Part 11 (2/2)

”The angle at the centre of a circle is double that of the angle at the circumference upon the same base--that is, upon the same part of the circule subtended by it at the centre of the circle is double the angle subtended by it at the circules in the sament of a circle are equal to one another”

[Illustration: Fig 40--Section and Plan of Prisard to this last proposition (Euclid III, 21), it will be observed that in the case of Fig 37 it would not have been possible to locate the point C by reading the angle A C B alone, as such point ht be amywhere on the circumference of a circle of which A B was the chord The usual andobject fro angles with a sextant, or box-sextant, between three fixed points on shore in two operations Let A B C, Fig 41, be the three fixed points on shore, the positions of which are measured and recorded upon an ordnance map, or checked if they are already there Let D be the floating object, the position of which is required to be located, and let the observed angles from the object be A D B 30 and B D C 45 Then on the les = 90 - 30 = 60, and they will intersect at point E, which will be the centre of a circle, which h A B, and the point D will be soles = 90-45 = 45, which will intersect at point F, which will be the centre of a circle of radius F B, which will pass through points B C, and point D will be somewhere on the circumference of this circle also; therefore the intersection of the two circles at D fixes that point on the les in the triangle A B E are together equal to two right angles (Euclid I 32), therefore the angle A E B = 180 - 2 x (90 - 30) = 600, so that the angle A E B is double the angle A D B (Euclid III, 20), and that as the angles subtending a given chord from any point of the circumference are equal (Euclid III, 21), the point that is common to the two circumferences is the required point When point D is inked in, the construction lines are rubbed out ready for plotting the observations froe of A, a new fixed point will be required on shore beyond C, so that B, C, and the new point will be used together Another approximate method which may so paper and draw froth, which shall forles If, now, this piece of paper is moved about on top of the ordnance h the corresponding fixed points on shore, then the point from which the lines radiate will represent the position of the boat

[Illustration: Fig 41 Geo Observation Point Afloat]

The general appearance of a box-sextant is as shown in Fig 42, and an enlarged diagra 43 It is about 3 in in diameter, and isapproxi poles at their extre 43, A is the sight- hole, B is a fixedone-half silvered and the other half plain; C is a mirror attached to the same pivot as the vernier arht frole forh the clear part of ht from the other pole would fall on to mirror C, which should be moved until the pole is reflected on the silvered part of mirror B, exactly in line, vertically, with the pole seen by direct vision, then the angle between the two poles would be indicated on the vernier Take the case of a single pole, then the angle indicated should be zero, but whether it would actually be so depends upon circumstances which may be explained as follows: Suppose the pole to be fixed at E, which is extremely close, it will be found that the arrow on the vernier ar to what may be called the width of the base line of the instrument If the pole is placed farther off, as at F, the rays of light from the pole will take the course of the stroke-and-dot line, and the vernier arm will require to be shi+fted nearer the zero of the scale After a distance of two chains between the pole and sextant is reached, the rays of light from the pole to B and C are so nearly parallel that the error is under one minute, and the instrument can be used under such conditions without difficulty occurring by reason of error To adjust the box-sextant the slass slide should be drawn over the eyepiece, and then, if the sun is sighted, it should appear as a perfect sphere when the vernier is at zero, in whatever position the sextant le formed by the lines froh the plain glass, whichthe instrule to be read between two stations exceeds 90, an interle taken in two parts, as in viewing large angles the mirror C is turned round to such an extent that its own reflection, and that of the ieways in the42--Box-sextant]

It should be noted that the box-sextant only reads angles in the plane of the instruhted is lower than the other, the angle read will be the direct angle between theiven by a theodolite

The sa the position of an object in the water when the observations have to be taken at some distance from it To illustrate this, use raphical surveying given at the Royal Naval College, Incidentally, it shows onethe observations The question was as follows:--

[Illustration: Fig 43--Diagrauard, Mound bore N 77 W (true) 045 of astations were taken to fix a shoal on which the sea breaks too heavily to risk the boat near:--

Mound 60 CG 47 Mill

[Greek: phi]

Centre of shoal Mound 55 CG 57 30' Mill

[Greek: phi]

Centre of shoal

Project the positions on a scale of 5 in = athe centre of the shoal” It should be noted that the sign [Greek: phi] signifies stations in one line or ”in transit,” and C G indicates coastguard station The order of lettering in Fig 44 shows the order of working

[Illustration: Fig 44--Method of Locating Point in Water When Observations Have to Be Taken Beyond It]

The base lines A B and A C are set out froiven; then, when the boat at D is ”in transit” with the centre of the shoal and the coastguard station, the angle formed at D by lines frole forles of 90 - 60 are set up at A and B, their intersection at E will, as has already been explained, give the centre of a circle which will pass through points A, B, and D Siles of 90-47 at A and C, a circle is found which will pass through A C and D The intersection of these circles gives the position of the boat D, and it is known that the shoal is situated soht line from D to A The boat was then moved to G, so as to be ”in transit” with the centre of the shoal and the le A G C 57 30' By a similar construction to that just described, the intersection of the circles will give the position of G, and as the shoal is situated somewhere in the line G B and also in the line A D, the intersection of these two lines at K will give its exact position