In 1811, after four years of preparation, a plan was unveiled that mapped out the future development of the island of Manhattan. Called the Commissioners' Plan, this scheme imposed a rectangular grid of avenues and streets without regard for the existing topography of the island.^{1,2}
The result, of course, is the most consistent urban design in the world. Other cities had been designed around grids before, but never so relentlessly. The first rule that any visitor to Manhattan learns is that numbered avenues run north-south and numbered streets run east-west (at least above Fourteenth Street and excluding Broadway).
The grid is so ubiquitous that New Yorkers tend to associate the layout of the streets with the points of the compass. Uptown is synonymous with north; downtown with south. There's a lower east side, an upper west side, the East Side Kids and a West Side Story. When describing the location of homes or hot restaurants, we refer to the north and south sides of streets, the east and west sides of avenues, and corners that are northwest, northeast, southwest, and southeast.
However, the orientation of the city's avenues was fixed to be parallel with the axis of Manhattan Island and has only a casual relationship to true north and south. Maps that are oriented to true north (like the one at the right) show the island at a significant tilt. In truth, avenues run closer to northeast and southwest than north and south.
Several years ago, a small garden was built on the southeast corner of 9^{th} Street and 3^{rd} Avenue that also highlighted the difference. The garden lies on a triangle of land created by Stuyvesant Street, which cuts through the block from 9^{th} Street and 3^{rd} Avenue to 10^{th} Street and 2^{nd} Avenue. This was the site of Peter Stuyvesant's manor, and the street that remains was once part of a larger grid of streets oriented to truth north. Most of these streets were destroyed to enforce the grid. A structure in the garden shows the points of the compass lined up with Stuyvesant Street — not with 3^{rd} Avenue.
What is the exact deviation of the city's avenues from true north? This seems like the sort of information that every New Yorker should know, similar to the "twenty blocks to the mile" rule. The truth is, however, that nobody I asked had ever heard of such a figure, or even thought about it, or even (despite my inquisitiveness) believed it to be of any importance whatsoever.
One way to determine the deviation is, of course, to take an actual compass out on the streets of Manhattan and try to find a place where it wouldn't be affected by the many metal structures in the city. But that seemed too inaccurate, even after adjusting for the difference between true north and magnetic north. Likewise, I wanted something more sophisticated than applying a protractor to a map. I didn't want to read a figure from a dial: I wanted a number to pop out of a calculation.
As we all know, positions on the surface of the Earth can be specified with two numbers: longitude and latitude. The latitude is the angular distance north or south of the equator. The longitude is the angular distance east or west of the prime meridian, which passes through Greenwich, England. Determination of one's longitude and latitude was once a complex procedure involving careful observations of the stars and (in the case of longitude) an accurate clock. Today, we can read the longitude and latitude from a hand-held GPS (Global Positioning System) device.
Or, we can make use of maps via the Internet. The maps on this page are generated using the recent (as of July 2005) Google Maps API, which defines several JavaScript classes. These maps are interactive: By clicking the little arrows, you can move in four directions, or zoom in and out. You can also switch between Map view and Satellite view. Whenever you see a little push-pin marker, you can click it to obtain the longitude and latitude at the marker.
In the paragraphs that follow, I'll be describing a position as the number pair (longitude, latitude). I've expressed longitude and latitude to five decimal places, equivalent to about four feet accuracy.^{3}
Here's Stuyvesant Street in the East Village:
Suyvesant Street is supposed to run exactly east-west but the longitude and latitude at the two markers reveals that it's not quite exact.
For the remaining calculations, I've chosen to focus on 5^{th} Avenue because it is traditionally considered to be the center of Manhattan and divides street addresses into east and west. I found the longitude and latitude of 5^{th} Avenue at two positions: Where it begins at Washington Square Park (specifically, Waverly Place, one block south of 8^{th} Street):
and where it meets the corner of Central Park at 59th street:
Remember, you can click the push pins to see the longitude and latitude of these two points.
More severely graphically, the two points look something like this:
There are a couple different approaches to dealing with this information. One approach is to assume that 5^{th} Avenue occupies an area small enough to be approximated with plane trigonometry. However, some simple spherical trigonometry is required as well. In particular, in spherical trigonometry, a great circle is the equivalent of a straight line: It represents the shortest distance between two points on the surface of the sphere. Every great circle divides the sphere into two halves of equal size.
Because distances on a sphere are segments of a great circle, the distances can be expressed in angles measured from the center of the sphere. To convert an angle expressed in radians to a linear distance, multiply the angle by the sphere's radius. To convert an angle expressed in degrees, multiply the angle by 2πR (the circumference) and divide by 360°.
