Stony Brook University MAT 336 History of Mathematics
Spherical Trigonometry and Navigation
1. Spherical Trigonometry deals with spherical triangles. On a sphere,
the role of straight lines is played by great circles. Each of these
is the intersection of the sphere with a plane passing through its
center. Line segments become great circle arcs. If the sphere has
radius R, so will all its great circles; so the length of an arc
is exactly R times the radian measure of the central
angle it intersects (or R times (π/180) times the
measure of that angle in degrees).
In navigation, for most purposes the
surface of the Earth can be considered a sphere. The distance between
points X and Y on the Earth is the length of the
great-circle arc between them. This length can be expressed essentially
as an angle through the convention that 1 nautical mile
is 1 minute (1') of arc (i.e. 1/60 of one degree) along a great circle.
For example, the distance from the North Pole
to a point on the Equator is 1/4 of a great circle: 90o
or 90x60=5400' which gives 5400 nautical miles. The radius of the
Earth is built into the conversion factor between nautical miles and
kilometers: 1 nautical mile = 1.852 km.
The upshot is that calculating the distance
between two points amounts to calculating the central angle they
determine on the great circle that passes through both of them.
This is where spherical trigonometry becomes useful.
2. "In book XI of [Ptolemy's] Almagest the principles of spherical
trigonometry are stated in the form of a few simple and useful lemmas;
plane trigonometry does not receive systematic treatment..." . This
is because of the great importance of spherical trigonometry for
astronomy. We have seen how Aryabhata (456-550) tabulated sines. The
next great progress came with Al-Battani (or, Albategnius; c.868-929),
the greatest Islamic mathematician and astronomer
of his time. He used the function sinvers defined by
sinvers(α) = 1 - sin(π/2 - α) essentially equivalent
to our cosine, in astronomical calculations; but despite insinuations to
the contrary (e.g. in ), he did not formulate the law of cosines
for a spherical triangle. The first formulation we know,
still in terms of sinvers, is due to
Regiomontanus (1464). Details from  are summarized in
The True History of the Law
The the spherical law of cosines
states, for a spherical triangle with surface angles
A, B, C opposite sides corresponding to central angles
a, b, c, that
cosa = cosb cosc + sinb sinc cos A.
3. This law has the planar law of cosines as limit when
a, b, and c tend to zero. Use
the approximations cosa = 1 - (1/2)a2, etc.,
and sinb = b, etc. valid for small a, b, c, and
discard the higher-order term (1/4)b2c2.
4. When we locate a point X
on the Earth by giving its latitude and longitude,
we are giving angular coordinates.
Latitude is the size of
the central angle, along a meridian (the great circle through X
and the North Pole P), between X and the equator. The latitude
is specified as North (N) for points above the equator, and South
(S) for points below, and runs from 0o to 90o.
New York City is at latitude 40o47'N, Moscow is at
latitude 55o45'N and
Buenos Aires is at latitude 34o35'S. (Sometimes
latitude is counted negative for points in the Southern Hemisphere).
is the size of the surface angle, at the North Pole, between the
"Prime Meridian" (the meridian through Greenwich, England) and the
meridian through X. It is measured East (E) or West (W) and
runs in each direction from 0o to 180o.
New York City is at longitude 73o58'W, while Moscow
is at 37o42'E and Beijing
is at 113o20'E.
5. Suppose we want to calculate the air distance from New York
to Moscow. We know their coordinates.
New York: Latitude l1 = 40o47'N,
Longitude L1 = 73o58'W
Moscow: Latitude l2 = 55o45'N,
Longitude L2 = 37o42'E. We can mark
these coordinates on a globe:
New York (N), Moscow (M) and the North Pole (P)
are the vertices of a spherical triangle; we label the sides (central
angles) oppsite the vertices by n, m, p . We know side m:
m = 90o - l1 = 49o13'.
Similarly side n is
n = 90o - l2 = 34o15'.
We also know the face angle at P: it is the sum of
the two longitudes, since one is East and the other is West:
P = L1 + L2 = 111o40'.
In this context, the law of cosines gives:
cosp = cosm cosn + sinm sinn cos P.
Using a calculator, we find
cosp = 0.6532x0.8266 + 0.7572x0.5629x(-0.3692) = 0.3826, and
arccos(0.3826) = 67.51o=4050.4'. This gives the great-circle
New York to Moscow as 4050 nautical miles, or 7501km.
 Ernest William Hobson, Entry "Trigonometry" in the Encyclopedia
Brittanica, 11th Edition (1910-1911).
- Calculate the great-circle distance from New York to London
(Latitude = 51o32'N, Longitude = 0o5'W). Note
that in this case the longitudes are both West and must be subtracted.
- Calculate the great-circle distance from Anchorage, Alaska
(Latitude = 61o10'N, Longitude = 150o1'W)
- What is the maximum latitude along the New York-Moscow great-circle
route? (Hint: it occurs at the vertex V of the route, the point
where the route makes a right angle with a meridian).
You will need to solve for surface angle M by a second application
of the law of cosines; then use the
spherical law of sines
sinA/sina = sinB/sinb = sinC/sinc
to solve the rectangular spherical triangle MPV .
-  Find the initial course and the distance for a voyage along a great
circle from Los Angeles (l = 34o03'N, L =
118o15'W) to Aukland (l = 41o18'S, L =
174o51'E). [The initial course is "the angle of departure,
measured from the north around through the east [i.e. measured
clockwise] from 0o to
Answer: Course =
224o 8' 48", distance = 5832 miles.
-  Find the distance by great circle from New York
(l1 = 40o40'N, L1 =
4h 55m 54"W) to Cape of Good Hope
(l2 = 33o56'S, L2 =
1h 13m 55"E). [Longitude is given here in
hours, minutes and seconds of time. Because of the rotation of the
Earth with respect to the Sun, 1o of longitude corresponds
to a 4m time difference.] Answer: 6779.9 miles.
 Kells, Kearn and Bland, Plane and Spherical Trigonometry,
McGraw-Hill, New York, 1940. This textbook was used at the United
States Naval Academy.
 Anton von Braunmühl, Vorlesungen über Geschichte der
Trigonometrie, Vol. 1, 1900.
Corrected Dec 6, 2006.