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.