The two pages on display involve the construction of the trammel of Archimedes (an ellipsograph). It is a mechanical device which draws an ellipse as two pivots slide along two rails. The device with perpendicular rails was known to Proclus (5th century AD), and possibly even Archimedes (3rd century BC), and the device with oblique rails probably dates back to the 15th or 16th century AD.
In Figure 1 (page 206), the trammel is the rod CBH (so the distances between C,B,H are fixed once and for all), and the pivots C and B are free to move along the horizontal rail DE and the vertical rail FG respectively. As the pivots B,C move, the third point H of the rod CBH will trace out an ellipse. For example, MKL is another possible position of the trammel CBH.
Figure 3 shows a trammel BCH with oblique rails DAE and FAB. The proof that H traces out an ellipse when the rails are oblique is rather tricky by early 17th century mathematics, that is without using Cartesian coordinates. Indeed, the author must go through a long succession of similarities between triangles in order to prove the relationship LI^2 / (DI x IE) = HG^2 / DE^2 (in Figure 3), which in turn will imply that the point L lies on a certain ellipse (here KML is the general position of the trammel BCH). Figures 4-6 illustrate the construction of Figure 3 as L moves along the ellipse.
The modern approach to solving the oblique trammel problem is to simply reduce it to the case of orthogonal rails via an affine transformation. But such an argument would only became routine in the very late 19th century.