HOW TO ESTABLISH A TRUE MERIDIAN.

[Footnote: A paper read before the Engineers' Club of Philadelphia.]

By PROFESSOR L. M. HAUPT.

INTRODUCTORY.

The discovery of the magnetic needle was a boon to mankind, and has been of inestimable service in guiding the mariner through trackless waters, and the explorer over desert wastes. In these, its legitimate uses, the needle has not a rival, but all efforts to apply it to the accurate determination of permanent boundary lines have proven very unsatisfactory, and have given rise to much litigation, acerbity, and even death.

For these and other cogent reasons, strenuous efforts are being made to dispense, so far as practicable, with the use of the magnetic needle in surveying, and to substitute therefor the more accurate method of traversing from a true meridian. This method, however, involves a greater degree of preparation and higher qualifications than are generally possessed, and unless the matter can be so simplified as to be readily understood, it is unreasonable to expect its general application in practice.

Much has been written upon the various methods of determining, the true meridian, but it is so intimately related to the determination of latitude and time, and these latter in turn upon the fixing of a true meridian, that the novice can find neither beginning nor end. When to these difficulties are added the corrections for parallax, refraction, instrumental errors, personal equation, and the determination of the probable error, he is hopelessly confused, and when he learns that time may be sidereal, mean solar, local, Greenwich, or Washington, and he is referred to an ephemeris and table of logarithms for data, he becomes lost in "confusion worse confounded," and gives up in despair, settling down to the conviction that the simple method of compass surveying is the best after all, even if not the most accurate.

Having received numerous requests for information upon the subject, I have thought it expedient to endeavor to prepare a description of the method of determining the true meridian which should be sufficiently clear and practical to be generally understood by those desiring to make use of such information.

This will involve an elementary treatment of the subject, beginning with the

DEFINITIONS.

The celestial sphere is that imaginary surface upon which all celestial objects are projected. Its radius is infinite.

The earth's axis is the imaginary line about which it revolves.

The poles are the points in which the axis pierces the surface of the earth, or of the celestial sphere.

A meridian is a great circle of the earth cut out by a plane passing through the axis. All meridians are therefore north and south lines passing through the poles.

From these definitions it follows that if there were a star exactly at the pole it would only be necessary to set up an instrument and take a bearing to it for the meridian. Such not being the case, however, we are obliged to take some one of the near circumpolar stars as our object, and correct the observation according to its angular distance from the meridian at the time of observation.

For convenience, the bright star known as Ursæ Minoris or Polaris, is generally selected. This star apparently revolves about the north pole, in an orbit whose mean radius is 1° 19' 13",[1] making the revolution in 23 hours 56 minutes.

[Footnote 1: This is the codeclination as given in the Nautical Almanac. The mean value decreases by about 20 seconds each year.]

During this time it must therefore cross the meridian twice, once above the pole and once below; the former is called the upper, and the latter the lower meridian transit or culmination. It must also pass through the points farthest east and west from the meridian. The former is called the eastern elongation, the latter the western.

An observation may he made upon Polaris at any of these four points, or at any other point of its orbit, but this latter case becomes too complicated for ordinary practice, and is therefore not considered.

If the observation were made upon the star at the time of its upper or lower culmination, it would give the true meridian at once, but this involves a knowledge of the true local time of transit, or the longitude of the place of observation, which is generally an unknown quantity; and moreover, as the star is then moving east or west, or at right angles to the place of the meridian, at the rate of 15° of arc in about one hour, an error of so slight a quantity as only four seconds of time would introduce an error of one minute of arc. If the observation be made, however, upon either elongation, when the star is moving up or down, that is, in the direction of the vertical wire of the instrument, the error of observation in the angle between it and the pole will be inappreciable. This is, therefore, the best position upon which to make the observation, as the precise time of the elongation need not be given. It can be determined with sufficient accuracy by a glance at the relative positions of the star Alioth, in the handle of the Dipper, and Polaris (see Fig. 1). When the line joining these two stars is horizontal or nearly so, and Alioth is to the west of Polaris, the latter is at its eastern elongation, and vice versa, thus:

