1048 



NAVIGATION. 



[SOLCTIOS OF Till ANGLES. 



Hence the diameter of the earth, u deduood from 

 thM observations, U 7962) mile.. 



Otitencitc by a laMe of natural cotinet. 



BCoo.. A 

 Since, u shown above, AD i-pog. A' we m *^ Om " 



pate by the table of natural tines and cosines as fol- 



t. eon. 2 13' 1 - -Ot' 



3 



1 - -099246 - -000754 )2 *>97738( 3970 



2 



7l>52 miles the diameter. 

 C78G 



6713 

 5278 



4358 

 4524 



3. Wanting to know the breadth of a river, I measured 

 a base of 500 yards in a straight line close to its edge, 

 and at each end of it I found the angles subtended by the 

 other end, and a tree close to the opposite margin of the 

 rivc-r to be 63 and 79 12 1 : what was the breadth of the 

 river 1 Ans. 529-5 yards. 



4. Two ships of war intending to cannonade a fort sepa- 

 rated from each other 500 yards, as near to the shore as 

 possible : the angles subtended by each ship and the fort 

 were observed from the other ship to be 38 16' and 37 9 : 

 required the distance of the fort from each place of ob- 

 servation. Ans. 312 yards and 320 yards. 



5. The peak of Tenerifle is said to be 2^ miles above the 

 surface of the sea : the angle formed at the top between 

 a plumb-line, which, of course, hangs perpendicular to 

 the horizon, and the line from the eye to touch the sea 

 at the remotest visible point, is found to be 87 58' : re- 

 quired the diameter of the earth. Ans. 7936 miles. 



6. Three objects, A, B, C, whose distances apart were 

 A C 8 miles, B C = 7i miles, and A B =-- 12 miles, were 

 visible from a station D, in the line joining A and B ; and 

 the angle at that station, subtended by AC, was observed 

 to be 107 56' 13' ; required the distance of the three ob- 

 j jets from D. 



Ans. AD = 5 miles, D B 7 miles, D C=4'89 miles. 



We here conclude the introductory chapter on the 

 solution of plane triangles : it is intended solely to famil- 

 iarise the learner with the business of practical calcula- 

 tion, and to illustrate the best and shortest methods of 

 arranging his arithmetical operations. Many instances 

 have been given, more especially in what concerns right- 

 angled triangles, of the advantage of making more use of 

 the table of natural sines and cosines than is at present 

 customary. Valuable as logarithms unquestionably are, 

 they do not always effect a saving of time or trouble, 

 even in cases well fitted for their application ; and we 

 would, therefore, recommend that the example we have 

 here set him be followed by the learner that he would 

 exercise his own deliberate judgment as to which kind of 

 table will aid him in reaching his result in the readiest 

 way, and not in all cases resort to logarithms, as is almost 

 invariably done in works of this kind. 



But whatever table he uses he should use with delibe- 

 ration and care : tabular numbers should never be tran- 



scribed in a hurry, as everything depends on the accuracy 

 with which they are written dm* u. It should also be the 

 habitual practice of the calculator to make all the use he 

 can of his table when it is in his hand ; and when it is 

 not in use, he should advance his work as much as prac- 

 ticable before he refers to it In order to this, as : 

 as possible of the argument*, as they are called that U, 

 of the quantities with which the tabular numbers are to 

 be connected should be written down before the tables 

 are touched : thus, in Kxamplc 1, for instance, page l<Mi;, 

 the entire skeleton of the operation should be formed be- 

 fore the table of logarithms i referred to, to till it up. 

 Again, in Example 2, page 1045, even- item, 111 both 

 columns of the work, should be written down while the 

 tables are in hand, :md they should be again referred to 

 only after the amount of each column is found. 



In the chapters on TRIGONOMETRY in the mathematical 

 section, the in.-iiu object has been to develop the analy- 

 tical theory of angular magnitude, and not to enter at any 

 length into arithmetical details ; the present chapter, in- 

 dependently of the special purpose it is intended to serve 

 in this section, may, therefore, bo acceptable to the stu- 

 dent as a praxis on certain theoretical principles- there 

 delivered. This collateral object has not been lost sight 

 of in the preparation of the foregoing pages ; and we 

 have, accordingly, introduced examples in sufficient num- 

 ber and variety to supply the learner with all n. 

 materials for exercise, in the ordinary calculations of 

 Practical Trigonometry. 



Before closing the introductory portion of our subject, 

 it may be well to collect into oae place the several for- 

 mula; for the solution of oblique triangles. By thus 

 bringing them together, future reference to them will be 

 facilitated ; and the learner, by having the whole more 

 frequently under his eye, will, at length, ^et them so 

 impressed on his mind as eventually to dispense alto- 

 gether with a formal reference to them. This amount of 

 familiarity with his tools is a necessary qualification in a 

 good workman. 



Formula for the Solution of Plane Triangles. 

 I. " : 6 : : sin. A : sin. B 



II. o+ 6 : a </> 6 : : tan. i (A + B) : tan. I (A^B) 

 sin. 4 (A </) B) : sin. i (A + B) : : a </> b : c 

 cos. \ (A c B) : cos. \ (A + B) : : a + 6 : c 

 Us. 12. 



Lc 





26c 



~ L 



The but of these expressions is to be computed by com- 

 mon arithmetic, and the cosine of A to be found in the 

 table of natural sines and cosines ; all the other formula; 

 are adapted to logarithmic computation. Whenever the 

 sides a, 6, c, have a factor common to all, it may he can- 

 celled, and the results used instead, in any of the for- 

 mulae III. 



CHAPTER II. 



NAVIGATION. 



RAL NOTIOXH OF TUB FlOURK AND ROTATION Ot 



THB EAKTH. The surface of the sea is very nearly that 

 of a perfect sphere or globe. For all the purposes of 

 Navigation, it may be regarded as accurately so ; for if 



the very small deviation from this figure wore to bo re- 

 moved, no sensible difference would be mode in any of 

 the rules and operations by which the sailings of a ship 

 are regulated, ami its position determined. 



