Biirgi’s Cossa

What does Biirgi’s Cossa tell us about the sides of the Pentagon?

We obtain the number 5j - 5iiJ + Iv in the previous part.

This is not equal to any chord.

If 5 quantities are constructed in continuous proportion, the first of them being the side of the Pentagon; the proportion being that of the side of the Pentagon to the semidiameter; then five lines equal to the first proportional plus one equal to the fifth will be equal to five equal to the third.

The heptagon does not tell us how to construct the continuous proportion for which this relationship will hold.

It does not express the lengths of the proportionals in terms of things already known.

But it tells us, once the system of continuous proportion is set out, what relationship will follow.

So I am instructed to represent the relationship (affectio), for it will then come about that I obtained the proportion also.

But how am I to represent the relationship, by what Geometrical procedure?

No other means of doing it, save using the proportion I seek. There is a circular argument.

The unhappy Calculator, robbed of all Geometrical defenses, held fast in the thorny thicket of Numbers, looks in vain to his algebra (cossa).

This is one distinction between Algebraic (Cossicas) and Geometrical determinations.

Another is that all this reasoning of Biirgi’s depends on the nature (essentia) of a discrete quantity, namely that of numbers.

It divides the diameter into precise small parts, as many times and as far as he wishes, generally into two parts; on which number [sc. of division] the whole process depends.

It would be changed if the Diameter were given another value (nomen), or a different number of parts.

But Geometry does not deal with figures in this way.

Though it does designate sides Expressible in length by Numbers.

But inexpressible ones it in no way attempts to capture with numbers, but states their magnitudes according to their particular kinds, so that it is clear that we are dealing not with discrete quantities but with continuous ones, that is with lines and surfaces.

3. So far, both the side of the Figure and the side of its related star, each had a precise description in this Algebraic Analysis.

The most surprising thing is that (although this may especially frighten the Geometer) there is no one way to produce what we are asked for.

All the same this is not entirely without a pattern of its own, but, as I started to explain above, the number of numbers making up what is required is the same as the number of chords or Diagonals of different lengths comprised in the figure, so that in the pentagon there are two, in the heptagon three, one for the side [of the figure], the remainder for chords subtending an angle [i.e. diagonals].

So that whatever is stated concerning the particular proportion of the figure holds for the proportions of all lines to the diameter.

4. Assuming that a single proportion would suffice to define what is required; I am not told how to bring the matter to a conclusion but only how to stalk the quarry, from a distance.

For since the kinds of line, according to their [degree of] knowledge, are found among the Inexpressibles (that is, they are not numerable but reject numbers), there will accordingly be no multiplicity of numbers that can exhaust the ratio without leaving some uncertainty in it.

On the other hand, this ratio, as mentioned in our second point above, takes no refuge except in numbers, but repeatedly divides the diameter in various ways into many Myriads of Myriads of parts, to make [the numerical expression for] the ratio more and more exact.

But this never gives a completely exact value; and, in short: this is not to know the thing itself but only something close to it, either greater or less than it; and some later calculator (computator) can always get closer to it [still]; but to none is it ever given to arrive at it exactly. Such indeed are all quantities which are only to be found in the properties of matter of a definite amount; and they do not have a knowable construction by which in practice they might be accessible to human knowledge.

5. Let us concern ourselves specifically with the heptagon and following figures of this type {genus) [ 5 ?c], as they follow one another in order the [series of lines in] continuous proportion will grow longer as the number of sides increases: so if the one of most interest were the last one, as, for the heptagon, the seventh of the proportionals; it would, all the same, not be possible to use it to find the intermediate proportionals. For between two [lines], which are not in the propor­ tion of two numbers of the continuous proportion, such as that one is the cube or the fifth power^^’’ and so on of the other, it is not C ons t ruct i on R egular F i gures

possible geometrically to set up any number of intermediate magni­ tudes in continuous proportion but only one or three or seven or fifteen, and so on, while in the plane it is not possible to set up two or four, five, six, eight, nine, and so on^^®; since here we are considering plane figures.

Between the semidiameter, of magnitude 1, and the seventh proportional, of magnitude Ivij, in the [system of] proportion relating to the heptagon, there are six mean proportionals, and the ratio of 1 to IviJ is not that of a number to a number in a continuous [system of] proportion that is equally long; that is to say, the proportion of the semidiameter to the side of the heptagon is not like that of 2 numbers, that is, it is not Expressible.

For if it were Expressible it would fall into one of the categories {species) already discussed, [those] belonging to the earlier classes, and the seven angles would not be seven but [instead] three or four, which involves a contradiction.

For the proportion of the sides of the first figures {primarum figurarum) was [deduced] from their angles.

Thus it would have been necessary to construct all six mean proportionals in a single step, that is [the mean proportionals] between 1 and Ivij.

On the other hand, if Ivij were given in magnitude; then there would be five mean proportionals between 1 and Ivj.

