Division of Arcs

The division of any arc of a circle into 3, 5, 7, and so on, equal parts, and in any ratio which is not obtainable by repeated doubling from the ones which have been shown above, cannot be carried out in a Geometrical manner which produces knowledge.

The division of an arc into 2, 4 and 8 parts, and so on, that is into a repeatedly doubled number of parts, can be carried out Geometrically, and has been used so far.

Not only can the complete circle be cut into 3 parts by the Trigon, but also the Semicircle, as for the Hexagon.

A quarter of the circle, as for the Dodecagon; and also a fifth, as for the Pentehaedecagon;

also the arc of 135 degrees, as in the Octagon; and

also the arc of 108 degrees, as in the Decagon.

Similarly:

• the complete circle can be cut into 5 parts, by the Pentagon
• the semicircle, as for the Decagon
• 1/3 of the circle, as for the Pentehaedecagon
• the arc of 150 degrees, as for the Dodecagon.

The same is true for:

• the halves of these arcs
• the quarters
• all other parts obtainable by successive halving.

But this happens:

• by chance, and
• because of the other properties of the shapes, already discussed.

It does not come about because of a characteristic of Trisection and Quinsection.

But in the general case trisection, or division in any other proposed ratio not obtainable by repeated doubling, is impossible, as can be seen by comparison with the possibility of bisection.

For that, the means used to bisect the arc, and the angle that it measures, is the straight line subtended by the arc, which [ie. the straight line] can be divided into 2 equal parts Geometrically: since from the equality of these 2 parts it follows that the parts of any arc are equal, whether it be large or small with respect to the whole circle:

From this starting point, we may also deduce that in a Triangle one may argue from the equality of sides to the equality of the angles opposite them.

This means [i.e. the division of the subtended line] is lacking to us in other types of section.

A straight line, subtended by an arc, can be divided into any number of [equal] parts Geometrically.

Yet from any proportion of the parts of the subtended line (after the proportion of equality) it is impossible to deduce a corresponding proportion of the parts of the arc.

In the same way in the Triangle, one may not argue from some proportion among the sides (apart from the proportion of equality alone) to the same proportion among the angles opposite them.

For, if the subtended line were, say, divided into three equal parts; if the lines dividing it have been drawn perpendicular to the chord, the middle part of the arc will be smaller than the ones to either side; if the dividing lines come out from the center of the arc [i.e. the center of the circle of which the arc forms part], the middle part of the arc will be larger than the side ones.

Therefore, between the infinite distance and the center of the circle, there is a point such that, if 2 lines were drawn from it, they would divide the subtended line and its arc into three equal parts.

In fact, this point is always further from the arc of the circle as the arc of the circle that is to be trisected becomes smaller, but not in constant proportion.

Thus since the arcs of the circle can be made indefinitely small (minui possunt in infinitum), the distance of this point can also increase indefinitely (excurret in infinitum): now there is no knowledge possible of something unbounded or of unbounded variation.

This difficulty besets even Trisection, which is still simpler and closer to equality [i.e. bisection].

Much greater difficulty will arise in the following divisions of a general arc, say into 5, 7, 9, 11, etc. equal parts.

For there can no longer be a single point from which the lines are drawn which cut the chord into the required equal parts while the same lines also cut the arc into equal parts.

Whatever techniques we can bring to bear to carry out division in the general case, techniques depending on the number that defines the division, these techniques must be general and apply equally to the lines subtended by any arc, both for a large arc which is very different from the line it subtends, and for a small arc, which differs little from the line. But leaving vague the ratio of the parts of the chord to the parts of its arc is definitely not a determination that yields knowledge. And let this be noted particularly for Trisection or quinsection etc. as carried out by Biirgi’s analytical method, which we discussed at length in the preceding proposition.

However all the things said there apply here also.

Moreover, some of the things said there are more appropriate here and become clearer and more significant in the division of arcs than they were in the division of the complete circle.

