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A TEXT-BOOKOF

FIELD ASTRONOMYFOR ENGINEERS.

GEORGE C. COMSTOCK,Director of the Washburn Observatory,

Professor of Astronomy in the University of Wisconsin.

FIRST ED TTTON .FIRST THOUSAND.

NEW YORKJOHN WILEY & SONS.

London : CHAPMAN & HALL, Limited.1902.


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Copyright, 1902,BY

GEORGE C. COMSTOCK.

/fsfroHM

.

ROBKRT DRUMMOND. PRINTER, NEW YORK.


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PREFACE.

The present work is not designed for professionalstudents of astronomy, but for another and larger classfound in technical colleges. For many years it has beenthe author's duty to teach to students of engineeringthe elements of practical astronomy, and the experiencethus acquired has gradually produced the unconventionalviews that find expression in the present text and which,to the author's mind, are justified by the followingconsiderations

In the engineering curriculum, work in astronomy isa part of a course of technical and professional trainingof students who have no purpose to become astronomers.Under these circumstances it seems the duty of theinstructor to select for presentation those parts ofastronomical practice most closely related to the workof the future engineer and, with reference to the narrowlimits of time allotted the subject, to keep in the back-ground many collateral matters that are of primary in-terest and importance to the student of astronomy as ascience.

The parts of astronomical practice most pertinent to


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PREFACE.engineering instruction seem to the author to be (a) Train-ing in the art of numerical computation ; (6) Training inthe accurate use of such typical instruments of precisionas the sextant and the theodolite, with special refer-ence to the elimination of their errors from the results ofobservation; (c) Determinations of time, latitude, andazimuth, with portable instruments, as furnishing sub-ject-matter through which a and b may be convenientlyrealized. If this work is to be done during the singlesemester usually allowed for the subject, the time givento its theoretical side, spherical astronomy, must bereduced to the minimum amount compatible with thestudent's intelligent use of his apparatus and formulae,and in the present work this pruning of the theoretical sidehas been carried to an extent that would be unpardon-able in the training of an astronomer, but which appearsnecessary and proper in this case.

Since many engineering students acquire from themathematical curriculum little or no knowledge ofspherical trigonometry and its numerical applications,the first chapter of the work is devoted to a brief presen-tation of the elements of this subject with special refer-ence to its astronomical uses and to the student's acqui-sition of good habits in the conduct of numerical work.The astronomical problems presented in the followingchapters are those that have been indicated by experienceas best adapted to the author's own pupils, and whilemany of the methods given for their solution arc notcontained in the current text-books, in every case theseare either methods in use in the best geodetic surveys,


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PREFACE. Vor such as have been repeatedly tested with students andfound well suited to their use. These methods are classi-fied in the text as rough, approximate, and precise, withrespect to their precision and the corresponding amountof time and labor required for their application, and thestudent is advised not to use the refined and laboriousmethods when only a rough result is required.

As a rule, in the development of formulas no attempthas been made to deal with the general case when thesolution of a particular case would suffice for the prob-lem in hand; e.g., the earth's compression is ignoredin treating of the effect of parallax, since its influenceis vanishingly small in the great majority of cases thatthe student will ever encounter, and cases in which thisinfluence is of sensible amount should be avoided by theinstructor. A more serious omission, but one requiredby the general plan of the work, is found in the theoryof the transit instrument, Chapter IX, where brokentransits, thread intervals, curvature of a star's apparentpath, flexure, etc., are passed by without treatment oreven suggestion. They are not required for the begin-nings of work with a transit instrument, and thereforeconstitute a part of more advanced study than is herecontemplated. As a partial guide to such study thereis given upon a subsequent page a list of references toworks that may be consulted with profit by the studentwho seeks a more complete knowledge of the processesof practical astronomy.

The adopted notation follows, with only slight devia-tions, that of Chauvenet, to whose elaborate treatise


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vi PREFACE.upon Spherical and Practical Astronomy the author isunder obligations that are common to every present-daywriter upon those subjects. His thanks are also dueto many of his former pupils, and in particular to Dr.S. D. Townley and Mr. Joel Stebbins, who have readand criticised portions of his manuscript.


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TABLE OF SYMBOLS.

The following table contains a brief explanation of theprincipal symbols employed in the text, with referencesto the page at which they are respectively defined. Thereare omitted from the table a considerable number ofsymbols employed only in immediate connection withtheir definition.

Mathematical.2 p. 154 Summation symbol.11 The enclosed number is a logarithm.Coordinates, etc.

Altitude.Zenith distance. Complement of h.Azimuth, reckoned from south.Azimuth,, reckoned from north.Hour angle.DeclinationRight ascension.Latitude.Longitude.Sun's semi-diameter.Horizontal parallax.

Time.Sidereal time.Mean solar time.

44 Time shown by a chronometer, whether right or wrong.Chronometer correction.Chronometer rate.Equation of time.Date of an observation.Date of conjunction, mean sun with vernal equinox,

vii

h


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a


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TABLE OF CONTENTS.

CHAPTER I. PAGEIntroductory iSpherical trigonometry. Approximate formula?. Numeri-

cal computations. Logarithmic tables. Limits of accuracy.

CHAPTER II.Coordinates 22

Fundamental concepts. Definitions. Notation. Table ofcoordinates. Transformation of coordinates.

CHAPTER III.TlME 35

Three time systems. Longitude. Conversion of time.Chronometer corrections. The almanac.

CHAPTER IV.Corrections to Coordinates 49

Dip of horizon. Refraction. Semi-diameter. Parallax.Diurnal aberration.

CHAPTER V.Rough Determinations 59

Latitude from meridian altitude. Time and azimuth fromsingle altitude. Meridian transits for time. Orientation andlatitude by Polaris.


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TABLE OF CONTENTS.

CHAPTER VI. PAGHApproximate Determinations 79

Circum-meridian altitudes for latitude. Time from singlealtitude. Azimuth observations at elongation. Time andazimuth from two stars.

CHAPTER VII.Instruments 99

The spirit-level. Value of half a level divis-ion. Theory ofthe theodolite. Repetition of angles. The sextant. Chro-nometers.

CHAPTER VIII.Accurate Determinations 141

Time by equal altitudes. Precise azimuth with theodolite.Zenith-telescope latitudes.

CHAPTER IX.The Transit Instrument 16S

Preliminary adjustments. Theory of the transit. Ordinarymethod for time determinations. Personal equation. Methodsand accuracy of observation. Time determination withreversal on each star. Azimuth of terrestrial mark.

Bibliography 196Orientation Tables 197


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FIELD ASTRONOMY.

CHAPTER I.INTRODUCTORY.

i. Spherical Trigonometry.Any three points on thesurface of a sphere determine a spherical triangle, whosesides are the arcs of great circles joining these points,and whose angles are the spherical angles included be-tween these arcs; e.g., on the surface of the earth,assumed to be spherical in shape, the north pole, the cityof St. Louis, and the borough of Greenwich, England, arethree points making a spherical triangle, two of whosesides are the arcs of meridians joining St. Louis and Green-wich to the pole ; the third side being the arc of a greatcircle connecting St. Louis and Greenwich, and measur-ing by its length the distance of one place from the other.The spherical angle at the pole between the two meridiansis the longitude of St. Louis, while the angle at St. Louisbetween its meridian and the third side of the trianglerepresents the direction of Greenwich from St. Louis, acertain number of degrees east of north. The particularnumber of degrees in this angle is to be found by solving


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2 FIELD ASTRONOMY.the triangle, i.e., determining the magnitude of its un-known parts by means of the known parts, and in thiscase we may suppose these known parts to be the differ-ence of longitude between the two places, and the distanceof each place from the north pole, i.e., the complementof its latitude.

The formulae required for the solution of a sphericaltriangle are best derived by the methods of analyticalgeometry, and in Fig. i we assume a spherical triangle,

ABC, situated on the surface of a sphere whose centre isat 0, and we adopt as the origin of a system of rect-angular coordinates, in which the axis OX passes throughthe vertex, A, of the triangle, OY lies in the plane AOB,and OZ is perpendicular to that plane. From the vertexC let fall upon the plane 0.4B the perpendicular CP, andfrom P draw PS perpendicular to OX and join the pointsC, S, thus obtaining the right-angled plane triangle CPS.


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INTRODUCTORY. 6The lines OS, SP, PC are respectively the x, y, and zcoordinates of the point C, and OC, which we shall repre-sent by the symbol r, is the radius of the sphere.