Therefore, if the ratio of 1 to Ivj were then to be that of a cubic number to another cubic number, then first it would be possible to construct lij and liiij in a single step, afterwards, in three steps, three mean proportionals between 1 , lij, liiij and Ivj.

However, if Iv were given in magnitude, again all four intermediate magnitudes would have to be constructed in a single step; which can not be done, unless the proportion concerned is Expressible, as above.

The other [examples] are all subsumed under these.

So we conclude that these Algebraic (Cossicas) Analyses make no contribution to our present concerns; nor do they set up any degree of knowledge that can be compared with what we discussed earlier.

Now it is appropriate to put a word in here for Metaphysicians in connection with this algebraic treatment: let them consider if they In case it should be supposed that these

Can take anything over from it to explain its comments are blasphemous.

One of myAxioms, since they say that which does not exist friends, a very practiced mathematician, [a Non-entity] has no characteristics and no prop- thought they could be left out. But nothing is more habitual among Theologians than to erties.2^®

For here we are concerning our claim that things are impossible if they selves with Entities susceptible of knowledge; and involve a contradiction: and that God’s

we correctly maintain that the side of the Hepta­ knowledge does not extend to such impossible things, particularly since these gon is among Non-Entities that is not susceptible formal ratios of Geometrical entities are of knowledge. For a formal description of it is im­ nothing else but the Essence of God; because whatever in God is eternal, that possible; thus neither can it be known by the hu­ thing is one inseparable divine essence: so it man mind, since the possibility of being con­ would be to know Himself as in some way structed is prior to the possibility of being known: other than He is if He knew things that are incommunicable as being communicable. nor can it be known by the Omniscient Mind by And what kind of subservient respect would a simple eternal act: because by its nature it is it be, on account of the inexpert who will not read the book, to defraud the rest. among unknowable things. And yet this which is not a knowable entity has some properties which are susceptible of knowledge; just as if [they were] Entities with characteristics. For if there were a Heptagon inscribed in a circle, the proportion of its sides [to the semidiameter] would have such properties. Let this in­ dication suffice.

There are also other untrue propositions put forward by Geometers concerning the sides of figures like this, but which someone relalively experienced in the Mechanical [art] would reject though because they are Mechanical they are pressed on the young‘^^b as when Albrecht Diirer puts the side of the Heptagon, AC, equal to half of AB, the side of the Trigon drawn in the same circle.^^^ That this is in fact considerably too short is apparent even from Mechanics-^^^: however, lest anyone be misled by a rather crude practical trial; he can recognize its falsity even by this reasoning alone, without any manual procedure.

From the number of its angles the side of the Trigon is proved to be Expressible in square: therefore so is half of it. The side of the Hepta­ gon is not Expressible in square, precisely because it belongs to the Heptagon: and because seven is not six, nor five, nor three. For prime numbers give rise to sides of [particular] kinds; but these kinds [of line] are incommensurable with one another, and no one of them is the same as another.

For the fallacies put forward by Carolus Marianus of Cremona and Francois de Foix, Comte de Candale, concerning the Heptagon see Christopher Clavius, Practical Geometry Book 8, proposition 30, and his commentary on Euclid Book IV, proposition 16.

This contest also spurred into action the Most Illustrious Lord the Marchese de Malaspina, who in 1614 was the Ambassador of the

Most Serene Duke of Parma to the Imperial court; and whose most ingenious diagram beat all the descriptions put forward by everyone else; estimating that the chord subtended by three fourteenths of the circle was equal to five quarters of the semidiameter, and thus express­ ible in length: so expertly was the apparatus of proof deployed that even Euclid himself might have failed to notice that something had been assumed without proof.

For the side of the Endecagon the following description is in circu­ lation; In a circle, let there be drawn from the same point A, the side of a Tetragon AC, in one direction, the side of a Trigon AD in the opposite one, and the side of the Hexagon AB, AF in each direction.

Let the angle FAB contained by the two Hexagon sides subtend another Trigon side, BF, which will cut the first Trigon side, AD, in G: let there also be drawn from the end C of the Tetragon side the diameter CE, passing through I, the center of the circle, and from the other end of the diameter, E, through the point of intersection, G, of the 2 Trigon sides, let there be drawn the straight line EG, cutting the Tetragon side AC in H: the line GH between these two points of intersection is said to be the side of the Endecagon.

It is too long, as even practical methods (Mechanica) show.

But an expert (sollers) Geometer will bear in mind the kind of line that is involved, which necessarily has something in common with the sides of the Trigon and the Tetragon, though it belongs to a remote degree. But, all the same, the number 11, being a Prime, does not in any way lead one to these figures, for since it is a Prime it has nothing [sc. no factors] in common with 3 or 4. So the Geometer is confident that the description [just given for the Endecagon] is incorrect; and he may easily dispense with the labor of [checking this by] computation.

It remains, therefore, that for all these objections, for all the frustrated attempts by all these scholars, the sides of figures of this kind are by their very Nature unknown and unknowable.

So it is not to be wondered at that what could not be found in the Archetype of the World is not expressed either in the structure of the parts of that World.