[Folio 0170 | p. 164]

PROPOSITION 16.

If four magnitudes be proportional, they will also be proportional alternately.

Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D ;[1]

I say that they will also be so alternately, that is, as A is to C, so is B to D.

For of A, B let equimultiples E, F be taken, and of C, D other, chance, equimultiples G, H.

Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [V. 15]

therefore, as A is to B, so is E to F.

But as A is to B, so is C to D ;

[V. 11]therefore also, as C is to D, so is E to F.

Again, since G, H are equimultiples of C, D,

[V. 15]therefore, as C is to D, so is G to H.

But, as C is to D, so is E to F ;

[V. 11]therefore also, as E is to F, so is G to H.

But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth ; if equal, equal ; and if less, less. [V. 14]

Therefore, if E is in excess of G, F is also in excess of H, if equal, equal, and if less, less.

[Folio 0171 | p. 165]

Now E, F are equimultiples of A, B, and G, H other, chance, equimultiples of C, D ;

[V. Def. 5]therefore, as A is to C, so is B to D.

Therefore etc. Q. E. D.

[1]

“Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D.” In a number of expressions like this it is absolutely necessary, when translating into English, to interpolate words which are not in the Greek. Thus the Greek here is : Ἔστω τέσσαρα μεγέθη ἀνάλογον τὰ Α, Β, Γ, Δ, ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς τὸ Δ, literally “Let $A$, $B$, $C$, $D$ be four proportional magnitudes, as $A$ to $B$, so $C$ to $D$.” The same remark applies to the corresponding expressions in the next propositions, V. 17, 18, and to other forms of expression in V. 20—23 and later propositions : e.g. in V. 20 we have a phrase meaning literally “Let there be magnitudes…which taken two and two are in the same ratio, as $A$ to $B$, so $D$ to $E$,” etc. : in V. 21 “(magnitudes)…which taken two and two are in the same ratio, and let the proportion of them be perturbed, as $A$ to $B$, so $E$ to $F$,” etc. In all such cases (where the Greek is so terse as to be almost ungrammatical) I shall insert the words necessary in English, without further remark.

Dependency Graph

Required for

V.19, V.23, V.25, VI.4, VI.5, VI.6, VI.7, VI.8, VI.13, VI.18, VI.19, VI.20, VI.22, VI.24, VI.25, VI.26, VI.27, VI.28, VI.29, VI.30, VI.31, VI.32, X.6, X.7, X.9, X.10, X.11, X.12, X.13, X.14, X.17, X.18, X.19, X.20, X.21, X.22, X.23, X.24, X.25, X.26, X.27, X.28, X.29, X.30, X.31, X.32, X.33, X.34, X.35, X.36, X.37, X.38, X.39, X.40, X.41, X.42, X.43, X.44, X.45, X.46, X.47, X.48, X.49, X.50, X.51, X.52, X.53, X.54, X.55, X.56, X.57, X.58, X.59, X.60, X.61, X.62, X.63, X.64, X.65, X.66, X.67, X.68, X.69, X.70, X.71, X.72, X.73, X.75, X.76, X.77, X.78, X.79, X.80, X.81, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.92, X.93, X.94, X.95, X.96, X.97, X.98, X.99, X.100, X.101, X.102, X.103, X.104, X.105, X.106, X.107, X.108, X.109, X.110, X.111, X.112, X.113, X.114, X.115, XI.23, XII.1, XII.2, XII.4, XII.5, XII.6, XII.7, XII.8, XII.10, XII.11, XII.12, XII.13, XII.14, XII.15, XII.17, XII.18, XIII.6, XIII.8, XIII.9, XIII.10, XIII.11, XIII.13, XIII.16, XIII.17, XIII.18

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See Commentary on Proposition V.16.