Let's begin by constructing a right triangle where the hypotenuse is the stretch of 5th Avenue between the two parks:
The line labeled a is a line of latitude; the line labeled b is a long of longitude; we're trying to find the angle α where:
Of course, this formula is from plane trigonometry. At some point we have to determine if this approximation is valid.
Because the endpoints of line b have the same longitude, line b is on a meridian, which means that it's on a great circle that passes through the North and South poles.
It's fairly easy to calculate the physical distance between two points on a meridian because the difference in latitudes are easily converted to a distance. If R is the radius of the Earth, then the total length of the meridian is the circumference of the Earth, which is 2πR. The length of a line on a meridian with endpoints at latitudes φ_{1} and φ_{2} is the total circumference of the Earth multiplied by the difference in latitudes divided by 360°:
Circles of equal latitude are in general not great circles. As you go further north or south from the equator, circles of equal latitude become smaller and smaller. However, it's easy to demonstrate that the radius of these circles is the cosine of the latitude φ. Thus, the length of a line on a circle of equal latitude with endpoints at longitudes λ_{1} and λ_{2} is:
So, if the tangent of angle α is a divided by b, then:
or in this case:
And the answer is:
Normally I'd be happy rounding the result to 29. I'm showing it to such precision to make a point. You see, an alternative method of constructing a right triangle with the same hypotenuse is like so:
If you calculate the angle based on this triangle, then the answer is:
That these numbers are so close is a good indication that our use of plane trigonometry is a satisfactory approximation.
But let's see if we can do better.
In spherical geometry, a triangle is defined by the intersection of three great circles. Triangles have three interior angles and three lengths. As in plane geometry, if you know any three of these quantities, you can determine the other three using appropriate formulas. In plane trigonometry, the interior angles of a triangle always sum to 180°. In spherical trigonometry, the interior angles of a triangle always sum to greater than 180°
Let's construct another triangle using that same stretch of Fifth Avenue, but this time using principles of spherical trigonometry. The two constructed sides are arcs of great circles that meet at the North pole (indicated by N):
Of course, this map is not to scale. By assuming that Fifth Avenue is on a great circle, I'm implying that the avenues of Manhattan get closer together as they get more northern, which may be a little disturbing.
Once again we're trying to find angle α. Of the three sides and three interior angles, we immediately know the lengths of two sides and one angle, which is theoretically sufficient information to get the other side and the other two angles (although the math may be messy).
We know the lengths of the two constructed sides because the latitudes imply distances from the North pole. Assuming a mean radius of the Earth of 3,960 miles, the distance from Washington Square Park to the North pole is:
The distance from the southeast corner of Central Park to the North pole is:
The angle at N between these two sides is equal to the difference of longitude between the two points: 0.02395°.
Solving for d, we get:
Because this distance along Fifth Avenue comprises 52 blocks, the 20-blocks-to-a-mile rule looks pretty good.
In spherical trigonometry, the law of sines says that for any triangle, the ratio of the sine of any side of triangle to the sine of the opposite angle is constant. That means we can determine α like this:
Solving for alpha we get:
The remaining angle — the angle at Central Park — is 151.088°, implying that Fifth Avenue deviates from true north at this point by 180° − 151.088° or:
As a great circle, Fifth Avenue would deviate from true north the more north it got.
I've been assuming that the Earth is a sphere, which it is not. It is actually closer to an oblate ellipsoid. A cross section of the Earth would look like an ellipse with the major axis along the equater and the minor axis between the poles. A more accurate formula for calculating the distance between two points is available.^{5}, but the difference is small,and it's not clear to me how these distances can be converted to angles for use in spherical trigonometry formulas.
I am thus satisfied with a figure rounded to the nearest degree. The avenues of Manhattan deviate from true north by
whether anybody else cares or not.
^{1}Edwin G. Burrows and Mike Wallace, Gotham: A History of New York City to 1898 (New York: Oxford University Press, 1999), 419-422.
^{2}"Grid Plan" in Kenneth T. Jackson, ed., The Encyclopedia of New York City (New Haven: Yale University Press, 1995), 510.
^{3}The mean circumference of the Earth is approximately 25,000 miles. One degree of any great circle around the Earth is thus about 69 miles. One minute of arc is about 1.15 miles, which is called a nautical mile. (A nautical mile is set at 6,076 feet as opposed to 5,280 feet.) One second of arc is about 100 feet. Five decimal places of latitude implies an accuracy of 0.00001 degrees, which is 0.00069 miles or 3.6 feet.
^{4}Robin M. Green, Spherical Astronomy (Cambridge: Cambridge University Press, 1985), 12-14.
^{5}Jean Meeus, Astronomical Algorithms (Richmond, Virginia: Willmann-Bell, Inc., 1991), 80-82.
© 2005, Charles Petzold (www.charlespetzold.com)
First Posted: July 2005
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