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[2008/07/06][2002/09/07][2006/05/05] [2000/07/18][2001/05/26][2006/05/11] [2003/10/30][2005/09/27][2001/05/28] [2006/05/18][1999/02/16][2003/01/31] [2002/03/18][2008/11/18] 176^∘°fancy[C][R]plain [C] [R]xobeysp @processline-0.25in@line@[email protected]@licensingevenheadProject Gutenberg Licensing. Transcriber's Notes A small number of minor typographical errors and inconsistencies have been corrected. See the DPtypo command in the source for more information. [1]FrontispieceFrontispiecefrontispieceFRONTISPIECE. The Great Telescope of the Lick Observatory, Mt. Hamilton, Cal. Object-Glass made by A. Clark & Sons:Aperture, 36 in.; Focal Length, 56 ft. 2 in. Mounting by Warner & Swasey. A TEXT-BOOKOFGENERAL ASTRONOMYFORCOLLEGES AND SCIENTIFIC SCHOOLS[2]BY CHARLES A. YOUNG, Ph.D., LL.D., Professor of Astronomy in the College of New Jersey (Princeton). [2][2]Boston, U.S.A., and London: GINN & COMPANY, PUBLISHERS. 1889.Entered at Stationers' Hall.
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Copyright, 1888, by CHARLES A. YOUNG,All Rights Reserved. Typography by J. S. Cushing & Co., Boston, U.S.A. [4cm] Presswork by Ginn & Co., Boston, U.S.A. [1]PrefacePrefaceCHAPTER: PREFACE.PREFACE. The limitations of time are such in our college course thatprobably it will not be possible in most cases for a class totake thoroughly everything in the book. The fine print is tobe regarded rather as collateral reading, important to anythinglike a complete view of the subject, but not essential tothe course. Some of the chapters can even be omitted incases where it is found necessary to abridge the course asmuch as possible; e.g., the chapters on Instruments and onPerturbations. While the work is no mere compilation, it makes no claimsto special originality: information and help have been drawnfrom all available sources.
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The author is under great obligationsto the astronomical histories of Grant and Wolf, andespecially to Miss Clerke's admirable “History of Astronomy inthe Nineteenth Century.” Many data also have been drawnfrom Houzeau's valuable “Vade Mecum de l'Astronomie.” ProfessorTrowbridge of Cambridge kindly provided the originalnegative from which was made the cut illustrating the comparisonof the spectrum of iron with that of the sun. Warner& Swasey of Cleveland and Fauth & Co. of Washington havealso furnished the engravings of a number of astronomicalinstruments. Professors Todd, Emerson, Upton, and McNeill have givenmost valuable assistance and suggestions in the revision of theproof; as indeed, in hardly a less degree, have several others. Princeton, N. J., August, 1888. In this issue of the book a number of errors which appeared in thefirst impression have been corrected. [1]Table of ContentsTable of ContentsCHAPTER: TABLE OF CONTENTS. [1]IntroductionIntroductionCHAPTER: INTRODUCTION.
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INTRODUCTIONINTRODUCTION. Astronomy (>'astron n�mos)is the science which treats of theheavenly bodies. As such bodies we reckon the sun and moon, theplanets (of which the earth is one) and their satellites, comets andmeteors, and finally the stars and nebul�.We have to consider in Astronomy:—[(a)] * The motions of these bodies, both real and apparent, and thelaws which govern these motions. Their forms, dimensions, and masses. * Their nature and constitution. The effects they produce upon each other by their attractions,radiations, or by any other ascertainable influence. At the same time it should be said at once that, even from thelowest point of view, Astronomy is far from a useless science. Theart of navigation depends for its very possibility upon astronomicalprediction. Take away from mankind their almanacs, sextants, andchronometers, and commerce by sea would practically stop. It need hardly be said that Astronomy is not separated from kindredsciences by sharp boundaries.
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It would be impossible, for instance,to draw a line between Astronomy on one side and Geologyand Physical Geography on the other. Many problems relating tothe formation and constitution of the earth belong alike to all three. Astronomy is divided into many branches, some of which, asordinarily recognized, are the following:—1. Descriptive Astronomy.—This, as its name implies, is merelyan orderly statement of astronomical facts and principles. It is sometimes called Gravitational Astronomy,because, with few exceptions, gravitation is the only force sensiblyconcerned in the motions of the heavenly bodies. The above-named branches are not distinct and separate, butthey overlap in all directions. Spherical Astronomy, for instance,finds the demonstration of many of its formul� in GravitationalAstronomy, and their application appears in Theoretical and PracticalAstronomy.
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Usually it will be best to proceed in the Euclidean order, by firststating the fact or principle in question, and then explaining itsdemonstration. The frequent references to “Physics” refer to the “ElementaryText-Book of Physics,” by Anthony & Brackett; 3d edition, 1887. Wiley & Sons, N.Y.INTRODUCTION IThe Doctrine of the SphereTHE “DOCTRINE OF THE SPHERE,” DEFINITIONS, AND GENERALCONSIDERATIONS. Astronomy, like all the other sciences, has a terminology of itsown, and uses technical terms in the description of its facts andphenomena. Or if we sometimes feel that the stars andother objects in the sky really differ in distance, we still instinctivelyimagine an immense sphere surrounding and enclosing all. This celestial sphere may be regarded in either of two differentways, both of which are correct and lead to identical results. Each observer, in this wayof viewing it, carries his own sky with him, and is the centre of hisown heavens.
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In what follows we shall usethis conception of the celestial sphere.[To most persons the sky appears, not a true hemisphere, but a flattened vault, as if the horizon were more remote than the zenith. This is a subjective effect due mainly to the intervening objects between us and the horizon. The sun and moon when rising or setting look much larger than when they are higher up, forthe same reason. Linear and Angular Dimensions.—Lineardimensions are such as maybe expressed in linear units; i.e., inmiles, feet, or inches; in metres ormillimetres. Angular dimensions areexpressed in angular units; i.e., inright angles, in radians,[A radian is the angle which is measured by an arc equal in length to radius. Hence, to reduce to seconds of arc an angle expressed in radians, we must multiply it by the number 206264.8; a relation of which we shall have to make frequent use. See Halsted's Mensuration, p. 25.]or (more commonly in astronomy) in degrees,minutes, and seconds.
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Thus, for instance, the linear semi-diameterof the sun is about 697,000 kilometers (433,000 miles),while its angular semidiametersemi-diameter is about 16', or a little more thana quarter of a degree. Obviously, angular units alone can properlybe used in describing apparent distances and dimensions in the sky. For instance, one cannot say correctly that the two stars which areknown as “the pointers” are two or five or ten feet apart: theirdistance is about five degrees. It is sometimes convenient to speak of “angular area,” the unitof which is a “square degree” or a “square minute”; i.e., a smallsquare in the sky of which each side is 1� or 1'. Thus we may comparethe angular area of the constellation Orion with that of Taurus,in square degrees, just as we might compare Pennsylvania and NewJersey in square miles. In the case of the moon, R = about 239,000 miles; and r, 1081miles. Hence s = 1081/239000 = 1/221 of a radian, which is a little morethan 1/4 of a degree.
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It may be mentioned here as a rather curious fact that most persons saythat the moon appears about a foot in diameter; at least, this seems tobe the average estimate. Vanishing Point.—Any system of parallel lines produced inone direction will appear to pierce the celestial sphere at a singlepoint. The different points, therefore,coalesce into a spot of apparently infinitesimal size—the so-called“vanishing point” of perspective. Thus the axis of the earth andall lines parallel to this axis point to the celestial pole. In order to describe intelligibly the apparent position of an objectin the sky, it is necessary to have certain points and lines from whichto reckon. We proceed to define some of these which are mostfrequently used. If the earth were exactly spherical, the zenith might also be definedas the point where a line drawn from the centre of the earth upwardthrough the observer meets the sky. The Nadir.—The Nadir is the point opposite the zenith—underfoot, of course.
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Both zenith and nadir are derived from the Arabic, which languagehas also given us many other astronomical terms. Horizon.—The Horīzon[ Beware of the common, but vulgar, pronunciation, H�rīzon. ]is a great circle of the celestialsphere, having the zenith and nadir as its poles: it is thereforehalf-way between them, and 90� from each. Many writers make a distinction between the sensible and rationalhorizons. It is evident, however, that on the infinitely distant surfaceof the celestial sphere, the two traces sensibly coalesce into one singlegreat circle, which is the horizon as first defined. In strictness,therefore, while we can distinguish between the two horizontal planes,we get but one horizon circle in the sky. The Visible Horizon is the line where sky and earth meet. On land it is an irregular line, broken by hills and trees, and of noastronomical value; but at sea it is a true circle, and of great importancein observation.
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It is not, however, a great circle, but,technically speaking, only a small circle; depressed below the truehorizon by an amount depending upon the observer's elevation abovethe water. This depression is called the Dip of the Horizon, and willbe discussed further on. Vertical Circles.—These are great circles passing throughthe zenith and nadir, and therefore necessarily perpendicular to thehorizon—secondaries to it, to use the technical term. Parallels of Altitude, or Almucantars.—These are small circlesparallel to the horizon: the term Almucantar is seldom used. The points and circles thus far defined are determined entirely bythe direction of gravity at the station occupied by the observer. The path thus daily described by a star is called its“diurnal circle.” The line joining these poles is, of course, theaxis of the celestial sphere, about which it seems to rotate daily.
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The exact place of the pole may be found by observing some starvery near the pole at two times 12 hours apart, and taking the middlepoint between the two observed places of the star. The definition of the pole just given is independent of any theoryas to the cause of the apparent rotation of the heavens. If, however,we admit that it is due to the earth's rotation on its axis, thenwe may define the poles as the points where the earth's axis producedpierces the celestial sphere. The pole is very nearlyon the line joining Polaris with the star Mizar (ζ Urs. Maj., at thebend in the handle of the dipper), and at a distance just about one-quarterof the distance between the pointers, which are nearly 5�apart. The southern pole, unfortunately, is not so marked by any conspicuousstar. 3.—The Pole Star and the Pointers. The Celestial Equator, or Equinoctial Circle.—This is a greatcircle midway between the two poles, and of course 90� from each.
