TO FIND THE LATITUDE BY DOUBLE

ALTITUDES.

FORM I.-By double altitudes of the sun.

WHEN (by reason of clouds, or from other causes) a meridian altitude cannot be obtained, the latitude may be found by two altitudes of the sun, taken at any time of the day, the interval or elapsed time between the observations being measured by a good watch or chronometer, noticing the seconds, if possible, or estimating the times to a third or a quarter of a minute, if the watch is not furnished with a second-hand. The observed altitudes of the sun must be corrected, as usual, for the semidiameter, dip, refraction, and parallax, in the same manner as in finding the latitude by a meridian altitude. When great accuracy is required, the declination must be found at the time of each observation, using the third method of solution hereafter given; but when the sun's declination varies slowly, or the elapsed time is small, it will in general be sufficiently accurate to find the sun's declination for the middle time between two observations, and to consider it as invariable during the observations, computing the latitude by the first or second method.

This manner of finding the latitude is, in general, most to be depended upon where the sun's meridian zenith distance is great. If the sun passes the meridian near to the zenith, much greater care must be taken in measuring the altitudes and noting the times, than would be necessary under other circumstances. The nearer the sun is to the meridian, at the time of one of the observations, the more correct the result will commonly be. In general, the elapsed time ought to be as great, or greater, than the time of the nearest observations from noon. Similar remarks may be made upon every one of the following forms.

In all these observations it is supposed that the watch moves uniformly according to apparent time, measuring twenty-four hours from the time of the sun's passing the meridian on two successive days at the same place of observation. If the watch gain or lose on apparent time, supposing the observer to be at rest, a correction must be applied for the gain or loss during the time elapsed between the observations, so as to obtain accurately the elapsed time or hour angle. It is not required that the watch should be regulated so as to give precisely the hour of observation; the only thing required is to find the elapsed time with all possible accuracy.

FORM II.-Double altitudes of a star.

Double altitudes of a fixed star may be used in finding the latitude, and the calculation is almost identical with that of double altitudes of the sun; the only difference consists in adding a small correction to the elapsed mean solar time between the observations, on account of the daily acceleration of 3′ 56′′ in the time a star comes to the meridian on successive days; in other words, the elapsed time (or hour angle) must be reckoned in sideral time, of which we have already spoken in the second note on page 147. Now, as a chronometer is usually adjusted to mean solar time, and the observations marked by it, we must add to the mean time, elapsed between the observations, the correction given in Table LI., to reduce it to sideral time. Thus, if the interval in mean solar time be 3, the corresponding correction in this table is 29.6, making the interval in sideral time (or the correct hour angle) 3h 00m 29.6, which is to be used in the rest of the calculation.

In observations of a fixed star, the altitudes are to be corrected for dip and refraction, as in finding the latitude by a meridian altitude. The declination of the star is to be found in Table VIII.* With these altitudes, the declination, and the hour

*Or more accurately in the Nautical Almanac, if any one of the bright stars is observed whose

angle, the calculation is to be made by either of the three methods hereafter given.

The chief difficulty, in observations of this kind, with a fixed star, is the want of a good horizon in the night-time. The method, however, might sometimes be used with success, soon after the dawn of day, or late in the evening twilight, at a time when the horizon is well defined, and the star sufficiently bright to bring its reflected image to the horizon. Sometimes a good horizon is produced by the aurora borealis, in which case a good observation might be made with stars in the northern horizon; but a single observation of the polar star will answer the same purpose, and will be much more simple.

FORM III.-Double altitudes of a planet.

Double altitudes of a planet (particularly Jupiter and Venus, on account of their great brightness) may sometimes be used with success. The observed altitudes must be corrected for dip and refraction. The parallax and semidiameter, being small, may be neglected, except in cases where extreme accuracy is required. The declination of the planet is to be found, in the Nautical Almanac, for the supposed time at Greenwich. The daily variation of the time of coming to the meridian is also to be found in the same page; and thus the time elapsed between the passage of the planet over the meridian on two successive days is found; then the corrected elapsed time, or hour angle, is obtained by the following rule:

RULE. As the interval of time between two successive passages of the object over the meridian is to twenty-four hours, so is the elapsed mean time between the observations to the corrected elapsed time, or hour angle.