It is evident from the construction that the points.0, S, A, and C all lie in the same plane. Also, 0, S, A,B, and P lie in another plane, and the angle between thesetwo planes is measured both by the spherical angle BACand by the plane angle CSP, and these angles must there-fore be equal each to the other. We may now expressthe coordinates of the point C in terms of the sides, a, b, c,and angles, A, B, C, of the spherical triangle as follows i

OS = x = r cos b,SP=y = r sin b cos A, (i)PC = z = r sin b sin A .

If the axis of x, instead of passing through A , had beenmade to pass through B, as is shown by the broken lineOX', the axis of Z remaining unchanged, we should havehad for the coordinates of C in this system,

x' = r cos a,y' = r sin a cos B, (2)z' = r sin a sin B.

For the sake of simplicity each angle of the triangle ABChas been made less than 90 , and the point P, therefore,falls between the axes OX, OX' , thus giving y and y'opposite signs, as shown above.

It is evident from the figure that the relations betweenx, x', y, y', are those furnished by the formulas for the-


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4 FIELD ASTRONOMY.transformation of coordinates in a plane, when the originremains unchanged and the axes are revolved through anangle, which in this case is measured by the side c of thespherical triangle. We have, therefore,

i /J 6,y' =y cose x sine, (3)x' = y sin c + x cos c ;

and introducing into these equations the values of thecoordinates above determined and dividing through byr, we obtain the following relations among the sides andangles of the triangle

sin a sin B = sin b sin A ,sin a cos B = cos b sin c sin b cos c cos A (4)

cos a = cos b cos c + sin b sin c cos AThese are the fundamental equations of spherical

trigonometry and hold true not only for the particulartriangle for which they have been derived, but for everyspherical triangle, whatever its shape or size.

2. Numerical Applications of Equations 4.We proceedto apply these equations to the logarithmic solution ofthe triangle above described, premising that in this solu-tion the signs of all the trigonometric functions must becarefully heeded, since upon them depend the quadrants,first, second, third, or fourth, in which the unknown partsof the triangle are to be found. , In this connection weshall reserve the signs + and for natural numbers andplace after a logarithm the letter n whenever the numbercorresponding to the logarithm is negative. The student


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INTRODUCTORY.should accustom himself to this practice, since it is theone in general use.

The assumed data of the problem areAngular distance, Greenwich to Pole. . . . b = 38. 5,Angular distance, St. Louis to Pole = 51. 4.Spherical angle at North Pole A =90.

and these data we treat as follows

Logarithms.sin A =0.000sin 6=9.794cos .4 =7.84472cos c =9.795

sin 6 cos .4 =7.638^cos 6=9.894sin c =9.893

SOLUTION.Numbers.

cos b sin c =4-0.613sin b cos c cos A = 0.003

cos b cos c= 4-0.489sin b sin c cos A = 0.003

log cos 0= 9.686

Logarithms.sin a sin B =9.794

9-854sin a cosB =9.785B=45-6

sin a = 9.940a=6o.a*

In the solution printed above, the student shouldexamine the orderly manner of the arrangement. Eachnumber is labelled to show what it is, and from theselabels we see that the first column contains the logarithmsof the several trigonometric functions that appear in thesecond members of Equations 4. The second columncontains natural numbers representing the values of theseveral terms contained in these second members. Theseare obtained by adding the proper logarithms shown inthe first column, and looking out the corresponding num-bers in the tables. An expert computer will do this work"in his head" without writing down a figure that is notshown in the printed solution.

At the bottom of the second column is given log cos a,obtained by looking out the logarithm of the sum of thetwo numbers that stand just above it. This sum beingpositive shows that the side a lies in either the first or


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6 FIELD ASTRONOMY.fourth quadrant, but it alone cannot decide between thesetwo possibilities. We must now have recourse to thethird column, which gives the logarithms of the products,sin a sin B and sin a cos B, as derived from the first andsecond columns, and indicates that these products arepositive quantities, since no n is appended to either of thelogarithms. The products being positive, the factorssin a, sin B, and cos B must all have like signs, and assum-ing, temporarily, that sin a is a positive quantity wefind that B must lie in the first quadrant, since sin B andcos B are positive numbers. To obtain its numericalvalue we divide sin a sin B by sin a cos B (subtractmentally the corresponding logarithms) and find, as theresult of the division, log tan B = 0.009. This furnishesthe value of B given in the solution, and fixes as thedirection of Greenwich from St. Louis, N. 45.6 E.

Now, looking up in the logarithmic tables the valueof sin B (log sin B = 9.854), and dividing it out fromsin a sin B, we obtain the value, 9.940, given in the solu-tion for sin a. This number might equally have beenobtained by looking up in the tables the value of cos B( = 9.845) and dividing it out from sin a cos B, and withreference to this double possibility the label for the linebetween sin a sin B and sin a cos B is omitted, it beingunderstood that either sin B or cos B, whichever is thegreater of the two, will be entered here and used in theproper manner to obtain sin a. The value of log cos ais given in the middle column, and both sin a and cos abeing positive numbers, a is to be taken in the first quad-rant. The agreement between the numerical values of a


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8 FIELD ASTRONOMY.by means of which the angle A may be computed withthe given data, and similar equations may be written forthe other angles.

By transformations more tedious than difficult, andinvolving the introduction of two auxiliary quantities,defined below by Equations 6, we may change Equation 5.into a form more convenient for computation when allthree of the angles are to be determined (see any treatiseon spherical trigonometry for the analytical processesinvolved) . As a result of these transformations we havethe following auxiliaries and the solution involving them

s = (a+ b + c), k = yj-r- sm s'sm(s-a) sin(s-b) sm(s c)'cot \A=k sin {s a), (7)

with corresponding expressions for the other angles,

cot \B=k sin (s b),cot \C = k sin (s c).

Right-angled Spherical Triangles.Since Equations 4hold true for all spherical triangles, we may apply themto the special case of a triangle right-angled at A, i.e., onein which the angle A equals 90 . We shall then havesin A = i, cos A =0, and with these special values weobtain by substitution the following equations, whichshould be compared with the corresponding formulaeof plane trigonometry


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INTRODUCTORY.

From the first equation, sin B =From first and second equations, tan B =From second and third equations, cos B

sin atan bsin c ' (8)tan ctan a

From the third equation, cos a = cos b cos c.These equations together with those derived in thepreceding sections, while far from covering the whole field

of spherical trigonometry, will be found sufficient for thepurposes of this work.

4. Approximate Formulae.In an important class ofcases all the preceding formulas may be greatly simplifiedfor numerical use as follows : It is shown, in treatises onthe differential calculus, that the trigonometric functionsmay be developed in series; e.g.,

sin xx -7-H etc.,6 120x 2%tanx=x + +r + etc, (9)3 J 5

;2 x 4- +2 24

x< , x 9cosx = i --7-7- etc.,

where x is expressed in radians (one radian = 57. 3,= 3437'-75> =206264". 8). When the angle x does notexceed a few minutes of arc, x radians is a small fraction,and its powers, x 2 , x 3 , etc., are still smaller quantities, sothat in these series we may suppress all terms save thefirst, or all terms save the first two, and the error pro-duced by neglecting these terms of a higher order, as


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10 FIELD ASTRONOMY.they are called, is approximately measured by the firstterm thus neglected. For illustration we assume x=i,and turning this into radians find the results shown inthe following short table

Radians. Arc.X=o.0174533+ = 1* 2 =o.0001523+ = 31".42 ^x z =0.0000009 + = o".i8^ 4 =o.ooooooo+ = o".oo

It appears from the value of %x 3 , here given, that if weare prepared to tolerate in our work an error of one partin a million we may, for an arc of i, substitute the arcitself in place of its sine, in any formula where the latteroccurs ; and similarly (from the value of hx 2 ) we may sub-stitute unity in place of the cosine of an arc of 1 , if we arewilling to admit an error of one part in seven thousand.Expressed in arc these errors are as shown in the table,o".2 and 31".4 respectively, and with reference to thesenumbers we may establish the approximate relations:the square of a degree equals a minute; the cube of adegree equals a second ; and find readily, from these rela-tions, the square and cube of any small arc, and thusdecide whether, in a given case, these quantities may ormay not be neglected. For example: if x = 2, we findx 2 = 4 r , x 3 = &", and for any work in which the data canbe depended upon to the nearest minute only, we mayassume sin x = x, but we cannot assume cos x = 1 withoutsacrificing some of the accuracy contained in the data.

It must be constantly borne in mind that the seriesgiven above are expressed in radians, and that whenapplied numerically, x and its powers must be trans-


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INTRODUCTORY. 11formed from arc into radians by dividing by the appro-priate factors given above; e.g.,

%"x (radians) =- ( 1 1206264.8The divisor given above is numerically equal to the

reciprocal of the sine of 1" , and in place of the precedingequation it is customary to write

x (radians) =x" sin 1". (12)As these numerical factors are of frequent use, we recordhere their values

log sin 1" =4.6855749-10,206264. 8 = [5.3144251].

