NUMBERS, OLD APPELLATIONS OF. The student of books verging on the middle ages will occasionally meet with some designa tions of numbers, or rather of the ratios of numbers to numbers, which may need explanation in a work of reference. Corresponding terms are found in the Greek writers, particularly in those on music, and they seem to have obtained universal curreucy in the middle ages by means of the work of Boethius on Arithmetic, which was a general object of study up to the middle of the 16th century.
These words illustrate a fault which has been avoided in our day by the adoption of the opposite extreme. The ancients often overloaded a subject with terms ; the moderns cannot prepare them as fast as they want them. The higher analysis now abounds with objects of thought for which there are no names except complex algebraical symbols ; the old arithmeticians strove to find names fur all the varieties of numerical ratio.
In describing these words, we shall, where we can, use the English form of the Latin adjectives, to avoid overloading our article with Latiu words : the adjectives accompany the word numerus, not ratio. The ratio of the greater integer number to the less was one of the following five, multiple, superparticular, superpartient, multiple super particular, or multiple superpartient. The ratio of the less to the greater was either submultiple, subsuperpartient, multiple subsuperparticular, or multiple subsuperpartient.
The term multiple has been preserved, and its species, duple (double), triple, quadruple, quintuple, he. Thus 10 to 2 is a multiple ratio, namely quintuple : that of 2 to 10 is submultiple, namely subquintuplo.
Superparticular ratio (part, that is, aliquot part, over) is when the greater contains the less and a submultiple of the less : its varie ties are sescuplo or sesquialter, sesquitertius, sesquiquartus he. Thus the following ratios are superparticular : 15 to 10, which is sesquialter ; 16 to 12, which is sesquitertius ; 15 to 12, sesquiquartus ; and so on. But the ratio of 12 to 15 is subsuperparticular, namely subsesquiquartus. One of these names is still preserved in our language, in the sesquialter stop of an organ. The ratio of 3 to 2 (a sesquialter ratio) is that of the length of a pipe to the length of a pipe which sounds the fifth above the note of the first. Accordingly
when a stop was made to sound with the ordinary stop, but a fifth above it, the name sesquialter was given to the stop which gave the higher note.
Superpartient ratio, according to Boethius, is that in which the major term is twice the minor all but au aliquot part. Its varieties are superbipartient (ratio of 5 to 3), supertripartient (ratio of 7 to 4), superquadripartient (of 9 to 5), and so on. Thus the ratio of one and four-fifths to one is super-quadri-partient. According to Boethius, then, the intermediate ratios of one and two-fifths and one and three fifths to one have no names. Some of his followers extend the name of superpartient to these, and some would invent the adjectives super biquintue and supertriquiutus to signify them • other used superbi partiens quintas and supertripartieus quintas. Multiple superparticular and multiple superpartient ratios have in the major term a multiple of the minor together with the fraction which gives the remaining adjectives. Thus, the ratio of 71 to 1 is multiple superparticular, being septuplus sesquiquartus ; and 71 to 1 is multiple superpartient, being septuplus supertripartiens. The preposition sub serves as before, prefixed to super, to denote the inverse ratios. The reader may fancy for bimself, if he can, the beauty of a treatise in which the ratio of 4 to 11 is expressed by duplus subsupertripartiens. Not that the writers of these works were ignorant of more simple phrases ; and we remem. her one place in which the example of Aristotle is brought forward to show that it would not be wrong to use them.
Of means, or medieties, Boethius discusses ten, to which Jordanus added an eleventh. The first three bear the names which have descended to arithmetic, geometric, and harmonic; and all are as follow. Let a, b, c, be three numbers, of which a is the greatest and c the least.
The works of Boethius and his followers consist iu dissertations on what would now be called the most obvious properties of the ratios of numbers, equiched with comments of every species, from numerical to – — theological. It is not right that they should be utterly lost sight of, for they form a dark background on which the merits of Sacrobesco, Breda-a/dine, Regiomontanua, and afterwards Tonga and Recorde, are very distinctly seen.