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It may also be defined as the intersection of the plane of the earth'sequator with the celestial sphere. It derives its name from the factthat, at the two dates in the year when the sun crosses this circle—aboutMarch 20 and Sept. 22—the day and night are equal in length. This crossing occurs twice a year, once in September and once inMarch, and the Vernal Equinox is the point on the equator wherethe sun crosses it in the spring. It is sometimes called the Greenwichof the Celestial Sphere, because it is used as a reference pointin the sky, much as Greenwich is on the earth. Its position is notmarked by any conspicuous star. Why this point is also called the “First of Aries” will appearlater, when we come to speak of the zodiac and its “signs.”18. Hour-Circles.—Hour-circles are great circles of the celestialsphere passing through its poles, and consequently perpendicularto the celestial equator.
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The Meridian and Prime Vertical.—The Meridian is the greatcircle passing through the pole and the zenith. Since it is a greatcircle, it must necessarily pass through both poles, and through thenadir as well as the zenith, and must be perpendicular both to theequator and to the horizon. The Prime Vertical is the Vertical Circle (passing through thezenith) at right angles to the meridian; hence lying east and weston the celestial sphere. The Cardinal Points.—The North and South Points are thepoints on the horizon where it is intersected by the meridian. TheEast and West Points are where it is cut by the prime vertical, andalso by the equator. The North Point, which is on the horizon, mustnot be confounded with the North Pole, which is not on the horizon,but at an elevation equal (see Art.30) to the latitude of the observer. With these circles and points of reference we have now the meansto describe intelligibly the position of a heavenly body, in severaldifferent ways.
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We may give its altitude and azimuth, or its declination and hour-angle;or, if we know the time, its declination and right ascension. Either of these pairs of co-ordinates, as they are called, will defineits place in the sky.21. Altitude and Zenith Distance (illo004Fig. 4).—The Altitude of aheavenly body is its angular elevation above the horizon, and is measuredby the arc of the vertical circle passing through the body, andintercepted between it and the horizon. The Zenith Distance of a body is simply its angular distance fromthe zenith, and is the complement of the altitude. Altitude + ZenithDistance = 90�.22. Azimuth and Amplitude (illo004Fig. 4).—The Azimuth of a bodyis the angle at the zenith, between the meridian and the vertical circle,which passes through the body. It is measured also by the arc of thehorizon intercepted between the north or south point, and the footof this vertical.
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The word is of Arabic origin, and has the samemeaning as the true bearing in surveying and navigation.illo004Fig. 4.—The Horizon and Vertical Circles. O, the place of the Observer. M, some Star. OZ, the Observer's Vertical. ZMH, arc of the Star's Vertical Circle. Z, the Zenith; P, the Pole. TMR, the Star's Almucantar. SENW, the Horizon. Angle TZM, or arc SWNEH, Star's Azimuth. SZPN, the Meridian. Arc HM, Star's Altitude. EZW, the Prime Vertical. Arc ZM, Star's Zenith Distance. The Amplitude of a body is the complement of the azimuth. There are various ways of reckoning azimuth. Many writers express itin the same manner as the bearing is expressed in surveying; i.e., so manydegrees east or west of north or south; N. 20� E., etc. The more usualway at present is, however, to reckon it in degrees from the south point clearround through the west to the point of beginning: thus an object in theSW.
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would have an azimuth of 45�; in the NW., 135�; in the N., 180�; inthe NE., 225�; and in the SE. For example, to find a star whoseazimuth is 260�, and altitude 60�, we must face N. 80� E., and then lookup two-thirds of the way to the zenith. The object in this case has anamplitude of 10� N. of W., and a zenith distance of 30�. Evidently boththe azimuth and altitude of a heavenly body are continually changing, exceptin certain very special cases. In illo004Fig. 4, SENW represents the horizon, S being the south point,and Z the zenith. The angle SZM, which numerically equals thearc SH, is the Azimuth of the star M; while EZM, or EH is itsAmplitude. MH is its Altitude, and ZM its Zenith Distance. Declination and Polar Distance (illo005Fig. It is reckonedpositive (+) north of the celestial equator, and negative (-) southof it. Evidently it is precisely analogous to the latitude of a placeon the earth.
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The north-polar distance of a star is its angular distancefrom the North Pole, and is simply the complement of thedeclination. Declination + North-Polar Distance = 90�. The declination of a star remains always the same; at least, theslow changes that it undergoes need not be considered for ourpresent purpose. “Parallels of Declination” are small circles parallelto the celestial equator. The Hour-Angle (illo005Fig. 5).—The Hour-Angle of a star is theangle at the pole between the meridian and the hour-circle passingthrough the star. Of course the hour-angle of an object is continually changing,being zero when the object is on the meridian, one hour, or fifteendegrees, when it has moved that amount westward, and so on. Right Ascension (illo005Fig. 5).—The Right Ascension of a staris the angle at the pole between the star's hour-circle and the hour-circle(called the Equinoctial Colure), which passes through the vernalequinox.
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It may be defined also as the arc of the equator, interceptedbetween the vernal equinox and the foot of the star's hour-circle. It is always reckoned from the equinox toward the east; sometimesin degrees, but usually in hours, minutes, and seconds of time. The right ascension, like the declination, remains unchanged by thediurnal motion. Sidereal Time (illo005Fig. =2em SENW, the Horizon.=2em POP', line parallel to the Axis of the Earth. =2em P and P', the two Poles of the Heavens. =2em EQWT, the Celestial Equator, or Equinoctial. =2em X, the Vernal Equinox, or “First of Aries.” =2em PXP' the Equinoctial Colure, or Zero Hour-Circle. =2em Ym, the Star's Declination; Pm, its North-polar Distance. =2em Angle mPR = arc QY, the Star's (eastern) Hour-Angle; = 24^h minus Star's (western) Hour-Angle. =2em Angle XPm = arc XY, Star's Right Ascension, Sidereal time at the moment = 24^h minus angle XPQ.
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The Sidereal Time at any moment may be defined as the hour-angleof the vernal equinox. On account of the annual motionof the sun among the stars, the Solar Day, by which time is reckonedfor ordinary purposes, is about 4 minutes longer than the siderealday. The exact difference is 3^m 56^s.394 (sidereal), or just one dayin a year; there being 3661/4 sidereal days in the year, as against3651/4 solar days. In the observatory, this definition of right ascensionis the most natural and convenient. The first of them (illo004Fig. 4) represents the horizon, meridian, andprime vertical, and shows how the position of a star is indicated byits altitude and azimuth. The other figure (illo005Fig. 5) represents the system of points andcircles which depend upon the earth's rotation, and are independentof the direction of gravity. At the poles of the earth, which are, mathematically speaking, “singular”points, the definitions of the Meridian, of North and South, etc., breakdown.
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There the pole (celestial) and zenith coincide, and any number of circlesmay be drawn through the two points, which have now become one. Thehorizon and equator coalesce, and the only direction on the earth's surfaceis due south (or north)—east and west have vanished. A single step of the observer will, however, remedy the confusion: zenithand pole will separate, and his meridian will again become determinate. One of the verticals, the Meridian, is singled out from the rest bythe circumstance that it passes through the pole of the sky, markingthe North and South Points where it cuts the horizon. Altitude and Azimuth (or their complements, Zenith Distanceand Amplitude) are the co-ordinates which designate the positionof a body by reference to the Zenith and the Meridian. Similarly, the direction of the earth's axis (which is independentof the observer's place on the earth) determines the Poles, theEquator, the Parallels of Declination, and the Hour-Circles.
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Declination and Hour-Angle are the co-ordinateswhich refer the place of a star to the Pole and the Meridian;while Declination and Right Ascension refer it to the Pole and EquinoctialColure. The Altitudeof the pole, or its height in degrees above the horizon, is always equalto the Latitude of the observer. Indeed, the German word for latitude(astronomical) is Polh�he; i.e., simply “Pole-height.” This will be clear from illo006Fig. 6. The latitude of a place isthe angle between its plumb-line and the plane of the equator; theangle ONQ in the figure. [If the earth were truly spherical, Nwould coincide with C, the centre of the earth. The ordinarydefinition of latitude given in the geographies disregards the slightdifference.] illo006Fig. 6.—Relation of Latitude to the Elevation of the Pole. This fundamental relation, that the altitude of the celestial pole isthe Latitude of the observer, cannot be too strongly impressed on thestudent's mind.
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The usual symbol for the latitude of a place is ϕ.31. The Right Sphere.—If the observer is situated at theearth's equator, i.e., in latitude zero (ϕ = 0), the pole will be in thehorizon, and the equator will pass vertically overhead through thezenith. The stars will rise and set vertically, and their diurnal circles willall be bisected by the horizon, so that they will be 12 hours aboveit and 12 below. This aspect of the heavens is called the RightSphere. The Parallel Sphere.—If the observer is at the pole of theearth (ϕ = 90�), then the celestial pole will be in the zenith, andthe equator will coincide with the horizon. Stars in the Southern Hemisphere, on the other hand, would neverrise to view. As the sun and the moon move in such a way thatduring half the time they are alternately north and south of theequator, they will be half the time above the horizon and half thetime below it. The moon would be visible for about a fortnight at atime, and the sun for six months.
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7.—The Oblique Sphere and Diurnal Circles. The Oblique Sphere (illo007Fig. 7).—At any station between thepole and equator the stars will move in circles oblique to the horizon,SENW in the figure. On the other hand, stars within the same distance of thedepressed pole will lie within the “Circle of Perpetual Occultation,”and will never rise above the horizon. A star exactly on the celestial equator will have its diurnal circleEQWQ' bisected by the horizon, and will be above the horizon justas long as below it. The north pole of the globe must be elevated to an angle equal to the latitude of the observer, which can be done by means of the degrees marked on the brass meridian. CHAPTERIIIAstronomical InstrumentsASTRONOMICAL INSTRUMENTS. Sometimes we wish merely to examine its surface, tomeasure its light, or to investigate its spectrum; and for all thesepurposes special instruments have been devised. We propose in this chapter to describe very briefly a few of themost important.35.
