To this important proposition there is one corollary of particular interest, namely, _The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter_. By means of this corollary we can easily construct a line whose numerical value is the square root of any number we please. Thus we may make _AD_ = 2 in., _DB_ = 3 in., and erect _DC_ [perp] to _AB_. Then the length of _DC_ will be [sqrt]6 in., and we may find [sqrt]6 approximately by measuring _DC_. [Illustration] Furthermore, if we introduce negative magnitudes into geometry, and let _DB_ = +3 and _DA_ = -2, then _DC_ will equal [sqrt](-6). In other words, we have a justification for representing imaginary quantities by lines perpendicular to the line on which we represent real quantities, as is done in the graphic treatment of imaginaries in algebra. It is an interesting exercise to have a class find, to one decimal place, by measuring as above, the value of [sqrt]2, [sqrt]3, [sqrt]5, and [sqrt]9, the last being integral. If, as is not usually the case, the class has studied the complex number, the absolute value of [sqrt](-6), [sqrt](-7), ..., may be found in the same way. A practical illustration of the value of the above theorem is seen in a method for finding distances that is frequently described in early printed books. It seems to have come from the Roman surveyors. [Illustration] If a carpenter's square is put on top of an upright stick, as here shown, and an observer sights along the arms to a distant point _B_ and a point _A_ near the stick, then the two triangles are similar. Hence _AD_ : _DC_ = _DC_ : _DB_. Hence, if _AD_ and _DC_ are measured, _DB_ can be found. The experiment is an interesting and instructive one for a class, especially as the square can easily be made out of heavy pasteboard. THEOREM. _If two chords intersect within a circle, the product of the segments of the one is equal to the product of the segments of the other._ THEOREM. _If from a point without a circle a secant and a tangent are drawn, the tangent is the mean proportional between the secant and its external segment._ COROLLARY. _If from a point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn._ These two propositions and the corollary are all parts of one general proposition: _If through a point a line is drawn cutting a circle, the product of the segments of the line is constant_. [Illustration]