Turner's four axioms of right triangles.

     Reverend Turner states four axioms, or principles, that can be used to solve right-angle triangles.  A combination of these theorems can be applied to solve any right triangle in which three or more parts of the triangle are already known.*  Although written in prose, Turner's text can be deciphered into the modern mathematical notation seen below [1]. 

The four axioms are as follows:

1.   A number known as the Natural Radius (NR) can be used to find a leg of the triangle (A) given the corresponding angle (a), the angle's complement (b), and the hypotenuse (C).   First, find the Natural Radius. Then find the length of the leg (A) of the triangle.

                                                                                              

2.   Known today as the Pythagorean Theorem, this can be used to find the hypotenuse (C) given the two legs, (A) and (B).  In words, this means that the sum of the squares of the shorter legs of the right triangle is equal to the square of the longer side.

   

3.   A reversed Pythagorean Theorem to find leg (A) given leg (B) and hypotenuse (C).

   

4.   Turner uses this formula to find an angle (a) when given its corresponding leg (A), leg (B), and the hypotenuse (C).

   

These four axioms are the basis for Turner's whole system of solving triangles.  In the next sections, Turner lays out a series of example cases showing the reader how they are used.

*Turner mentions that at least one of the three known components must be the length of a side of the triangle.  When all three angles of a right triangle are known and none of the side lengths are known, the lengths of the sides cannot be determined because they could vary.