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Telescopes in General.—Telescopes are of two kinds, refractingand reflecting. The former were first invented, and are muchmore used, but the largest instruments ever made are reflectors. Inboth the fundamental principle is the same. In the form of telescope, however, introduced by Galileo,[In strictness, Galileo did not invent the telescope. Its first invention seems to have been in 1608, by Lipperhey, a spectacle-maker of Middleburg, in Holland; though the honor has also been claimed for two or three other Dutch opticians. But on account of the smallness of thefield of view, and other objections, this form of telescope is never used whenany considerable power is needed. Simple Refracting Telescope.—This consists essentially asshown in the figure (illo008Fig. 8), of a tube containing two lenses: a singleconvex lens, A, called the object-glass; and another, of smaller sizeand short focus, B, called the eye-piece.
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The more exact theory of Gauss and later writers would require some slight modifications in our statements, but none of any material importance. For a thorough discussion, see Jamin, “Trait� de Physique,” or Encyc. Britannica,—Optics. 8.—Path of the Rays in the Astronomical Telescope. Similarlywith the rays which meet at b. The observer, therefore, willsee the top of the moon's disc in the direction ck, and the bottom inthe direction cl. It will appear to him inverted, and greatly magnified;its apparent diameter, as seen by the naked eye and measuredby the angle aob (or its equal b_0oa_0); having been increased to acb. Object and image subtend equal angles. Distinctness of Image.—This depends upon the exactnesswith which the lens gathers to a single point in the focal image allthe rays which emanate from the corresponding point in the object.
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A single lens, with spherical surfaces, cannot do this very perfectly,the “aberrations” being of two kinds, the spherical aberration andthe chromatic. The former could be corrected, if it were worth while,by slightly modifying the form of the lens-surfaces; but the latter,which is far more troublesome, cannot be cured in any such way. The object-glass was mountedat the top of a high pole and the eye-piece was on a separate stand below. With such an “aerial telescope,” of six inches aperture and 120 feet focus,Huyghens discovered the rings of Saturn. His object-glass still exists, andis preserved in the collection of the Royal Society in London. The convex lens is usually made of crown glass, theconcave of flint glass. r0ptt]c 1 illo009a Clark t]c 2 illo009b Gauss t]c 3 illo009c Littrow *Fig. 9.—Different Forms of the Achromatic Object-glass. These object-glasses admit of considerable variety of forms. Formerlythey were generally made, as in illo009cFig.
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9, No. 3, having the two lenses closetogether, and the adjacent surfaces of the same, or nearly the same, curvature. In small object-glasses the lenses are often cemented together withCanada balsam or some other transparent medium. In all these forms the crown glass is outside;Steinheil, Hastings, and others have constructed lenses with the flint-glasslens outside. Secondary Spectrum.—It is not, however, possible with thekinds of glass at present available to secure a perfect correction of thecolor. Our best achromatic lenses bring the yellowish green rays toa focus nearer the lens than they do the red and violet. In consequence,the image of a bright star is surrounded by a purple halo,which is not very noticeable in a good telescope of small size, butis very conspicuous and troublesome in a large instrument. This imperfection of achromatism makes it unsatisfactory to use an ordinarylens (visually corrected) for astronomical photography.
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A 13-inch object-glass of this constructionat Cambridge performs admirably. Much is hoped from the new kind of glass now being made at Jena. The size of this disc-and-ring system can becalculated from the known wave-lengths of light and the dimensionsof the lens, and the results agree very precisely with observation. The diameter of the “spurious disc” varies inversely with the apertureof the telescope. According to Dawes, it is about 4”.5 for a1-inch telescope; and consequently 1” for a 41/2-inch instrument, 0”.5for a 9-inch, and so on. This circumstance has much to do with the superiority of large instrumentsin showing minute details. If, however, the magnifying poweris more than about 50 to the inch of aperture, the edge of the disc will beginto appear hazy. There is seldom any advantage in the use of a magnifyingpower exceeding 75 to the inch, and for most purposes powers ranging from20 to 40 to the inch are most satisfactory.44.
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Usually it is best to employ for the eye-piecea combination of two or more lenses which will give a more extensivefield of view. Eye-pieces belong to two classes, the positive and the negative. Theformer, which are much more generally useful, act as simple magnifying-glasses,and can be used as hand magnifiers if desired. The focalimage formed by the object-glass lies outside of the eye-piece. In the negative eye-pieces, on the other hand, the rays from theobject-glass are intercepted before they come to the focus, and theimage is formed between the lenses of the eye-piece. Such an eye-piececannot be used as a hand magnifier. The simplest and most common forms of these eye-pieces are theRamsden (positive) andHuyghenian (negative). Each is composed of twoplano-convex lenses, butthe arrangement andcurves differ, as shownin illo010aFig. It is freefrom all internal reflections, which in other eye-pieces often produce “ghosts,”as they are called. r0ptb]cRamsden (Positive)
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illo010a b]cHuyghenian(Negative) illo010b b]cSteinheil `Monocentric'(Positive)illo010c*Fig. 10.—Various Forms of Telescope Eye-piece. There are numerous other forms of eye-piece, each with its own advantagesand disadvantages. The erecting eye-piece, used in spy-glasses, isessentially a compound microscope, and gives erect vision by again invertingthe already inverted image formed by the object-glass. It is obvious that in a telescope of any size the object-glass is the mostimportant and expensive part of the instrument. Its cost varies from a fewhundred dollars to many thousands, while the eye-pieces generally cost onlyfrom $5 to $20 apiece. Reticle.—When a telescope is used for pointing, as in mostastronomical instruments, it must be provided with a reticle of somesort. As spider-threads are veryfragile, and likely to get broken or displaced, it is often better to substitutea thin plate of glass with lines ruled upon it and blackened.
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Of course,provision must be made for illuminating either the field of view or thethreads themselves, in order to make them visible in darkness. r0pt1 illo011a2 illo011b3 illo011c*Fig. 11.—Different Forms of Reflecting Telescope. 1. The Herschellian; 2. The Newtonian; 3. With this instrument one looks directly at thestars as with a refractor, and the image is erect. Formerly the greatmirror was always madeof a composition of copperand tin (two partsof copper to one of tin)known as “speculummetal.” At present it isusually made of glasssilvered on the front surface,by a chemical processwhich deposits themetal in a thin, brilliantfilm. These silver-on-glassreflectors, when new,reflect much more lightthan the old specula, butthe film does not retain its polish so long. It is, however, a comparativelysimple matter to renew the film when necessary. The largest telescopes ever made have been reflectors.
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At the head of thelist stands the enormous instrument of Lord Rosse, constructed in 1842, witha mirror six feet in diameter and sixty feet focal length. Relative Advantages of Refractors and Reflectors.—There hasbeen a good deal of discussion on this point, and each construction has itspartisans. In favor of the reflectors we may mention,—First. Ease of construction and consequent cheapness. The concave mirrorhas but one surface to figure and polish, while an object-glass has four. This makes it vastly easier to get the material for a largemirror than for a large lens. Second (and immediately connected with the preceding). The possibilityof making reflectors much larger than refractors. Lord Rosse's great reflectoris six feet in diameter, while the Lick telescope, the largest of all refractors,is only three.Third. This is unquestionably a very great advantage,especially in photographic and spectroscopic work.
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But, on the whole, the advantages are generally considered to lie with therefractors. In their favor we mention:—First. Great superiority in light. No mirror (unless, perhaps, a freshlypolished silver-on-glass film) reflects muchmore than three-quarters of the incidentlight; while a good (single) lens transmitsover 95 per cent. In a good refractorabout 82 per cent of the lightreaches the eye, after passing throughthe four lenses of the object-glass andeye-piece. In a Newtonian reflector, inaverage condition, the percentage seldomexceeds 50 per cent, and morefrequently is lower than higher. illo012Fig. 12.—Effect of Surface Errors in a Mirror and in a Lens. But since the index of refraction of glass is about 1.5 thechange in the direction of the refracted ray from R to r will only be abouttwo-thirds of aPb.
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Moreover, so far as distortions are concerned, when a lens bends a littleby its own weight, both sides are affected in a nearly compensatory manner,while in a mirror there is no such compensation. As a consequence, mirrorsvery seldom indeed give any such definition as lenses do. The lens, once made, and fairly taken care of,suffers no deterioration from age; but the metallic speculum or the silverfilm soon tarnishes, and must be repolished every few years. This aloneis decisive in most cases, and relegates the reflector mainly to the use ofthese who are themselves able to construct their own instruments. Time-Keepers and Time-Recorders.—The Clock, Chronometer,and Chronograph.—Modern practical astronomy owes its developmentas much to the clock and chronometer as to the telescope. It is true that the Arabian astronomer Ibn Jounis had made someuse of the pendulum about the year 1000 a.d., more than 500 yearsbefore Galileo introduced it to Europeans.
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But it was not untilnearly a century after Galileo's discovery that Huyghens applied itto the construction of clocks (in 1657). So far as the principles of construction are concerned, there is nodifference between an astronomical clock and any other. Of course it is constructed with extreme care inall respects. The pendulum itself is usually suspended by a flat spring, andgreat pains should be taken to have the support extremely firm: thisis often neglected, and the clock then cannot perform well. Compensation for Temperature.—In order to keep perfect time,the pendulum must be a “compensation pendulum”; i.e., constructed in such a waythat changes of temperature willnot change its length. An uncompensated pendulum, with steel rod,changes its daily rate about one-third of a secondfor each degree of temperature (centigrade). A wooden pendulum rod is much less affectedby temperature, but is very apt to be disturbedby changes of moisture. 13.Compensation Pendulums.
