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Euclid Book 5 Definitions
Statement
A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
Original statement
μέρος ἐστὶ μέγϵθος μϵγέθους τὸ ἔλασσον τοῦ μϵίζονος, ὅταν καταμϵτρῇ τὸ μϵῖζον.
Statement
The greater is a multiple of the less when it is measured by the less.
Original statement
πολλαπλάσιον δὲ τὸ μϵῖζον τοῦ ἐλάττονος, ὅταν καταμϵτρῆται ὑπὸ τοῦ ἐλάττονος.
Statement
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
Original statement
λόγος ἐστὶ δύο μϵγϵθῶν ὁμογϵνῶν ἡ κατὰ πηλικότητά ποια σχέσις.
Statement
Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
Original statement
λόγον ἔχϵιν πρὸς ἄλληλα μϵγέθη λέγϵται, ἃ δύναται πολλαπλασιαζόμϵνα ἀλλήλων ὑπϵρέχϵιν.
Statement
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Original statement
ἐν τῷ αὐτῷ λόγῳ μϵγέθη λέγϵται ϵἶναι πρῶτον πρὸς δϵύτϵρον καὶ τρίτον πρὸς τέταρτον, ὅταν τὰ τοῦ πρώτου καὶ τρίτου ἰσάκις πολλαπλάσια τῶν τοῦ δϵυτέρου καὶ τϵτάρτου ἰσάκις πολλαπλασίων καθ᾽ ὁποιονοῦν πολλαπλασιασμὸν ἑκάτϵρον ἑκατέρου ἢ ἅμα ὑπϵρέχῃ ἢ ἅμα ἴσα ᾖ ἢ ἅμα ἐλλϵίπῃ ληϕθέντα κατάλληλα.
Statement
Let magnitudes which have the same ratio be called proportional.
Original statement
τὰ δὲ τὸν αὐτὸν ἔχοντα λόγον μϵγέθη ἀνάλογον καλϵίσθω.
Statement
When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.
Original statement
ὅταν δὲ τῶν ἰσάκις πολλαπλασίων τὸ μὲν τοῦ πρώτου πολλαπλάσιον ὑπϵρέχῃ τοῦ τοῦ δϵυτέρου πολλαπλασίου, τὸ δὲ τοῦ τρίτου πολλαπλάσιον μὴ ὑπϵρέχῃ τοῦ τοῦ τϵτάρτου πολλαπλασίου, τότϵ τὸ πρῶτον πρὸς τὸ δϵύτϵρον μϵίζονα λόγον ἔχϵιν λέγϵται, ἤπϵρ τὸ τρίτον πρὸς τὸ τέταρτον.
Statement
A proportion in three terms is the least possible.
Original statement
ἀναλογία δὲ ἐν τρισὶν ὅροις ἐλαχίστη ἐστίν.
Statement
When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.
Original statement
ὅταν δὲ τρία μϵγέθη ἀνάλογον ᾖ, τὸ πρῶτον πρὸς τὸ τρίτον διπλασίονα λόγον ἔχϵιν λέγϵται ἤπϵρ πρὸς τὸ δϵύτϵρον.
Statement
When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.
Original statement
ὅταν δὲ τέσσαρα μϵγέθη ἀνάλογον ᾖ, τὸ πρῶτον πρὸς τὸ τέταρτον τριπλασίονα λόγον ἔχϵιν λέγϵται ἤπϵρ πρὸς τὸ δϵύτϵρον, καὶ ἀϵὶ ἑξῆς ὁμοίως, ὡς ἂν ἡ ἀναλογία ὑπάρχῃ.
Statement
The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.
Original statement
ὁμόλογα μϵγέθη λέγϵται τὰ μὲν ἡγούμϵνα τοῖς ἡγουμένοις τὰ δὲ ἑπόμϵνα τοῖς ἑπομένοις.
Statement
Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
Original statement
ἐναλλὰξ λόγος ἐστὶ λῆψις τοῦ ἡγουμένου πρὸς τὸ ἡγούμϵνον καὶ τοῦ ἑπομένου πρὸς τὸ ἑπόμϵνον.
Statement
Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
Original statement
ἀνάπαλιν λόγος ἐστὶ λῆψις τοῦ ἑπομένου ὡς ἡγουμένου πρὸς τὸ ἡγούμϵνον ὡς ἑπόμϵνον.
Statement
Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.
Original statement
σύνθϵσις λόγου ἐστὶ λῆψις τοῦ ἡγουμένου μϵτὰ τοῦ ἑπομένου ὡς ἑνὸς πρὸς αὐτὸ τὸ ἑπόμϵνον.
Statement
Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
Original statement
διαίρϵσις λόγου ἐστὶ λῆψις τῆς ὑπϵροχῆς, ᾗ ὑπϵρέχϵι τὸ ἡγούμϵνον τοῦ ἑπομένου, πρὸς αὐτὸ τὸ ἑπόμϵνον.
Statement
Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
Original statement
ἀναστροϕὴ λόγου ἐστὶ λῆψις τοῦ ἡγουμένου πρὸς τὴν ὑπϵροχήν, ᾗ ὑπϵρέχϵι τὸ ἡγούμϵνον τοῦ ἑπομένου.
Statement
A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes; Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.
Original statement
δι' ἴσου λόγος ἐστὶ πλϵιόνων ὄντων μϵγϵθῶν καὶ ἄλλων αὐτοῖς ἴσων τὸ πλῆθος σύνδυο λαμβανομένων καὶ ἐν τῷ αὐτῷ λόγῳ, ὅταν ᾖ ὡς ἐν τοῖς πρώτοις μϵγέθϵσι τὸ πρῶτον πρὸς τὸ ἔσχατον, οὕτως ἐν τοῖς δϵυτέροις μϵγέθϵσι τὸ πρῶτον πρὸς τὸ ἔσχατον: ἢ ἄλλως: λῆψις τῶν ἄκρων καθ᾽ ὑπϵξαίρϵσιν τῶν μέσων.
Statement
A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.
Original statement
τϵταραγμένη δὲ ἀναλογία ἐστίν, ὅταν τριῶν ὄντων μϵγϵθῶν καὶ ἄλλων αὐτοῖς ἴσων τὸ πλῆθος γίνηται ὡς μὲν ἐν τοῖς πρώτοις μϵγέθϵσιν ἡγούμϵνον πρὸς ἑπόμϵνον, οὕτως ἐν τοῖς δϵυτέροις μϵγέθϵσιν ἡγούμϵνον πρὸς ἑπόμϵνον, ὡς δὲ ἐν τοῖς πρώτοις μϵγέθϵσιν ἑπόμϵνον πρὸς ἄλλο τι, οὕτως ἐν τοῖς δϵυτέροις ἄλλο τι πρὸς ἡγούμϵνον.