Updated 20 June 2023
Part of the Theology of Arithmetic website.
arithmetic
The mathematical science of absolute quantity (see quadrivium). The term is based on the Greek word arithmos, translated "number," but with plurality implied: one was for the ancient Greeks, not a number, but the source or principle of number. Arithmetic, then, treats the progression of numbers, their basic division into odd and even (the two fundamental classes of numbers in antiquity), and basic operations such as addition, subtraction, multiplication, and division.
arithmology
This term was reintroduced by A. Delatte, who observed it in the title of an anonymous treatise preserved in an eighteenth century manuscript. The word appears to be a hapax legomenon, and late in Greek literature, but it is useful nonetheless to describe a genre of literature that proceeds one through ten, expounding on the nature of the number through its mathematical properties, its role in ancient myth, and other types of symbolism.
astronomy
The mathematical science of relative magnitude (see quadrivium), more conventionally, the mathematical science of the motions and relations of the heavenly bodies. Some ancient authors combine, others separate, stereometry from astronomy. Ancient astronomers did not distinguish astrology from astronomy, although they seem to have used the same science to different ends, thereby forming the basis for the distinction, which first appears around the sixth century A.D. As the fourth mathematical discipline, it is also the most complex.
gematria
See isopsephy.
geometry
The mathematical science of absolute magnitude (see quadrivium). The term is based on a combination of Greek words meaning "land" and "measure," probably indicating its roots in land surveying. Ancient geometry usually dealt with objects in one or two dimensions. Some authors subsume three-dimensional objects under geometry, but others create a separate category, stereometry. Geometry also included trigonometry and the science of proportions between line segments, as seen in Euclid.
isopsephy
Better known today as gematria, this literary device is built upon the ancient convention of assigning the letters of the alphabet numerical values. Letters, words, or entire sentences could then be composed or interpreted based upon the sum of their numerical values. Isopsephy first emerged as a literary phenomenon in Greek during the Hellenistic period, not much before earlier than the first century A.D. It was adopted from the Greeks in the first or second century A.D., and later used in Arabic. There is no clear evidence that the practice caught on in Latin. The term isopsephy derives from Greek terms that mean literally "equal count," and refers to the specific practice of taking a verse or phrase and finding another one equal in count to it. It is also correct to refer to the practice as psephy and the numerical value of a word or phrase as its psephic value. The term gematria is a Hebrew calque on the Greek grammateia, and was initially applied to different kinds of techniques for literary interpretation, but in the sixth century A.D., the term began to apply specifically to isopsephy. N.B., many scholars would like to assign the linguistic origins of isopsephy to Hebrew, and its origin in Biblical times. I argue in my dissertation, and will argue in future studies, that there is no basis for this claim.
mathematics
The modern definition of the term encompasses a great variety of pure sciences, ranging from simple arithmetic to string theory. The Greek word upon which ours is based, mathema, refers generally to something learned, but more specifically to sciences concerned with quantity and magnitude. Under the scheme of mathematics that would become the basis of the medieval quadrivium, these four mathemata were arithmetic, geometry, music, and astronomy (sometimes called stereometry). Throughout this website, I use mathematics in the classical sense (and in the sense used in English until the eighteenth century), to refer to all four sciences, and not just arithmetic, geometry, and other disciplines thought of today as constituting mathematics.
music
The mathematical science of relative quantity (see quadrivium; but note that music failed to make it into many ancient lists of what constituted mathematics), but more conventionally, the mathematical science of tonality. Musical tones were thought of in terms of relations between two numbers, e.g., the octave, which exhibits the ratio two to one. Of the four branches of ancient mathematics, music is the least well preserved.
number symbolism
The use of numbers to represent other realities, or to bring together them into relation with each other. This is often achieved through epithets given numbers. For instance, seven is called Athena or ever-virgin because it is neither the product nor the factor of any number ten or less. In a case like this, the arithmetical properties of seven illustrate the properties of virginity, and vice versa. To take another example, Platonic texts often use the numerical connection between the seven planets and the seven notes in the scale to discuss the music of the spheres. In this case, the number seven serves a connective tie between otherwise disconnected realities. Compare numerology.
numerology
Loosely, any non-mathematical use or interpretation of numbers. Webster calls it "the study of the occult significance of numbers," which is ambiguous, meaning either the study of esoteric numbers, or the use of numbers for occult practices such as divination or magic (analogous to astrology). Because the term is used in so many different ways, authors should define exactly what they mean, and be aware of the term's pejorative overtones.
quadrivium
This is the Latin term, first coined by Boethius (under the slight change of spelling, quadruium), to describe the four mathematical disciplines (see mathematics). Although the term was developed late, the quadrivium as a cornerstone of ancient pedagogy goes back at least to late antiquity, if not before. Evidence from the era of Plato, Aristotle, and their predecessors suggest that the number and order of the mathematical disciplines was not standardized. It first took on the traditional fourfold order—arithmetic, geometry, music, and astronomy—under the popular influence of pseudo-Pythagorean writings circulated and epitomized in textbooks composed in the second century A.D. and later. In this scheme, especially popularized by Nicomachus of Gerasa, there is the science of quantity, and that of magnitude. Each of these subdivide further into the study of quantities/magnitudes as either absolutes, or in relation to each other (numbers as properties or as relations). Arithmetic is the science of absolute quantity; music, of relative quantity; geometry, of absolute magnitude; astronomy, of relative magnitude.
stereometry
See geometry.
tetraktys
This is a Doric Greek term for the first four numbers (one, two, three, and four), the sum of which is ten. It was often thought of in the form of ten pebbles arranged in the shape of an isosceles triangle. The ancient Pythagoreans regarded the term as sacred, and used it for oath taking.