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Graham's mercurial pendulum (illo013Fig. 13) is theone most commonly used. It consists simply of ajar (usually steel), three or four inches in diameter,and about eight inches high, containing forty or fiftypounds of mercury, and suspended at the end of asteel rod. When the temperature rises, the rodlengthens (which would make the clock go slower);but, at the same time, the mercury expands, fromthe bottom upwards, just enough to compensate. This pendulum will perform well only when notexposed to rapid changes of temperature. Underrapid changes the compensation lags. A compensation pendulum, constructed on theprinciple of the old gridiron pendulum of Harrison,but of zinc and steel instead of brass and steel, isnow much used. The heavy pendulum-bob, a lead cylinder, is hungat the end of a steel rod, which is suspended from the top of a zinc tube,and hangs through the centre of it.
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This tube is itself supported at the bottomby three or four steel rods which hang from a piece attached to the pendulumspring. The standard clock at Greenwich has a pendulum of this kind.52. It varies considerably, however, with differentpendulums. In theGreenwich clock a magnet is raised or lowered by the rise or fall of themercury in a barometer attached to the clock-case. There are several other contrivances for the same purpose. Therate of a clock is the amount of its daily gain or loss; plus (+) whenthe clock is losing. Sometimes the hourly rate is used, but “hourly”is then always specified. A perfect clock is one that has a constant rate, whether that ratebe large or small. They can be dropped into place or knocked off withoutstopping the clock or perceptibly disturbing it. But this isexceptional performance. In a run as long as that, most clocks wouldbe liable to change their rate as much as half a second or more, and to doit somewhat irregularly.
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The Chronometer.—The pendulum-clock not being portable,it is necessary to provide time-keepers that are. It usuallybeats half-seconds. It is not possible to secure in the chronometer-balanceas perfect a temperature correction as in the pendulum. Theyare simply indispensable at sea. Never turn the hands of a chronometerbackward. 14.—A Chronograph by Warner and Swasey.55. Before the invention of the telegraph it was customary to notetime merely “by eye and ear.” The Chronograph.—This is the instrument which carries themarking-pen and moves the paper on which the time-record is made. The paper is wrapped upon a cylinder, six or seven inches in diameter,and fifteen or sixteen inches long. This cylinder is made to revolveonce a minute, by clock-work, while the pen rests lightly uponthe paper and is slowly drawn along by a screw-motion, so that itmarks a continuous spiral. 15.—Part of a Chronograph Record.
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The beginning of a new minute (the 60th sec.) is indicated eitherby a double mark as shown, or by the omission of a mark. Of course the minutes when the chronograph wasstarted and stopped are noted by the observer on the sheet, and soenable him to identify the minutes and seconds all through the record. Many European observatories use chronographs in which the record ismade upon a long fillet of paper, instead of a sheet on a cylinder. Theinstrument is lighter and cheaper than the American form, but much lessconvenient. The regulator of the clock-work must be a “continuous” regulator, workingcontinuously, and not by beats like a clock-escapement. Clock-Breaks.—The arrangements by which the clock is made tosend regular electric signals are also various. One of the earliest and simplestis a fine platinum wire attached to the pendulum, which swings through adrop of mercury at each vibration. Clocks with thegravity escapements have a decided advantage in this respect.
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Usually a wheel on theaxis of the scape-wheel is made to give the electric signals by touching alight spring with one of its teeth every other second. Chronometers are now also fitted up in the same way, to be used with thechronograph. The signals sent are sometimes “breaks” in a continuous current, andsometimes “makes” in an open circuit. Usage varies in this respect, andeach method has its advantages. Meridian Observations.—A large proportion of all astronomicalobservations are made at the time when the heavenly bodyobserved is crossing the meridian, or very near it. The Transit Instrument is the instrument used, in connectionwith a clock or chronometer, and often with a chronograph also, toobserve the time of a star's “transit” across the meridian. Vice versa, if the right ascension is known, the error or correctionof the clock will be determined. The instrument (illo016Fig. 16) consists essentially of a telescope mountedupon a stiff axis perpendicular to the telescope tube.
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This axis isplaced horizontal, east and west, and turns on pivots at its extremities,in Y-bearings upon the top of two fixed piers or pillars. Asmall graduated circle is attached, to facilitate “setting” the telescopeat any designated altitude or declination. 16.—The Transit Instrument (Schematic). One ortwo wires also cross the field horizontally. Theobject in having a number of wires is, of course, simply to gainaccuracy by taking the mean of a number of observations instead ofdepending upon a single one. illo017Fig. 17.—Reticle of the Transit Instrument. The proper construction and grinding ofthese pivots, which are usually of hard bell metal (sometimes of steel),taxes the art of the most skilful mechanician. The level, also, is a delicateinstrument, and difficult to construct. Provision is made, of course, for illuminating the field of view at nightso as to make the reticle wires visible.
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Usually one (or both) of the pivotsis pierced, and a lamp throws light through the opening upon a small mirrorin the centre of the tube, which reflects it down upon the reticle. The Y's are used instead of round bearings, in order to prevent anyrolling or shake of the pivots as the instrument turns. Fig. 18 shows a modern transit instrument (portable) as actually constructedby Fauth & Co. Another form of the instrument is much used, which is oftendesignated as the “Broken Transit.” It is veryconvenient and rapid in actual work, but the observations require aconsiderable correction for flexure of the axis. Focus and verticality of wires. The first thing to do after the instrument is set on its supports andthe axis roughly levelled, is to adjust the reticle. If this adjustment is correctly made, motion ofthe eye in front of the eye-piece will not produce any apparent displacementof the object in the field, with reference to the wires.
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There are screws provided to turn the reticle alittle, so as to effect this adjustment. illo018Fig. 18.—A 3-inch Transit, with reversing apparatus. When the wires have been thus adjusted for focus and verticality, thereticle-slide should be tightly clamped and never disturbed again. The eye-piececan be moved in and out at pleasure, to secure distinct vision for differenteyes, but it is essential that the distance between the object-glass and thereticle remain constant. If the middlewire, after reversal, points just as it did before, the “collimation” is correct;if not, the middle wire must be moved half way towards the object by thescrews. Collimator.—It is not always easy to find a distant object on which tomake this adjustment, and a “collimator” may be substituted with advantage. The instrument furnishes us a mark optically celestial, butmechanically within reach of our finger-ends for illumination, adjustment,etc.
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If the pier on which it is mounted is firm, the collimator cross is in allrespects as good as a star, and much more convenient. The adjustment for level is made by setting a stridinglevel on the pivots of the axis, reading the level, then reversing the level(not the transit) and reading it again. If the pivots are round and of thesame size, the difference between the level-readings direct and reversed willindicate the amount by which one pivot is higher than the other. One ofthe Y's is made so that it can be raised and lowered slightly by means of ascrew, and this gives the means of making the axis horizontal. If thepivots are not of the same size (and they never are absolutely), the astronomermust determine and allow for the difference.Fourth. In order that the instrument may indicate the meridiantruly, its axis must lie exactly east and west; i.e., its azimuth must be 90�.
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This adjustment must be made by means of observations upon the stars, andis an excellent example of the method of successive approximations, whichis so characteristic of astronomical investigation. (a) After adjusting carefullythe focus and collimation of the instrument, we set it north and southby guess, and level it as precisely as possible. By looking at the pole star,and remembering how the pole itself lies with reference to it, one can easilyset the instrument pretty nearly; i.e., within half a degree or so. The middlewire will now describe in the sky a vertical circle, which crosses the meridianat the zenith, and lies very near the meridian for a considerable distanceeach side of the zenith. (b) We must next get an “approximate” time; i.e., set our clock orchronometer nearly right. We nowhave the time within a second or two. (c) Next turn down the telescope upon some Almanac star, which issoon to cross the meridian within 10� of the pole. It will appear to movevery slowly.
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A repetition of the operation may possibly be needed to secure all thedesired precision. The accuracy of this azimuth adjustment can then beverified by three successive “culminations” or transits of the pole star, orany other circumpolar. The final test of all the adjustments, and of the accurate goingof the clock, is obtained by observing a number of Almanac stars ofwidely different declination. 19.—The Meridian Circle (Schematic). Nor are observationsever absolutely accurate. It can be more nearly donethan one might suppose. But the discussion of the subject belongs toPractical Astronomy, and cannot be entered into here. It is then called a Prime Vertical Transit. The Meridian Circle.—Inorder to determine the Declinationor Polar Distance of anobject, it is necessary to havesome instrument for measuringangles; mere time-observationswill not suffice.
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The instrumentmost used for this purpose is theMeridian Circle, or Transit Circle,which is simply a transit instrument,with a graduated circleattached to its axis, and revolvingwith the telescope. Sometimesthere are two circles, oneat each end of the axis. Fig. 19 represents the instrument“schematically,” showing merely the essential parts. illo020Fig. 20is a meridian circle, with a 4-inch telescope, constructed by Fauth& Co.illo020Fig. =2em A, B, C, D, the Reading Microscopes. =2em K, the Graduated Circle. =2em H, the Roughly Graduated Setting Circle. =2em I, the Index Microscope. This is usually, however, placed half way between A and D. =2em F, the Clamp. G, the Tangent Screw.=2em LL, the Level, only placed in position occasionally. =2em M, the Right Ascension Micrometer. =2em WW, Counterpoises, which take part of the weight of the instrument off from the Y's.
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In acircle of forty inches diameter, 1” is a little less than 1/10000 of an inch,(20/206265 inch), so that the necessity of fine workmanship is obvious. The Reading Microscope (illo021Fig. 21).—This consists essentiallyof a compound microscope, which forms a magnified image ofthe graduation at the focus of its object-glass, where this image isviewed by a positive eye-piece. One revolution of the screwcarries the wire 1' of arc, which makesone division of the screw-head 1”, thetenths of seconds being estimated. 21.—The Reading Microscope. The adjustment of the microscope for“runs,” as it is called (that is, to make onerevolution of the micrometer screw exactlyequal to 1'), is effected as follows. If they overrun, it shows that the imageof the graduation formed by the microscope objective is too small to fit thescrew, and vice-versa. The reading of the circle is as follows: An extra index-microscope,with low power and large field of view, shows by inspection the degreesand minutes.