With this hour angle, the declination, and corrected altitudes, the latitude may be found by either of the three following methods of calculation.

FORM IV.-Double altitudes of the moon.

Double altitudes of the moon may also be used in finding the latitude. These observations may be easily and very accurately made; but the calculation is much more complex than any of the preceding methods, on account of the great change in the moon's declination and right ascension during the elapsed time between the observations. If, however, by the times of observation, and the longitude of the ship, (or else by a chronometer,) the time at Greenwich can be obtained within a few minutes, we may, from the Nautical Almanac, find the corresponding declination, semidiameter and horizontal parallax of the moon for each of these observations. With the horizontal parallax, and the moon's apparent altitude, find the correction in Table XIX., which, being subtracted from 59′ 42′′, leaves the correction of the moon's altitude for parallax and refraction; this is to be added to the corresponding observed altitude, corrected for semidiameter and dip, to obtain the moon's correct central altitude. This is to be done at each observation. Lastly, the time of the moon's passing the meridian on successive days, given in the Nautical Almanac, shows the interval of time between two successive passages of the moon over the meridian,† and this time is to twenty-four hours as the elapsed time between the observations is to the corrected elapsed time or hour angle. With this hour angle, the correct central altitudes, and the declinations, the latitude may be found by the fourth of the following methods of calculation, it being very rare that the other methods can be used, on account of the great change in the moon's declination.

FORM V.-By altitudes of two different objects, taken at the same time.

The latitude may be obtained by observing, at the same moment of time, the altitudes of two heavenly bodies; as, for example, (1) The sun and moon; (2) The moon and a fixed star or planet; ‡ (3) A planet and a fixed star; (4) Two planets; (5) Two fixed

* When extreme accuracy is not required, we may find the correction for parallax and refraction from Table XXIX., which, the altitudes are large, will not vary much from the truth.

This time is given to tenths of a minute, which in general is sufficient, because, if the elapsed time be small, the effect of this correction will be only a few seconds. It might be obtained more accurately by means of the right ascensions of the sun and moon, using the second differences, as taught in the Appendix.

A particular case of this method occurs in taking a lunar observation, which will be treated of separately, because, the distance of the two bodies being known, the calculation becomes more simple.

stars. In these methods the altitudes are to be corrected, as in the preceding Forms, for dip and refraction; also for parallax and semidiameter when necessary, as is always the case in observations of the moon and sun. The declinations of the bodies are to be found for the supposed time of observation, reduced to the meridian of Greenwich, by means of the Nautical Almanac, or by Table VIII. for the fixed stars, as before taught. Then the difference of the right ascensions of the bodies (or that difference subtracted from 24 hours, if it exceed 12 hours) will be the hour angle, which is to be used, with these declinations and corrected altitudes, in finding the latitude, by either of the three first methods, if the declinations should be equal, or differ but one or two minutes; otherwise by the fourth method, which, in fact, may be considered as the only method to be used in this kind of observations, because, in almost all cases, the declinations of the objects differ considerably.

FORM VI.-By altitudes of two different objects, taken within a few minutes of each other, by one observer.

It may sometimes happen, for want of two good instruments, or from not having two observers, that the preceding Form V. cannot be employed. In this case the whole of the observations may be made by one person, noticing the interval between the observations, and making the calculation as in the following Form VII. But it is in general much better to make the observations as near to each other as possible, and then, by a very simple process, the calculation may be reduced to that of Form V., in which the observations are taken at the same moment. This is done by observing the first object twice, before and after observing the second object. For if the intervals of time between these three observations be equal, (as, for example, one minute, or two minutes,) the half-sum of the two altitudes of the first object may be taken for the altitude corresponding to the time of observing the second altitude, and the calculation may then be made as in Form V. Thus, suppose at 10h 2m, A. M., per watch, the altitude of Sirius was 17° 54', at 10h 4m per watch the altitude of Capella 60° 45', and at 10h 6m per watch the altitude of Sirius was again observed and found to be 17° 58'. In this case, the intervals of time are exactly two minutes; therefore the half-sum of the altitudes of Sirius is to be taken 17° 56, and combined with the altitude of Capella 60° 45', supposing both to have been observed at 10 4 per watch. This is the most simple form in which an observation of this kind can be made by one observer.

