John Napier (1550–1617) discovered a way to reduce 10 equations in spherical trig down to 2 equations and to make them easier to remember.

Draw a right triangle on a sphere and label the sides *a*, *b*, and *c* where *c* is the hypotenuse. Let A be the angle opposite side *a*, B the angle opposite side *b*, and C the right angle opposite the hypotenuse *c*.

There are 10 equations relating the sides and angles of the triangle:

sin *a* = sin *A* sin *c* = tan *b* cot *B*

sin *b* = sin *B* sin *c* = tan *a* cot *A*

cos *A* = cos *a* sin B = tan *b* cot *c*

cos *B* = cos *b* sin A = tan *a* cot *c*

cos *c* = cot *A* cot *B* = cos *a* cos *b*

Here’s how Napier reduced these equations to a more memorable form. Arrange the parts of the triangle in a circle as below.

Then Napier has two rules:

- The sine of a part is equal to the product of the tangents of the two adjacent parts.
- The sine of a part is equal to the product of the cosines of the two opposite parts.

For example, if we start with *a*, the first rule says sin *a* = cot *B* tan *b*. (The tangent of the complementary angle to *B* is the cotangent of *B*.) Similarly, the second rule says that sin *a* = sin *c* sin *A*. (The cosine of the complementary angle is just the sine.)

For a more algebraic take on Napier’s rules, write the parts of the triangle as

(*p*_{1}, *p*_{2}, *p*_{3}, *p*_{4}, *p*_{5}) = (*a* , *b*, co-*A*, co-*c*, co-*B*).

Then the equations above can be reduced to

sin* p*_{i} = tan *p*_{i-1} tan *p*_{i+1} = cos *p*_{i+2} cos *p*_{i+3}

where the addition and subtraction in the subscripts is carried out mod 5. This is just using subscripts to describe the adjacent and opposite parts in Napier’s diagram.

**Source**: Heavenly Mathematics

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Eric: Yes.