Observe the peculiar notation of the last line. Thebrackets indicate that the number placed within them isa logarithm, and the equation asserts that this bracketednumber is the logarithm of the first member of the equa-tion. This use of the brackets is very common andshould be remembered.

We may apply to the equations of spherical trigo-nometry the principles here developed, and assuming thatthe sides, a, b, c, do not much exceed i, i.e., that for atriangle on the surface of the earth the vertices of theangles are not more than sixty or seventy miles apart, weshall find that Equations 8 become

b c bsm.B = -, cos B = -, tan .> = -, a 2 = b 2 + c 2 .a a c


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12 FIELD ASTRONOMY.These are the formulae of plane trigonometry, and indicatethat small spherical triangles may be treated as if theywere plane.

The use of these approximate relations is not limitedto the solution of triangles, but they may be applied tothe trigonometric functions of any small angle whereverfound, and we shall have frequent occasion to use themin the following pages.

5. Numerical Computations.Engineer and astronomeralike should acquire the art of rapid and correct compu-tation, and as a means to that end there will be foundon subsequent pages examples of numerical work whichshould be studied with reference to their arrangement andthe order in which the several processes were executed.Often the order in which this work was done is not theorder in which the numbers appear upon the printedpage, although their arrangement upon the page alwaysfollows exactly the original computation, and in no caseis to be regarded as a mere summary of results, pickedout and rearranged after the actual ciphering had beenperformed. For illustration we revert to the exampleof 2, and note that sin A is the first number written inthe solution and sin b stands second. But the secondnumber actually written down in the computation wascos A, instead of sin b, for, having found the place inwhich to look up sin A, it is more convenient and moreeconomical to look up cos A at once, while the tables areopen at the right place, rather than to turn away forsomething else and then have again to find the pageand place corresponding to the angle A. Having finished


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INTRODUCTORY. 13with the required functions of A , sin 6 was next lookedout and was followed by cos b, although this requiredthe computer to skip two intervening lines of the com-putation and, temporarily, to leave them blank.

The general principle here observed is: When a tableis open at a given place, look up, before leaving it, allthat is to be taken from that place. In order to do thisit is necessary to block out the computation in advance,and this was done in the case under consideration, everylabel, from the initial sin A of the first column to theconcluding a of the last column, being written in itsappropriate place before a number was set down or thelogarithmic table opened. The form of computationthus prearranged is called a schedule, and it is to bestrongly urged upon the student as a measure of econ-omy and good practice, that he should draft, at thebeginning of each computation, a complete schedule, inwhich every number to be employed shall be assignedthe place most convenient for its use. In general thebeginner will not be able to do this without assistancefrom an instructor, or from models suitably chosen, andfor the purposes of the present work the numerous exam-ples contained in the text may be taken as such models.

Some cardinal points in the arrangement of a goodschedule are as follows:

(A) Make it short but complete. Do as much of thework "in your head" as can be done without undulyburdening the mind, and write upon paper only thethings that are necessary. But all things that are to bewritten should have places assigned them in the schedule.


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14 FIELD ASTRONOMY.No side computations, upon another piece of paper,should be allowed, and the entire work should be soarranged and labelled that a stranger can follow it andtell what has been done.

(B) When the same quantity is to be used severaltimes in a computation (sin b appears as a factor in threedifferent terms of the preceding example) the scheduleshould be so arranged that the number need be writtenonly once, e.g., since sin b is to be multiplied by bothsin A and cos A it is placed between these numbersin the schedule, and for a similar reason the productsin b cos A is placed between cos c and sin c. In addingthe logarithms to form the product sin b sin c cos Acover cos b with a pencil or penholder and the additionwill be as easily made as if the intervening number werenot present.

(C) Frequently, several similar computations are tobe made with slightly different data, e.g., it may be re-quired to find the direction and distance of half a dozenAmerican cities from Greenwich. A single scheduleshould then be prepared and the several computationsshould be carried on simultaneously, in parallel columns,all placed opposite the same schedule; e.g., look outsin A and cos A for all six places before proceeding tofind sin b for any of them, etc. In this particular casesin b and cos b, depending on the latitude of Greenwich,are the same for all the solutions, and instead of writingtheir values in each column, they should be written uponthe edge of a slip of paper and moved along from columnto column as needed. As a memorandum for future


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INTRODUCTORY. 15reference they should also be written in one column ofthe computation. Practise this device whenever thesame number is to be used in several different places.See 36 and 40 for examples of two computations de-pending upon a single schedule.

6. The Trigonometric Functions.There is opened tothe inexperienced computer an abundant opportunity forerror in looking out from the tables the trigonometricfunctions of angles not lying in the first quadrant. Thebest mode of guarding against such errors is the acqui-sition of fixed habits of procedure, so that the same thingshall always be done in the same way, and to this endthe following simple rules may be adopted

(1) For any odd-numbered quadrant, first, third, etc.Reduce the given angle to the first quadrant by castingout the nines from its tens and hundreds of degrees (addthese digits together and repeat the addition until thesum is reduced to a single digit, less than nine), and lookup the required function of the reduced arc.

(2) For any even-numbered quadrant, second, fourth,etc. Reduce the angle to the first quadrant, as above,and look out the function complementary to the onegiven. The algebraic sign of the function is, of course,in all cases determined by the quadrant in which theoriginal angle falls.

See the following applications of these rules:Quadrant. Required. Equivalent. Process2d, Even cos 144 29' = -sin S4 293d, Odd tan 264 33' = +tan 84 334th, Even sin 316 51' = cos 46 515th, Odd cot 4 1 18' = +cot 54 186th, Even tan 499 49' = cot 49 49

i+4=S2+6=83 + I= 44 + i=5Reject the 9

etc. etc. etc. etc.


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16 FIELD ASTRONOMYWe may readily formulate a corresponding rule for

the converse process, of passing from the function tothe angle, as follows:

(i) When the arc lies in an odd quadrant. Look out,in the first quadrant, the angle that corresponds to thegiven function and add to it the required even multipleof 90 , i.e., o or 180 .

(2) When the arc lies in an even quadrant. Changethe name of the function (for cos read sin, for tan readcot, etc.). Look out, in the first quadrant, the corre-sponding angle and add to it the required odd multipleof 90 , i.e., 90 or 270 .

See the following examples, in which we representthe required angle by z and suppose that there is giventhe numerical value of its tangent, e.g., log tan 2 = 9.654.The process of looking out in the several quadrants theangle corresponding to this tangent is as follows

Quadrant.


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INTRODUCTORY. 17z = o 33' 1 7". The difficulty comes from the rapid varia-tion of the function, large and changing tabular differ-ences. On the other hand, log cos z changes slowly andmay be readily and accurately interpolated. If we takethe converse case and suppose the logarithmic functionto be given and the corresponding angle required, weshall obtain the opposite result. The angle will be accu-rately determined by the sine or tangent and very poorlydetermined by the cosine, e.g., log cos o 33' 17" =9.99998and every angle between o 29' and o 36' has thiscosine, thus leaving a possible error of several minutesin the value of the angle determined from this function,while the log sin, if correctly given to five decimal places,will determine the same angle within a small fraction ofa second.

In the interest of precision an angle should always bedetermined from a function that changes rapidly (largetabular differences), while a quantity that is to be foundfrom a given angle is best determined through a functionthat changes slowly. In the example of 2, sin a mighthave been determined through sin B or cos B, and theformer was used for this purpose because it varied themore slowly. In cases of this kind, and they are verycommon, use the function that stands on the right-handside of the page, in the tables, and subtract it from thelarger of the two numbers, sin a sin B or sin a cos B, andit will be then unnecessary to consider whether it is sineor cosine that is employed.

The angle a, in this example, was determined throughits tangent (log tan a = log sin a log cos a) , since the


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18 FIELD ASTRONOMY.tangent always varies more rapidly than either sine orcosine and should generally be preferred for this purpose.After obtaining a, its sine and cosine were looked outfrom the tables and compared with the numbers obtainedin the solution, for the sake of the "check" thus fur-nished upon the accuracy of the numerical work. Insubsequent pages other checks will be shown, and theseshould be applied to test the accuracy of numerical workwhenever they are available. The mental strain accom-panying a long computation is, under the best of circum-stances, considerable, and a check properly satisfiedserves to relieve this tension and facilitate the subse-quent work.

