(Note: This piece was written in 2008.)

I've heard that the cities of Boston, New York, Philadelphia, Baltimore, and Washington DC were laid out by the original designers to line up with each other (i.e., be on the same "great circle") and with Stonehenge. I was curious as to the extent, so I worked it out.

The main steps of the calculation are as follows:

A. Obtain the latitude and longitude of those locales
B. Define a coordinate system for the earth
C. Convert latitude and longitude to x-y-z coordinates
D. Determine the "great circle" that goes through two of the locales (that is, a plane that goes through those two locales and the center of the earth)
E. Test roundoff error in the plane equation by substituting coordinates of Washington DC and Boston
F. Determine the closeness of the other locales to that great circle
G. Conclusion

A. Obtain the latitude and longitude of those locales

I found the following information online for latitude and longitude. These figures are given in degrees and minutes (1 minute = 1/60 of a degree = 1 nautical mile = 6,080 feet).

Washington DC - 38^o54' N lat., 77^o2' W long.
Baltimore - 39^o17' N, 76^o37' W
Philadelphia - 39^o57' N, 75^o10' W
New York - 40^o45' N, 74^o0' W
Boston - 42^o22' N, 71^o4' W

Because there are no universally-agreed coordinates for Stonehenge, I used 51^o11' N, 1^o47' W, which are extrapolated from its being supposedly 13 km north of Salisbury (51^o4' N, 1^o47' W).

B. Define a coordinate system for the earth
Define the earth as a unit sphere (that is, having radius 1). I visualize it as having the plane of equator parallel to the ground (that is, no "tilt"), held so that the point along the equator at 90^oW longitude is closest to me. Now define x, y, and z axes as follows:

The center of the earth is the origin (0,0,0).
The point nearest me is (0,1,0).
The point along the equator at 0^o long. is (1,0,0).
The North Pole is at (0,0,1).

C. Convert latitude and longitude to x-y-z coordinates
x-coordinate: take the cosine of the longitude and multiply by the cosine of the latitude (to account for the diameter of the latitudinal line)
y-coordinate: take the sine of the longitude and multiply by the cosine of the latitude
z-coordinate: take the sine of the latitude

(Note: the table of trig functions I referred to has values only for 0^o to 45^o. To compensate, sines and cosines for angles greater than 45^o were expressed as cosine or sine, respectively, of the complementary angle.)

Washington DC: 38^o54' N lat., 77^o2' W long.
x-coordinate: cos(77^o2') * cos(38^o54') = sin(12^o58') * (.77824) = (.22438) * (.77824) = .1746214
y-coordinate: sin(77^o2') * cos(38^o54') = cos(12^o58') * (.77824) = (.97450) * (.77824) = .7583948
z-coordinate: sin(38^o54') = .62796
(.1746214, .7583948, .62796)

Baltimore: 39^o17' N, 76^o37' W
x-coordinate: cos(76^o37') * cos(39^o17') = sin(13^o23') * (.77402) = (.23146) * (.77402) = .1791546
y-coordinate: sin(76^o37') * cos(39^o17') = cos(13^o23') * (.77402) = (.97284) * (.77402) = .7529976
z-coordinate: sin(39^o17') = .63316
(.1791546, .7529976, .63316)

Philadelphia: 39^o57' N, 75^o10' W
x-coordinate: cos(75^o10') * cos(39^o57') = sin(14^o50') * (.76661) = (.25601) * (.76661) = .1962598
y-coordinate: sin(75^o10') * cos(39^o57') = cos(14^o50') * (.76661) = (.96667) * (.76661) = .7410588
z-coordinate: sin(39^o57') = .64212
(.1962598, .7410588, .64212)

New York: 40^o45' N, 74^o0' W
x-coordinate: cos(74^o0') * cos(40^o45') = sin(16^o0') * (.75756) = (.27564) * (.75756) = .2088138
y-coordinate: sin(74^o0') * cos(40^o45') = cos(16^o0') * (.75756) = (.96126) * (.75756) = .7282121
z-coordinate: sin(40^o45') = .65276
(.2088138, .7282121, .65276)

Boston: 42^o22' N, 71^o4' W
x-coordinate: cos(71^o4') * cos(42^o22') = sin(18^o56') * (.73885) = (.32447) * (.73885) = .2397346
y-coordinate: sin(71^o4') * cos(42^o22') = cos(18^o56') * (.73885) = (.94590) * (.73885) = .6988782
z-coordinate: sin(42^o22') = .67387
(.2397346, .6988782, .67387)

Stonehenge: 51^o11' N, 1^o47' W,
x-coordinate: cos(1^o47') * cos(51^o11') = (.99952) * sin(38^o49') = (.99952) * (.62683) = .6265291
y-coordinate: sin(1^o47') * cos(51^o11') = (.03112) * sin(38^o49') = (.03112) * (.62683) = .0195069
z-coordinate: sin(51^o11') = cos(38^o49') = .77916
(.6265291, .0195069, .77916)

D. Determine the "great circle" that goes through two of the locales (that is, a plane that goes through those two locales and the center of the earth)

We define a plane based on Washington DC, Boston, and the origin (0,0,0). The first two points are chosen because they are the farthest apart of any pair of cities for which we have precise latitude and longitude, which will tend to mitigate the inherent error.

