Old Math

I am a philosophy nerd who likes to think about ancient & medieval mathematics in her spare time. This blog is where I drop my ideas about such things. The title makes no false pretenses-- the theories and proofs of the mathematicians considered are as dusty as they are venerable. But I know that they are worthy of contemplation.

[luciafroula]

A Belated Thursday Definition

Spiral, n. If a straight line drawn in a plane revolve at a uniform rate about one extremity which remains fixed and return to the position from which it started, and if, at the same time as the line revolves, a point move at a uniform rate along the straight line beginning from the extremity which remains fixed, the point will describe a spiral in the plane. (Definition offered by Archimedes in "On Spirals")

[luciafroula]

On the Difference Between the Unit and the Point

In his general consideration of arithmetic and geometry in his Introduction to Arithmetic II, Nicomachus draws a parallel between their principles. "Unity, then, occupying the place and character of a point, will be the beginning of intervals and of numbers, but not itself an interval or a number, just as the point is the beginning of line, or an interval, but is not itself line or interval." (Ch VI, paragraph 3) Here we see the main similarity between the point and the unit, namely, that each is a principle, an archê, of its science. 

The point is a beginning of geometry because as it has no dimension. Nicomachus calls dimension "that which is conceived of as between two limits." (Introduction to Arithmetic II, Ch VI, paragraph 3) Now, as Euclid says, "A point is that which has no part." (Euclid’s Elements, Book I Definition 1) This lack of dimension makes the point a fitting limit for the very first dimension, which is made up of lines. These lines then define surfaces, which are outlined by two dimensions. In turn, these surfaces outline solids, which possess three dimensions and are the last and highest sort of continuous quantity with which geometry concerns itself.

The unit also has no dimension in itself (for two limits are found first in the dyad), but it is a beginning of arithmetic in a very different way than it is a beginning of geometry. Nicomachus defines number as "limited multitude or a combination of units or a flow of quantity made up of units." (Introduction to Arithmetic I, Ch VII, paragraph 1) Here it is clear that a unit, unlike the point, "makes up", or, is a part of, number. 

Thus, the unit is even more intimately tied to arithmetic than the point is to geometry. Both are causes of their sciences, insofar as they are principles, archai. However, a unit's role moves beyond that of a point in that it not only is a principle but is an element of its science. For Nicomachus defines "element" as "[that which] is said to be, and is, the smallest thing which enters into the composition of an object and the least thing into which it can be analyzed." (Introduction to Arithmetic II, Ch I, paragraph 1) 

Whether a point is "the least thing into which" geometry can be analyzed may be the subject of debate, but it is certain that it cannot "enter into the composition of" geometry in the way that the unit enters into the composition of arithmetic. For if a point did enter into the composition of geometry, it would seem to enter into it by composing the first dimension under which it lies, namely, the line. If it composed the line, the line would not be infinitely divisible. But it belongs to continuous quantity to be infinitely divisible, so then geometry would no longer be a study of continuous quantity, but of something else. 

A unit, on the other hand, does compose number, the discrete quantity with which arithmetic is concerned. In a way it is amazing that the unit should compose number without itself being number in any way, but in another way it is unsurprising because the nature of the whole and the part must be different. 

One might object that unity, multiplied by itself, creates nothing new, just as when a point is added to a point. However this is just a consequence of the fact that a unit is just one limit, and one limit taken by one limit creates no new dimension. Yet a limit taken not by itself but in relation to another single limit does create a new dimension. There is no avoiding the fact that the dyad is composed of units, just as is the triad, and the number four, etc. 

Now we have seen another way in which arithmetic is prior to geometry: one might say that it is nobler, and simpler, because its principle is the same as its element. 

[luciafroula]

Definition Thursday

Chiliagon, n. A figure with 1000 sides. In "The Sand Reckoner" Archimedes inscribes a chiliagon inside the sphere of the universe.

[luciafroula]

There Are Numbers of Shapes But No Shapes of Numbers

In Nicomachus' Introduction to Arithmetic II, Ch. VII, he introduces the concept of plane numbers. These are numbers which, in their composition, somehow resemble the composition of geometric planes. For example, 3 is the first actual triangular number, because it is the first that fulfills Nichomachus' definition: "Now a triangular number is one which, when it is analyzed into units, shapes into triangular form the equilateral placement of its parts in a plane." (Arithmetic II, Ch. VIII, paragraph 1) The sides of this triangular number are one unit in length. To illustrate*: 

*Note: In the sets of plane and solid numbers, Nicomachus presents the unit as the first potential plane or solid number, as opposed to the actual plane and solid numbers which are composed of several units. 

Nicomachus goes on to build squares, pentagons, hexagons, etc. out of numbers, after which he progresses to "solid" numbers, e.g., the first actual cube made up of the parts of the number 8.

Looking at these plane and solid numbers naturally brings to mind the question of how they participate in geometry. For Nicomachus says that arithmetic is prior to geometry in Ch. IV, paragraph 2 of his Introduction to Arithmetic I: "It is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them. For example, 'animal' is naturally antecedent to 'man,' for abolish 'animal' and 'man' is abolished; but if 'man' be abolished, it no longer follows that 'animal' be abolished at the same time. [...] Thus since it has the property of abolishing the other ideas with itself, it is likewise the older." From this it is evident that if the number three were to be "abolished", then the triangle would cease to exist, for Euclid defines it by giving "being contained by three [sides]" as the species of trilateral figure underneath the genus "rectilineal figure" (Book 1, Definition 19). 

However, if the three-sided plane figure were to be destroyed, it would by no means follow that the number three would be destroyed. This is because, as Nicomachus has said, arithmetic is prior to geometry according to consequence of being. It is necessary for arithmetic to exist for geometry to exist, but it is not necessary for geometry to exist for arithmetic to exist. 

From this we see that "plane" and "solid" are not integral to the triangular number 3 and the cube number 8, for 8 and 3 would retain all of their properties if the ideas of "plane" and "solid" were destroyed. The predication of "plane" and "solid" of "number" is an imposition designed to facilitate (or perhaps happening to reflect) our thought and imagination about such numbers. This is illustrated by the passage in Introduction to Arithmetic II, Ch XVI, paragraph 2: 

"Such solid figures, in which the dimensions are everywhere unequal to one another, are called scalene in general. Some, however, using other names, call them 'wedges' for carpenters', house-builders' and blacksmiths' wedges and those used in other crafts  having unequal sides in every direction, are fashioned so as to penetrate; they begin with a sharp end and continually broaden out unequally in all the dimensions. 

Some also call them sphekiskoi, 'wasps,' because wasps' bodies also are very like them, compressed in the middle and showing the resemblance mentioned. 

From this also the sphekoma, 'point of the helmet,' must derive its name, for where it is compressed it imitates the waist of the wasp. 

Others call the same numbers 'altars,' using their own metaphor, for the altars of ancient style, particularly the Ionic, do not have the breadth equal to the depth, nor either of these equal to the length, not the base equal to the top, but are of varied dimensions everywhere."

  Nicomachus gives his interesting etymological account of the names for scalene numbers like 24 (2 by 3 by 4) as the reason for each of the names the similarity in inequality of dimensions of some thing we see with our own eyes to the numbers we contemplate in our minds. In reality and considered on their own, these numbers have no connection to wasps, altars, shims, and helmets. In a similar way, we name "plane" and "solid" numbers by their similarity to shapes, things more easily grasped by our imagination. Whether this is done unconsciously or with the purpose in mind of ease of imaginative manipulation (or whether these things, in the end, are helpful at all) are questions for another day.