The Works of Euclid, Vol. 3 (1818) - Page 19

But the rectangle AB $\times$ B$Γ$ is given (lemma), as well as the rectangle B$Γ$ $\times$ BA; the ratio of AB $\times$ B$Γ$ to B$Γ$ $\times$ BA is therefore given. Let the ratio of AB $\times$ B$Γ$ to B$Γ$ $\times$ BA be the same as that of $m$ to $o$,

We shall have AB $\times$ B$Γ$ : B$Γ$ $\times$ BA

$m$ : $o$.

But AB $\times$ B$Γ$ : B$Γ$ $\times$ BA

AB : BA ;

Therefore AB : BA

$m$ : $o$ ;

Therefore AB² : BA²

$m^2$ : $o^2$

$m$ : $p$.

But B$Γ$ $\times$ A$Γ$ : AB²

$m$ : $n$ (A) ;

Therefore B$Γ$ $\times$ A$Γ$ : BA²

$m^2$ : $n \times p$

$m$ : $q$ ;

Therefore 4 B$Γ$ $\times$ A$Γ$ : BA²

4 $m$ : $q$ ;

Therefore 4 B$Γ$ $\times$ A$Γ$ + BA² : BA²

4 $m$ + $q$ : $q$

$m^2$ : $s^2$.

But 4 B$Γ$ $\times$ A$Γ$ + BA² = (B$Γ$ + A$Γ$)² (book II, prop. VIII) ;

Therefore (B$Γ$ + A$Γ$)² : BA²

$m^2$ : $s^2$ ;

Therefore B$Γ$ + A$Γ$ : BA

$m$ : $s$ ;

Therefore B$Γ$ + A$Γ$ + BA, that is to say 2 B$Γ$ : BA

$m + s$ : $s$ ;

Therefore (B) B$Γ$ : BA

$\frac{m + s}{2}$ : $s$

$m^2$ : $t^2$.

But B$Γ$ : BA

B$Γ$ $\times$ BA : BA² ;

Therefore (C) B$Γ$ $\times$ BA : BA²

$m^2$ : $t^2$.

But B$Γ$ $\times$ BA is given; BA² is therefore also given; the straight line BA is thus given; the straight line B$Γ$ is therefore also given. But AB $\times$ B$Γ$ is given, as well as the angle B; the straight line AB is therefore also given; the straight lines AB, B$Γ$ are thus given.

It is evident, from this, that one will have the values of the unknowns AB, B$Γ$ by means of the two proportions B and C. Indeed, substituting, in the proportion C, the given area $a^2$ for the rectangle B$Γ$ $\times$ BA, we shall have BA = $\frac{at}{m}$, and substituting this value of BA in the proportion B, we shall have B$Γ$ = $\frac{am}{t}$.

In the books of Hypsicles, I have removed a multitude of gross errors that were obvious, and which nevertheless were found in the three manuscripts 190, 2342, 2343 *, and in the editions of Basel and Oxford. (See the Variants.)

* These three manuscripts, if one excepts 2342, are defective and full of lacunae.