8. Accuracy of Logarithmic Computation.The exam-ple of 2 was solved with logarithms extending only tothree places of decimals, and corresponding to this use ofa three-place table the results are given to the nearesttenth of a degree. If it were required to obtain resultscorrect to the nearest minute or nearest second, a greaternumber of decimals must be employed (four-, five-, or six-place tables). The labor of using these tables increasesvery rapidly as the number of decimals is increased, anda compromise is always to be made between extra laboron the one hand and limited accuracy on the other.

As the choice of a proper number of decimal placesis usually an embarrassing one for the beginner, there isgiven below for his guidance a formula intended to repre-sent, at least approximately, the limit of error to be ex-pected in the results of computation on account of theinherent imperfections of logarithms (neglected deci-


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INTRODUCTORY. 19mals, etc.). The actual error may fall considerablyshort of this limit or may overstep it a little. It isevident that the limit will be greater for a long compu-tation than for a short one, and if we measure the lengthof a computation by the number, n, of logarithms thatenter into it and represent by m the number of decimalplaces to which these logarithms are carried, there maybe derived from the theory of probabilities the followingexpression, in minutes of arc, for the limit of probableerror:

Limit = 2800' . s/~n . \o~m .Applying this formula to the example of 2 we mayput w = i6, ^ = 3, and find 10' as the limit of unavoid-able error; corresponding well with the one-tenth of adegree to which the results were carried. If the datawere given to the nearest minute and it were requiredto preserve this degree of accuracy in the results, weshould write,

.i / = 28oo' . 4.10-'*,and solving, find ^ = 4.0, i.e., a four-place table is re-quired for this purpose.

Let the student verify by means of the above equa-tions the following precepts:

To obtain UseTenths of degrees Three-place tables.Minutes Four-place tables.Seconds Five- or six-place tables.Tenths of seconds Seven-place tables.


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20 FIELD ASTRONOMY.If the results are to be expressed in linear instead of

angular measure, the limit of error must be representedas a fractional part of the quantity, x, that is to bedetermined, and corresponding to this case we have

Limit = o . 8 x . y/n . io~.Corollary. Do not attempt to obtain from a table

more than it is capable of furnishing; e.g., do not inter-polate hundredths of a degree in the example of 2,and in connection with linear quantities do not, as a rule,interpolate more than three significant figures from athree-place table, four from a four-place table, etc.

9. Logarithmic Tables.There exists a great varietyof logarithmic tables of different degrees of accuracy,from three to ten places of decimals, and having deter-mined the number of decimal places required in a givencomputation, the choice among the corresponding tablesis largely a matter of personal taste. The beginner,however, will do well to observe the following rules fordistinguishing good tables from bad ones:

(A) Wherever the tabular differences exceed 10,a good table should furnish proportional parts, PP,in the margin of each page, so that the logarithms maybe interpolated "in the head."

(B) The tables should be accompanied by tables ofaddition and subtraction logarithms. For an explana-tion of these, their purpose and use, the student is re-ferred to the tables themselves, but we note here thatby their aid the example of 2 might have been much


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INTRODUCTORY. 21more conveniently solved, as is illustrated in a similarproblem in 15.The most generally useful tables are those of fivedecimal places, but computers find it to their advantageto have and use at least one table of each kind, from threeto six or seven places. In the examples solved in thepresent work the following tables have been used

Three-place, Johnson. New York.Four-place, Slichter. New York. Gauss. Berlin.Five-place, Albrecht. Berlin.Six-place, Albrecht's Bremiker. Berlin.As a very useful supplement to the logarithmic tables

a slide-rule and the extended multiplication tables ofCrelle and Zimmermann are highly esteemed.


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CHAPTER II.COORDINATES.

10. Fundamental Concepts. For most purposes ofpractical astronomy the stars may be considered asattached to the sky, i.e., to the blue vault of the heavens,which is technically called the celestial sphere, and is re-garded as of indefinitely great radius but having theearth at its centre, so that a plane passing through anyterrestrial point intersects this sphere in a great circle,and parallel planes passing through any two terrestrialpoints intersect the sphere in the same great circle.

If the axis about which the earth rotates be producedin each direction, it will intersect the celestial sphere intwo points, called respectively the north and south poles.If a plumb-line be suspended at any place, P, on theearth's surface, and be produced in both directions, itwill intersect the celestial sphere, above and below, inthe zenith and nadir of the place. The direction thusdetermined by the plumb-line is called the vertical of theplace.

The figure (shape) of the earth is such that the verticalof any place, when produced downward, intersects therotation axis, and a plane may therefore be passed through

22


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COORDINATES. 23this axis and the vertical. This plane, by its intersectionwith the celestial sphere, produces a great circle whichpasses through the poles, the zenith and nadir, and iscalled the meridian of the place, P.

A plane passed through P perpendicular to the direc-tion of the vertical produces by its intersection with thesphere the horizon of P. Any plane passing through thevertical is called a vertical plane and produces by its inter-section with the sphere, a vertical circle. That verticalcircle whose plane is perpendicular to the meridian iscalled the prime vertical.

With exception of the poles, all of the terms abovedefined depend upon the direction of the vertical, andas this direction varies from place to place upon theearth's surface each such place has its own meridian,horizon, zenith, etc., while the poles of the celestialsphere are the same for all places.

A plane passed through the centre of the earth per-pendicular to the rotation axis produces by its inter-section with the earth's surface the terrestrial equator,and by its intersection with the celestial sphere it pro-duces the celestial equator.

Owing to the motion of the earth in its orbit we seeanything within the orbit from different points of viewat different seasons of the year, and by the earth's motionthe sun is thus made to describe an apparent path amongthe stars, making the complete circuit of the sky in ayear. This path is a great circle intersecting the celestialequator in two points diametrically opposite to eachother, and that one of these points through which the


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24 FIELD ASTRONOMY.sun passes on or about March 2 2 of each year, is calledthe vernal equinox.

11. Systems of Coordinates. Most of the problemsof practical astronomy require us to deal with the appar-ent positions and motions of the heavenly bodies asseen projected against the sky, and for this purposethere are employed several systems of coordinates basedupon the concepts above defined, and three of thesesystems we proceed to consider. These are all systemsof polar coordinates having the following characteristicsin common:

(1) The origin of each system is at the centre of thecelestial sphere.

(2) Each system has a fundamental plane and con-sists of an angle measured in the fundamental plane;an angle measured perpendicular to the fundamentalplane; and a radius vector. The first of these anglesis frequently called the horizontal coordinate, and thesecond the vertical coordinate, of the system. Latitudesand longitudes on the earth furnish such a system ofcoordinates. The longitude of St. Louis (horizontalcoordinate) is measured by an angle lying in the planeof the equator, which is the fundamental plane of thissystem. The latitude of St. Louis (vertical coordinate)is measured by an angle lying in a plane perpendicularto the equator, and the radius vector of St. Louis is itsdistance from the centre of the earth, which latter pointis taken as the origin of coordinates.

(3) In each system the horizontal coordinate ismeasured from a fixed direction in the fundamental


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COORDINATES. 25plane, called the prime radius, through 360 . The ver-tical coordinate is measured on each side of the funda-mental plane from o to 90 .

(4) Those vertical coordinates are called positivethat lie upon the same side of the fundamental planewith the zenith of an observer in the northern hemi-sphere of the earth. Those that lie upon the oppositeside of the fundamental plane are negative.

(5) It is frequently convenient to measure a verticalcoordinate from the positive half of a line perpendicularto the fundamental plane instead of from the funda-mental plane itself, (e.g., in 2 we take as the verticalcoordinate of St. Louis its distance from the pole insteadof from the equator) . In such cases this coordinate isalways positive and is included between the limits oand 180 . If h represent any vertical coordinate meas-ured in the manner first described and z be the corre-sponding coordinate measured in the second way, weshall obviously have the relation, z = go li.

The several systems of astronomical coordinatesdiffer among themselves in the following respects(a) Different fundamental planes for the systems.(b) Different positions of the prime radii in the fun-

damental planes.(c) Different directions in which the horizontal co-

ordinates increase.The data which completely define each system of

coordinates are given in the following table togetherwith the names of the several coordinates, the lettersby which these are usually represented, and the point


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26 FIELD ASTRONOMY.of the heavens, called the pole of the system, towardwhich the positive half of the normal to the fundamentalplane is directed. The terms east and west are usedin this table with their common meaning, to indicatethe direction toward which the horizontal coordinateincreases. The letters associated with the several co-ordinates are conventional symbols that should becommitted to memory.

SYSTEMS OF COORDINATES.System

Fundamental planePrime radius points towardHorizontal coordinates increasetowardNormal points towardName of horizontal coordinate. . .Name of vertical coordinate ,

HorizonMeridianWestZenith

Azimuth=/2Altitude=/i

EquatorMeridianWestNorth Pole

Hour angle=^Declination= S

III.

