he built a telescope through which he could study the skies。为什么不能用that？

through为介词，定语从句中介词后的关系代词不能用that。

通常不用that的情况
1. 在非限制性定语从句中
St Petersburg, which was once called Leningrad, is a very beautiful city.
2. 在介词后面常用which/whom
This is the room in which we lived last year.
3. 先行词为指人的one, ones，those，anyone，he时常用who
Those who break the law must be punished.

通常用that做关系代词的情况
1当先行词是指物的all, everything, anything, nothing, much, little, none, one等不定代词时(something除外)
Pay attention to everything that I do.
2当先行词被all, every, no, some, any, little, much等修饰时
He has little time that he can spare.
3先行词为序数词adj.最高级或the only/ last或被其修饰时
This is the best novel (that) have read.
4先行词既有人又有物时
They were talking about the person and things that they remembered in school.
5在疑问词who, which, what开头的特殊疑问句中，为了避免重复用that
Which of the students that knows something about history.
6当定语从句为there be 句型时，关系代词只能用that，也可以省略
This is the fastest train (that) there is to Beijing.
7当指人或物的关系代词在定语从句中做表语时，关系代词只能用that，也可以省略
China is not the country (that) it was.

这是一个定语从句，一般是介词+which。而through是介词。
所以这里是不能用that的。
译出来就是，通过前面这种方式能够做什么事。。（through which）

介词+which=that
而这里已经有了介词through，所以后面只能用which
望采纳，谢谢

关系词前有介词through ，只可以用which。