Using Lagrange's four-square theorem and geometry to solve for square root.

SquareRoot.java

public class SquareRoot {
	public static double sqrt(int n){
		int[] v = vector(n);
		return magnitude(v);
	}
	public static double magnitude(int[] v){
		double a=v[1],b=v[0];
		if(v.length>2){
			int[] u = new int[v.length-1];
			System.arraycopy(v, 1, u, 0, v.length-1);
			a = magnitude(u);
		}
		b = v[0];
		return a/Math.sin(Math.atan(a/b));
	}
	public static int[] vector(int n){
		int[] r = new int[4];
		for(r[0]=0;r[0]*r[0]<=n;r[0]++){
			for(r[1]=r[0];r[0]*r[0]+r[1]*r[1]<=n;r[1]++){
				for(r[2]=r[1];r[0]*r[0]+r[1]*r[1]+r[2]*r[2]<=n;r[2]++){
					for(r[3]=r[2];r[0]*r[0]+r[1]*r[1]+r[2]*r[2]+r[3]*r[3]<=n;r[3]++){
						if(r[0]*r[0]+r[1]*r[1]+r[2]*r[2]+r[3]*r[3]==n){
							return r;
						}
					}
				}
			}
		}
		return new int[4];
	}
}