A PROBLEM IN PRACTICAL SURVEYING: DEMONSTRATED BY MEANS OF TRANSVERSALS. BY W. M. GILLESPIE, Professor of Civil Engineering in Union College. FROM THE AMERICAN JOURNAL OF SCIENCE AND ARTS, VOL. XXIII, MARCH, 1957. ON PRACTICAL SURVEYING. LET A and B represent two points, inaccessible, and invisible from one another. Let it be required to find a third point, C, in the line of A and B, but invisible from them. It is supposed that no means of measuring either distances or angles are at hand. The problem may be solved thus. Set three stakes, D, E, F, in a straight line. Set a stake, G, in the line of DB and EA; set a stake, H, in the line of FA; and a stake, J, in the line of FB, and at the same time in the line of GH. Then range out the lines DH and EJ, which will meet in a point, C, which will be the one required. Any number of such may be similarly obtained to verify the work. This problem is given in a recent number of the Vienna Engineer's Journal (Zeitschrift des Oesterreichischen Ingenieur Vereines, 1856, p. 245) by an Austrian mathematician, who represents it as employed by practical surveyors, but as not having any known geometrical proof. He proceeds to give an analytical investigation of it, saying, "I have in vain tried to prove the problem in the synthetic way, by pure geometry." The "Theory of Transversals," however, the foundation of the "Recent Geometry," or "Geometry of Segments," (too little cultivated beyond On Practical Surveying. 3 a small circle of French geometers) will furnish a simple and perfect demonstration, as follows. The theorem to be proved is equivalent to the assertion that if A, B, C, and D, E, F, lie respectively in two straight lines, and lines be drawn as in the figure, then will the intersections G, H, J, lie in one and the same straight line. Conceive the two given lines produced to meet in Z, beyond the limits of this figure. The triangle BFZ is so cut by the transversal CE as to give the equality BJXFEXZC JFXEZ×BC.* The triangle AFZ, cut by CD, gives FHXAC×DZ=HA×CZ>FD.* The triangle ABG, cut by CD, gives BCXALXGD-BDXCAXGL.* The triangle DEG, cut by CZ, gives GAXEZXBD=EA×DZXBG.* The triangle DEG, cut by AF, gives GKXDFXAE=KDEF×AG.* The triangle AGK, cut by HD, gives KDXGLXAH=GDXAL>KH.* Multiplying together the corresponding members of these six equations, we get a new one containing eighteen factors on each side. Of these, fifteen factors in each member can be cancelled, and we have left BJ×FHXGK=JF>BG>KH. This shews,† that the points G, H, J, lie in a straight line, which is a transversal to the triangle BFK. What Poinsot said, forty years ago, that "The simple and fruitful principles of this ingenious theory of transversals deserve well to be admitted into the number of the elements of geometry,” is even more true and desirable at the present time. * By the theorem "If a straight line be drawn so as to cut any two sides of a triangle and the third side, one or all being prolonged, thus dividing them into six segments (the prolonged sides and the prolongations being taken as segments) then will the product of any three of those segments whose extremities are not contiguous, be equal to the product of the other three segments." † By the converse of the preceding theorem. PART IV. TRANSIT AND THEODOLITE SURVEYING: By the Third Method. CHAPTER I. THE INSTRUMENTS. (324) THE TRANSIT and THE THEODOLITE (figures of which are given on the next two pages) are Goniometers, or Angle-Measurers. Each consists, essentially, of a circular plate of metal, supported in such a manner as to be horizontal, and divided on its outer circumference into degrees, and parts of degrees. Through the centre of this plate passes an upright axis, and on it is fixed a second circular plate, which nearly touches the first plate, and can turn freely around to the right and to the left. This second plate carries a Telescope, which rests on upright standards firmly fixed to the plate, and which can be pointed upwards and downwards. By the combination of this motion and that of the second plate around its axis, the Telescope can be directed to any object. The second plate has some mark on its edge, such as an arrow-head, which serves as a pointer or index for the divided circle, like the hand of a clock. When the Telescope is directed to one object, and then turned to the right or to the left, to some other object, this index which moves with it, and passes around the divided edge of the other plate, points out the arc passed over by this change of direction, and thus measures the angle made by the lines imagined to pass from the centre of the instrument to the two objects. |