Commentary on Book IV Definitions

Back to Book IV Definitions.

DEFINITIONS 1—7.

I append, as usual, the Greek text of the definitions.

  1. Σχῆμα εὐθύγραμμον εἰς σχῆμα εὐθύγραμμον ἐγγράφεσθαι λέγεται, ὅταν ἑκάστη τῶν τοῦ ἐγγραφομένου σχήματος γωνιῶν ἑκάστης πλευρᾶς τοῦ, εἰς ὃ ἐγγράφεται, ἅπτηται.

  1. Σχῆμα δὲ ὁμοίως περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἑκάστης γωνίας τοῦ, περὶ ὃ περιγράφεται, ἅπτηται.

  2. Σχῆμα εὐθύγραμμον εἰς κύκλον ἐγγράφεσθαι λέγεται, ὅταν ἑκάστη γωνία τοῦ ἐγγραφομένου ἅπτηται τῆς τοῦ κύκλου περιφερείας.

  3. Σχῆμα δὲ εὐθύγραμμον περὶ κύκλον περιγράφεσθαι λέγεται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἐφάπτηται τῆς τοῦ κύκλου περιφερείας.

  4. Κύκλος δὲ εἰς σχῆμα ὁμοίως ἐγγράφεσθαι λέγεται, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης πλευρᾶς τοῦ, εἰς ὃ ἐγγράφεται, ἅπτηται.

  5. Κύκλος δὲ περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης γωνίας τοῦ, περὶ ὃ περιγράφεται, ἅπτηται.

  6. Εὐθεῖα εἰς κύκλον ἐναρμόζεσθαι λέγεται, ὅταν τὰ πέρατα αὐτῆς ἐπὶ τῆς περιφερείας ᾖ τοῦ κύκλου.

In the first two definitions an English translation, if it is to be clear, must depart slightly from the exact words used in the Greek, where “each side” of one figure is said to pass through “each angle” of another, or “each angle” (i.e. angular point) of one lies on “each side” of another (ἑκάστη πλευρά, ἑκάστη γωνία).

It is also necessary, in the five definitions 1, 2, 3, 5 and 6, to translate the same Greek word ἅπτηται in three different ways. It was observed on III. Def. 2 that the usual meaning of ἅπτεσθαι in Euclid is to meet, in contradistinction to ἐφάπτεσθαι, which means to touch. Exceptionally, as in Def. 5, ἅπτεσθαι has the meaning of touch. But two new meanings of the word appear, the first being to lie on, as in Deff. 1 and 3, the second to pass through, as in Deff. 2 and 6; “each angle” lies on (ἅπτεται) a side or on a circle, and “each side,” or a circle, passes through (ἅπτεται) an angle or “each angle.” The first meaning of lying on is exemplified in the phrase of Pappus ἅψεται τὸ σημεῖον θέσει δεδομένης εὐθείας, “will lie on a straight line given in position”; the meaning of passing through seems to be much rarer (I have not seen it in Archimedes or Pappus), but, as pointed out on III. Def. 2, Aristotle uses the compound ἐφάπτεσθαι in this sense.

Simson proposed to read ἐφάπτηται in the case (Def. 5) where ἅπτηται means touches. He made the like suggestion as regards the Greek text of III. 11, 12, 13, 18, 19; in the first four of these cases there seems to be MS. authority for the compound verb, and in the fifth Heiberg adopts Simson’s correction.