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Thus in illo022Fig. 22, the reading of the microscope is3' 22”.1, the 3' being given by the scale in the field, the 22”.1 by thescrew-head. illo022Fig. 22.—Field of View of Reading Microscope. Method of observing a Star.—A minute or two before the starreaches the meridian the instrument is approximately pointed, so thatthe star will come into the field of view. The microscopes are then read, and their mean result isthe star's “circle-reading.” Frequently the star is bisected, not by moving the whole instrument, butby means of a “micrometer wire,” which moves up and down in the field ofview. The micrometer reading then has to be combined with the readingof the microscope, to get the true circle-reading. Zero Points.—In determining the declination or meridianaltitude of a star by means of its circle-reading, it is necessary toknow the “zero point” of the circle.
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For declinations, the “zeropoint” is either the polar or the equatorial reading of the circle; i.e.,the reading of the circle when the telescope is pointed at the poleor at the equator. The “polar point” may be found by observing some circumpolarstar above the pole, and again, twelve hours later, below it. Whenthe two circle-readings have been duly corrected for refraction andinstrumental errors, their mean will be the polar point. The polar distance of the star would be the half-difference of thetwo readings, or 3� 23' 24”.8.67. Nadir Point.—The determination of the polar point requirestwo observations of the same star at an interval of twelve hours. It isoften difficult to obtain such a pair; moreover, the refraction complicatesthe matter, and renders the result less trustworthy. Accordinglyit is now usual to use the nadir or the horizontal reading as thezero, rather than the polar point. The Collimating Eye-Piece.
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This peculiar illuminationis commonly effected by means of Bohnenberger's “collimatingeye-piece,” shown in illo023Fig. 23. In the simplest form it is merely a commonRamsden eye-piece, with a hole in one side, and a thin glass plate insertedat an angle of 45�. A light from one side, entering through the hole, will be(partially) reflected towards the wires, and will illuminate them sufficiently. The horizontal point of course differs just 90� from the nadir point. But the method of the collimating eye-piece is fully as accurateand vastly more convenient. illo024Fig. 24.—Altitude and Azimuth Instrument. The meridian circle is said to be used “differentially”when thus treated. The outstandingerrors ought not to exceed a second or two. But when thetenths of a second are in question, the case is different. Mural Circle.—This instrument is in principle the same as themeridian circle, which has superseded it.
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It consists of a circle, carrying atelescope mounted on the face of a wall of masonry (as its name implies)and free to revolve in the plane of the meridian. The wall furnishes a convenientsupport for the microscopes.illo025Fig. 25. The Equatorial (Schematic). Altitude and Azimuth Instrument.—Since the transit instrumentand meridian circle are confined to the plane of the meridian,their usefulness is obviously limited. Meridian observations, whenthey are to be had, are better and more easily used than any others,but are not always attainable. We must therefore have instrumentswhich will follow an object to any part of the heavens. The altitude and azimuth instrument is simply a surveyor's theodoliteon a large scale. It has a horizontal circle turning upon a verticalaxis, and read by verniers or microscopes. Upon this circle, andturning with it, are supports which carry the horizontal axis of thetelescope with its vertical circle, also read by microscopes.
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Obviouslythe readings of these two circles, when the instrument is properlyadjusted and the zero points determined, will give the altitudeand azimuth of the body pointed on. illo024Fig. 24 represents a small instrumentof this kind.illo026Fig. 26.—The 23-inch Princeton Telescope. Sometimes, also, it iscalled the Right Ascension Circle. Uponthis polar-axis are secured the bearingsof the declination axis, which is perpendicularto the polar axis, and carries thetelescope itself and the declination circle. Fig. 25 exhibits schematically the ordinary form of equatorialmounting, of which there are numerous modifications. illo026Fig. 26 is the23-inch Clark telescope at Princeton, and illo027Fig. 27 is the 4-footMelbourne reflector. The frontispiece is the great Lick telescopeof thirty-six inches diameter. 27.—The Melbourne Reflector. The advantages of the equatorial mounting for a large telescopeare very great as regards convenience.
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This motion, since it is uniform,can be, and in all large instruments usually is, given by clock-work,with a continuous regulator of some kind, similar to that usedin the chronograph. The instrument once directed and clamped,and the clock-work started, the object will continue apparently immovablein the field of view as long as may be desired. This hour-angle, it will be remembered, is simply the difference betweenthe sidereal time and the right ascension of the object. The hour-angleis east if the right ascension exceeds the time; west, if it is less. When the telescope is thus set, the object will be found (with a low magnifyingpower) in the field of view, unless it is near the horizon, in whichcase refraction must be taken into account. 28.—The Filar Position-Micrometer. The Micrometer.—Micrometers of various sorts are employedfor the purpose.
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The most common and most generally useful is theso-called “filar position-micrometer,” illo028Fig. 28, which is an indispensableauxiliary of every goodtelescope. It is a small instrument, muchlike the upper part of the readingmicroscope, but more complicated. It usually contains areticle of fixed wires, two orthree parallel to each other, andcrossed at right angles by asecond set. This “position angle” is read on agraduated circle, which forms part of the instrument. Means ofillumination are provided, giving at pleasure either dark wires in abright field, or vice versa. illo029Fig. 29.—Construction of the Micrometer. With this instrument one can measure the distance (in seconds ofarc), and the direction between any two stars which are near enoughto be seen at once in the same field of view. A new form of equatorial, known as theEquatorial Coud�, or Elbowed Equatorial, has beenrecently introduced at the Paris Observatory.
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Moreover, the revolving dome, which isusually erected to shelter a great telescope, is an exceedinglycumbrous and expensive affair. In the Equatorial Coud�, illo030Fig. 30, these difficulties are overcome bythe use of mirrors. The observer sits always in one fixed position,looking obliquely down through the polar axis, which is also thetelescope tube. The Paris instrument has an object-glass about ten inches indiameter, and performs very satisfactorily. The two reflections,however, cause a considerable loss of light, and some injury to thedefinition. The mirrors, and the consequent complications, also addheavily to the cost of the instrument. 30 is from a photographof this instrument. 30.—The Equatorial Coud�.75. All the instruments so far described, except the chronometer,are fixed instruments; of use only when they can be set upfirmly and carefully adjusted to established positions. Not one ofthem would be of the slightest use on shipboardship-board.
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We have now to describe the instrument which, with the help ofthe chronometer, is the main dependence of the mariner. The Sextant.—The graduated limb of the sextant is carriedby a light framework, usually of metal, provided with a suitable handleX. The arc is about one-sixth of a circle, as the name implies, andis usually from five to eight inches radius. The best instruments read to 10”. Just over the centre of motion, the “index-mirror” M, abouttwo inches by one and one-half in size, is fastened securely to theindex-arm, so as to be perpendicular to the plane of the limb. Only half of the horizon-glass is silvered, the upper half being lefttransparent. When the two mirrors areparallel, and the vernier reads zero, the two images coincide, providedthe object is at a considerable distance. The principal use of the instrument is in measuring the altitudeof the sun. On land the visible horizon is of no use, and we have recourse to an“artificial horizon,” as it is called.
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This is merely a shallow basin of mercury,covered, when necessary to protect it from the wind, with a roof made ofglass plates having their sides plane and parallel. In this case we measure the angle between the sun's image reflected in themercury and the sun itself. But its portability andapplicability at sea render it absolutelyinvaluable. First, from the law of reflection, we have, SMP = HMP, or Similarly, MHE = 2 � MHQ. Similarly, from the triangle HMQ, we have HQM = PMH - MHQ,which is half the value just found for HEM, and proves the proposition. Besides the instruments we have described, there are manyothers designed for special work, some of which, as the zenith telescope,and heliometer, will be mentioned hereafter as it becomesnecessary. There is also a whole class of physical instruments,photometers, spectroscopes, heat-measuring appliances, and photographicapparatus, which will have to be considered in due time.
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But with clock, meridian circle, and equatorial and their usualaccessories, all the fundamental observations of theoretical and sphericalastronomy can be supplied. The chronometer and sextant arepractically the only astronomical instruments of any use at sea. CHAPTERIIIIICorrections to Astronomical ObservationsCORRECTIONS TO ASTRONOMICAL OBSERVATIONS, DIP OF THEHORIZON, PARALLAX, SEMI-DIAMETER, REFRACTION, ANDTWILIGHT. The amount of this dipdepends upon the size of the earthand the height of the observer's eye abovethe sea-level.illo033Fig. 33.—Dip of the Horizon. In illo033Fig. 33, C is the centre of the earth,AB a portion of its level surface, and O theobserver, at an elevation h above A. Theline OH is truly horizontal, while the tangentline, OB, corresponds to the line drawnfrom the eye to the visible horizon. This is obviously equal to the angle OCBat the centre of the earth, if we regard the earth as spherical, as wemay do with quite sufficient accuracy for the purpose in hand.
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From the right-angled triangle OBC we have directly cos OCB = BC/CO. Putting R for the radius of the earth, and Δ for the dip, this becomes cosΔ = R/R + h. This formula is exact, but inconvenient, because it gives the small angleΔ by means of its cosine. Since, however, 1- cosΔ = 2 sin^2 1/2Δ, we easilyobtain the following:— sin12Δ = √(h/2(R + h)). This gives the true depression of the sea horizon, as it would be if theline of sight, drawn from the eye to the horizon line, were straight. Onaccount of refraction it is not straight, however, and the amount of this“terrestrial refraction” is very variable and uncertain. It is usual todiminish the dip computed from the formula by one-eighth its whole amount. An approximate formula[ This approximate formula may be obtained thus:— 2 sin^2 (12Δ) = h/R+h = ( h/R) + ( 1 + h/R).
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But since hR is a very small fraction, it may be neglected in the divisor ( 1 + hR), and the expression becomes simply, 2 sin^2 12Δ = h/R; whence sin12Δ = √(h/2R). Since Δ is a very small angle, Δ = sinΔ = 2 sin12Δ, so that Δ (in radians) = 2 √(h/2R) = √(h/1/2R). To reduce radians to minutes, we must multiply by 3438, the number of minutes in a radian. Accordingly, Δ' (in minutes of arc) = 3438 √(h/1/2R). In fact, the refraction makes so much difference that after taking the numerical factor, 34383231, as unity, the formula still gives Δ' about 1/20 part too large. The formula Δ' = √( 3 h (metres)) is yet more nearly correct. ] for the dip is Δ (in minutes of arc) = √(h (feet));or, in words, the square root of the elevation of the eye (in feet) givesthe dip in minutes. This gives a value about 1/20 part too large.
