Proper and Improper Construction of Shapes

by Johannes Kepler

Part 8: Measuring Shapes

The quantity of a line is knowable if it is either itself immediately measurable.

The quantity of a surface is knowable if its square is measurable by the diameter.

Part 9: Recreating Measures

The construction of a quantity which is either to be described or to be known is its deduction from the diameter, by permitted means, in Greek [these are called] Kopipa, “practicable.”

So constriction generally yields either description or knowledge. But Description declares mere quantity, whereas knowledge also in addition declares quality or a definite quantity.

A line can be geometrically determined, in Greek raKtri [“fixed”], even though its quality is not yet known intellectually.

On the other hand, a line or lines may be known qualitatively, but that does not yet determine them or make them determinate, that is to say if their quality is common to many other things which are different in quantity. So for such lines description is easy, knowledge very difficult. Einally, many things can be described by some Geometrical means or other; but cannot be knowable by their nature: as knowledge has been defined above.

Part 10: Proper Construction

This is when the number either of the angles of the shape itself, or of the shape related to it by having either double or half its number of sides, forms the middle term in finding the ratio of the side to the Diameter.

Every regular figure is either:

• itself a triangle or
• can be resolved into triangles by drawing diagonals.

Every such triangle has its 3 angles equal to 2 right angles.

• In the elementary triangle of the Trigon, the angle is 1/3
• In the elementary triangle of the Tetragon, the smallest angle is 1/4
• In the Pentagon, 1/5
• In the Heptagon, 1/7 etc.

Each fraction is that of 2 right angles. It is from the size of the angle that the construction begins.

Part 11: Improper Construction

This is when the ratio of the side to the diameter cannot immediately be determined Geometrically from the number of the angles, unless the side of another figure is brought in. This extra side is not from the figure with double or half the number of sides [of the original figure].