The Rule of Thirds is what artists call the grid that appears on your cell phone. It helps you compose and edit.
A variation of this is used in Practical Geometry.
Sebastiano Serlio used this diagram in his book, On Architecture, published in 1545. He writes simple instructions for the reader; he says to construct the 'lines'.
Note that the triangle (with its base at the bottom of the drawing) intersects the diagonals at the the upper corners of the door. The width of the square is divided into thirds.
Check how the division into thirds in the square above this drawing lines up with those intersections. Serlio is using a a variation of the Rule of Thirds.
Then he explains how to solve problems. He does not show how he knows where to draw the lines shown above. He assumes the reader knows.
Here are the instructions:
Draw a square;
Add the diagonals to your square. Where they cross in the center. You have point 1.
Divide one side of your square in half. Now you have points 1 and 2.
With 2 points you can draw a line.
Add diagonals in each new rectangle.
Add the diagonals from the square.
If you were drawing this for a construction project on wood, on masonry, or on paper, you would not have separate squares. All lines would be on your first square. I have drawn each step without the extra lines for clarity.
Do you see that the center line does not pass through the intersection of the diagonals? If you were the builder you would know that your diagonals will match when the line in centered. In this diagram they don't. So you would move your center line.
This is the diagram for Serlio's drawing for the door.
For the Rule of Thirds (as we know it today) add the diagonals for the rectangles on both sides of the square.
Note that you have intersections (4 points) not just where the lines divide the square into smaller squares, but where the diagonals cross those lines. 2 points above the horizontal center line and 2 points below. Or: 2 on the right side of the vertical center line and 2 on the left.
I have deliberately not added black points where the lines cross. You who are reading this will see it more clearly if you find those points yourself.
Connect those new points and extend the lines across the square.
You have drawn the Rule of Thirds.
Similar diagonals could be drawn from the left to the right side and vice versa.
I drew all the diagonals on graph paper to make it easier to follow. The next lines to add would be the diagonals of the small squares.
The line does not come back to its beginning until it has continued through the complete pattern.
A post on Serlio. https://www.jgrarchitect.com/2017/04/serlio-writes-about-practical-geometry.html