If, from any cause whatever, the observations cannot be taken at exactly equal intervals, the altitude of the first object, at the time of observing the second object, may be found by proportion, supposing the altitudes to vary uniformly during the few minutes of the observations. Thus, in the preceding example, suppose the altitudes and the two first-noted times to remain unaltered, but the last observation of Sirius to have been at 10h 10m per watch, instead of 10h 6m. In this case, during the eight minutes of time elapsed between 10h 2m and 10h 10m, Sirius would have risen 4', (from 17° 54′ to 17° 58';) therefore, by proportion, it is found that in two minutes (the time elapsed between 10h 2m and 10h 4m) the star would have risen l', and the altitude would have increased from 17° 54′ to 17° 55'; therefore, at the time 10h 4m per watch, the altitude of Sirius must be taken at 17° 55', the altitude of Capella 60° 45', and with these quantities, considered as observed at this last-mentioned time 10h 4m, the calculation must be made as in Form V.

There are several advantages attending these two last forms V., VI., since no allowance is necessary for the change of place of the ship; the observations can be immediately made, in a short interval of fair weather, when the common method of double altitudes might fail from the intervention of clouds; the time can also be obtained at the same operation, &c.

FORM VII.-By altitudes of two different objects, taken at different times.

This method differs but very little from the two last. The altitudes are to be corrected, in the same manner, for dip and refraction; also for parallax and semidiameter, when necessary. The right ascension and declination of each object is to be found for the supposed time of observing that object reduced to the meridian of Greenwich. Then the apparent elapsed time between the observations, is to be turned into sideral time, which may be done, as in Form II., by adding the correction in Table LI. corresponding to this time; the difference between this sum and the right ascension of the body last observed is the hour angle. This, with the

declinations and corrected altitudes, is to be used in finding the latitude by the third or fourth of the following methods of calculation, it being very rarely the case that the first or second methods can be used, on account of the difference of the declinations. These three last forms, when a fixed star or planet is used, are restricted very much from the want of a good horizon in the night; they are best adapted to the morning and evening twilight.

GENERAL REMARKS.

Having thus explained several of the different forms of making these observations, and the manner of finding in each form the hour angle, the declinations, and the correct central altitudes, we shall now give four different methods of calculating the latitude, and shall illustrate the rules by proper examples. In the first and second methods, the declination is supposed to be the same at both observations, which is true as it respects observations of a fixed star, and is in general sufficiently correct for common observations of double altitudes of the sun. The first of these methods is direct and simple, not embarrassed with much variety of cases, requiring only ten openings of the Table XXVII., without any halving or doubling of the logarithms, or the use of natural or versed sines. This method is in fact nearly, if not fully, as short as the second or approximative method invented by Mr. Douwes, and which was exclusively used in the former editions of this work. This second (or Douwes') method is liable to the objection that the calculation must sometimes be repeated several times before a true solution can be obtained, and then it becomes extremely troublesome. This difficulty does not occur in the first method; and on this account, as well as for its remarkable simplicity, the first method is always to be preferred.

The third method is applicable to cases where there is a small variation in the declination of the object, during the elapsed time between the observations, as most commonly happens when the sun is used. This method is short and simple, and is much facilitated by the use of Table XLVI., which I have computed.

The fourth method embraces the general solution of the problem in the case where any variation whatever of declination is noticed. This increases the labor considerably, and renders the solution more complex in its cases. It is, however, believed, that this method, drawn up in its present form by the author of this work, will be easily understood by navigators, and that they will thus be enabled to determine the latitude with considerable accuracy in cases where it might be of the utmost importance to know it, and where other methods could not be resorted to on account of bad weather. This method is nearly, if not quite, as short as that published by Dr. Brinkley in the Nautical Almanac of 1825, and does not require, like his method, a second or third (or even a greater number) of operations.

