Napier's Rules for Right Spherical Triangles

Theorem

Let $\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$ such that the angle $\sphericalangle C$ is a right angle.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

Let the remaining parts of $\triangle ABC$ be arranged according to the interior of this circle, where the symbol $\Box$ denotes a right angle.

Let one of the parts of this circle be called a middle part.

Let the two neighboring parts of the middle part be called adjacent parts.

Let the remaining two parts be called opposite parts.

Then:

The sine of the middle part equals the product of the tangents of the adjacent parts.

The sine of the middle part equals the product of the cosines of the opposite parts.

Tangents

We are given that $\sphericalangle C$ is a right angle.

Let the remaining parts of $\triangle ABC$ be arranged according to the interior of the circle above, where the symbol $\Box$ denotes a right angle.

$\sin a$

$\ds \cos a \cos C$

$=$

$\ds \sin a \cot b - \sin C \cot B$

Four-Parts Formula on $B, a, C, b$

$\ds \leadsto \ \ $

$\ds \cos a \times 0$

$=$

$\ds \sin a \cot b - 1 \times \cot B$

Cosine of Right Angle, Sine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \sin a \cot b$

$=$

$\ds \cot B$

$\ds \leadsto \ \ $

$\ds \sin a$

$=$

$\ds \tan b \cot B$

multiplying both sides by $\tan b = \dfrac 1 {\cot b}$

$\ds \leadsto \ \ $

$\ds \sin a$

$=$

$\ds \tan b \, \map \tan {\Box - B}$

Tangent of Complement equals Cotangent

$\Box$

$\sin b$

$\ds \cos b \cos C$

$=$

$\ds \sin b \cot a - \sin C \cot A$

Four-Parts Formula on $a, C, b, A$

$\ds \leadsto \ \ $

$\ds \cos b \times 0$

$=$

$\ds \sin b \cot a - 1 \times \cot A$

Cosine of Right Angle, Sine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \sin b \cot a$

$=$

$\ds \cot A$

$\ds \leadsto \ \ $

$\ds \sin b$

$=$

$\ds \tan a \cot A$

multiplying both sides by $\tan a = \dfrac 1 {\cot a}$

$\ds \leadsto \ \ $

$\ds \sin b$

$=$

$\ds \tan a \, \map \tan {\Box - A}$

Tangent of Complement equals Cotangent

$\Box$

$\map \sin {\Box - A}$

$\ds \sin a \cos C$

$=$

$\ds \cos c \sin b - \sin c \cos b \cos A$

Analogue Formula for Spherical Law of Cosines for angle $A$

$\ds \leadsto \ \ $

$\ds \sin a \times 0$

$=$

$\ds \cos c \sin b - \sin c \cos b \cos A$

Cosine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \sin c \cos b \cos A$

$=$

$\ds \cos c \sin b$

$\ds \leadsto \ \ $

$\ds \cos A$

$=$

$\ds \cot c \tan b$

dividing both sides by $\sin c \cos b$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - A}$

$=$

$\ds \map \tan {\Box - c} \tan b$

Sine of Complement equals Cosine, Tangent of Complement equals Cotangent

$\Box$

$\map \sin {\Box - c}$

$\ds \cos C$

$=$

$\ds -\cos A \cos B + \sin A \sin B \cos c$

Spherical Law of Cosines for Angles for angle $C$

$\ds \leadsto \ \ $

$\ds 0$

$=$

$\ds -\cos A \cos B + \sin A \sin B \cos c$

Cosine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \sin A \sin B \cos c$

$=$

$\ds \cos A \cos B$

$\ds \leadsto \ \ $

$\ds \cos c$

$=$

$\ds \cot A \cot B$

dividing both sides by $\sin A \sin B$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - c}$

