Finding a Square Root Geometric Algebra

How to Begin

In order to see how this was done by Euclid, you will need two tools – a compass and a ruler.

Using your ruler draw a straight line.

Choose an arbitrary point P on your line.

P

Next, using your ruler, add points Q and R such that the line segment RP has length 1 and line segment PQ has length S, where S is the integer that you want to find the square root of.

Bisect the line segment RQ, thereby finding it’s center. Call the center point C.

Using the compass, draw a semi-circle of radius equal to the length of RC. This semi-circle will go from point R to point Q.

R P Q

1 S

R P C Q

Raise a perpendicular line from the original point P to intersect the semicircle.

And then…..

R P Q

The length of the vertical line from point P to the semi-circle is the square root of the number S.

S1€

S

How to find the square rootof a rational number that is NOT an integer

This procedure can be extended to find the roots of rational fractions.

Say we want to find the square root of S = m/n.

We redefine the area to the left of point P as having n subunits.

The area to the right will still be m/n = S in length.

Proceed as usual.

R P Q

1 S = m/n

Why Does This Work? S

R P Q

Angle SPQ is a right angle because it lies on the circumference of a circle with diameter RQ.

Angle SPQ and SPR are similar and share a side SP.

S1€

S

S

R P Q

RP (=1) is to SP as SP is to RQ (=S).

Therefore SP x SP = S which implies that the length of line SP = square root of S.

S1€

S

Background on the Method

This method was first outlined in Euclid’s Elements Book VI, proposition 13.

Book 13 constructs the five regular Platonic solids inscribed in spheres, calculates the ratio of their edges to the radius of the sphere, and proves that there are no further regular solids.

Proposition 13 finds a mean proportional to two given straight lines. The square root calculation is a byproduct of this proportion.

Using geometric algebra, can you find….

The square root of 9? The square root of 10? The square root of 5/2? The square root of 2/3? What problem

do you encounter here?

Post your work to the DB for this topic.