A plane is defined by (or so I've read)

Ax + By + Cz + D = 0

where
A = y₁(z₂-z₃) + y₂(z₃-z₁) + y₃(z₁-z₂)
B = z₁(x₂-x₃) + z₂(x₃-x₁) + z₃(x₁-x₂)
C = x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)
-D = x₁(y₂z₃-y₃z₂) + x₂(y₃z₁-y₁z₃) + x₃(y₁z₂-y₂z₁)

Taking Washington DC as point 1 (x₁, y₁, z₁), Boston as point 2, (x₂, y₂, z₂), and the origin as point 3 (x₃, y₃, z₃) yields the following (taking the coordinates previously derived for these locales):

Washington DC
x₁ = .1746214
y₁ = .7583948
z₁ = .62796

Boston
x₂ = .2397346
y₂ = .6988782
z₂ = .67387

origin
x₃ = 0
y₃ = 0
z₃ = 0

Substituting and solving for A,

A = .7583948(.67387-0) + .6988782(0-.62796) + 0(.62796-.67387)
= .5110595 + (-.4388675) + 0
= .072192

Substituting and solving for B,

B = .62796(.2397346-0) + .67387(0-.1746214) + 0(.1746214-.2397346)
= .1505437 + (-.1176721) + 0
= .0328716

Substituting and solving for C,

C = .1746214(.6988782-0) + .2397346(0-.7583948) + 0(.7583948-.6988782)
= .122039 + (-.1818134) + 0
= -.0597744

Substituting and solving for D,

-D = x₁[(y₂)(0)-(0)(z₂)] + x₂[(0)(z₁)-(y₁)(0)] + 0(y₁z₂-y₂z₁)
= 0

Substituting these coefficients into the equation for a plane yields

.072192x + .0328716y -.0597744z + 0 = 0
.072192x + .0328716y -.0597744z = 0

E. Test roundoff error in the plane equation by substituting coordinates of Washington DC and Boston

To gauge the roundoff error in our calculations, we substitute the coordinates for Washington DC and Boston, which we know to be on the plane because their coordinates were used to define it. .

.072192x + .0328716y - .0597744z = 0

Washington DC (.1746214, .7583948, .62796)

.072192(.1746214) + .0328716(.7583948) - .0597744(.62796) =
.0126062 + .0249296 - .0375359 =
-.0000001

The roundoff error seems negligible.

Boston (.2397346, .6988782, .67387)

.072192(.2397346) + .0328716(.6988782) - .0597744(.67387) =
.0173069 + .0229732 - .0402801 =
0

This result of "0" means that to the precision of our calculations, Boston lies directly on the plane.

F. Determine the closeness of the other locales to that great circle

To determine how well a given locale aligns with the plane that was just calculated, we substitute its coordinates into the plane's equation and calculate the result (which ideally should be close to zero).

.072192x + .0328716y - .0597744z

Then, to determine the distance between the given point and the plane, the result just obtained is divided by the following expression:

(A² + B² + C²)^1/2 =
[ (.072192)² + (.0328716)² + (-.0597744)² ]^1/2 =
(.0052116 + .0010805 + .0035729)^1/2 =
(.009865)^1/2 =
.0993227, which we will round to .1 to simplify the calculation.

Recall that we assumed the earth to have a radius of 1 unit, which therefore represents about 4,000 miles. To determine the distance from the plane represented by -.0000001, the result from substituting the coordinates of Washington DC into the plane equation, we would divide its absolute value by .1, yielding .000001. This would constitute (¹/_1,000,000) x 4,000 miles, or .004 miles (about 20 feet).

Baltimore (.1791546, .7529976, .63316)

.072192(.1791546) + .0328716(.7529976) - .0597744(.63316) =
.0129335 + .0247522 -.0378467 =
-.000161

Taking the absolute value and dividing by .1, we obtain a distance of .00161, which translates to about 6.5 miles.

Philadelphia (.1962598, .7410588, .64212)

.072192(.1962598) + .0328716(.7410588) - .0597744(.64212) =
.0141683 + .0243597 - .0383823 =
.0001457

Taking the absolute value and dividing by .1, we obtain a distance of .001457, which translates to about 5.8 miles.

New York (.2088138, .7282121, .65276)

.072192(.2088138) + .0328716(.7282121) - .0597744(.65276) =
.0150746 + .0239374 - .0390183 =
.0000063

Taking the absolute value and dividing by .1, we obtain a distance of .000063, which translates to about one-fourth of a mile. This could be attributed solely to the inherent imprecision of the coordinates, which are to the nearest nautical mile.

Stonehenge (.6265291, .0195069, .77916)

.072192(.6265291) + .0328716(.0195069) - .0597744(.77916) =
.0452303 + .0006412 - .0465738 =
-.0007023

Taking the absolute value and dividing by .1, we obtain a distance of .007023, which translates to about 28 miles. This large discrepancy is partly attributable to the distance from Boston to England being several times larger than that between Boston and Washington DC. A given angular error in the plane would tend to affect distance-from-the-plane in proportion to a locale's distance beyond the plane's two anchor cities.

G. Conclusion

The value for distance-from-the-plane for non-anchor cities was under 7 miles in each case and essentially 0 for New York (because New York's value was less than the inherent imprecision in that city's coordinates). This seems small enough, especially given the size of cities and that the location of a city's center sometimes changes, that the original proposition of the cities being designed to line up with each other seems feasible (and virtually certain in the case of New York).

The distance-from-the-plane for Stonehenge was about 28 miles, or more than 4 times that for Baltimore. Nonetheless, the mitigating factors discussed above - Stonehenge's distance from the anchor cities and its lack of universally-agreed coordinates - suggest that these calculations are not cause to reject Stonehenge as the alignment point for the other cities discussed here.