EquatorVernal EquinoxEastNorth Pole

Right Ascension = cDeclination= 8

Exercises.Let the student define in his own language the severalquantities above represented by the letters A, t, a, h, and 8.

i . What is the azimuth of the north pole ?2. What do the hour angle and altitude of the zenith respectively

equal?3 What are the azimuths of the prime vertical ?4. What are the declinations of the points in which the horizoncuts the prime vertical?5 Does S in the second system differ in any way from S in the third

system ?

The directions of the prime radius as above definedfor systems I and II, are ambiguous, since the meridiancuts the fundamental plane of each of those systemsin two points. Either of these points may be usedto determine the direction of the prime radius, but ingeneral that one is to be employed which lies south ofthe zenith.


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COORDINATES. 27Let the student show the relation between the coordinates furnished

in System I by adopting each of the possible positions for the primeradius.

12. Uses of the Three Systems.It is well to considerhere, very briefly, the reasons for using more than onesystem of coordinates, and the relative advantages anddisadvantages of these systems.

The coordinates of System I are well adapted toobservation with portable instruments, e.g., an engineer'stransit, since the horizon is more easily identified withsuch an instrument than is any other reference plane,and the circles of the instrument may be made to read,directly, altitudes and azimuths. The horizon has beendefined by reference to the direction of a plumb-line,but in practice a spirit-level, or the level surface of aliquid at rest, are more frequently used to determineits position.

System I possesses the disadvantage that, throughthe earth's rotation about its axis, both the altitude andazimuth of a star are constantly changing in a compli-cated manner, and in this respect System II possessesa marked advantage. Since the normal to its funda-mental plane coincides with the earth's axis, rotationabout this axis has no effect upon the vertical coordi-nates, declinations, which remain unchanged, while thehorizontal coordinates, hour angles, increase uniformlywith the time, 15 per hour, and are therefore easilytaken into account and measured by means of a clock.

Suppose a watch to have its dial divided into twenty-four hours,instead of the customary twelve. If this watch be held with its dialparallel to the plane of the equator, the hour hand, in its motion around


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28 FIELD ASTRONOMY.the dial, will follow and keep up with the sun as it moves across thesky. If the watch be turned in its own plane until the hour handpoints toward the sun, the time indicated upon the dial by this handwill be approximately the hour angle of the sun, and the zero of thedial will point toward the meridian, i.e., sotith.

Let the student compare the ideal case above considered with thefollowing rough rule sometimes given for determining the directionof the meridian by means of a watch with an ordinary, twelve-hour,dial: Hold the watch with its dial as nearly parallel to the plane of theequator as can be estimated. (See 13 for the position of this plane.)Revolve the watch in this plane until the hour hand points towardthe sun, and the south half of the meridian will then cut the dial mid-way between the hour hand and the figure XII.

A further advantage is gained in the third systemof coordinates, since here the prime radius shares inthe apparent rotation of the celestial sphere about theearth's axis, and both the horizontal and vertical coor-dinates are therefore unaffected by this motion. In-struments have been devised for the measurement ofthe coordinates in each of these systems, but we shallbe mainly concerned with those that relate to the firstsystem, and shall consider System III as employedchiefly to furnish a set of coordinates independent of theearth's rotation and of the particular place upon theearth at which the observer chances to be. Thesefeatures make it suited to furnish a permanent recordof a star's position in the sky, and it is so used in theAmerican Ephemeris (see 21) and other nautical alma-nacs, where there may be found, tabulated, the rightascensions and declinations of the sun, moon, planets,and several hundred of the brighter stars.13. Relations between the Systems of Coordinates.A problem of frequent recurrence is the transformationof the coordinates of a star from one system to another;


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COORDINATES. 29indeed most of the problems of spherical astronomy are,analytically, nothing more than cases of such trans-formation, and as an introduction to these problemswe shall examine the relative positions of the funda-mental planes and prime radii of the several systems.

The plane of the equator intersects the plane of thehorizon in the east and west line, and the angle betweenthe two planes is called the colatitude, since it is thecomplement of the geographical latitude of the place


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30 FIELD ASTRONOMY.and planes passed through its centre, it is apparentthat the latitude,


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COORDINATES. 31ine the triangle projected against the sky and the threepoints to be visible in their true positions.

PZ is an arc of a great circle passing through the poleand zenith and must therefore be a part of the observer'scelestial meridian, and in Fig. 2 we have already seenthat this arc of the meridian is equal in length to thecomplement of the latitude, 90 0. The broken line

Fig. 3.The Astronomical Triangle.HH' in the figure, is an arc of a great circle, every partof which is 90 distant from Z. But the great circle90 distant from the zenith is the horizon, and the arcHS that measures the distance of 5 from AA' must bethe star's altitude, h, and the side SZ of the astronom-ical triangle, being the complement of this arc, is equalto go h. In like manner, EE', drawn 90 distantfrom P, is an arc of the celestial equator; the a-x SE,that measures the distance of 5 from EE' , is the star'sdeclination, d, and the side PS of the triangle equals9o-d.


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32 FIELD ASTRONOMY.The star's hour angle, i.e., horizontal coordinate

lying in the equator, is measured by the arc EE'', andthis arc, by a theorem of spherical geometry, is numer-ically equal to the spherical angle, t, included betweenEP and E'P, which is therefore the star's hour angle.In like manner the spherical angle SZE' is shown to beequal to the star's azimuth, A, and the angle SZP ofthe astronomical triangle is equal to i8o A. Thethird angle of the triangle, marked q in the figure, iscalled the parallactic angle.

To apply to the astronomical triangle the fundamentalformulas of spherical trigonometry derived in i, wereplace the general symbols used in Equations 4 bythe particular values which they have in the astronomicaltriangle, as follows

a = go-h A=tb = 9o-d B = i8o-Ac = 90

and introducing these values into Equations 4 we obtainthe required formulas for transforming altitudes andazimuths into declinations and hour angles, as follows:

cos h sin A = + cos d sin t,cos h cos A = cos


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COORDINATES. 33hour angle of the vernal equinox. Similarly the arcVE and its corresponding angle at P are equal to theright ascension of the star, and from the figure we thenobtain the required relations,

a+ t= 0, d = d (i 4a)The transformation between the first and third systemsis best made through the second system; i.e., by usingboth groups of formulae 14 and 14a.15. Problem in Transformation of Coordinates. Atthe sidereal time 13 11 2 2 m 49 s . 3 the altitude and azimuthof a star were measured at a place in latitude 43 4' 36",as follows: h = 6 1 19' 36", A =253 9' 42". Requiredthe right ascension and declination of the star.

The required transformation formulas may be obtainedfrom the astronomical triangle in the same manner asEquations 14, and are:

cos d sin t = + cos h sin A ,cos d cos t = + cos h cos A sin d> + sin h cos 0,

sin d = cos h cos A cos


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CHAPTER III.TIME.

16. In astronomical practice, time is measured bywatches and clocks that differ in no essential respectfrom those in common use, but in addition to the com-mon system of time reckoning, astronomers employseveral others, of which we shall have to consider thefollowing

Sidereal Time, already referred to in 13.True Solar Time, which is frequently called Appar-

ent Solar Time.Mean Solar Time, which is the common system of

every-day life.These three systems possess the following features

in common : In each system that common phrase ' ' thetime of day" means the hour angle of a particular pointin the heavens, which we shall call the zero point of thesystem. The unit of time is called a day and is, in everycase, the interval between consecutive returns of thezero point to a given meridian; i.e., consecutive transitsof a given meridian past the same zero point. This unitis subdivided into aliquot parts called hours, minutes,and seconds. Each day begins at the instant when thezero point is on the meridian; i.e., on the upper half of

35


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36 FIELD ASTRONOMY.the meridian (noon) in astronomical practice, on thelower half of the meridian (midnight) in civil affairs. Inastronomical practice the hours from the beginning ofthe day are reckoned consecutively, from o to 24; incivil practice from o to 12, and then repeated to 12 again,with the distinguishing symbols a.m. and p.m. In con-sequence of the different epochs at which the day begins,the astronomical date in the a.m. hours is one day be-hind the civil date; e.g.,

Civil Time May 10, 5 h a.m. equalsAstronomical Time May 9, 17 11 .

In the p.m. hours the dates agree.Since an hour angle must be reckoned from a deter-

minate meridian, this meridian must be specified in orderto make "the time" a determinate quantity, and thisspecification of the meridian should be included in thename assigned to the time; e.g., Local Time denotes thehour angle of the zero point reckoned from the observer'sown (local) meridian. Greenwich Time is the hour angleof the zero point reckoned from the meridian of Green-wich. Standard Time is the hour angle of the zero pointreckoned from some meridian assumed as standard;e.g., in the United States and Canada the meridians 75 ,qo, 105 , and 120 west of Greenwich are called standard,and Eastern, Central, Mountain, and Pacific StandardTimes are hour angles reckoned from these meridians.

