This tools helps you solve Thaumcraft research notes by showing you how to connect one aspect to another. Simply select two spects from the drop down, select the minimum number of aspects between them, and click the Find Connections button. It will show you all the ways you can build a path between the two aspects.
If there is more than one path, you can use the Next and Previous buttons to move through the possibilities. The tool tries to prioritize the path that consumes the fewest number of points, but that may not always be the best option for a given note.
The Accept button is used to accept a path. Accepting a path subtracts all the points necessary to build that connection from your points list. In this way, you can continue solving a research note with multiple paths within the constraints of the points that would be available to you.
On the top left, you'll see your available aspect points. This will limit your connection path to things you can build. You can add points to this list by clicking on an aspect from the aspects list at the top right. Selecting a primal aspect will add one point of that aspect to your points list. Selecting a compound aspect will build that aspect using the points in your points list. If you click an aspect in the points list, it will break down that aspect. Primal aspects get removed, and compound aspects get broken down into their components. You can also shift click on a primal aspect in the points list to add a point of that primal.
The save slots allow you to store and load a points list. You can save your current point list by clicking the Save button. To load those points back, just click the load button. The slot gives you more than one place to save a point list. This makes it easier to experiemnt with paths and go back if you get stuck.
You can reset your points list back to a given number of primal aspect points using the Reset button. It will reset your points list to just primal aspects with that number of points for each.