The Works of Euclid, Vol. 3

But the rectangle AB $\times$ B$Γ$ is given (lemma), and likewise the rectangle B$Γ$ $\times$ B$Δ$; therefore the ratio of AB $\times$ B$Γ$ to B$Γ$ $\times$ B$Δ$ is given. Let, however, the ratio of AB $\times$ B$Γ$ to B$Γ$ $\times$ B$Δ$ be the same as the ratio of m to o;

Therefore AB $\times$ B$Γ$ : B$Γ$ $\times$ B$Δ$

AB : B$Δ$;
Therefore AB : B$Δ$ m² : o² m : n (A);
Therefore B$Γ \times ΔΓ$ : B$Δ^2$ m : q;
Therefore 4 B$Γ \times ΔΓ$ : B$Δ^2$ 4 m + q : q m² : s²;
Therefore B$Γ + ΔΓ$ : B$Δ$ m + s : s;
Therefore (B) B$Γ$ : B$Δ$ m² : t².
But B$Γ$ : B$Δ$ m² : t².

But B$Γ \times$ B$Δ$ itself is given; therefore B$Δ^2$ itself is given; the straight line B$Δ$ is therefore given; wherefore B$Γ$ itself is also given. But AB $\times$ B$Γ$ itself is given, and likewise angle B; wherefore AB itself is also given; the straight lines AB, B$Γ$ are therefore given.

From this it is manifest that we shall have the values of the unknown straight lines AB, B$Γ$ by means of the two proportions B and C. Indeed, if in proportion C we substitute the given surface a² for the rectangle B$Γ \times$ B$Δ$, we shall have B$Δ = \frac{at}{m}$, and if we substitute this value of B$Δ$ into proportion B, we shall have B$Γ = \frac{am}{t}$.

From the books of Hypsicles, I have removed several very gross errors, most evident at a single glance, which are nevertheless found in the three codices 190, 2342, 2345 *, and in the editions of Basel and Oxford. (See the various readings.)

---

* These three codices, with the exception of codex 2342, are defective and teeming with lacunae.