DEFINITION XIX.

Ὅρος ιθ'.-Τεταγμένη ἀναλογία ἐστὶν, ὅταν ᾖ ὡς ἡγόμενον πρὸς ἑπόμενον οὕτως ἡγούμενον πρὸς τὸ ἑπόμενον, ᾖ δὲ καὶ ὡς ἑπομένον πρὸς ἄλλό τι, ὅντως ἑπόμενον πρὸς ἄλλό τι.

Arranged proportion is when antecedent is to consequent as antecedent to consequent, and it comes to pass that as the [first] consequent is to something else, so is the [second] consequent to something else.

are said to be in ordered proportion. This is commonly called ex æquo ordinate by the commentators.

DEFINITION XX.

Ὅρος κ'.—Τεταραγμένη δὲ ἀναλογία ἐστὶν, ὅταν, τριῶν ὄντων μεγεθῶν καὶ ἄλλων αὐτοῖς ἴσων τὸ πλῆθος, γίνεται, 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, ὅντως ἐν τοῖς δευτέροις μεγέθεσιν ἄλλό τι πρὸς ἡγούμενον.

Disturbed proportion is when there are three magnitudes, and others equal to them in number, and it comes to pass that, as in the first magnitudes, ante

cedent is to consequent, as antecedent to consequent in the second magnitudes; so, in the first magnitudes, as the consequent is to something else, so in the second magnitudes something else is to the antecedent.

the magnitudes a, b, c, and x, y, z, are said to form a disturbed or disordered proportion. This is commonly called ex equo perturbate by the commentators.

Both the ἀναλογία τεταγμένη and the ἀναλογία τεταραγμένη are cases of the λóyos ditoov; because it can be proved that in both cases the first is to the last of the first magnitudes as the first to the last of the second magnitudes. The definitions of both kinds of proportion may readily be extended to any number of magnitudes, compared with an equal number of other magnitudes.

PROPOSITION I.—THEOREM.

Πρότασις α'.--Εὰν ᾧ οποσαοῦν μεγέθη ὑποσωνοῦν μεγεθῶν ἴσων τὸ πλῆθος, ἕκαστον ἑκάστου ἰσάκις πολλαπλάσιον· ὁσαπλάσιόν ἐστιν ἓν τῶν μεγεθῶν ἑνὸς, τοσαυταπλάσια ἔσται καὶ τὰ παντὰ τῶν πάντων.

If there be any number of magnitudes, equimultiples of as many other magnitudes, each of each; whatever multiple one of the magnitudes is of one, the same multiple shall all be of all.

Statement.-Let any number of magnitudes AB and CD be equimultiples of as many others E and F, each

AG B

C

H D

of each. Whatsoever multiple AB is of E, the same multiple is AB and CD together, of E and F together. Construction.-Divide AB into magnitudes each equal to E, viz.: AG and GB; and CD into CH and HD, each equal to F.

E

F

Proof. Because the number of the magnitudes CH and HD is equal to the number of the others AG and GB; and AG is equal to E, and CH to F. Therefore AG and CH together are equal to E and F together; and because GB is equal to E, and HD to F; therefore GB and HD together are equal to E and F together. Wherefore as many magnitudes as AB contains each equal to E, so many do AB and CD together contain each equal to E and F together. Therefore, whatsoever multiple AB is of E, the same multiple is AB and CD together, of E and F together. Therefore, &c. Q. E. D.

Annotation.

Algebraical Statement.—This Proposition of Euclid is proved by the algebraists as follows:

Let a, b, c, &c. denote any magnitudes; and x, y, z, &c., as many other magnitudes; and let

where m is a whole number; then adding these equations together, we find

a + b + c + &c. = m (x + y + z + &c.)

PROPOSITION II.-THEOREM.

Q. E. D.

Πρότασις β'.—Εὰν πρῶτον δευτέρου ισάκις ἢ πολλαπλάσιον · καὶ τρίτον τετάρτου, ὦ δὲ καὶ πέμπτον δευτέρου ἰσάκις πολλαπλάσιον και ἕκτον τετάρτου και συντεθὲν πρῶτον καὶ πέμπτον δευτέρου ἰσάκις ἔσται πολλαπλάσιον καὶ τρίτον καὶ ἕκτον τε τάρτου.

If the first be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then the first and fifth together shall be the same multiple of the second, that the third and sixth are of the fourth.

Statement. Let AB the first, be the same multiple of C the second, that DE the third, is of F the fourth; and BG the fifth, the same multiple of C the second, that EH the sixth, is of F the fourth. I say that AG, the first together with the fifth, is the same multiple of C the second, that DH, the third together with the sixth, is of F the fourth.

is of F, AB con

tains as many magnitudes each equal to C, as DE contains each equal to F. For the same reason, BG contains as many each equal to C, as EH contains each equal to F. Therefore the whole AG contains as many each equal to C, as the whole DH contains each equal to F. Therefore AG is the same multiple of C that DH is of F; that is, AG, the first and fifth together, is the same multiple of the second C, that DH, the third and sixth together, is of the fourth F.

If, therefore, the first be the same multiple, &c. Q. E. D.

Annotation.

Corollary. From this it is plain, that if any number of magnitudes AB, BG, GH, be multiples of another C; and as many DE, EK, KL, be the same multiples of F, each of each; the whole of the first, viz., AH, is the same multiple of C, that the whole of the last, viz., DL, is of F.

Algebraical Statement.-Let a, b, x, y, be the first, second, third, and fourth magnitudes, and let c and z be the fifth and sixth:—

m and n being whole numbers; adding together, we find―

a + c = (m + n) b,

x + z = (m + n) y.

PROPOSITION III.-THEOREM.

Q. E. D.

Πρότασις γ ́.—Εὰν πρῶτον δευτέρου ισάκις ᾖ πολλαπλάσιον καὶ τρίτου τετάρτου, λῆφθῇ δέ ἰσάκις πολλαπλάσια τοῦ πρῶτου καὶ τρίτου· καὶ διΐσου τῶν ληφθέντων, ἑκάτερον ἑκατέρου ισάκις ἔσται πολλαπλάσιον, τὸ μὲν τοῦ δευτέρου, τὸ δὲ τοῦ τετάρτου.

If the first be the same multiple of the second, which the third is of the fourth; and if of the first and third there be taken equimultiples; these are equimultiples, the one of the second, and the other of the fourth.

Statement.—Let A the first, be the same multiple of B the second, that C the third, is of D the fourth; and of A and C let equimultiples EF and GH be taken. EF is the same multiple of B, that GH is of D.

Construction.-Di

vide EF into the mag

E

nitudes EK and KF, each equal to A; and AGH into GL and LH, Beach equal to C.

Proof.-Because the number of the magnitudes EK and KF is equal to the number of the others GL and LH; and A is the same multiple of B, that C is of D, and EK is equal to A, and GL equal to C; therefore EK is the same multiple of B, that GL is of D. For the same reason, KF is the same multiple of B, that LH is of D. Since then the first EK is the same multiple of the second B, which the third GL is of the fourth D, and the fifth KF is the same multiple of the second B, which the sixth LH is of the fourth D. Therefore,