But since the star at either elongation is off the meridian, it will be necessary to determine the angle at the place of observation to be turned off on the instrument to bring it into the meridian. This angle, called the azimuth of the pole star, varies with the latitude of the observer, as will appear from Fig 2, and hence its value must be computed for different latitudes, and the surveyor must know his latitude before he can apply it. Let N be the north pole of the celestial sphere; S, the position of Polaris at its eastern elongation; then N S=1° 19' 13", a constant quantity. The azimuth of Polaris at the latitude 40° north is represented by the angle N O S, and that at 60° north, by the angle N O' S, which is greater, being an exterior angle of the triangle, O S O. From this we see that the azimuth varies at the latitude.

We have first, then, to find the latitude of the place of observation.

Of the several methods for doing this, we shall select the simplest, preceding it by a few definitions.

A normal line is the one joining the point directly overhead, called the zenith, with the one under foot called the nadir.

The celestial horizon is the intersection of the celestial sphere by a plane passing through the center of the earth and perpendicular to the normal.

A vertical circle is one whose plane is perpendicular to the horizon, hence all such circles must pass through the normal and have the zenith and nadir points for their poles. The altitude of a celestial object is its distance above the horizon measured on the arc of a vertical circle. As the distance from the horizon to the zenith is 90°, the difference, or complement of the altitude, is called the zenith distance, or co-altitude.

The azimuth of an object is the angle between the vertical plane through the object and the plane of the meridian, measured on the horizon, and usually read from the south point, as 0°, through west, at 90, north 180°, etc., closing on south at 0° or 360°.

These two co-ordinates, the altitude and azimuth, will determine the position of any object with reference to the observer's place. The latter's position is usually given by his latitude and longitude referred to the equator and some standard meridian as co-ordinates.

The latitude being the angular distance north or south of the equator, and the longitude east or west of the assumed meridian.

We are now prepared to prove that the altitude of the pole is equal to the latitude of the place of observation.

Let H P Z Q¹, etc., Fig. 2, represent a meridian section of the sphere, in which P is the north pole and Z the place of observation, then H H¹ will be the horizon, Q Q¹ the equator, H P will be the altitude of P, and Q¹ Z the latitude of Z. These two arcs are equal, for H C Z = P C Q¹ = 90°, and if from these equal quadrants the common angle P C Z be subtracted, the remainders H C P and Z C Q¹, will be equal.

To determine the altitude of the pole, or, in other words, the latitude of the place.

Observe the altitude of the pole star when on the meridian, either above or below the pole, and from this observed altitude corrected for refraction, subtract the distance of the star from the pole, or its polar distance, if it was an upper transit, or add it if a lower. The result will be the required latitude with sufficient accuracy for ordinary purposes.

The time of the star's being on the meridian can be determined with sufficient accuracy by a mere inspection of the heavens. The refraction is always negative, and may be taken from the table appended by looking up the amount set opposite the observed altitude. Thus, if the observer's altitude should be 40° 39' the nearest refraction 01' 07", should be subtracted from 40° 37' 00", leaving 40° 37' 53" for the latitude.

TO FIND THE AZIMUTH OF POLARIS.

As we have shown the azimuth of Polaris to be a function of the latitude, and as the latitude is now known, we may proceed to find the required azimuth. For this purpose we have a right-angled spherical triangle, Z S P, Fig. 4, in which Z is the place of observation, P the north pole, and S is Polaris. In this triangle we have given the polar distance, P S = 10° 19' 13"; the angle at S = 90°; and the distance Z P, being the complement of the latitude as found above, or 90°--L. Substituting these in the formula for the azimuth, we will have sin. Z = sin. P S / sin P Z or sin. of Polar distance / sin. of co-latitude, from which, by assuming different values for the co-latitude, we compute the following table:

