里得时空的历史求和。

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。

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欧几里德几何外， 无补救。

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.

在欧几里得方式中，虚时宇宙历史是一个四维曲面，就像地球表面，多了两个维度。

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.

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

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.

古典理论家类似于欧几里得世界中的欧几里得几何学家，他们发现在经验中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。

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，当太阳终于升入深翠鸟蓝的天空时，他又觉得不太舒服了。