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Parallax.—In the most general sense, “parallax” is the changeof a body's direction resulting from the observer's displacement. Its position as seen from C would be determined in thesame way by producing CP to which OX is drawn parallel. Theangle POX, therefore, or its equal, OPC, is the parallax of P foran observer at O.illo034Fig. Obviously, from the illo034figure, we may also give the following definitionof the parallax. It is the angulardistance (number of seconds ofarc) between the observer's station andthe centre of the earth's disc, as seenfrom the body observed. The moon'sparallax at any moment for me is myangular distance from the earth's centre,as seen by “the man in the moon.” When a body is in the zenith itsparallax is zero, and it is a maximumat the horizon. In all cases itdepresses a body, diminishing thealtitude without changing the azimuth.
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The “law” of the parallax is, that it varies as the sine of the zenith distancedirectly, and inversely as the linear distance (in miles) of the body. This gives us sinp = R/D ·sinζ; This gives us or, since p is always a small angle, p” = 206265” R/D ·sinζ. 83. Horizontal Parallax.—When a body is at the horizon (P_h inthe illo034figure), then ζ becomes 90�, and sinζ = 1. In this case the parallaxreaches its maximum value, which is called the horizontal parallaxof the body. Taking p_h as the symbol for this, we have sin p_h A glance at the illo034figure will show that we may define thehorizontal parallax, OPC, of any body, as the angular semi-diameterof the earth seen from that body. To say, for instance, that the sun'shorizontal parallax is 8”.8, amounts to saying that, seen from the sun,the earth's apparent diameter is twice 8”.8, or 17”.6.84. Relation between Horizontal Parallax and Distance.—Since we have sin p_h
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= R/D,it follows of course that D = R � sin p_h; or, (nearly) D = 206265”/p_h” � R. or, (nearly) If the sun's parallax equals 8”.8, = 206265/8.8 � R = 23439 R.85. Itis agreed to take as the standard the equatorial horizontal parallax;i.e., the earth's equatorial semi-diameter as seen from the body. Parallax.—The parallax we have been discussing issometimes called the diurnal parallax, because it runs through all itspossible changes in one day. When the sun, for instance, is rising, its parallax is a maximum, and bythrowing it down towards the east, increases its apparent right ascension. At noon, when the sun is on the meridian, its parallax is a minimum, andaffects only the declination. At sunset it is again a maximum, butnow throws the sun's apparent place down towards the west. Althoughthe sun is invisible while below the horizon, yet the parallax,geometrically considered, again becomes a minimum atmidnight, regaining its original value at the next sunrise.
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The qualifier, “diurnal,” is seldom used except when it isnecessary to distinguish between this kind of parallax and theannual parallax of the fixed stars, which is due to theearth's orbital motion. Smallness of Parallax.—The horizontalparallax of even the nearest of the heavenly bodies isalways small. In the case of the moon the average value is about57', varying with her continually changing distance. Excepting nowand then a stray comet, no other heavenly body ever comes within adistance a hundred times as great as hers. Venus and Mars approachnearest, but the parallax of neither of them ever reaches 40”. Semi-Diameter.—In order to obtain the truealtitude of an object it is necessary, if the edge, or“limb,” as it is called, has been observed, to add ordeduct the apparent semi-diameter of the object.
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In most cases thiswill be sensibly the same in all parts of the sky, but the moon isso near that there is quite a perceptible difference between herdiameter when in the zenith and in the horizon. A glance at illo034Fig. 34 shows that in the zenith the moon's distance isless than at the horizon, by almost exactly the earth's radius—thedifferencebetween the lines OZ and OP_h. Now this is very nearly one-sixtieth partof the moon's distance, and consequently the moon, on a night whenits apparent diameter at rising is 30', will be 30”larger when near the zenith. This measurable increase of the moon's angular diameter at highaltitudes has nothing to do with the purely subjective illusion whichmakes the disc look larger to us when near the horizon. That it is amere illusion may be made evident by simply looking through a darkglass just dense enough to hide the horizon and intervening landscape. The moon or sun then seems to shrink at once to normaldimensions.
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illo035Fig. 35.—Atmospheric Refraction. So far asthe action is regular, theeffect is to bend the raysdirectly downwards, andthus to make the objectsappear higher in the sky. Refraction increases thealtitude of a celestial objectwithout altering theazimuth. Like parallax,it is zero at the zenithand a maximum at thehorizon; but it follows adifferent law. It is entirely independent of the distance of theobject, and its amount varies (nearly) as the tangent of the zenithdistance—not as the sine, as in the case of parallax.90. This approximate law of the refraction is easily proved. Suppose in illo035Fig. 35 that the observer at O sees a star in the direction OS,at the zenith distance ZOS or ζ. The light has reached him from S' by apath which was straight until the ray met the upper surface of the air at A,but afterwards curved continually downwards as it passed from rarer todenser regions.
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We may therefore apply the law of refraction directly at A, and writesin Z'A S' = n sin BAC ( = ZOS ), or sin( ζ +r ) = n sinζ;AC being drawnparallel to OS. Developing the first member, we have sinζcosr + cosζsinr = n sinζ. But r is always a small angle, never exceeding 40'; we may therefore takecos r = 1. Doing this and transposing the first term, we get cosζsinr= n sinζ- sinζ= ( n - 1 ) sinζ. Whence, sinr = (n- 1) tanζ; or, r” =(n-1) 206265 tanζ (nearly). The index of refraction for air, at zero centigrade and a barometricpressure of 760^mm, is 1.000294; whence, r” = .000294 � 206265 � tanζ= 60”.6 tanζ. This equation holds very nearly indeed down to a zenith distanceof 70�, but fails as we approach the horizon. At the horizon, where ζ = 90� and tanζ = infinity, the formulawould give sin r = infinity also; an absurdity, since no sine canexceed unity.
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The refraction there is really about 37', under thecircumstances of temperature and pressure above indicated. Effect of Temperature and Barometric Pressure.—The indexof refraction of air depends of course upon its temperature and pressure. As the air grows warmer, its refractive power decreases; as it growsdenser, the refraction increases. Hence, in all precise observations ofthe altitude (or zenith distance), it is necessary to note both thethermometer and the barometer, in order to compute the refractionwith accuracy. For rough work, like ordinary sextant observations,it will answer to use the “mean refraction,” corresponding to anaverage state of things. No amount of care in observation can evade this difficulty; the only remedyis a sufficient repetition of observations under varying atmospheric conditions. Observations at an altitude below 10� or 15� are never much to betrusted.
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Lateral Refraction.—When the air is much disturbed, sometimes objectsare displaced horizontally as well as vertically. Indeed, as a generalrule, when one looks at a star with a large telescope and high power, it willseem to “dance” more or less—the effect of the varying refraction whichcontinually displaces the image. Of course, sunset is delayed by the sameamount, and thus the day is lengthened by refraction from fourto eight minutes, at the expense of the night. Effect on the Form and Size of the Discs of the Sun and Moon.—Nearthe horizon the refraction changes very rapidly. This distorts the disc into the form of an oval,flattened on the under side. In cold weather the effect is much moremarked. Determination of the Refraction.—1. Theory furnishes the law of astronomical refraction, though themathematical expression becomes rather complicated when we attemptto make it exact. It is difficult, however, tomake these determinations with the necessary precision.
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In fact, atpresent our knowledge of the constants of air rests mainly on astronomicalwork.2. By Observations of Circumpolar Stars. At an observatory whoselatitude exceeds 45� select some star which passes through the zenithat the upper culmination. (Its declination must equal the latitude ofthe observatory.) It will not be affected by refraction at the zenith,while at the lower culmination, twelve hours later, it will. With themeridian circle observe its polar distance in both positions, determiningthe “polar point” of the circle as described on pp. 46–47. As a first approximation, however, we may neglect the refractionat the pole, and thus obtain a first approximate lower refraction. By means of this we may compute an approximate polar refraction,and so get a first “corrected polar point.” But it would never be necessary to go beyond the third, asthe approximation is very rapid.
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If the star does not go exactlythrough the zenith, it is only necessary to compute each time approximaterefractions for its upper observation, as well as for the polarpoint. 3. By Observations of the Altitudes of Equatorial Stars made at an Observatorynear the Equator. We have only, then, to observe the apparent zenith distance of astar with the corresponding time, and the refraction comes out directly. The French astronomer Loewy has recently proposed a method whichpromises well. He puts a pair of reflectors, inclined to each other at a convenientangle of from 45� to 50� (a glass wedge with silvered sides), in frontof the object-glass of an equatorial. Twilight, the illumination of the sky which begins before sunrise,and continues after sunset, is caused by the reflection of light to theobserver from the upper regions of the earth's atmosphere. Suppose the atmosphere to have the depth indicated in the illo036figure.
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Then, if the sun is at S, illo036Fig. 36, it will just have set to an observer at 1,but all the air within his range of vision will still be illuminated. Nothing remains in sight on whichthe sun is shining. Duration of Twilight.—This depends upon the height of the atmosphere,and the angle at which the sun's diurnal circle cuts the horizon. The time required to reach this point in latitude 40� varies from twohours at the longest days in summer, to one hour thirty minutes about Oct.12 and March 1, when it is least. At the winter solstice it is about onehour and thirty-five minutes. In higher latitudes the twilight lasts longer, and the variation is moreconsiderable; the date of the minimum also shifts. Near the equator the duration is shorter, hardly exceeding an hour at thesea-level; while at high elevations (where the amount of air above theobserver's level is less) At Quito and Lima it issaid not to last more than twenty minutes.
