It is a leading characteristic of the St. Hilaire method of laying off position lines by means of “calculated altitudes," which is now coming into very general use, that it readily lends itself to the employment of tabular methods, whereby logarithmic calculation is either altogether dispensed with or very greatly simplified, as the case may be.
Such a method, devised by Lieut. Radler de Aquino, of the Brazilian Navy, was described at length in the PROCEEDINGS not long since. In the year 1910 Admiral Garcia Mansilla, of the Argentine Navy, published a volume of tables with the same object in view, and more recently Captain Guyou, of the French Navy, has brought out a compendious work of the same nature.
In each case the method followed presents novel and ingenious features of its own, but probably the most simple and easily intelligible work of the kind is that produced by Mr. F. Ball, a naval instructor in the British Navy. In this comprehensive work Mr. Ball gives the altitude corresponding to each complete degree of latitude and of hour angle between the values of 0° and 60° both of latitude and declination, the argument declination proceeding also by single degrees. And since in working by the St. Hilaire process the computer can take as his point of departure any position not far removed in latitude and longitude from the dead reckoning values, it is easy by a simple process of adjustment to obtain a point such that latitude and hour angle may each be represented by a complete degree. No interpolation is, therefore, required, except for the odd minutes of declination, which again is simplified by the use of a table specially designed for the purpose.
It happens, however, that a set of tables planned upon such an elaborate design as this is necessarily some what bulky, and that the consequent expense of production places it beyond the reach of mariners in general, and it has occurred to the writer that there might perhaps be room for a volume of tables, which, proceeding generally upon the lines followed by Mr. Ball, would be of smaller dimensions, and consequently be more generally accessible to seamen.
The general idea upon which the proposed table would be based is this: In the triangle of position PXZ, P is the pole, Z the zenith and X the place of the body observed. Let us suppose that c, the colatitude, and h, the hour angle, are regarded as constant, and that z is the zenith distance corresponding to a given value of p, the polar distance. Let p be increased by a certain amount (p), and let z be the zenith distance corresponding to the new value of the polar distance for the constant values c and h.
Then by Taylor's theorem,
z1 - z = dz/dp (?p) + ½d2z/dp2 sin 1’ (?p) 2 + 1/6 d3z/dp3 (sin2 1’) (?p)3 + …
Let us suppose that z has been calculated and tabulated for given values of p, c, h, and, for the present leaving out of consideration the third term of the expansion, let dz/dp (?p) be denoted by Correction A, and let ½d2z/dp2 sin1'(?p)2 be indicated by Correction B.
Then
z1 = z + A + B
We will assume, then, that dz/dp, ½d2z/dp2 sin 1’ have been calculated. It will be convenient to give the results in the form of logarithms. Thus
log A = log dz/dp + log (?p)
log B = log ½d2z/dp2 sinc 1’ + 2log (?p)
the logarithms of dz/dp and 1/2 d2z/dp2 sin 1’ being taken from the proposed table, and that of (?p) from the ordinary table of logarithms of numbers.
Taking into account only these two terms of the series it will be sufficient to tabulate for each 8° of declination, so that by including only 0°, 8°, 16°, 24° North and South, we should cover the whole range of declination from 28° N. to 28° S., thus effecting a very great saving in the size of the volume of tables necessary at the expense of a very little additional trouble in calculation.
It will first be necessary to obtain expressions for dz/dp and d2z/dp2.
TO FIND AN EXPRESSION FOR dz/dp.
From the triangle PXZ we have
cos h sin c sin p= cos z — cos p cos c.
Differentiating,
cos h sin c cos p = -sin z dz/dp + sin p cos c,
sin z dz/dp = cos c sin p – sin c cos p cos h,
= cos c sin p – sin c cos p cos z – cos p cos c / sin p sin c,
= cos c sin2p – sin c cos2p – cos p cos z/sin p,
dz/dp = cos c – cos p cos z / sin p sin z = cos PXZ.
TO FIND AN EXPRESSION FOR d2z/dp2.
Since dz/dp = cos X,
d2z/dp2 = -sin X dX/dp.
We have, therefore, to find an expression ford dX/dp.
In the triangle PXZ the formula connecting X, p, h, c is
cot c sin p= cot X sin h+ cos h cos p.
Differentiating,
cot c cos p = -cosec2 X dX/dp sin h – cos h – cos h sin p,
dX/dp = -sin2 X cosec h(cot c cos p + cos h sin p),
= -sin2 X (cot c cos p/sin h + cot h sin p).
