It is an interesting exercise to have a class find, to one decimal place, by measuring as above, the value of √2, √3, √5, and √9, the last being integral. If, as is not usually the case, the class has studied the complex number, the absolute value of √(-6), √(-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.
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.
If P is within the circle, then xx' = yy'; if P is on the circle, then x and y become 0, and 0 · x' = 0 · y' = 0; if P is at P3, then x and y, having passed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; if P is at P4, then y = y' and we have xx' = y2, or x : y = y : x', as stated in the proposition.