Draw a straight line. Choose a point O on the line. Using the compass as dividers, mark off a unit length to the left of the point and P unit lengths to the right, where P is the number whose square root is to be found. Label the endpoints L and R respectively.
If the number A is not an integer but is rational, we have A=M/N and so mark off N sub-units to the left and M sub-units to the right of the chosen point. Example : 1.5 =3/2. Then just redefine the distance marked off to the left to be the unit distance and the distance marked off to the right will be the rational number A (=1.5 in our example).
Bisect the line segment between these end-points, finding the centre of the line-segment. You do that by drawing arcs of radius equal to the line-segment-length about each end-point, then drawing a straight vertical line between the two arc-intersections. It intersects the horizontal line-segment in its centre.
Using this centre, draw a semicircle from one end of the line-segment to the other.
Raise a vertical from the original point O to intersect the semicircle at S (as explained
above in the paragraph about bisection).
The length of this OS line is the square root of the number A.
This method can be extended to irrational numbers like root(2) by first constructing a line of such irrational length geometrically, since such an irrational line cannot be counted off a priori with dividers.
Congratulations, you have now calculated a square root geometrically using only Euclid's tools, a compass and a straightedge!
Proof : The angle LSR is a right angle because it is on the circumference of the circle with diameter LR. Therefore the triangles LOS and ROS are similar ( have the same internal angles) and share the side OS. thus LO (=1) is to OS as OS is to OR (=A). Thus OS*OS=A. thus OS = root(A).
Just for the sake of completeness, the two constructions shown below demonstrate how to
do multiplication and division respectively, just using geometry.
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