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OF the figures which the geometers call σχήματα there are two kinds, “plane” and “solid.” These the Greeks themselves call respectively ἐπίπεδος and στερεός. A “plane” figure is one that has all its lines in two dimensions only, breadth and length; for [p. 93] example, triangles and squares, which are drawn on a flat surface without height. We have a “solid” figure, when its several lines do not produce merely length and breadth in a plane, but are raised so as to produce height also; such are in general the triangular columns which they call “pyramids,” or those which are bounded on all sides by squares, such as the Greeks call κύβοι, 1 and we quadrantalia. For the κύβος is a figure which is square on all its sides, “like the dice,” says Marcus Varro, 2 “with which we play on a gaming-board, for which reason the dice themselves are called κύβοι” Similarly in numbers too the term κύβος is used, when every factor 3 consisting of the same number is equally resolved into the cube number itself, 4 as is the case when three is taken three times and the resulting number itself is then trebled. Pythagoras declared that the cube of the number three controls the course of the moon, since the moon passes through its orbit in twenty-seven days, and the ternio, or “triad,” which the Greeks call τριάς, when cubed makes twenty-seven. Furthermore, our geometers apply the term linea, or “line,” to what the Greeks call γραμμή. This is defined by Marcus Varro as follows: 5 “A line,” says he, “is length without breadth or height.” But Euclid says more tersely, omitting “height” : 6 “A line is μῆκος ἀπλατές, or 'breadthless length.'” ᾿απλατές cannot be expressed in Latin by a single word, unless you should venture to coin the term inlatabile.
1 See Euclid, Elementa I, Definitions, 20, cubs autem est aequaliter aequalis aequaliter, sive qui tribus aequalibus numeris comprehenditur.
2 Fr. p. 350, Bipont.
4 That is, is an equal factor in the cube number.
5 Fr. p. 337, Bipont.
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