296 PROCEEDOGS OF SECTION A. 



ellipse, and with the declination of the circle. The method of 

 using them is simply to note where they cut the proper ellipse of 

 latitude, and this -will be the position of the body when on the 

 horizon. 



The centres of these circles may be found by the following 

 rule: — Subtract twice the declination from 90, and the remainder 

 will be the centre of a circle on the meridian, using the declination 

 scale, radius being equal to the distance from this point to the 

 focus of the diagram. The reasoning by which I arrived at this 

 rule I am not at present prepared to give, but its correctness may 

 be proved in several ways, und it gives the same results as are 

 obtained by calculation. An illustration of its correctness at one 

 point may, however, be given on the diagram itself. If an observer 

 were in latitude 60° S., and a celestial body were in declination 30*^ 

 S., the body would not set at all, but would simply touch the south 

 point of the horizon and again commence to rise, and similarly at 

 any place where the declination of the body is the complement of 

 the latitude and of the same name it would simply touch the horizon 

 as described ; and this, it will be observed, is exactly what happens 

 on the diagram. 



In the preceding problems we have again four elements to work 

 with, viz., latitude, declination, time, and bearing; and, given any 

 tico of these, the other two can be found by the diagram, provided 

 that one of the known quantities is either latitude or declina- 

 tion. 



So far it may be observed that I have said nothing about 

 altitude^ although it plays a very important part in nautical 

 astronomy. The diagram may, ho-\vever, be used for workintj out 

 problems in which altitude is one of the elements, by the help of 

 a pair of compasses and scales of cosines, laid down separately. 

 Having found the radius of the circle representing the horizon, 

 with a given latitude and declination, as previously explained, 

 apply it to the scales of cosines, and find with which cosine of 0° 

 it corresponds. The altitxxde of any point within this circle may be 

 found by measuring its distance from the centre, and this distance 

 applied to the proper scale will give its altitude. 



Here we have five elements to work with, viz., latitude, declina- 

 tion, altitude, hour angle, and azimuth ; and, given three of these, 

 the other two can be found, provided that either latitude or declina- 

 tion is included amongst the known quantities. 



I will not further intrude on the time of the meeting by going 

 into the variety of ways in which the diagram may be used, but 

 will content myself with laying before you a list of problems 

 Avhich I have succeeded in solving by the use of ihe diagram; and 

 I may mention that there are a few which I have failed to solve, 

 although I have no doubt they can be solved, and probably there 

 are a good many which I have not thought of, but which can be 

 worked out by its use. 