Substituting this value in the formula above,
d2z/dp2 = sin3 X cot c cos p/sin h + cot h sin p)
and
log for Cor B = log d2z/dp2 + log (1/2 sin 1’)
With regard to the algebraic signs of A and B, Correction A is additive for actual polar distance greater than tabular, subtractive for polar distance less, so long as the angle X lies in the first quadrant.
The rule is reversed when X is greater than 90°, and the cosine, therefore, becomes negative. This can only occur in the exceptional case when latitude of place is of same name as the declination, and smaller in amount. Correction B will be always additive.
THE METHOD OF TABULATION.
The arguments of the table will be three in Umber, viz., declination; proceeding by differences of 8° from 0° to 24°; the latitude, by single degrees from 0° to 60°; the hour angle, by differences of single degrees, i.e., of 4 minutes of time.
The quantities tabulated would also be three in number, viz., the zenith distance, corresponding to the tabulated values of declination, latitude and hour angle.
The logarithm for Correction A, which, as has been shown, is simply L cos X.
The logarithm for Correction B, obtained from the expression given above.
THE ADJUSTMENT OFTHE DEAD RECKONING POSITION.
It is a feature common to tables of this nature that the point worked from upon the chart, that is, the point of origin for the intercepts, should be such that its latitude is an even degree. The longitude of the point also must be such as to give for the hour angle an even degree, or 4 minutes of time.
In the case of the latitude, no difficulty occurs, and it is only necessary to select the degree nearest to the D. R. latitude. To find the longitude the following artifice is adopted:
Find the hour angle of the body at Greenwich, i.e., in the case of the sun the Greenwich apparent time, and express this hour angle in arc.
To this apply such an amount of arc, adding if the longitude is east, subtracting if it is west, as will make the hour angle resulting an exact degree, the amount of arc so applied being as near as possible to the D. R. longitude.
The arc thus applied will be the longitude of the position from which to work.
The following example will serve to illustrate the procedure:
In D. R. position lat. 28° 47' N. longitude 27° 3' W., when Greenwich apparent time was 3h 47m 238, an altitude of the sun was observed. Find the latitude and longitude from which to work. The latitude is 29°N. For the longitude,
Greenwich apparent time (in arc) 56° 53'
Subtract 26° 53'
Hour angle 30° 00'
Thus 26° 53' W., differing from the D.R. longitude by gives the exact value of 2h for the hour angle.
DESCRIPTION OF TABLES I AND II.
To illustrate practically the method proposed, two specimen tables are appended. For each table the hour angle 2h is adopted. Table I gives the values of the zenith distances and of the angle PXZ for declination 0°, 8° and 12° of same name as latitude, and for each degree of latitude from 0° to 30°. It has not been thought necessary to proceed to higher latitudes than 30° because the higher the latitude the more exact are the results furnished by the method. The latitudes 0° to 30°, therefore, supply a sufficient test.
In Table II are given the logarithms for the calculation of corrections A, B and C for the three values of the declination, except that in the case of declination 8° the logarithm for C is omitted, since this value of the declination would only be included in the tables under the system where Correction C is left out of account, the computation being effected by the use of Corrections A and B only.
It should also be mentioned that although, for greater clearness of exposition, the value of the angle X is shown in Table I, this actual value is not really necessary, the value of L cos X, given in Table II as the "Log for Cor A" being all that is required. By adding the column of zenith distances to Table II, therefore, we should be able to dispense with Table I and give all the necessary details in a single table.
PRACTICAL RULES FOR THE USE OF TABLES I AND II.
To obtain a required zenith distance by means of the tables we should proceed as follows:
FOR CORRECTION A.
Find the difference, expressed in minutes, between the polar distance of body, measured from the elevated pole, and the nearest tabular value.
TABLE I.
ZENITH DISTANCES AND VALUES OF THE ANGLE X.
Hour Angle 2h.
{chart}
TABLE II.
LOGARITHMS FOR CALCULATION OF CORRECTIONS.
Hour Angle 2h.
{chart}
To the log of this difference add the "Log for Cor A" taken from the table.
The sum, rejecting 10, is the logarithm of Correction A.
FOR CORRECTION B.
