Individual Prime numbers

Individual Prime numbers of sides define individual classes of shapes.

Shapes are counted as belonging to classes which have a number of sides obtained by repeated doubling of the Prime [that defines the class].

This follows from the definition in section X of this book. For, if all figures such that the numbers of their sides can be obtained by repeated doubling of one given number of sides of one of them have the same form of proper construction: then they all belong to the same Class on account of their construction.

Because bisection [of the sides of a figure] does not alter the type or class [to which thefigure belongs], when it is associated with individualfigures; because of the simplicity and quality of the Parts, both together: for from the individual arcs of the former figure [the process of bisection] makes only two parts, which are equal.

But by trisection, or Quinsection, or division into more parts, you cannot avoid either obtaining unequal parts if there are to be only two of them, or many parts, that is more than two, if they are to be equal. Thus in trisecting an arc [of length] 3 it is either cut into 2 and 1, two unequal parts, or into 1, 1, 1, equal parts but many.

The foregoing proposition is proved thus. Constructibility depends on the number of sides [of the figure], by X of this book. Now prime numbers do not have any numerical part [i£. factor] in common, for unity, which they do have in common, does not determine a form of division and is thus not a numerical part or number.

So the demonstrations constructed by means of these numbers [primes] have nothing in common. Therefore the classes determined by individual primes are distinct. The first of these is that which contains the figures (or sort-offigures) with these numbers of sides: 2, 4, 8,16,32, and so on indefinitely: the second has 3, 6, 12, 24, 48, 96, and so on indefinitely: the Third has 5, 10, 20, 40, 80, 160, 320, and so on indefinitely.^*^ And there are indefinitely many others.

Proposition 31

Individual Numbers which are the lowest common multiples of two Primes (excluding two) define individual classes of Figures. This follows from the definition in section XI of this book.

For if such a figure does not employ the number of its angles in the construction of its sides: then the form of its construction is different from all the above, and therefore its class is also different.

The number two was indeed excluded from producing a new class when multiplied by any Prime: because the bisection of any angle is Geometrical^^ and in fact is the process whereby individual classes are each extended indefinitely: if this were not so there would be no classes, but only individual figures. The first [of the classes to which the proposition refers] is 13, 30, 60, 120, 240, 480, etc. multiplying 3 by 5. The second is 21, 42, 84, etc. multiplying 3 by 7. Indefinitely many others follow, as for 5 times 7. Whence we obtain 33, 70, 140, etc.

Proposition 32

But both the squares of Prime numbers, except the square of Two, and the products of these squares with another Prime or the square of a Prime also give rise to individual classes distinct from the preceding ones.

Now the square of a Prime number does not make the same class as the Prime [itself], because since the Prime itself makes a new class offigures, those which divide the whole circle,^’^ by section XXX of this book: now the same Prime, dividing not all but only a part of the circle will give a completely different construction,"^*^^ if indeed it is possible [to give one]: since a Part of a circle is very different from the Whole [circle], different that is in kind, and in its absolute configuration: Let us now concern ourselves with this configuration, since it determines the proof of the construction. Now, the square of two is again excluded; for the reason that the figure that has twice two angles, that is, the Tetragon, falls into the first class: if the number four is multiplied by a Prime, it [sc. the figures with that number of sides] falls into the class of the Prime, because four is twice two: and every figure with twice the Number of sides belongs to the same [class] as the figure with the orig­ inal Number of sides.

The first [of the classes to which this proposition refers] contains the figures with 9, 18, 36, 72, 144, 288 sides and so on indefinitely.

The second contains figures with 23,50,100,200,400, and so on indefinitely.

The third contains 49, 98, and so on indefinitely.

There are indefinitely many other classes derived from squared [primes].

Ai 27, 34, 108, 216, 432, and so on indefinitely, from 3 and 9.

As 73, 130, 300, and so on indefinitely, from 3 and 23.

As 147, 294, and so on indefinitely, from 3 and 49.

As 43, 90, 180, 360, and so on indefinitely, from 3 and 9.

As 123, 230, 300, 1000, and so on indefinitely, from 3 and 23.

As also 225, 450, 900, and so on indefinitely, from 9 and 25, two squares.

There are indefinitely many more classes, from Primes multiplied by squares [of primes], or by squares of Primes multiplied by themselves.