CHAMBERS'S INFORMATION FOR THE PEOPLE. 



TRIGONOMETRY LAND-SURVEYING. 



Trigonometry signifies literally the art of measur- 

 ing triangles, but with the progress of science the 

 meaning of the word has been much extended. 

 Trigonometry is divided into plane and spherical, 

 according as it is directed to the investigation of 

 plane or of spherical triangles. 



We can only speak here of Trigonometry in the 

 original sense of the term, and as applied to plane 

 triangles. 



The mensuration of triangles in the cases where 

 only lines and areas enter into the calculation, 

 is treated in the foregoing pages, and requires 

 nothing beyond the simple operations of arith- 

 metic. But where sides are to be found from 

 angles, or angles from sides, an apparatus of 

 trigonometrical tables is required ; and for their 

 use and their application to all the cases that 

 occur, we must refer to larger treatises on the 

 subject, such as Practical Mathematics, and 

 Mathematical Tables, in Chambers's Educational 

 Course. All that can be done here is to explain 

 why tables should be necessary a question that 

 frequently occurs to beginners, but the answer to 

 which is not always forthcoming. 



We shall take the case of a right-angled triangle 

 as the simplest ; and suppose that, in the triangle 

 oeb, the hypotenuse ob (= 370 yards) and the 

 angle boe (= 57) are given, to find the side be. 

 From the numerical value of the angle, we can- 

 not directly get the length of the side, because 



the two things are of different natures ; an angle 

 is not any kind of length, but merely an amount 

 of opening or divergence ; and the angle and the 

 side do not vary in the same proportion. The 

 relation between the two can only be got at in a 

 roundabout and laborious way, as thus : Let a 

 circle be described with any convenient radius ; 

 draw two diameters at right angles ; make the 

 angle AOB equal to o, and draw BE perpen- 

 dicular to OA. In the language of trigonometry, 

 BE is the sine of the arc AB, or of the angle AOB, 

 of which AB is the measure. For the purpose of 

 calculation, the radius OB is either taken as unity 

 with decimal subdivisions, or it is supposed to be 

 divided into 1 0,000,000,000 equal parts ; and the 

 quadrant AC is divided into 90 equal parts called 

 degrees. Supposing that AB contains 57 of those 

 degrees, then AB is said to be an arc of 57, 

 and AOB, or eob, an angle of 57. From the 



640 



number of degrees in an arc or angle, we cannot 

 directly deduce the sine, except in one or two 

 particular cases. One of these is when the angle 

 is one-third of aright angle, that is, 30; the sine is 

 then equal to one- half the radius. This is mani- 

 fest on looking at a hexagon inscribed in a circle 

 (p. 621), and supposing lines drawn from the centre 

 to the extremities of a side and to its middle 

 point. It might seem to the beginner that this 

 one sine furnished a bridge of transition to any 

 other sine ; and that he had only to put 30 : 57 

 : : sine of 30 : sine of 57. But the sines are not 

 in proportion to the arcs ; a little inspection 

 makes it manifest that when an arc is doubled in 

 size, the sine does not become double too. There 

 are, however, theorems and processes which en- 

 able us from the sine of 30 to deduce the sine 

 of any other angle. But the processes are long 

 and intricate ; and if the calculations had to be 

 made for every individual triangle, trigonometry 

 would be a laborious business indeed. The cal- 

 culations of the different sines have, therefore, 

 been made once for all, and the results recorded 

 for subsequent use. Taking the radius at the 

 number of units above mentioned, mathematicians 

 have computed on the same scale the lengths of 

 the sines of all arcs or angles from a minute 

 fraction of a degree up to 90, and arranged 

 them in a table. Turning to this table, we find 

 the sine of 57 put down as 8,386,706,000 ; and 

 from this, the principle of similar triangles en- 

 ables us to deduce the line we want. For the 

 triangles OEB and oeb being similar, 



OB (radius) : BE (sine) : : ob be 

 or 10,000,000,000 : 8,386,706,000 : : 370 : answer. 



As the working out of this would involve a long 

 operation of multiplication and division, the prac- 

 tice is to use the logarithms of the numbers ; 

 which reduces the process to simple addition and 

 subtraction. Accordingly, trigonometrical tables 

 give the logarithms of the sines, as well as the 

 sines themselves, which are called the natural 

 sines. These tables contain also the logarithms 

 of other lines connected with arcs (tangents, co- 

 sines, &c.), which are used in calculation in a 

 similar way. 



Land-surveying is the method of measuring and 

 computing the area of any small portion of the 

 earth's surface, as a field, a farm, an estate, or 

 district of moderate extent. There are three dis- 

 tinct operations in the art of land-surveying, all of 

 which require the surveyor to possess a competent 

 knowledge of arithmetic, algebra, and geometry. 

 In the first place, the several lines and angles 

 must be measured ; secondly, they must be pro- 

 tracted or laid down on paper, so as to form a 

 plan or map of the district; and, thirdly, the 

 whole area of the district must be computed on 

 the principles above explained. For details of the 

 various operations we must refer to the work on 

 Practical Mathematics already named. 



