[239] Cantor believes that the use of the abacus had been forgotten before Isidore’s time, _cf._ “calculator a calculis, id est a lapillis minutis quos antiqui in manu tenentes numeros componebant.” _Etym._, 10, 43. See Cantor, _Vorlesungen über Geschichte der Mathematik_ (Leipzig, 1894–1900), vol. i, p. 774. Isidore’s source in the _De Arithmetica_ was Cassiodorus,[240] whom he copies with little change; while Cassiodorus’ work was apparently a bare abstract of Boethius’ translation of Nicomachus. Isidore’s account is of great brevity and contains a number of unexplained technical terms. [240] Isidore adds to the account as found in Cassiodorus a few remarks about numbers in the Scriptures, some derivations of numbers, and the sections on the means and on infinity. EXTRACTS PREFACE. Mathematics is called in Latin _doctrinalis scientia_. It considers abstract quantity. For that is abstract quantity which we treat by reason alone, separating it by the intellect from the material or from other non-essentials, as for example, equal, unequal, or the like. And there are four sorts of mathematics, namely, arithmetic, geometry, music and astronomy. Arithmetic is the science of numerical quantity in itself. Geometry is the science of magnitude and forms.[241] Music is the science that treats of numbers that are found in sounds. Astronomy is the science that contemplates the courses of the heavenly bodies and their figures, and all the phenomena of the stars. These sciences we shall next describe at a little greater length in order that their significance may be fully shown. [241] Du Breul has _magnitudinis et formarum_; Arevalo, _magnitudinis formarum_. Chapter 1. On the name of the science of arithmetic. 1. Arithmetic is the science of numbers. For the Greeks call number ἀριθμός. The writers of secular literature have decided that it is first among the mathematical sciences since it needs no other science for its own existence. 2. But music and geometry and astronomy, which follow, need its aid in order to be and exist. Chapter 2. On the writers. 1. They say that Pythagoras was the first among the Greeks to write of the science of number, and that it was later described more fully by Nicomachus, whose work Apuleius first, and then Boethius, translated into Latin. Chapter 3. What number is. 1. Number is multitude made up of units. For one is the seed of number but not number. _Nummus_ (coin) gave its name to _numerus_ (number), and from being frequently used originated the word. _Unus_ derives its name from the Greek, for the Greeks call _unus_ ἕνα, likewise _duo_, _tria_, which they call δύο and τρία. 2. _Quattuor_ took its name from a square figure (_figura quadrata_). _Quinque_, however, received its name from one who gave the names to numbers not according to nature but according to whim. _Sex_ and _septem_ come from the Greek.