To twice the logarithm of the difference between actual and tabular polar distance add the "Log for Cor B" from Table II. The sum, rejecting 10, is the logarithm of Correction B. Apply Corrections A, B to the zenith distance tabulated in Table I, in accordance with the precepts at the foot of the respective tables.
EXAMPLES IN USE OF TABLES.
Example 1.—Find the sun's zenith distance for hour angle 2h, in latitude 3° N., when the declination is 4° 6' N. Here 8° is the nearest tabular value for the declination, the difference of polar distance (?p) being 3° 54', or 234'. Actual polar distance greater than tabular.
A | B | ||
Log for A (from Table II) | 9.13355 | Log for B (from Table II) | 6.3600 |
Log 234 | 2.36922 | 2 Log 234 | 4.7384 |
(Sum) | 1.50277 | (Sum) | 1.0984 |
Cor A | 31.8’ - | Cor B | 12.5’ + |
Cor B | 12.5’ + | Zenith distance (from Table I) | 30° 16' |
| 19.3’ - | Corrections | 19.3’ - |
|
| Zenith distance required | 29° 56.7' |
Here Correction A is subtractive for actual P. D. greater, because the "Log for A" is above the "bar," which, in Table II, for declination 8°, is drawn between the logarithms for 6° and 70 of latitude. This bar indicates the point at which the angle X passes from one quadrant to another, so that the cosine of the angle changes its sign, and the algebraic sign of the correction is altered accordingly. Reference to Table I shows that for latitude 6° the value of X is 91° 53', and for latitude 7° it is 89° 53'.
Example 2.—Find the zenith distance of Procyon (Dec. 5° 27' N.) for H. A. 2h in latitude 25° N.
The nearest tabular value is again 8°, and the difference of polar distance 2° 33' or 153'. Actual polar distance greater.
A | B | ||
Log for A (from Table II) | 9.75110 | Log for B (from Table II) | 6.1771 |
Log 153 | 2.18469 | 2 Log 153 | 4.3693 |
(Sum) | 1.93579 | (Sum) | .5464 |
Cor A | 86.2’ - | Cor B | 3.5’ + |
Cor B | 3.5’ + | Zenith distance (from Table I) | 33° 16.5' |
| 89.7’ - | Corrections | 1° 29.7’ |
|
| Zenith distance required | 34° 46.2’ |
Here actual polar distance greater gives an additive Correction A, because latitude 25° is below the bar, and X consequently less than 90°.
THE THIRD CORRECTION C.
We have so far considered only two terms of the series for z, the tabulation of which would enable us to employ a difference of declination of 8°, the values tabulated being 0°, 8°, 16°, 24° north and south, seven values in all. But if we include an other term of the series, and thus obtain the logarithm for a third Correction C, it would suffice to proceed by a common difference of 12° of declination, as 0°, 12°, 24°, and the total number of values for the declination employed would thus be reduced from seven to five, with a considerable saving in the space occupied.
The next term of the expansion would be
1/6 d3z/dp3 sin 1’ (?p) 3,
so that if the values of 1/6 d3z/dp3 sin2 1’ are calculated and tabulated we have only to add three times the logarithm of (?p) to the tabular logarithm to obtain the logarithm of C. As three places of decimals are amply sufficient, this is not a very troublesome operation.
TO INVESTIGATE AN EXPRESSION FOR d3z/dp3.
It has already been shown that
d2z/dp2 = sin3 X (cot c cos p/sin h + cot h sin p),
and also that
dX/dp = - sin2 X cot c cos p/sin h + cot h sin p).
Differentiating the first of these expressions,
d3z/dp3 = 3sin2 X cos X dX/dp (cot c cos p/sin h + cot h sin p)
+ sin3 X (cot h cos p – cot c sin p/sin h).
But
cos h cos p – cot c sin p/sin h = - cos X,
sin3 X (cot h cos p – cot c sin p/sin h) = -sin2 X cos X
Thus
d3z/dp3 = 3sin2 X cos X dX/dp (cot c cos p/sin h + cot h sin p) - sin2 X cos X,
Substituting for dZ/dp,
=-3sin2 X cos X dX/dp (cot c cos p/sin h + cot h sin p)-sin2 X cos X,
=-2sin2 X cos X{3sin2 X (cot c cos p/sin h + cot h sin p)2 + 1},
The final expression to be tabulated, therefore, is
-1/6sin2 X cos X{3sin2 X (cot c cos p/sin h + cot h sin p)2 + 1}
which has to be multiplied by (?p)3 to give the amount of the correction.