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Probably, also, in mountainregions the clearness of the air, and its purity contribute to the effect. 1 C � 9�, or R+H = R � 9�; whence H = R ( 9� - 1) = 0.0125 R, or almost exactly fifty miles. This mustbe diminished about one-fifth part on account of the curvature ofthe lines 1 m and 5 m by refraction, making the height of the atmosphereabout forty miles. The result must not, however, be accepted too confidently. There is abundant evidence from the phenomena of meteors that theatmosphere extends to a height of 100 miles at least, and it cannotbe asserted positively that it has any definite upper limit. Theaberration of light is an apparent displacement of the object observed, dueto the combination of the earth's orbital motion with the progressive motionof light. It can be better discussed, however, in a different connection (seeCHAPTERVIChap. VI.), and we content ourselves with merely mentioning it here.
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CHAPTERIIIIVProblems of Practical AstronomyPROBLEMS OF PRACTICAL ASTRONOMY, LATITUDE, TIME,LONGITUDE, AZIMUTH, AND THE RIGHT ASCENSION ANDDECLINATION OF A HEAVENLY BODY. The first of these problems is that of the§ LATITUDE. It mayalso be defined, from the mechanical point of view, as the angle betweenthe plane of the earth's equator and the observer's plumb-line or vertical. Determination of Latitude.—First:By Circumpolars. Each of the observations must be corrected for refraction,and then the mean of the two corrected altitudes will be the latitude. This method has the advantage of being an independent one; i.e., it doesnot require any data (such as the declination of the stars used) to be acceptedon the authority of previous observers. But to obtain much accuracyit requires considerable time and a large fixed instrument. In low latitudesthe refraction is also very troublesome.102. By the meridian altitude or zenith distance of abody of known declination.
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In illo037Fig. 37 the semicirclesemi-circle AQPB is the meridian, Q and P beingrespectively the equator and the pole, and Z the zenith. QZ isthe declination of the zenith, or the observer's latitude (=PB=ϕ). Suppose now that we observe Zs (=ζ_s), the zenith distance of astar s (south of the zenith), as it crosses the meridian, and that itsdeclination Qs (=δ_s) is known; then evidently ϕ = δ_s + ζ_s. In the same way, if the star were at n, between zenith and pole,ϕ = δ_n - ζ_n. But we have to get our stardeclinations out of catalogues made by previous observers, and so the methodis not an independent one.illo037Fig. 37.—Determination of Latitude. At Sea the latitude is usually obtained by observing with the sextantthe sun's maximum altitude, whichof course occurs at noon. Atfirst the altitude will keep increasing, but immediately after noon itwill begin to decrease.
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It would take us a littletoo far to explain the method of reduction, which is given with the necessarytables in all works on Practical Astronomy. The meridian-circle cannotbe used, as the instrument must be such as to make extra-meridianobservations possible. Usually the sextant or universal instrument isemployed. This method is much used in the French and German geodeticsurveys. After thisreversal and adjustment, the telescope tube is then evidently elevatedat exactly the same angle, ζ, as before, but on the opposite side of thezenith. From illo037Fig. 37 we have for star south of zenith, ϕ = δ_s + ζ_s; for star north of zenith, ζ_n.Adding the two equations and dividing by 2, we have ϕ = ( δ_s + δ_n/2) The star catalogue gives us the declinations of the two stars( δ_s + δ_n ); and the difference of the zenith distances ( ζ_s - ζ_n) is determinedwith great accuracy by the micrometer measures. illo039Fig. 39.—Latitude by Prime Vertical Transits.
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Evidently thelimit of accuracy depends upon the exactnesswith which the level measures the slight, butinevitable, difference between the inclinationsof the instrument when pointed on the two stars. Fifth:By the Prime Vertical Instrument (p. fig:illo020).—We observesimply the moment when a known star passes the prime vertical on theeastern side, and again upon the western side. Half the interval will givethe hour-angle of the star when on the prime vertical; i.e., the angle ZPSin illo039Fig. 39, where Z is the zenith, P the pole, and SZS' the prime vertical. The distance PS of the star from the pole is the complement of the star'sdeclination; and PZ is the complement of the observer's latitude. whenceIf δ nearly equals ϕ, t will be small, and a considerable error in the observationof t will then produce very little change in its secant or in the computedlatitude. It also entirely evades the difficulties caused by refraction. 40.—Latitude by the Gnomon.
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By the Gnomon.—The ancients had no instrumentssuch as we have hitherto described, and of course could not use anyof the preceding methods of finding the latitude. They were, however,able to make a very respectable approximation by means of thesimplest of all astronomical instruments, the gnomon. The height AB being given,we can easily compute inthe right-angled triangle theangle ABC, which equalsSBZ, the sun's zenith distancewhen farthest north. Again observe the length AD of theshadow at noon of the shortest day in winter, and compute the angleABD, which is the sun's corresponding zenith distance when farthestsouth. The method is an independent one, like that by the observation of circumpolarstars, requiring no data except those which the observer determinesfor himself. Evidently, however, it does not admit of much accuracy, sincethe penumbra at the end of the shadow makes it impossible to measure itslength very precisely.
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It is supposed, indeed known, that many of the Egyptian obelisks wereerected primarily to serve as gnomons, and were used for that purpose. 108. Possible Variations of the Latitude.—It is an interesting questionwhether the position of the earth's axis is fixed with reference to itsmass and surface. Theoretically it is hardly possible that it should be,because any change in the arrangement of the matter of the earth, bydenudation, subsidence, or elevation, would almost necessarily disturb it. If so disturbed, the latitudes of places toward which the pole approachedwould be increased, and those on the opposite side would be decreased. § TIME AND ITS DETERMINATION. One of the most important problems presented to the astronomeris the determination of Time. In practice, three kinds of time arenow recognized, viz., sidereal time, apparent solar time, and meansolar time. 0^h 00^m 00^s, at each transit of the vernal equinox.
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Apparent Solar Time.—Just as sidereal time is the hour-angleof the vernal equinox, so at any moment the apparent solar time isthe hour-angle of the sun. It is the time shown by the sun-dial, andits noon is when the sun crosses the meridian. December23d is fifty-one seconds longer from (apparent) noon to noon thanSept. 16th. For this reason, apparent solar time is not satisfactoryfor scientific use, and has long been discarded in favor of meansolar time. Until within about a hundred years, however, it was theonly kind of time commonly employed, and its use in the city of Pariswas not discontinued until the year 1816.112. It is mean noon when this “fictitioussun” crosses the meridian, and at any moment the hour-angleof this “fictitious sun” is the mean time for that moment. Sidereal time will not answer for business purposes, because its noon (thetransit of the vernal equinox) occurs at all hours of night and daylight indifferent seasons of the year.
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Apparent solar time is scientifically unsatisfactory,because of the variation in the length of its days and hours. Andyet we have to live by the sun; its rising and setting, daylight and night,control our actions. In mean solar time we find a satisfactory compromise,an invariable time unit, and still an agreement with sun-dial time close enoughfor convenience. It is the time now used for all purposes except in certainastronomical work. The nautical almanac furnishes data by means of which the siderealtime may be deduced from the corresponding solar, or vice versa, bya very brief and simple calculation.113. The methods most in use by astronomers are the following:—First. By means of the transit instrument. Practically,it is usual to observe a number of stars (from eight to ten),reversing the instrument once at least, so as to eliminate the collimationerror (Art.60).
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This personal equation differs for different observers, but is reasonably(though never strictly) constant for one who has had much practice. In thecase of observations with the chronograph, it is usually less than ± 0^s.2. If mean time is wanted, it can be deduced from the sidereal timeby the data of the almanac. The advantage of this method is that the errors of graduation ofthe sextant have no effect, nor is it necessary for the observer toknow his latitude except approximately. 41.—Determination of Time by a Single Altitude. Per contra, there is, of course, danger that the afternoon observationsmay be interfered with by clouds; and, moreover, both observationsmust be made at the same place. By a single altitude of the sun, the observer's latitudebeing known.—This is the method usual at sea. The difference between thisand that shown by the chronometeris the error of thechronometer. In the triangleZPS all three of the sides are given, viz.:
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The formula is sin12P = ( sin12[ ζ + ( ϕ - δ) ] sin12[ ζ - ( ϕ - δ) ] cosϕcosδ) ^1/2. In order to accuracy, it is desirable that the sun should be on theprime vertical, or as near it as practicable. It should not be near themeridian. Any slight error in the assumed latitude produces nosensible effect upon the result, if the sun is exactly east or west atthe time the observation is taken. The disadvantage of the methodis that any error of graduation of the sextant enters into the resultwith its full effect. Theshadow will then always fall upon the meridian line at apparent noon. The Civil and the Astronomical Day.—The astronomical daybegins at mean noon. The civil day begins at midnight, twelve hoursearlier. Astronomical mean time is reckoned round through the wholetwenty-four hours, instead of being counted in two series of twelvehours each. Thus, 10 a.m. of Wednesday, May 2, civil reckoning, isTuesday, May 1, 22^h by astronomical reckoning.
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Beginners need tobear this in mind in using the almanac. Having now methods of obtaining the true local time, we canattack the problem of longitude, which is perhaps the most importantof all the economic problems of astronomy. The great observatoriesat Greenwich and at Paris were established simply for the purposeof furnishing the observations which could be made the basis of theaccurate determination of longitude at sea. It is nowusually reckoned in hours, minutes, and seconds, instead of degrees. The methods of finding the longitude may be classed under threedifferent heads: First, By means of signals simultaneously observable at the placesbetween which the difference of longitude is to be found. Second, By making use of the moon as a clock-hand in the sky. Third, By purely mechanical means, such as chronometers and thetelegraph.
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Under the first head we may make use of[A] A Lunar Eclipse.—When the moon enters the shadow of theearth, the phenomenon is seen at the same moment, no matter wherethe observer may be. By noting, therefore, his own local time at themoment, and afterwards comparing it with the time at which the phenomenonwas observed at Greenwich, he will obtain his longitudefrom Greenwich. Unfortunately, the edge of the earth's shadow isso indistinct that the progress of events is very gradual, so thatsharp observations are impossible. (Now superseded by the telegraph.) [D] Artificial signals, such as flashes of powder and rockets, canbe used between two stations not too far distant. This method is now superseded by the telegraph. Second, the moon regarded as a clock. The simplest lunar method is, The method has been very extensively used, andwould be an admirable one were it not for the effects of personalequation.