Alike practice is followed in the use of the term Noon;e.g., Washington Noon is the instant at which the zeropoint is in the act of crossing the meridian of Washington.


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TIME. ' 3717. Longitude and Time.We have introduced above

a reference to the time at different meridians, and we havenow to note that since ' ' the time ' ' is defined as an hourangle, it is evident that the number of hours, minutes,and seconds expressing either time or hour angle willdepend upon the meridian from which the latter is meas-ured. The difference between the hour angles reckonedfrom two different meridians will equal the angle betweenthe meridians, i.e., their difference of longitude, so thatif T' and T" represent the times of any event (whethersidereal, mean solar, or true solar time) referred to twodifferent meridians whose difference of longitude is X,we shall have T T" = k. It is customary in astronom-ical practice to express differences of longitude in hoursrather than in degrees, since both members of the pre-ceding equation should be given in terms of the sameunits.

By transposition of one term in the preceding equa-tion we obtain

V = T" + A (16)and this extremely simple equation indicates that anygiven time referred to the second meridian may be re-duced to the corresponding time of the first meridianby addition of the difference of longitude, where this dif-ference, A, is to be counted a positive quantity when thesecond meridian is west of the first. A very commonblunder is to omit this reduction to the prime meridianwhen interpolating from the almanac (see 21). Bewareof it, and note that the hour and minute for which a quan-


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38 FIELD ASTRONOMY.tity is required to be interpolated are usually given inthe time of some meridian other than that of Greenwichor Washington, for which the almanac is constructed,and must therefore be reduced to one of these standardmeridians, by addition of the longitude, before they canserve as the argument for the tabular quantity sought.

18. The Three Time Systems.The several time sys-tems differ one from another chiefly in respect of theirzero points, and these we have now to consider.

Sidereal Time.As already indicated, the zero pointof this system is the vernal equinox, and since this is apoint of the heavens whose position with respect to thefixed stars changes very slowly, it measures well theirdiurnal motion. In colloquial language, "the stars runon sidereal time," and this system is chiefly used in con-nection with their apparent diurnal motion.

Solar Time.As their names indicate, both TrueSolar Time and Mean Solar Time have zero points thatdepend upon the sun, and before drawing any distinctionbetween the two systems we recall that, owing to theearth's annual motion in its orbit, the sun's positionamong the stars changes from day to day (we see it fromdifferent standpoints). While this change in the sun'sposition is not an altogether uniform one and takes placein a plane inclined to that of the earth's rotation (ecliptic,and equator), its net result is that in each year the sunmakes one entire circuit of the sky, so that any givenmeridian of the earth, in the course of a year, makes oneless transit over the sun than over a star, or over thevernal equinox. The number of solar days in a year is


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TIME. 39therefore one less than the number of sidereal days; e.g.,for the epoch 1900, (according to Harkness,)

One (tropical) year = 366.242197 sidereal days= 365.242197 solar days. . (17)It appears from this relation that a sidereal unit of time(day, hour, minute) must be shorter than the corre-sponding solar unit, a relation that we shall have to con-sider hereafter.

Apparent, or True, Solar Time.This system has forits zero point the centre of the sun, and the hour angleof the sun's centre at any moment is, therefore, the truesolar time. This system is very convenient for use inconnection with observations of the sun, but owing tothe irregularities in the sun's motion, above noted,apparent solar days, hours, etc., are of variable length,a day in December being nearly a minute longer than onein September. When time is to be kept by an accuratelyconstructed clock or watch such irregularities are intoler-able, and for the sake of clocks and watches there isemployed for most purposes the third system, viz.,

Mean Solar Time.In this system the days and otherunits are of uniform length and equal, respectively, to themean length of the corresponding units of apparent solartime. The zero point of the system is an imaginarybody, called the mean sun, that is supposed to moveuniformly along the ecmator, keeping as nearly in thesame right ascension with the true sun as is consistentwith perfect uniformity of motion. The mean solartime at any moment is the hour angle of the mean sun


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40 FIELD ASTRONOMY.and, numerically, it differs from the corresponding truesolar time by the difference between the hour angles,or right ascensions, of the true and mean suns. Thisdifference is called the equation of time (the "sun fast"and ' ' sun slow ' ' of the common almanacs and calendars)and its value for each day of the year, at Greenwich noonand at Washington noon, is given in the AmericanEphemeris (see 21) and other almanacs.To change local solar time from one system to the otherwe have therefore to interpolate the equation of timefrom the almanac, with the argument the given localtime, reduced to the Greenwich or Washington meridianby addition of the longitude, and apply this differencewith its proper sign to the given local time. For exam-ple, let it be required to find for the meridian of Denver,and for the date May 10, 1905, the local apparent solartime corresponding to the mean solar time, M, givenbelow. The course of the computation is as follows

A. Denver west of Greenwich 6h 59 47 s . >M, Denver Mean Solar Time 3 5 10.5 2Greenwich Mean Solar Time 10 4 58 . I + 2Equation of Time +3 44-2 3Denver Apparent Solar Time 3 8 54 .7 2 + 3

19. Relation of Sidereal to Mean Solar Time. Sincethe sun makes the complete circuit of the heavens oncein each year it must, once in each year, have the sameright ascension, and therefore the same hour angle, asthe vernal equinox. At this particular moment, whichwe shall represent by the symbol V, sidereal and meansolar time will agree, but at any other moment they will


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TIME. 41differ, since, the sidereal units of time being shorter thanthe solar ones, sidereal time gains continuously and uni-formly upon mean solar time, with a daily rate that wemay represent by the letter a. Let D represent anygiven moment of the year, and let and M be the corre-sponding sidereal and mean solar times; we shall thenhave, at the instant D,

0-M = a{D-V), (18)where the interval D-V must be expressed in days,since a is a daily gain. This daily gain is clearly equalto the difference of length of the sidereal and solar day,and putting

i Sidereal Day = i Solar Day a"i Solar Day = i Sidereal Day -\-a'"

where a" and a'" are the values of a expressed respectivelyin solar seconds and in sidereal seconds, we readily find,from Equation 17,

a" = 2353.910, a'" =2363.556. (19)Where only a rough determination of the difference

between sidereal and mean solar time is required we mayassume in Equation 18,

V = March 22.6, a = a"' =4m[i- TVl,and obtain,

d=M + 4m(D- March 22.6) [i- TVl (20)This formula will furnish a result correct within one ortwo minutes.


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42 FIELD ASTRONOMY.If greater precision than the above is required we

must use more accurate values of V and a, and to thisend there is given in the American Ephemeris, for Wash-ington (and Greenwich) Mean Noon of every day of theyear, the value of the term a(D-V), which is therecalled the Sidereal Time of Mean Noon. We shall repre-sent this quantity by the symbol Q, and we note that forany other time than noon, or any other meridian thanthat of Washington, the difference between siderealand mean solar time, and M respectively, is equal toQ, plus the gain of one time upon the other in the intervalfrom Washington mean noon to the given d or M ; e.g., fora place whose longitude west of Washington is repre-sented by X we have,0-M = Q+(M+X)ar".In this equation, for any given place, the term Xa'" is aconstant, whose value may be determined once for alland written in the margin of the page containing thevalues of Q, so that we may take out from the almanacfor any given day, at a glance, and without interpola-tion, instead of 0, the sum,

Q+Xa'"=Qv (21)where Qx is for the local meridian what Q is for the Wash-ington meridian. We shall then have as the relationbetween the local and M,d=M + Q l + Ma'", (22)which is to be used for the accurate conversion of meansolar into sidereal time.


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TIME. 43For the converse process, converting sidereal into

mean solar time, we have the corresponding relationM = {d-Q l)-{d-Q l)a", (23)

where the last term is the equivalent of the Ma'" of thepreceding equation, but is expressed in sidereal units.The numerical values of {0 Q x)a" and Ma'" are mostconveniently to be obtained from Tables II and III, atthe end of the almanac. They give the values of theseterms for each minute and second of the twenty-fourhours, with the arguments Qt and M, expressed,respectively, in sidereal and mean solar time. Take Xa'",the constant correction to Q, from Table III, with A asthe argument.

To illustrate the actual process of changing timefrom one system to the other we shall, for an assumeddate given below, convert the Boston mean solar time,9h i9m 26 s .6;, into the corresponding sidereal time, andthen reconvert this sidereal time into mean solar time.The final result should, of course, be the same as theinitial value of M. The difference of longitude betweenBoston and Washington is assumed to be X = oh 24 i s .