As the amount of Correction C is never very large it will be found to be sufficient to calculate for each 5° of latitude, and interpolate by ordinary proportion. It will suffice also to take three places of decimals only, and to tabulate for each second degree. Interpolation for intermediate degrees could easily be effected at sight.
The negative sign of the correction indicates that C will be of opposite sign to A. In practice, the simplest plan will be to regard C as a deduction to be made from the value of A, and proceed accordingly.
PRACTICAL RULES FOR FINDING CORRECTION C.
Write down the log for Correction C from Table II. Under this write three times log (?p) in minutes.
The sum, less 10, is the logarithm of C.
EXAMPLES (CORRECTION C TAKEN INTO ACCOUNT).
Example 3.—Find the zenith distance of Procyon (Dec. 5° 27' N.) for H. A. 2h in latitude 25° N.
It will be noticed that this example has been already worked, taking as our starting point declination 8°, and making use of Corrections A and B only. It will be instructive to work it also with the aid of all three corrections, making use first of the tabular values for declination 0°, and again with those given under declination 12°.
WITH DECLINATION 0°.
The difference of polar distance (?p) is 5° 27' or 327'. Actual polar distance less than tabular.
A | B | C | |||
From Table II | 9.83378 | From Table II | 5.9924 | From Table II | 2.474 |
Log 393 | 2.51455 | 3 Log 393 | 7.544 | ||
(Sum) | 2.34833 | 2 Log 327 | 5.0291 | (Sum) | .018 |
A | 223’- | (Sum) | 1.0215 | C | 1’ + |
C | 1 + | B | 10.5’+ |
|
|
A-C | 222 - | Z.D. from Table I | 38° 17' |
|
|
B | 10.5 + | Correction | 3° 31.5' |
|
|
Correction | 211.5 - | Z.D. required | 34° 45.4’ |
|
|
WITH DECLINATION 12° N.
A | B | C | |||
From Table II | 9.68397 | From Table II | 6.2577 | From Table II | 2.674 |
Log 393 | 2.59439 | 3 Log 393 | 7.783 | ||
(Sum) | 2.27836 | 2 Log 393 | 5.1888 | (Sum) | .457 |
A | 189.8’+ | (Sum) | 1.4465 | C | 2.9’ |
C | 2.9 - | B | 28’+ |
|
|
A-C | 186.9+ | Z.D. from Table I | 31° 10.5' |
|
|
B | 28. + | Correction | 3° 34.9'+ |
|
|
Correction | 214.9+ | Z.D. required | 34° 45.4’+ |
|
|
Thus, proceeding by a wholly different route, we arrive at precisely the same result as before.
For a second example we take a case in the neighborhood of the "bar," which, in the column "Log Cor A," in Table II, forms the dividing line between values of A in the first and second quadrant. Example 4.—Find the zenith distance of the sun for H. A. 2h in latitude 10° N. when the declination is 18° N.
Difference of polar distance is 360'. Actual polar distance less than tabular.
A | B | C | |||
Log from Table II | 8.17130 | From Table II | 6.4081 | From Table II | 1.333 |
Log 393 | 2.55630 | 3 Log 393 | 7.669 | ||
(Sum) | .72760 | 2 Log 393 | 5.1126 | (Sum) | 9.002 |
A | 5.3’+ | (Sum) | 1.5207 | C | .1’ |
C | .1 - | B | 32.2’+ |
|
|
A-C | 5.2+ | Z.D. from Table I | 29° 30' |
|
|
B | 33.2 + | Correction | 38.4'+ |
|
|
Correction | 38.4+ | Z.D. required | 30.8° 4’ |
|
|
This case may be regarded as a somewhat special one because A and C, being on the point of vanishing and changing sign, B, which is not far from attaining a maximum value, becomes the most important element in determining the amount of the total correction.
It should be remembered that although provision is made in the tables for all three corrections, it is by no means necessary that the whole of these should be taken into account on every occasion. When actual declination, for instance, falls within 1°, or 60', of the tabular value, both B and C might very generally be both omitted, especially in the higher latitudes, and C need not be included when actual is within 30 or 40 of the tabular value of declination. A moment's inspection of the value for the logarithms of the corrections given in the table should show whether these are worth taking in to account. In the example just worked, for instance, although we were employing a difference of polar distance between actual and tabular of 360', which is a limiting value, the small amount of the "Log for C," viz., 1.333, would indicate at once that the resulting value of Correction B could only be a small fraction of a minute, and might, therefore, be left out of account.
