Maecianus starts with the subdivision of the coin known as the solidus, also called libra or as. The three alternative nomenclatures are the prelude to a systematic taxonomy, where each part of the as is introduced in turn as a numerical fraction, a name (the formula is ‘it is called’ (vocatur) or ‘its name is’ (nomen est))14 and a sign (‘its sign is’ (cuius nota)).15 The as is subdivided into halves (semisses), thirds (trientes), fourths
(quadrantes), sixths (sextantes), eighths (sescunciae), ninths (unciae duae sextulae) and twelfths (unciae) – the “elements, as it were’ of the first division (distributio). Maecianus perhaps alludes here to Euclid’s Elements, and hence to the fundamental, seminal nature of his present work. These elements, he says, ‘preserve equality’,16 unless they are added or subtracted to each other, in which case they sometimes produce equal, sometimes unequal parts. For example, if you add a sextans to a quadrans, you obtain a quincunx, equivalent to five unciae, i.e. 5/12; or, if you add a semis to a sextans, you obtain a bes, i.e. 8/12.17 Those are unequal parts. In general, the subdivisions of the as can be equal – a certain multiple of each subpart produces a whole as; for instance, six sextantes make an as, and so do two semisses, three trientes, and so on - or unequal. No multiple of a quincunx can produce a whole as – two will fall short of an as by a sextans, three will exceed an as by a triens.
The two parallel subdivisions are distinguished also by the fact that equal parts can only be characterized in one way, whereas unequal parts have several alternative definitions. For instance, a semis is, simply, one half, 1/2, and is obtained by dividing the as into two. A bes, on the other hand, can be obtained by adding 1/12 to 7/12, or by adding 1/2 to 1/6, or 5/12 to 1/4, or 1/3 to 1/3, and can be defined as eight unciae or four
sextants or two trientes or even an as minus a third.18 Even though unequal parts have a non-univocal nomenclature, and are characterized in a multiplicity of ways, their distinctive ‘signs’ (notae) remain the same. A bes, no matter how defined in terms of addition or multiplication of parts, is denoted by S =. The ‘signs’ of the unequal parts are in fact loaned from those of the equal ones: the sign of the bes reveals, and possibly privileges, one of its possible origins as the sum of a semis (denoted by S) and a sextans (denoted by =).
https://core.ac.uk/download/pdf/6581.pdf