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44 FIELD ASTRONOMY.parison only. It forms no part of the actual conversionof M into d.20. Chronometer Corrections. As already indicated,in actual practice the measurement of time is made byclocks or chronometers.

A chronometer does not differ essentially from anordinary watch, and like the latter is designed to showupon its face, at each moment, the mean solar time (orsidereal time) of some definite meridian, e.g., the merid-ian 90 west of Greenwich. Since the time indicatedby such an instrument is seldom correct, the error of thetimepiece must usually be taken into account, and inastronomical practice this is done through the equation,

6 = T + JT, or M = T + AT, (24)where T is the time shown by the chronometer (orwatch) and AT is the correction of the chronometer,i.e., the quantity which must be added, algebraically,to the watch time in order to obtain true time of thegiven meridian. When the chronometer is too slow Tis less than the true time at any moment, and AT is there-fore positive in this case and negative when the chro-nometer is too fast. While the symbol AT always repre-sents a chronometer correction its numerical value in agiven case depends upon the particular use required,i.e., whether the chronometer time is to be reduced tosidereal, or solar, local, or standard time. In the twoEquations 24, therefore, AT represents quite differentquantities, since and M are usually different one fromthe other, and in every case a special memorandum


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TIME. 45must be made showing whether the given AT relates tosidereal, mean solar, o~: apparent solar time.

If the chronometer gains or loses, it is said to have arate and AT will then change from day to day. If weassume a uniform rate, the relation between T and 6becomes

= T+ ATo+p(T-To), (25)where the subscript denotes the particular value of ATbelonging to the chronometer time T , and p is the rateof the chronometer per day or per hour, positive whenthe chronometer is losing time. The interval TTmust be expressed in the same unit as that for which pis given, hours for an hourly rate, etc. A similar equa-tion represents the relation between T and M,.but p andAT will be numerically different from the values requiredin Equation 2 5A sidereal chronometer, i.e., one intended to keepsidereal time, differs from a mean solar chronometeronly in the more rapid motion of its mechanism, and isin fact an ordinary timepiece for which p=3m 56 s .per day. Similarly a watch may be regarded as asidereal timepiece for which jO= + ios per hour. Asidereal chronometer is most convenient for use in obser-vations of stars, since their diurnal motion in hour angleis proportional to the lapse of sidereal time, but theseobservations may perfectly well be made with a watchor other mean solar timepiece, provided this is treatedas a sidereal chronometer with a large rate; e.g., if pdenote the hourly rate of the watch relative to mean solar


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46 FIELD ASTRONOMY.time, its hourly rate upon sidereal time will be p' = p + 10 s .Use Equation 25 in connection with this value of p'to determine from minute to minute the varying valueof J7.

21. The Almanac. The American Ephemeris andNautical Almanac is an annual volume issued by theU. S. Navy Department for the use of navigators, astron-omers, and others concerned with astronomical data.These data are for the most part quantities that varyfrom day to day and whose numerical values are givenat convenient intervals of Greenwich or Washingtonsolar time, e.g., the E and Q of the preceding sections,and the right ascensions and declinations of the sun,moon, planets, and principal fixed stars. The varia-tions of these quantities are due to many causes, orbitalmotion, precession, nutation, aberration, etc., that, ingeneral, lie beyond the scope of the present work, butwe shall have frequent occasion, as in 18 and 19, totake from the almanac numerical values of the quanti-ties above indicated, and these values are to be inter-polated for some particular instant of time, usually thatof an observation in connection with which they arerequired, as logarithms are interpolated to correspondto some particular value of the argument of the table.Since quantities are tabulated in the almanac for selectedinstants of Greenwich or Washington time, the time usedas the argument for their interpolation must be referredto one of these meridians (see 17).

For a detailed account of the way in which the almanacis to be used, consult the explanations given at the end of


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TIME. 47each volume, under the title, Use of the Tables. Inaddition to those explanations it should be noted thatunder the heading Fixed Stars, pages 304-399, thereare given three separate tables, from the last of which,bearing the subtitle Apparent Places for the UpperTransit at Washington, accurate coordinates of most ofthe stars may be obtained for use in the reduction ofobservations. For the remaining stars, five in numberand all very near the celestial pole, special provision ofthis kind is made in the second table, which bears thesubtitle Circumpolar Stars. Look here for the coor-dinates of Polaris. The first table, under the subtitleMean Places, etc., gives in very compact form, for allstars contained in the other two tables, right ascensionsand declinations, together with their Annual Variations,that may be consulted with advantage when only approx-imate values of these quantities are required, e.g., in thepreliminary selection of stars suitable to be observed.

In this connection the second column of the tableof Mean Places, entitled Magnitude, deserves especialnotice, since it furnishes an index to the brightness ofthe stars, which is an important element in decidingupon their availability for a given instrument. Thebrightness of each star is represented by a numberadapted, upon an arbitrary scale, to that brightness,so that a very bright star is represented by the numbero, one at the limit of naked eye visibility by 6,and intermediate degrees of brightness are representedby the intermediate numbers, carried to tenths of a mag-nitude. Polaris is of the magnitude 2.2, and is a con-


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48 FIELD ASTRONOMY.spicuous object in even a very small telescope, providedthe telescope is properly focussed. In the telescopeof an engineer's transit, stars of the magnitude 4.0 oreven 5.0 may be readily observed, while with a sextant,under ordinary conditions, the third magnitude maybe taken as the limit of availability.


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CHAPTER IV.CORRECTIONS TO OBSERVED COORDINATES.

It has already been pointed out that the problemsof practical astronomy are in great part cases of thetransformation of coordinates between systems havinga common origin but different axes, and it should benoted that the observed data for these transformationsfrequently require some correction before they can beintroduced into the equations furnished by the astro-nomical triangle. Aside from errors arising from de-fective adjustment or other purely instrumental causes,the observed coordinates of a celestial body may requireany or all of the following corrections.

22. Dip of the Horizon. This correction is requiredwhen an altitude is to be derived from a measurementof the angle of elevation of a body above the sea horizon.Owing to the spherical shape of the earth the visiblesea horizon always lies below the plane of the observer'strue horizon, and the amount of this depression mighteasily be determined from the geometrical conditionsinvolved, were it not that the rays of light coming tothe observer from near the horizon are bent by the at-mosphere (refraction), in a manner that does not admit

49


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50 FIELD ASTRONOMY.of accurate estimation in any given case, although itsaverage amount is fairly well known. We thereforeabstain from any formal investigation of this correc-tion, and expressing by, in feet, the observer's elevationabove the water, we adopt as a sufficient approximationto the observed amount of the depression, either of thefollowing formulas,

IT-y/eTlW*, #" = [I.7738V*- (26)The values of D given by these equations are expressedin minutes and seconds, respectively, but owing to varia-tions in the amount of the refraction the numericalvalues furnished in a given case may be in error byseveral per cent. As a correction D must always beso applied as to diminish the observed elevation abovethe horizon.

Note that if the depression of the visible horizonbe measured with a theodolite or other suitable instru-ment, Equation 26 will furnish an approximate valueof the elevation of the instrument above the water.

23. Refraction. In general the apparent directionof a star is not its true direction from the observer, sincethe light by which he sees it has been bent from itsoriginal course in passing through the earth's atmos-phere. The resulting displacement of the star from itstrue position is called refraction, and, like the similareffect noted in the previous section, its analytical treat-ment presents mathematical and physical problemswhose solution must be sought in more advanced worksthan the present. Some of the results of that solution


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CORRECTIONS TO OBSERVED COORDINATES. 51which we shall have occasion to use hereafter are asfollows: Save at very low altitudes, less than io, therefraction does not sensibly change the azimuth of astar, but its whole effect is to increase the altitude, sothat every star appears nearer to the zenith than it wouldappear if there wrere no refraction. The amount of thisdisplacement depends chiefly upon the star's distancefrom the zenith, but is also dependent in some measure:upon the temperature of the air and its barometricpressure.

If we represent by z' the star's apparent zenith dis-tance, as affected by refraction, by z the correspondingtrue zenith distance, by t the temperature, in degreesFahr., of the air surrounding the observer, and by Bthe barometric pressure in inches, the amount of therefraction, in seconds of arc, will be furnished by thefollowing two equations, which for all altitudes greaterthan 1 faithfully reproduce the refractions of thePulkowa Refraction Tables, and furnish values that maybe relied upon to within a small fraction of a secondof arc.

log F = [4.079 -io](353 + *) tan 2 2',, r -,-B tans' (27)s 2; =[2.0022 slv=r , nvv DJF 456 + 1

For most purposes of field astronomy the first of theseequations may be suppressed and the divisor, F, be putequal to unity; the resulting error in the computedrefraction will rarely be greater than 1".