Roughly speaking, C might probably be safely neglected in two-thirds of the whole number of cases.
Some little simplification would result also from the inclusion in the tables of a special table of "Logarithms of Numbers," occupying only a few pages of space. Such a table might furnish at sight the logarithm of a number, the logarithm of its square and of its cube. Risk of errors in multiplication would thus be eliminated.
The form for such a table is exhibited below.
It will be noticed that in the examples worked out in pairs, that the corrections once determined give us the total correction for an actual zenith distance greater as well as less than the tabular by a given number of minutes.
Example 5.—In latitude 27° N. from the tabular values for declination 0°, find the zenith distances at H. A. 2h for declination 40 N. and 4° S. respectively.
The results show that for a given hour angle the increase of zenith distance produced by an increase of polar distance (?p) will in general be greater than the decrease in zenith distance caused by a diminution (?p) in polar distance. In the exceptional case, however, when latitude being less than declination and of same name, the angle X lies in the second quadrant, the contrary will be the case.
A | B | ||
Log for Table II | 9.85349 | Log from Table II | 5.9359 |
Log 240 | 2.38021 | 2 Log 240 | 4.7604 |
(Sum) | 2.23370 | (Sum) | .6963 |
A | 171.3’ | B | 5’+ |
SOME SPECIAL CASES IN THE VALUES OF THE CORRECTIONS.
If the hour angle is 0°, the angle X is also 0°; Corrections B and C vanish, cos X becomes unity and the change of zenith distance is equal to (?p), the change in polar distance.
Correction B also vanishes if the body is in the horizon, since in that case
cot c cos p + cos h sin p= 0.
If X=90°, that is, if the body is at the position of "maximum azimuth," cos X=0°. In that case A and C must both vanish and the total correction depends upon B. This point has been already illustrated in the work of Example 4.
GENERAL OBSERVATIONS.
A few words may perhaps be added here as to the history of the system of Marcq Saint-Hilaire position-lines, in which this problem of calculating zenith distance plays such an important part. Originally proposed by the late Admiral Saint-Hilaire in the Revue Maritime, in July, 1875, the method attracted little attention at the time, or for some years after wards. Eventually, however, it came to be practiced extensively in the French Navy, and within the last few years it has received official recognition in the British Navy also, to the exclusion of all other methods. Still more recently the advantage of having one simple method of procedure, available in all cases, has come to be recognized more generally, and the practice of the Saint-Hilaire system is to-day steadily gaining ground in the mercantile marine.
Under these circumstances the problem of calculating a zenith distance, which, in the old days, was required chiefly for use in the identification of a star, and which commonly received only a casual degree of attention, and, indeed, in some cases, was not noticed at all in standard works on navigation, has acquired a largely increased amount of importance, so that navigators are naturally not unwilling to consider any device which seems to offer a simplification of the problem.
With regard to the system proposed in these pages the following points may be noticed.
(a) The method of procedure, by means of corrections applied to a tabulated zenith distance, is in many ways analogous to the recognized process for deducing latitude from an altitude of the pole-star, and, with many of the methods for reducing a zenith distance, taken upon a small bearing to the meridian.
(b) With the aid of such a table of logarithms of squares and cubes of numbers as has been suggested up to 360, with the logarithms of the numbers continued up to 1000, the whole of the calculations need not involve more than two openings of the tables.
(c) The rules for the application of the algebraic signs are of a very simple character, the only point requiring particular attention being the difference between the actual and tabular polar distance, in connection with the sign of Correction A, and this involves only a moment's consideration.
(d) The summation of only two logarithms at a time is substituted for the customary "long tot." Risk of error in the process of addition is thereby largely reduced.
In view of the growing popularity of the Marcq position-lines, navigators would seem to be standing at a "parting of the ways." They are gradually discarding the ancient methods of fixing position by the direct determination of a latitude or longitude in favor of a system which offers a single solution of the problem, available at all times, irrespective of special conditions of hour angle and azimuth, involving the simple calculation of a zenith distance in all cases a like. The processes offered by the standard text-books leave something to be desired, and amongst the multitude of new methods proposed the mariner will doubtless in due time make his selection and pick out that which is best suited to his needs. The wider, therefore, the field of choice the better.