Conclusion

Part 29: Conclusion A

There are in all:

• 12 shapes which will form congruences
• 8 basic or primary figures
• 4 augmented or star figures.

The degrees of congruence are distinct. The trigon and the tetragon are of the first degree, because they form congruences in space as well as in the plane, both among themselves, with figures all of one kind, and also when combined with other figures.

The pentagon and its star are of the second degree because:

• in 3D they will form congruences among themselves, with figures all of one kind
• in 2D they come to one another’s aid

But the pentagon is the more powerful of the two because it will also form congruences with some other figures, both in the plane and in space.

The hexagon is of the 3rd degree, because figures of this kind form:

• congruences in 2D
• congruences both in 2D and 3D in combination with other figures

The octagon and the decagon and their stars are of the 4th degree because:

• their basic shapes will form solid congruences with some other shapes
• the stars will form congruences with figures all of one kind at least to a limited extent.
• in 2D, all 4 shapes form congruences with others, the octagonal figures doing so in more ways and more perfectly.

The dodecagon and its star is of the fifth degree because:

• in 3D they do not form any congruences at all, it is only their size which prevents them forming congruences.
• in 2D they should be of the 4th degree in terms of congruence. They combine with other figures to form many different congruences.

The icosigon is of the last degree, because this figure will form congruences only in the plane and then only when combined with other figures. These congruences are imperfect.

So if we consider only 2D, the order of the figures will be:

1. Hexagon
2. Tetragon
3. Trigon
4. Dodecagon
5. Dodecagon star
6. Octagon
7. Octagon star
8. Pentagon
9. Pentagon star
10. Decagon
11. Decagon star
12. Icosigon

All other figures are incapable of forming congruences, though the figure that comes closest to doing so is the pentekaedecagon, because it begins to form congruences with other figures in the plane

but it is excluded, by Part 20, because, unlike the icosigon, it cannot be surrounded at all its angles in the same way. After that comes the figure with 16 sides and others like it, which do not form plane congruences with other regular figures because their angles are too large.

But the heptagon and similar figures do not form congruences for a quite different reason, namely because neither whole angles nor aliquot parts of an angle of such a figure are able to form congruences with other regular figures.

So congruence can be divided into three demonstrably distinct classes: the octagon class, the decagon class, and the icosigon class, together with a fourth, spurious, class in which there is no congruence.

These classes will find their application in the choice of Aspects in Book 4.

Part 30: Conclusion

From this, we see that there is a genuine difference between construction and congruence in respect of the width of the classes they form.

For

1. The degrees of proper construction extend to infinity from the octagon, decagon, and dodecagon to include all figures that can be obtained by successive doubling of the number of sides; congruence is confined to the degrees of the octagon, the icosigon, and the dodecagon.

2. In respect of construction and knowledge the pentagon and its star are less noble than the dodecagon; in respect of congruence in space they are much nobler.

3. In construction and knowledge the octagon ranks lower than the pentagon but takes precedence over it in congruence.

4. The hekkaedecagon was higher placed than the icosigon for construction, yet the former will not form congruences whereas the latter will, to a limited extent.

5. But the pentekaedecagon shows a pleasing uniformity of properties in these two respects: since it has no proper construction but only an accidental one and it will not form any complete congruences but only the beginning of a con­ gruence which does not surround the whole figure.

These properties are to be taken into account below in