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[B] Lunar-Distances.—At sea it is, of course, impossible toobserve the moon with a transit instrument, but we can observe itsdistance from the stars near its path by means of a sextant. From this the longitude can bedetermined. [C] Occultations.—Occasionally, in its passage through the sky,the moon over-runs a star, or “occults” it. [D] In the same way a solar eclipse may be employed by observingthe moment when the moon's limb touches that of the sun. There are still other methods, depending upon measurements ofthe moon's position by observations of its altitude or azimuth. Inall such cases, however, every error of observation entails a vastlygreater error in the final results. Lunar methods (excepting occultations)are only used when better ones are unavailable. Finally we have what may be called the mechanical methodsof determining the longitude. [A] By the chronometer; which is simply an accurate watch thathas been set to indicate Greenwich time before the ship leaves port.
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If the chronometer indicates true Greenwich time, the error deducedfrom the observation will be the longitude. Usually, however, the indicationof the chronometer face requires correction for the rate andrun of the chronometer since leaving port. Chronometers are only imperfect instruments, and it is important, therefore,that several of them should be used to check each other. It requiresthree at least, because if only two chronometers are carried and they disagree,there is nothing to indicate which one is the delinquent. But the method which, wherever it is applicable, has supersededall others, is that of The Telegraph. Theoperation is closed by another series of star observations. We have now upon each chronograph sheet an accurate comparisonof the two clocks, showing the amount by which the western clock isslow of the eastern. If the transmission of electric signals wereinstantaneous, the difference shown upon the two chronograph sheetswould agree precisely.
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Especial care must be taken to determinewith accuracy, or to eliminate, the personal equations of the observers. It is customary to make observations of this kind on not less than fiveor six evenings in cases where it is necessary to determine the difference oflongitude with the highest accuracy. Local and Standard Time.—In connection with time andlongitude determinations, a few words on this subject will be in place. Untilrecently it has always been customary to use only local time, each observerdetermining his own time by his own observations. In some such places the local time alsomaintains its place. In order to determine the standard time by observation, it is only necessaryto determine the local time by one of the methods given, and correctit according to the observer's longitude from Greenwich. But what noon?It was Monday when he started, and when he gets back to London, twenty-fourhours later, it is Tuesday noon there, and there has been no interveningsunset.
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When does Monday noon become Tuesday noon? The conventionis that the change of date occurs at the 180th meridian from Greenwich. Ships crossing this line from the east skip one day in so doing. If it isMonday forenoon when the ship reaches the line, it becomes Tuesday forenoonthe moment it passes it, the intervening twenty-four hours beingdropped from the reckoning on the log-book. Vice versa, when a vesselcrosses the line from the western side, it counts the same day twice, passingfrom Tuesday forenoon back to Monday, and having to do its Tuesday overagain. This 180th meridian passes mainly over the ocean, hardly touching landanywhere. There is a little irregularity in the date upon the differentislands near this line. When Alaska was transferred from Russia to the United States, it wasnecessary to drop one day of the week from the official dates. § THE PLACE OF A SHIP AT SEA.
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The methods employed are necessarily such that observationscan be made with the sextant and chronometer, the onlyinstruments available under the circumstances. The Latitude is usually obtained by observations of the sun'saltitude at noon, according to the method explained in Art.103. The Longitude is usually found by determining the error upon localtime of the chronometer, which carries Greenwich time. The necessaryobservations of the sun's altitude should be made when thesun is near the prime vertical, as explained in Art.116. In the case of long voyages, or when the chronometer has for anyreason failed, the longitude may also be obtained by measuring alunar-distance and comparing it with the data of the nautical almanac. By these methods separate observations are necessary for the latitudeand for the longitude. Sumner's Method.—Recently a new method, first proposedby Captain Sumner, of Boston, in 1843, has been coming largely intouse.
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In this method, each observation of the sun's altitude, with thecorresponding chronometer time, is made to define the position of theship upon a certain line, called the circle of position. Two such observationswill, of course, determine the exact place of the vessel atone of the intersections of the two circles. At any moment the sun is vertically over some point upon theearth's surface, which may be called the sub-solar point. An observerthere would have the sun directly overhead. In other words, the azimuth of the sun at the time of observationinforms him upon what part of the circle he is situated. Suppose a similar observation made at the same place a few hourslater. The sub-solar point, and the zenith distance of the sun, willhave changed; and we shall obtain a new circle of position, with itscentre at the new sub-solar point.
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The observer must be at one ofits two intersections with the first circle—which of the two intersectionsis easily determined from the roughly observed azimuth ofthe sun. If the ship moves between the two observations, the proper allowancemust be made for the motion. The intersection with the second circle thengives the ship's place at the time of the second observation. The only problem remaining is to find the position of the “sub-solarpoint” at any given moment. Now, the latitude of this point is obviouslythe declination of the sun (which is found in the almanac). If the sun's declination is zero, the sun is vertically over some pointupon the equator. If its declination is +20�, it is vertically oversome point on the twentieth parallel of north latitude, etc. The sub-solar point will then be(illo042Fig. 42) at a point in Africa having a latitude of +20�, and an east longitudeof 15�—at A in the illo042figure.
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And the radius of the “circle of position,”i.e., the distance from A to C—will be 50�. Again, a second observation is made three hours later, when the sun'saltitude is found to be 65�. The sub-solar point will then be at B, latitude20�, longitude 30� W., and the radius of the circle of position BC be25�, C being the ship's place. The observations need not betaken at any particular time. We are not limited to observations atnoon, or to the time when the sun is on the prime vertical. It is tobe noted, however, that everything depends upon the chronometer, asmuch as in the ordinary chronometric determination of longitude. It is desirable that it should be at leasta mile away from the observer, so thatany small displacement of the instrumentwill be harmless. The theodolite mustbe carefully adjusted for collimation, andespecial pains must be taken to have thetelescope perfectly level. illo043Fig. 43.—Determination of Azimuth.
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The next morning by daylight the observermeasures the angle or angles between the night-signal and theobjects whose azimuth is required. This can easily be done, as the right ascension and declination ofthis star are given in the almanac for every day of the year. This will come out inhours, of course, and must be reduced to degrees before making the computation. We thus have two sides of the triangle, viz., PS and PZ, withthe included angle at P, from which to compute the angle Z at the zenith. This is the star's azimuth. The altitude should not exceed thirty degreesor so. But the results are usually rough compared with these obtained bymeans of the pole star. § DETERMINATION OF THE POSITION OF A HEAVENLY BODY. The position of a heavenly body is defined by its rightascension and declination. These quantities may be determined—(1) By the meridian circle, provided the body is bright enough tobe seen by the instrument and comes to the meridian in the night-time.
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If the instrument is in exact adjustment, the sidereal timewhen the object crosses the middle wire of the reticle of the instrument isdirectly (according to Art.27) the right ascension of the object. In either case the declination can be immediatelydeduced, being the complement of the polar distance, and equal tothe latitude of the observer, minus the distance of the star south ofthe zenith. One complete observation, then, with the meridian circle,determines both the right ascension and declination of the object. In measuring this difference of right ascension and declination, we usuallyemploy a filar micrometer fitted like the reticle of a meridian circle. It carriesa number of wires which lie north and south in the field of view, andthese are crossed at right angles by one or more wires which can be movedby the micrometer screw. The observeddifference must be corrected for refraction and for the motion ofthe body, if it is appreciable.
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Other less complicated micrometers are also in use. One of them, calledthe ring micrometer, consists merely of an opaque ring supported in the fieldof view either by being cemented to a glass plate or by slender arms ofmetal. The observations are made by noting the transits of the comparisonstar and of the object to be determined across the outer and inner edges ofthe ring. If the radius of the ring is known in seconds of arc, we canfrom these observations deduce the differences both of right ascension anddeclination. There are also many other methods of effecting the same object.130. If it is verycold, with the barometer standing high, sunrise will be accelerated, or sunsetretarded, by a considerable fraction of a minute. The beginning and end of twilight may be computed in the same wayby merely substituting 108� for 90� 50'. pg:90Note to Art.125.—In the explanation of Sumner's method it isassumed that the earth is a perfect sphere.
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In the actual application of themethod certain corrections are therefore necessary to take into account theearth's ellipticity. CHAPTERIVVThe EarthTHE EARTH AS AN ASTRONOMICAL BODY. The facts are broadly these:—[1.] * The earth is a great ball, about 7918 miles in diameter. * It rotates on its axis once in twenty-four sidereal hours. * It is flattened at the poles, the polar diameter being nearlytwenty-seven miles, or one two hundred and ninety-fifth part less thanthe equatorial. * It is flying through space in its orbital motion around the sun,with a velocity of about nineteen miles a second; i.e., about seventy-fivetimes as swiftly as any cannon-ball. The Earth's Approximate Form and Size.—It is not necessaryto dwell upon the ordinary proofs of its globularity. * It can be circumnavigated. * The appearance ofvessels coming in from sea indicates that the surface is everywhereconvex.
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The fact that the sea-horizon, as seen from an eminence,is everywhere depressed to the same extent below the levelline, shows that the surface is approximately spherical. * The factthat as one goes from the equator toward the north, the elevation ofthe pole increases proportionally to the distance from the equator,proves the same thing. * The shadow of the earth, an seen uponthe moon at the time of a lunar eclipse, is that which only a spherecould cast. The earth is really relatively smoother androunder than most of the balls in a bowling-alley.illo044Fig. 44.—Curvature of the Earth's Surface.134. An approximate measure of the diameter is easily obtained. Erectupon a level plain threerods in line, a mile apart,and cut off their tops atthe same level, carefullydetermined with a surveyor'slevelling instrument. It will then befound that the line AC,Fig. 44, joining the extremitiesof the two terminal rods, passes about eight inches below B, thetop of the middle rod.
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