In a recent issue of the Proceedings, I outlined a method for computing the altitude from the dead-reckoning position using Martelli's tables. Those using these tables know that this was an important addition. No one has found additional uses for these tables until now, although the tables have been in use for many years, in fact the examples given in the 1932 edition are for the year 1888. 4No attempt will be made to compare these tables with any of the new methods, except, in a general way, to show their simplicity and accuracy for computing the altitude, time and altitude azimuth, amplitude, great circle course and distance, identifying unknown stars, and finding the local hour angle.
Table I of the present edition extends only to 71°-34', but I have extended this table to 89°-55'. No other changes in table construction have been made. An example in the different uses will readily show the simplicity and accuracy of this method
Time and altitude azimuth.—Take the example for computing the altitude of the sun as a sample problem. Convert hour angle to arc, either sun or star.
Great circle course and distance.—Required great circle course and distance between lat. 41° N., long. 700 W. and lat. 10° S., long. 10° W.
Comment.—The comparison of this method with the cosine-haversine method, when computing the altitude of the sun, was made in order that those who adhere to the cosine-haversine method may see that the accuracy of the computation is not impaired by short cuts. A similarity will be noted between Martelli's and the cosine-haversine formula.
For those who are not acquainted with these tables, a short description will be of some information. In the main, the book consists of 5 different tables occupying 49 pages.
Table I occupies 6 pages of logs for altitude and declination, with 11 degrees of 4-place logs to a page.
Table II is the logs for the sum or difference of latitude and declination, these logs expressed in minutes, seconds and tenths. The table occupies 9 pages with 10 degrees to a page.
Table III is the logs for angle of altitude and is constructed in the same manner as Table II.
Table IV is the auxiliary logs and occupies 17 pages. This table is constructed with the sum or difference of the quantities found by Tables II and III as the heading. Since these sums or differences are never less than 20' nor more than 36' 59", the table is between these limits.
Table V is the logs for hour angle from 00h00m05a to 23h59m558, in 5-second intervals. The table was constructed for use with the old method of time, with the top of the page named a.m., and the bottom p.m., but using the new method of time it is not necessary to refer to the top of page and the hour angle must always be read from the bottom of the page, this includes hour angle when computing altitude. When computing the altitude the only thing to remember is (1) latitude and declination same name, subtract and take difference to Table II and (2) latitude and declination opposite name, add and take sum to Table II.
While no arrangement has been made for interpolations, none is found to be necessary, as in most cases the value of the logs makes it possible to interpolate, especially the log for altitude can be found to within 15".
The computing of the azimuth when the altitude was computed has been purposely omitted; however, a method of computing the azimuth has been included under the heading of Time and Altitude Azimuth. This can be used to find the azimuth at the same time the altitude is found, if so desired.
The identification of an unknown star is the most extended problem found in working with these tables, as it requires the working of two complete problems. (1) It is necessary to compute a line of position sight to find the declination, and with this declination (2) a time sight must be worked in order to find the hour angle and then the right ascension. However, the tables are simple and require few functions of mathematics and the whole problem takes but little time. Great-circle distance is no more than a line of position sight and the resulting course is the same as an altitude azimuth.
An amplitude was described, for it was thought that while there is more arithmetic by this method, the fact that all the logs and the resulting bearing were taken from one handy table would offset the additional mathematics. These tables were constructed for finding time at sea by the time-sight method, and the problems for practice included in the present edition were calculated using the old method of time. Two problems showing the use of the new time were shown above. The similarity between the time sight and line of position will be noticed, especially in the use of Tables I and II. It will be noticed, too, that the sum of the logs for latitude and declination from Table L for line of position are always negative, while for time sight they are positive. This must be remembered when working a line of position sight with angles exceeding 71°" 34'. The logs in Table I is the remainder of 9.5000 subtracted from the log cosines, to four places and considered as whole numbers. The extension of this table to 89°-55' was accomplished by subtracting the log cosines from 9.5000. As the log cosines for angles greater than 71°-34' are less than 9.5000, these logs are prefixed with a minus sign and this sign must be always considered when working with the extension table. No greater difficulty will be experienced in working with Martelli’s than with any other method and all cases applicable to other methods are equally applicable to this system, as is shown by the following example taken from an article by Lieutenant A. A. Ageton, U. S- Navy, on the secant-cosecant method.
Problem: On May 15, 1928, about 8:00 p.m., the U.S.S. Mississippi, while in Deposition lat. 40°-43' N., long. 68°-30'W-, observed the star Vega as follows: W-j 7h36m12s, C-W,4h59m12“; chro. slow 1»1*; ht. eye, 34 ft.