$=$

$\ds \map \tan {\Box - A} \, \map \tan {\Box - B}$

Sine of Complement equals Cosine, Tangent of Complement equals Cotangent

$\Box$

$\map \sin {\Box - B}$

$\ds \sin b \cos C$

$=$

$\ds \cos c \sin a - \sin c \cos a \cos B$

Analogue Formula for Spherical Law of Cosines for angle $B$

$\ds \leadsto \ \ $

$\ds \sin b \times 0$

$=$

$\ds \cos c \sin a - \sin c \cos a \cos B$

Cosine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \sin c \cos a \cos B$

$=$

$\ds \cos c \sin a$

$\ds \leadsto \ \ $

$\ds \cos B$

$=$

$\ds \cot c \tan a$

dividing both sides by $\sin c \cos a$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - B}$

$=$

$\ds \map \tan {\Box - c} \tan a$

Sine of Complement equals Cosine, Tangent of Complement equals Cotangent

$\blacksquare$

Cosines

We are given that $\sphericalangle C$ is a right angle.

Let the remaining parts of $\triangle ABC$ be arranged according to the interior of the circle above, where the symbol $\Box$ denotes a right angle.

$\sin a$

$\ds \dfrac {\sin a} {\sin A}$

$=$

$\ds \dfrac {\sin c} {\sin C}$

Spherical Law of Sines for side $a$

$\ds \leadsto \ \ $

$\ds \dfrac {\sin a} {\sin A}$

$=$

$\ds \dfrac {\sin c} 1$

Sine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \sin a$

$=$

$\ds \sin A \sin c$

$\ds \leadsto \ \ $

$\ds \sin a$

$=$

$\ds \map \cos {\Box - A} \map \cos {\Box - c}$

Cosine of Complement equals Sine

$\Box$

$\sin b$

$\ds \dfrac {\sin b} {\sin B}$

$=$

$\ds \dfrac {\sin c} {\sin C}$

Spherical Law of Sines for side $b$

$\ds \leadsto \ \ $

$\ds \dfrac {\sin b} {\sin B}$

$=$

$\ds \dfrac {\sin c} 1$

Sine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \sin b$

$=$

$\ds \sin B \sin c$

$\ds \leadsto \ \ $

$\ds \sin b$

$=$

$\ds \map \cos {\Box - B} \map \cos {\Box - c}$

Cosine of Complement equals Sine

$\Box$

$\map \sin {\Box - A}$

$\ds \cos A$

$=$

$\ds -\cos B \cos C + \sin B \sin C \cos a$

Spherical Law of Cosines for Angles for angle $A$

$\ds $

$=$

$\ds -\cos B \times 0 + \sin B \times 1 \times \cos a$

Cosine of Right Angle and Sine of Right Angle as $C = \Box$

$\ds $

$=$

$\ds \sin B \cos a$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - A}$

$=$

$\ds \map \cos {\Box - B} \cos a$

Sine of Complement equals Cosine, Cosine of Complement equals Sine

$\Box$

$\map \sin {\Box - c}$

$\ds \cos c$

$=$

$\ds \cos a \cos b + \sin a \sin b \cos C$

Spherical Law of Cosines for side $c$

$\ds $

$=$

$\ds \cos a \cos b$

Cosine of Right Angle as $C = \Box$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - c}$

$=$

$\ds \cos a \cos b$

Sine of Complement equals Cosine

$\Box$

$\map \sin {\Box - B}$

$\ds \cos B$

$=$

$\ds -\cos A \cos C + \sin A \sin C \cos b$

Spherical Law of Cosines for Angles for angle $B$

$\ds $

$=$

$\ds -\cos A \times 0 + \sin A \times 1 \cos b$

Cosine of Right Angle as $C = \Box$

$\ds $

$=$

$\ds \sin A \cos b$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - B}$

$=$

$\ds \map \cos {\Box - A} \cos c$

Sine of Complement equals Cosine, Cosine of Complement equals Sine

$\blacksquare$

Also see

Source of Name

This entry was named for John Napier.

Sources

1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $10$. Right-angled and quadrantal triangles.
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Napier, John (1550-1617)
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Napier's rules of circular parts
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Napier's rules of circular parts