To measure an inaccessible diftance AB by the equilateral triangle. Fig. 4. plate 5. Place the equilateral triangle horizontally at E, so that you can see the point B along one fide of it; and, in the direction of the other side, mark out the line EDCF by poles, and leave a pole at E. Then meafure from E until you come to the point D, where EDB is a right angle, and mark down the length of ED. Measure forward in the fame right line until you come to the point C, where ACD is a right angle, and mark the length of DC. Measure on in the fame line until you come to the point F, (which must be found by trials), where the triangle being placed horizontally, one fide of it in the direction of FCD, you can see the point A along the other side of it, and mark down the length of CF. Then, if FC be equal to ED, the distance CD is equal to AB; which therefore is known; but, if FC be not equal to ED, calculate DB and CA thus; as 1000 is to 1732, fo is ED to DB; and fo is FC to CA. Take the difference between AC and BD, and the fquare of this difference added to the fquare of CD, gives the square of AB, which, by extracting the square root, will be known. This method of measuring an acceffible distance is both eafy and accurate, and may be applied with faccefs to the distances of mountains, and the fides of fields, where the ground does not admit of applying the measure to each fide; but is not fo proper for large distances. The equilateral triangle has been proposed as the moft fimple inftrument, which can be made in a fhort short time, with very little trouble; but, if we ufe a fquare, and divide two of its fides into fome number of equal parts, we may folve problems more conveniently. DESCRIPTION of the GEOMETRICAL SQUARE. Fig. 5. Plate 5. The Geometrical Square may be made of wood, brafs, or any other matter, either folid throughout, or confifting of four plain rulers joined together at right angles. A is the center, from which hangs a plummet ; each of the fides BE, DE, is divided into 100 or 1000 equal parts; C and F are two fights fixed on the fide AD; there is also an index AL, which turns round the center A; and thereon are two fights K and L. The fide DE is called the upright fide, and BE the reclining fide of the square. To measure an acceffible height AB by the Geometrical Square. Fig. 6. plate 5. Let BR be a horizontal plane, on which the high object AB stands perpendicular; and let BP be the distance of the obfervator =96 feet; and let the height of the obferver's eye be 5 feet. Set the fquare perpendicular to the horizon, fo that the eye being at D, and looking through the fights, fees A the top of the high object; and the plummet hanging freely, fuppofe the thread cuts off 80 equal parts from the upright fide, viz. DN. Then the tri angles LDN, ADC being fimilar, as ND: DL:: DC: CA; that is, as 80: 100::96: 120 CA. But, if the plum line fall on the angle opposite to the center, as at the ftation E; then, because LD is equal to DP, DC is equal to CA; for the triangles LDP, DCA are fimilar; therefore, CA will be known by measuring DC. If the diftance of the obfervator BF be greater than the altitude, such as 300 feet, then the plum-line will cut the inclined fide of the fquare in N: Suppofe NE 40 equal parts, the triangles LNE, DCA, are fimilar; for the angles LEN, DCA, are right, and the angle LNE is equal to the angle DAC, and confequently the angle ELN is equal to the angle ADC; therefore, as LE: EN :: DC: CA; that is, as 100: 40 :: 300: 120; to which add 5 feet, the height of the obfervator's eye, and AB=125 feet. To measure a distance AF, accessible at one extremity; by the Geometrical Square. Fig. 7. plate 5. Place the fquare horizontally on a fupport at A, fo that through the fights on the fide of the fquare you fee the point F; turn the index, until it coincides with the other fide of the fquare, and in the direction of that fide place a pole at B, any convenient distance. from A. Leave a pole at A, and measure AB. Then place the fquare horizontally at B, so that through the fixed fights you fee the pole at A; and turn the index until through the fights on it you fee the point F. Sup pose the index cuts the right fide of the fquare in G, then the triangles BDG, BAF are fimilar; and, as BD: DG :: BA: AF. But, if a large distance, as AH, is to be measured, the fquare being placed at B, the index will cut the reclined fide of the square, suppose at L; then the triangles BIL, BAH, are fimilar, for the angles at A and I are right; the angle BLI is equal to the angle ABH, and the angle IBL to the angle 'AHB; therefore, as LI: IB BA: AH. If the fquare be well made, and properly applied, moft problems may be folved by it; but, because it requires the same trouble and expence to fix the square on a fupport, and to fet it level or perpendicular to the horizon as any circular inftrument, without any peculiar advantage, it is now almost out of ufe... PROB. 8. To measure the diftances of any num ber of inacceffible objects from each other, by means of two stations, from whence the objects can be seen. The method of performing this is in all respects the fame as in the last problem; for, it is only taking a greater number of angles at each station, and folving a greater number of triangles. EXAM. Suppofe A, B, C, Fig. 52. are three inacceffible objects, whofe diftances from each other are required, and E and D two stations, whofe distance is known, and from whence the objects A, B, and C can be feen. The angles being measured, fuppofe them to be, 740 links; re And the distance of the ftations DE quired AB, BC, and AC? Ans. AB=1243, BC=1702, and AC 2822. PROB. 9. To measure the distance of any inacceffible high object from a given ftation, and its altitude above the level of that ftation. Find out the diftance by problem 6th; then meafure the angle of ascent at the given station, and, by means of this, and the distance before found, calculate the height. EXAM. Let BE, Fig. 53. be any high object, and A the given station from whence it can be feen; chuse another station C; then measure the angle BAC= 75° 12′, and the angle ACB=64° 29′, and let the distance of the stations AC=425 links. At A measure the angle of afcent EAB=11° 19′; required the diftance AB, and height BE? Ans. AB=592.7, and BE=118.6. PROB. 10. The distances of three objects A, B, C, in the fame plane being given, and the angles obferved at a fourth place as at the station S; to find the distances of these three objects from that station. CASE 1. If the ftation S be taken without the triangle ABC in one of its fides produced. Fig. 8. plate 5, |