An analysis of this table shows that the azimuth this year (1882) increases with the latitude from 1° 28' 05" at 26° north, to 2° 3' 11" at 50° north, or 35' 06". It also shows that the azimuth of Polaris at any one point of observation decreases slightly from year to year. This is due to the increase in declination, or decrease in the star's polar distance. At 26° north latitude, this annual decrease in the azimuth is about 22", while at 50° north, it is about 30". As the variation in azimuth for each degree of latitude is small, the table is only computed for the even numbered degrees; the intermediate values being readily obtained by interpolation. We see also that an error of a few minutes of latitude will not affect the result in finding the meridian, e.g., the azimuth at 40° north latitude is 1° 43' 21", that at 41° would be 1° 44' 56", the difference (01' 35") being the correction for one degree of latitude between 40° and 41°. Or, in other words, an error of one degree in finding one's latitude would only introduce an error in the azimuth of one and a half minutes. With ordinary care the probable error of the latitude as determined from the method already described need not exceed a few minutes, making the error in azimuth as laid off on the arc of an ordinary transit graduated to single minutes, practically zero.

REFRACTION TABLE FOR ANY ALTITUDE WITHIN THE LATITUDE OF THE UNITED STATES.

APPLICATIONS.

In practice to find the true meridian, two observations must be made at intervals of six hours, or they may be made upon different nights. The first is for latitude, the second for azimuth at elongation.

To make either, the surveyor should provide himself with a good transit with vertical arc, a bull's eye, or hand lantern, plumb bobs, stakes, etc.[1] Having "set up" over the point through which it is proposed to establish the meridian, at a time when the line joining Polaris and Alioth is nearly vertical, level the telescope by means of the attached level, which should be in adjustment, set the vernier of the vertical arc at zero, and take the reading. If the pole star is about making its upper transit, it will rise gradually until reaching the meridian as it moves westward, and then as gradually descend. When near the highest part of its orbit point the telescope at the star, having an assistant to hold the "bull's eye" so as to reflect enough light down the tube from the object end to illumine the cross wires but not to obscure the star, or better, use a perforated silvered reflector, clamp the tube in this position, and as the star continues to rise keep the horizontal wire upon it by means of the tangent screw until it "rides" along this wire and finally begins to fall below it. Take the reading of the vertical arc and the result will be the observed altitude.

[Footnote 1: A sextant and artificial horizon may be used to find the altitude of a star. In this case the observed angle must be divided by 2.]

ANOTHER METHOD.

It is a little more accurate to find the altitude by taking the complement of the observed zenith distance, if the vertical arc has sufficient range. This is done by pointing first to Polaris when at its highest (or lowest) point, reading the vertical arc, turning the horizontal limb half way around, and the telescope over to get another reading on the star, when the difference of the two readings will be the double zenith distance, and half of this subtracted from 90° will be the required altitude. The less the time intervening between these two pointings, the more accurate the result will be.

Having now found the altitude, correct it for refraction by subtracting from it the amount opposite the observed altitude, as given in the refraction table, and the result will be the latitude. The observer must now wait about six hours until the star is at its western elongation, or may postpone further operations for some subsequent night. In the meantime he will take from the azimuth table the amount given for his date and latitude, now determined, and if his observation is to be made on the western elongation, he may turn it off on his instrument, so that when moved to zero, after the observation, the telescope will be brought into the meridian or turned to the right, and a stake set by means of a lantern or plummet lamp.

It is, of course, unnecessary to make this correction at the time of observation, for the angle between any terrestrial object and the star may be read and the correction for the azimuth of the star applied at the surveyor's convenience. It is always well to check the accuracy of the work by an observation upon the other elongation before putting in permanent meridian marks, and care should be taken that they are not placed near any local attractions. The meridian having been established, the magnetic variation or declination may readily be found by setting an instrument on the meridian and noting its bearing as given by the needle. If, for example, it should be north 5° east, the variation is west, because the north end of the needle is west of the meridian, and vice versa.

Local time may also be readily found by observing the instant when the sun's center[1] crosses the line, and correcting it for the equation of time as given above--the result is the true or mean solar time. This, compared with the clock, will show the error of the latter, and by taking the difference between the local lime of this and any other place, the difference of longitude is determined in hours, which can readily be reduced to degrees by multiplying by fifteen, as 1 h. = 15°.

[Footnote 1: To obtain this time by observation, note the instant of first contact of the sun's limb, and also of last contact of same, and take the mean.]

APPROXIMATE EQUATION OF TIME.