If the observer should change his place or station, during the elapsed time between the observations, a correction must be applied to one of the altitudes on this account. The manner of doing this is shown in the following examples.

It may be observed that in like manner as there are two latitudes corresponding to the same meridian altitude of the sun, according as the zenith is north or south of the sun when on the meridian, so in double altitudes there are generally two latitudes, corresponding to the proposed altitudes, according as the zenith and north pole are on the same side, or on different sides, of the arc or great circle passing through the two observed bodies, or through the two places of the same body; and it therefore becomes necessary to notice, at the time of observation, how the zenith and north pole are situated with respect to this great circle.

To estimate the effect of small errors in the observations.

When running in with the land, or crossing a dangerous parallel with no other means of obtaining the latitude than by double altitudes, it becomes a matter of great importance to ascertain the possible error of the latitude thus computed, arising from supposed errors in the observed altitudes, or in the elapsed time. The differential expressions in spherical trigonometry afford methods of doing this; but they are not adapted to the nature of this work, on account of the complication and variety of cases. The following method, though long, is general and infallible, and was once used by the writer in a case of great anxiety and danger.

RULE. After having computed the latitude by either of the four following methods, using the observed altitudes * and elapsed time, repeat the operation, varying

That is, the observed altitudes, corrected as usual for dip, refraction, parallax, and semidiameter, if necessary.

the altitude you suspect may be erroneous by 2 or 3, (or whatever you suppose the limit of the error in that altitude may be ;) the difference between this second latitude and that first computed, is the effect of the supposed error in that altitude. If you suspect the second altitude also to be erroneous, the operation may be again repeated, varying this second altitude 2 or 3', (or whatever the limit may be supposed,) but using the first observed altitude and elapsed time; comparing this third computed latitude with the first, the difference is the effect of this supposed error in the second altitude. Finally, if the elapsed time is supposed to be erroneous, the operation may be again repeated, using the observed altitudes and varying the elapsed time by 20 or 30 seconds, (or whatever the limit of this error may be supposed ;) the difference between this fourth latitude and that first computed is the effect of this supposed error of the elapsed time.

Thus, suppose the first-computed latitude was 30°, the second 30° 1', the third 30° 3, the fourth 30° 2; the error arising from the first altitude would be 1', that from the second altitude 3′, and that from the elapsed time 2. If all these errors existed at the same time, the greatest limit of the error would be the sum of these quantities (or 6), so that the true latitude would be 30° ± 6', or between 29° 54' and 30° 6'. In this way the limit of the error may be obtained in any case, and the degree of confidence that may be placed in the observation obtained. This examination is sometimes very necessary, because the objects may be so situated, that a small error in the observations might produce a considerable change in the computed latitude. It may be observed that the error of one observation is frequently corrected, in whole or in part, by the error of the other; the one tending to increase the latitude, the other to decrease it.

FIRST METHOD.

To find the latitude by double altitudes of the sun, or any other object, the declination being invariable.

In this method, the log. sines, cosines, &c., of Table XXVII. are used; and, for brevity, the word log. is omitted in the rule. For the convenience of writing down at once, in the same line, all the logarithms which occur at the same opening of the book, they are arranged in three columns, as in the following formula; and it will be very convenient to have one of these blanks prepared at the commencement of the operation, and then the logarithms may be written down, in their proper places, with great rapidity.

1. Find the elapsed time* in column P. M.; take out the corresponding cosecant, and put it in Col. 1.

2. Put the secant of the declination in Col. 1; its cosecant in Col. 3.

3. The sum of the logarithms in Col. 1 (rejecting 10 in the index) is the cosecant of the angle A, whose cosine is to be put in Col. 2 and Col. 3.†

4. The sum of the logarithms in Col. 3 (rejecting 10 in the index) is the cosecant of the angle B, (less than 90°,) which is to be named north or south, like the declination.

*If any other object than the sun is observed, the corrected elapsed time, or hour angle, found as before taught, is to be used.

The cosines of A and C are each written down twice, which reduces the number of logarithms in