Lateral Magnification in Thin Lens

For an object $OO'$ of height $h_1$ on the axis, the ray $OP$ through the lens's own pole $P$ travels completely undeviated, creating the similar triangles $\triangle POO'\sim\triangle PII'$ with the inverted real image $II'$ of height $h_2$. This gives $II'/OO'=PI/PO$, and applying the sign convention ($h_2=-II'$, $h_1=OO'$, $PI=v$, $PO=-u$) gives the lateral magnification $\boxed{m=h_2/h_1=v/u}$. Combined with the lens equation, this can equivalently be written as $m=\dfrac{f}{f+u}$ or $m=\dfrac{f-v}{f}$. A negative $m$ means a real, inverted image; a positive $m$ means a virtual, erect image -- for a converging lens the image is inverted when the object lies beyond the focus, and erect, virtual and magnified (the simple-microscope arrangement) …