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141 18 THE MILKY WAY LIES ALONG OUR EQUATOR Coordinate systems are devised to identify uniquely each and every point on a surface. They typically fix points on a plane, a left-right measure commonly labeled the x-coordinate and an up-down scale known as the y-coordinate. These kinds of systems are called rectangular or, more commonly, Cartesian coordinate systems, named after René Descartes (1596 –1650), the eminent French philosopher and mathematician . I suppose the best-known Cartesian system of coordinates is the following. The levels of the y measure might be labeled 4, 14, 23, 34, 42, 50, 57, etc., and the x lines might be 1, 2, 3, etc., or, more speci fically, 1, 2, 3, Lexington, Park, Madison, 5, 6, etc. Yes, this is the coordinate system designating the streets and avenues of New York City or, more specifically, Manhattan Island north of the Village. Any New Yorker recognizes the y-divisions listed above as the major stops along the IRT and BMT subway lines running in the north-south directions. Each point in the scheme is designated by coordinates (x,y) so that (5,42) uniquely describes Fifth Avenue and Forty-second Street. We ignore Broadway for the nonce, slicing as it does through the system at an oblique angle like a rampant bolt of lightning. Most systems have an origin described by (0,0) but in New York’s case, one would have to move a block to the east of First Avenue and four blocks south of Fourth Street to find it. Such a point exists, although it has no significance to most New Yorkers since it lies somewhere between Houston and Delancey Streets a little east of the Bowery ! Admittedly, the Manhattan coordinate system, like those of most cities of Cartesian design, doesn’t always maintain equal spacing between one block and another. In general, however, the pattern is spaced 6 blocks to the mile between the numbered avenues and 20 blocks to the mile in the north-south layout of the streets. Thus the Empire State Building at (5,34) and the Chrysler Building at (31 ⁄2,42— Lexington Avenue being midway between Third and Fourth [Park] Avenues) are separated by the hypotenuse of the triangle with legs of 11 ⁄2x (1/6) ⫹ 8x (1/20) miles; this amounts to sqrt (0.252 ⫹ 0.42 ) ⫽ 0.47 miles; hence, the separation between these two tallest towers is just under one-half mile. The other common type is the spherical coordinate system, a pattern used to designate points on a grid wrapped around a globe. To be sure, New York is part of the surface of a sphere, but it is a sufficiently small part of it that a plane representing the city is adequate. In the case of a ball, a fundamental great circle exists (a great circle is defined as a circle on a sphere whose center is coincidental to that of the sphere—it is also the largest possible circle that can be fit onto the sphere). The Earth’s equator is such a great circle, the most fundamental of all. Any great circle with an axis perpendicular to it passes through the north pole and the south pole. The axis connecting the poles is the axis of rotation of the Earth, and is the obvious choice for the fundamental reference frame on its surface. For centuries navigators have defined latitude as the angular north-south measure from the equator defined as the circle of latitude 0 and the poles as 90N and 90S, or ⫹90 and ⫺90, for the north and south poles, respectively. A point midway between the north pole and the equator lies in latitude 45N, and so forth. Now it is clear that we need a reference point for east-west measure, known as longitude, but no clear unequivocal point exists in nature, and we are required to define one arbitrarily. Actually, the requisite reference is not a point but a half great circle extending from one pole to the other, called a meridian. Any meridian will do, and seafaring nations established their own meridians, ones that usually passed through their capital cities. Thus France defined one that passed through the Observatoire de Paris and the Jardin du Luxembourg; Great Britain’s went through the Royal Observatory in Greenwich, an eastern suburb of London; and that of the United States passed along Sixteenth Street in Washington...

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