The readings of a mercurial barometer, B', do not


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52 FIELD ASTRONOMY.furnish immediately the barometric pressure, B, butrequire a "reduction to the freezing-point," i.e., a cor-rection to reduce the reading to what it would be if themercury were at the normal temperature assumed in thetheory of the barometer. This reduction may be ob-tained with sufficient accuracy from the equation

B'-B = B ' (T ~ 29\ i- T\-), (28)10 000

where T is the temperature of the mercury, in degreesFahr., and the barometer reading and its resulting cor-rection are expressed in inches.

24. Semi-diameter.Observations of the sun or otherbody presenting a sensible disk are usually made bypointing the instrument at the edge of the body, techni-cally called the limb, and the resulting altitude or azimuthis that of the limb observed, while the data furnishedby the almanac relate to the centre of the body. Thesemi-diameters of the sun, moon, and planets, i.e., theangles subtended at the earth by their respective radii,are given in the almanac at convenient intervals of time,and the interpolated values of these quantities may beused to pass from the observed coordinates of the limbto those of the centre of the body, e.g., the sun. Inthe case of the altitude or zenith distance we have thevery simple relation

h'=h"S, (29)where 5 denotes the semi-diameter and h" and h! are, re-spectively, the observed and the corrected altitude. The


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CORRECTIONS TO OBSERVED COORDINATES. 53sign of 5 depends upon whether the lower or the upperlimb was observed.

In the case of an azimuth the relation is morecomplicated. From the right-angled spherical triangleformed by the zenith, the sun's centre and that point ofthe limb at which the latter is tangent to a vertical circle(see Fig. 4) we obtain,

sin z sin (A' A") = sin S, (30)which determines the correction, A' A", for differenceof azimuth between centre and limb. Since S does notmuch exceed 15', we may in mostcases assume the arcs to be propor-tional to their sines and simplify thisrigorous equation to the form,

A' =A" S sec h, (3i)in which the positive sign is to beused for the following and the nega-tive for the preceding limb.

25. Parallax In the reductionof astronomical observations it isusually necessary to combine theobserved coordinates, azimuth, alti-tude, etc., with data obtained fromthe almanac, e.g., the right ascension and declina-tion of the body observed. But the origin to whichthese latter coordinates are referred is the centre of theearth, while the origin for the observed coordinates is

Fig. 4.Semi-diameter.


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54 FIELD ASTRONOMY.at the eye of the observer, and before combining theseheterogeneous data we must reduce them to a commonorigin, for which we select that used in the almanac.

In Fig. 5 let C represent the centre of the earth, Othe observer's position, and P the observed body, at therespective distances p and r from C. Neglecting the

Fig. 5.Parallax.earth's compression, i.e., its slight deviation from a trulyspherical form, the line OC is the observer's verticaland, therefore, OPC is a vertical plane and marks outupon the celestial sphere a vertical circle, against whichthe body P will appear projected whether seen fromor C. Its azimuth will, therefore, be the same for thetwo origins and requires no reduction to the centre of theearth.

The altitude, however, does require such a reduction,and to determine its amount we let OH in Fig. 5 repre-sent the plane of the observer's horizon and obtain as


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CORRECTIONS TO OBSERVED COORDINATES. 55the observed altitude of P the angle there marked h'As seen from the centre of the earth the altitude of Pwill be measured by the angle PIH, marked h in thefigure, and from principles of elementary geometry wehave

h=h'+ LOPC.This last angle is called the parallax in altitude, and rep-resenting it by P we find from the triangle OPC

p sin (90 + h') =r sin P.Since r is always much greater than p, P must be a smallangle and, applying the principles of 4, we may write,in place of the preceding equation,

P=h W = 206265 cos h' t (32)which is the required correction to reduce an observedaltitude to the corresponding coordinate referred to anorigin at the centre of the earth.

For the fixed stars this correction is absolutely insen-sible, less than o".oooi, on account of their great distancefrom the earth. For the sun and planets it amounts to afew seconds of arc, and in its computation the value of thecoefficient, 206265 , should be taken from the almanac,where it is given for each of these bodies and is calledtheir horizontal parallax, since it is the amount of theparallax in altitude when the body is in the horizon,h'=o. For the sun it is usually sufficient to assume


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56 FIELD ASTRONOMY.8". 8 as a constant value for its horizontal parallax.The moon's parallax is much greater, about i, and thesimple analysis given above neglects some factors thatare of sensible magnitude in this case, although for ordi-nary purposes they may be ignored in connection withevery other celestial body.

Since the effect of parallax is to make the body appearfarther from the zenith than it really is, the correctionsfor parallax and refraction will always have oppositesigns.

26. Example.In the application of the several cor-rections required by an observed altitude they shouldbe applied in the order in which they have been treatedabove, and each successive partially corrected altitudeshould be used as the value of h required in computingthe next correction. As an example of such correctionswe take the following observed angle between the sun'supper limb and a water horizon as seen by an observerat an elevation of 63 feet above the water. The data

REDUCTION OF AN OBSERVED ALTITUDE.Temp, t


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CORRECTIONS TO OBSERVED COORDINATES. 57furnished directly by the observation are placed at thetop of the first column. The tan z' used above is ofcourse equal to cot h' and was taken from the logarith-mic tables as a cotangent.

In accordance with general custom the symbol logis printed in the above schedule only when necessary toavoid misunderstanding, as at the bottom of the firstcolumn. Usually the figures themselves indicate whetherthey are logarithms or natural numbers; e.g., the severalnumbers marked Const, are clearly the logarithms ofconstant coefficients. For similar reasons of conveniencethe 10 that strictly should be placed after a logarithmwhose characteristic has been increased by 10 is usuallyleft to the imagination.

27. Diurnal Aberration. There is a very small cor-rection to observed data, arising from the fact that theobserver himself is not at rest relative to the stars, butis always in rapid motion toward the east point of hishorizon, carried along by the earth in its diurnal rotation.This correction is so small that it may usually be omittedand we therefore abstain from an analytical investigationof its effect, such as may be found in the larger treatisesupon spherical astronomy, and note as a result of thatinvestigation that all stars when near the meridian aredisplaced toward the east point of the horizon throughan angular distance equal to o".32 cos 4>, where de-notes the observer's latitude. As a result of this dis-placement each star comes a little later to the meridianthan it otherwise would come and since the rate of motion( f a star when measured in arc of a great circle is propor-


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58 FIELD ASTRONOMY.tional to the cosine of its declination, the amount of thisretardation, expressed in time, is os.o2i cos sec d.See the theory of the transit instrument for an exampleof the application of this correction, and see also the de-termination of precise azimuths for another case in whichdiurnal aberration is to be taken into account.


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CHAPTER V.ROUGH DETERMINATIONS OF TIME, LATITUDE, ANDAZIMUTH.

28. General Considerations. For the purposes offield astronomy, which are the only ones contemplatedin the present work, the most important astronomicalproblems relate to the determination of time, latitude,and azimuth.

A time determination implies the making and reduc-ing of astronomical observations which suffice to furnishthe correction, AT, of a chronometer or other timepiece,and for this purpose we obtain from 15 and 20 therelations

a + t = d = T + AT, (33 )where a and t represent the right ascension and hour angleof any star at the chronometer time T. The studentshould particularly note that the chronometer is notsupposed to be correctly set ; T is the time shown by thechronometer regardless of whether that time be right orwrong, since the AT fully compensates for any error ofthis kind. In the case of the sun we have, from the rela-tion between mean and apparent solar time,

E + t =M = T + AT, (34)where E denotes the equation of time at the instant T.

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60 FIELD ASTRONOMY.Since a and E may be obtained from the almanac, any-

observation which determines the hour angle of a celestialbody at the observed time T will suffice to determineAT, and such an observation when properly reducedconstitutes a time determination.

An azimuth determination may be required eitherfor fixing the true azimuth of the line joining two terres-trial points, or for determining the relation of a particularinstrument to the meridian; e.g., to determine the read-ing, K, to which the azimuth circle of a theodolite mustbe set, in order that the line of sight shall point due south.A theodolite is said to be oriented when its verniers havebeen set to read the true azimuth of the object towardwhich the line of sight is directed, i.e., when K = o.By a latitude determination we mean any set of ob-servations from which a knowledge of the observer'slatitude may be obtained.

For each of these determinations, time, azimuth,latitude, many methods have been devised and thesediffer greatly among themselves with respect to theinstrumental equipment and expenditure of time andlabor which they require, and with respect to the corre-sponding degree of accuracy furnished in their results.In any given case a choice must be made among thesemethods wit

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