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That means that p1 and -- where I am, the point number 1 and point number 2 were completely arbitrary.
因为1和，两个点是2,陆军上面。
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How to go from one reference point to the other with this property. This property, f which we're going to call f.
这两个参考点插值,得到不同温度时工作物质的特性，我们把这一特性记做。
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So I think that is one of those place where nature and the notion of God come together.
我想应该有一个点,自然和上帝这两个概念在这点汇合。
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And the problem is, that we want to build this abstract data type, but we'd like to basically know what kind of object is it, and what functions actually belong to it, how do we use them?
然后我对它们运行了这个方法，就是这一小段代码这个方法,这有意义么？,当然没有了，对不对？,因为你知道，当你把两个极坐标，表示的点进行相加运算的时候？
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I don't know if you know how a spark plug works, but for the electrical current to fly into operation in a spark plug, the two points of contact have to be gapped.
我不知道你们是否知道,火花塞是怎样工作的，在电流通入火花塞中使它运转之前，两个接触点之间需要有空隙。
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And he couldn't get values of velocity in between certain steps.
他得不出两个点之间,位移的速度。
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Just take a look: if this is 9, 5 divided by 9 is always going to be 0 point something, and if you thus have two integers and you're rounding down, which is what happens when you do integral math we're using this operator, I'm going to get zero times whatever.
稍微看一看：如果这是9，5除以9会得到,0点几，如果你用两个整型数,你舍去小数，这就是当你们,用整型数使用这个操作的所发生的事情，我将得到数值0乘以任何一个数字。
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It says you are on the ground on two occasions.
在两个点你是在地面上的
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We can say that, but I wonder if it's possible even for these similes to convince us of the easy coexistence of these two incredibly important theological categories, free will and divine foreknowledge.
我们可以说，但是我还是怀疑，这些比喻是否能够,使我们相信如此这种两个太过于重要的相异点,即自由意志和神圣天命，之间的共存性。
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So let's reflect these two points.
那我们就把这两个点对称过去
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Thing I've got a little procedure that could do that, but you know the simplest thing I could do is to say well, gee, wait a minute, why don't I just check to see if these are the same thing?
这两个点是同一个点么？,我已经写了一小块代码，可以提供这个功能,但是你知道的我可以做的，最简单的事情就是说好，嗯？,等等，我为什么不直接去看看,这两个东西相等不相等呢？
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Square root of 16, and my plot basically shows it crosses at 4 and-- well, I think that's minus 4.
我的图清楚的显示这个函数,在4和-4两个点和x轴交合。
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Now, suppose in fact these weren't x and y glued together, these were radius and angle glued together.
我实际也说过了,我在这里操作的是，和这两个点。
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If it was non-reversible, I would be allowed to put an initial point and a final point, but I wouldn't be allowed to put a path between them like this, connecting them together.
如果是不可逆过程，我可以画出过程的初态点,和末态点，但是我不能再像这样,画出连接这两个点的，路径曲线来。
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This is saying, that's a class, true or sorry, an instance of a class, and I'm going to get the x value associated with it.
那么两个点是同一个点，我会返回，注意格式，这儿的意思是,它是一个类，噢对不起。
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So the next little piece of code I've written here, just move down to it slightly.
好，我可能想做的一件事情是,这两个点是不是同一个点呢？,因此我在下面写的。
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They happen to be on the path.
任意的两个点。
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And, likewise, there will be a test running from 1:05 to 1:55: two different tests of comparable difficulty. I did this last year, and the difference in the scores on all three tests 1% was less than 1%.
还有，测试的时间会是,从1点5分到1点55分，两个不同的测试,但是难度差不多，我去年这样做过，3个测试的分数的区别,不到。
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Two important points.
未翻两个重要的点。
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So I want to take this idea classes now, and I want to generalize it.
一个能够用来处理一些，例如我想看看两个点是不是一个点。
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Two reference points, freezing and blood-warm, you know,the human body.
两个参考点，水的）冰点和温血，也就是人的体温。
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So this first little piece of code right here says, ok you give me 2 points, I'll create another 1 of these lists and I'll simply take the x, sorry I shouldn't say x, I'm going to assume it's the x, the x-values are the two points, add them together, just right there, the y-values, add them together and return that list.
好，为了来认识到这一点,让我们来看一个简单的小例子，在你们的课堂手册上，你可以看到我写了一个小程序,它假设我得到了，这些点中的一些，我想对它们做一些操作，例如我想把它们加到一起，那么这里的第一小片，代码的意思是，好给我两个点，我会再创建一个数组。
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It's easy to think of a point as just a list of an x- and a y- coordinate.
或者说有两个元素的数组,把一个点认为是含有x坐标。
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Because the difference in the vector between two times is this vector.
因为两个不同时间点上矢量的差,就是这个矢量 ΔR
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Another way of saying it is, I've actually said, what did I use there, and , It's basically saying there is the first point, there's the second point, add them together and I get that point.
我要去做的是-对不起，再运行一次-好，你可以看到我已经把，这两个点的坐标值加到一块儿了，然后返回了r的值,我会让大家看到我们，实际上得到的是什么，这看起来不错，好,我做了正确的操作。
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