If I pass over the points which the 2 cases have in common, namely that it is begging the question if we are told to do what it was required to find out how to do: that the properties of a continuous quantity cannot be given, in a way that produces knowledge, by discrete quantities or numbers; that whatever number is obtained for the side which determines the required part of the arc it cannot tell us more than that the side is either larger or smaller than it should be; that as [the relationship of] rough and unshaped matter is to something which has form, and as [the relationship of]an indeterminate and indefinite quantity is to a figure, so also is [the relationship of] the analytic method to geometrical determination (the former is particularly excellent and noble in this semimechanical Cossa, but base and degraded in geometry which produces knowledge); that whereas every single chord which is less than a diameter is associated with two unequal arcs of the circle, of which one is smaller than a semicircle, and the other greater, and therefore the chord of a fractional part of the smaller one is smaller, and a part which is an equal fraction of the greater one is greater: this Analytic [method] of Burgi’s tells us something general, not only about these two unequal chords but also about many other chords of a circle, which is useful for expressing [their lengths] in numbers. For example, for trisection the rule (lex) is this:

If the arc be given (let it be 48 degrees) and its chord and let it be required to divide this arc into three parts, each of 16 degrees; that is if it be required to find the chord of this part, or its proportion to the whole chord of length 48 degrees: then I am required to make it so that the proportion of the chord of the whole [arc] to the chord that is to be found, that of the part [of the arc], is to be equal to the proportion of this chord to the second, and of the second to the third:

now I am required to triple the chord of the part [of the arc], and from it to subtract the third proportional: the remainder is said to be equal to the whole chord. That is, from the given chord, one third is cubed, as a fraction, and the resulting number is added to the whole: the third part of this sum is a little less than the required chord. For if, instead, this [chord] itself, cubed, is added to the whole; a third part of the sum comes quite close to the correct value; and this can be repeated, indefinitely.

By this procedure one comes gradually closer to the chord subtended by 16 degrees.^^^ But if you set up the number that is to be cubed as greater, and in fact to have about the value that the compasses suggest should be a third part of the remainder of the circle when 48 degrees have been subtracted, namely 312 degrees, a third of which is 104: then in this way you will arrive at the chord of the arc of 104 degrees, and of its complement 256 degrees. Nor is this all; but if you add to 48 and 312 the complete circle, 360, you will also find the thirds of those .sums, 408 and 6 72, namely 136 and 224, through [i.e. given as the value of] the same Term in the Cossic relationship.

In general, if one subtracts two from the number that defines the section, the number of units remaining gives the number of times one may add the complete circle to the arc it is proposed to divide, so as to discover the chords of new arcs by means of the same algebraic relationship.

From which it is clear that there is a huge difference between these algebraic terms and the degrees of knowledge which I discussed above.

But would it not be possible to find a nobler art by which arcs may be divided into any number of parts? I reply that if all the chords of the arcs that are to be divided could all be considered in the same manner, and if we only have techniques applicable in common to all the required chords, as for dividing them in the required proportion with any number of means in continuous proportion: then no one will be able to devise anything nobler, and whoever takes any further pains about the matter is wasting his time; and in his confusion is setting up the opposite as a predicate.

For from what is common [to all cases], nothing can be deduced that is applicable in any particular case.

If, however, on the contrary we address ourselves to the specific differences among the lines subtended by the arcs which are to be divided: then the status of the question is changed, and in place of the problem of all kinds of division of an arc we substitute that of dividing the whole circle, using a Regular figure, which establishes a connection between the proposed chord and its own specific property:

We have already dealt with these Regular figures above and we shall deal with them more fully below: because in this particular investigation we were seeking a means by which we may be able to draw some of those figures.

So since such a means must by its nature be prior to the thing that is to be carried out by it; we should be assuming what we wish to prove if in order to procure our Means we were to seek the aid of the Regular figures.

It might here be argued against me: that Pappus of Alexandria, in Proposition 31 of Book 4 of his Mathematical Collection, gives a way of trisecting an angle using a Hyperbola; and in Proposition 35 a way of dividing an angle in any ratio using a (fuadratrix and a Spiral: and Clavius, in Proposition 25 of Book VIII of his Practical Geometry achieves the same using the Conchoid of Nicomedes.^