GRAPHICAL METHODS

Dominion, Volume 9, Issue 2705, 26 February 1916, Page 9

GRAPHICAL METHODS

A MATHEMATICAL PAMPHLET REVIEWED. To everyone who has to deal with the practical application of mathematics, in engineering, surveying, astronomy). or the hundred and_ one applied sciences, the use of graphical methods is of tlie first importance. The eyo at a glance can often discover, by the intersections of certain lines of a figure, )he nature of the solution to a problem, which calculation alono could only reveal after prolonged and arduous work. Mr. C. W. Adams, ifl a little pamphlet, recently published, explains a 'graphical method of solving spherical triangles, which may be useful to tho surveyor as a check on his calculations, The method has, indeed, only been applied to one case, but this is the "ambiguous case," where tho eye requires all the assistance it' can get in distinguishing the nature of the solution. In order to explain the procedure let us take a plane triangle first. The problem is to determiue the remaining sides and angles of a triangle when 1 wo sides, A, B, and the angle A opposite one of them are known. The diagram, which is to be found in any text-book on plane trigonometry, is constructed as follows Construct the angle BAG, and along AC measure off a distance AC equal to b. Then with centre C and radius equal to a draw a circle. Three cases may arise: (1) The circle inav fail to cut AB; (2) it may touch AB~; (3) it may cut AB in two points. In the third case thero may be two triangles satisfying th<s given conditions, while in As first caso there is no solution at all.

lii order to obtain a similar diagram for a spherical triangle, we observe that while in a plane triangle the sides aro proportional to the trigonometrical sines of the opposite angles, in a spherical triangle the sines of the sides aro proportional to the sines of '.ho opposite angles. This is known as the Law of Sines. Therefore, instead of > measuring off distances proportional to the sides, we measure off distances proportional to the sines of the sides, which, of course, like the angles, are given in angular measurement. The diagram is then constructed exactly as for a plane Wangle. Tho value of tho diagram' is that it shows at a glanco whether there are two solutions, only one, or >10110 at all. Die author has, however, claimed too much for his diagrams, and his rulo does not always give the true number of solutions. It may be relied upon when the angular magnitudes do not Bxceed a quadrant, hut in certain cases whore the diagram indicates two solutions both, havo to be rejected. The diagram may be used also for determining tho value of the angle B, opnosito to tho side b, for it obviously follows from the law of sines that tho angle B is correctly represented in tho figure; and tho author claims that lie ran determine the angle to within onofliird of a degreo. The method, however, j is of vcrv limited implication, and its utility for accurate calculation doubtful J