Ex. 1. From the foot, B, of the perp. AB, to C, I measure BC,—it is 40 feet; the index CD, cuts off 100 eq. pts., as ED;-EC also being 100 eq. pts.; required AB. Here 100 100 40: BA; ·. BA= 100 X 40 = 40 feet. 100 Ex. 2. The index cuts ED at 60; as before CB = 40 feet, EC or DF = 100 eq. pts.; required AB. 100 X 40 Here 60 100 40: AB; ·. AB= = 663 feet. 60 PROB VII. To find, by aid of the cross staff, or theodolite, the distance of A from B without approaching A. At rt. /s to AB, lay down the line BD; and in BD take a C, at which place a staff; from D, but on the other side of BD, lay down DE perp. to BD, and measure along DE, until E, C & A are in one st. line; then, the s at C are equal, _D = /B, and E ZA;.. the As ABC, CDE are equiangular ; .. CD: CB = DE : AB ; i. e. AB CB DE CD D C B E The measurement of BC 200 links; that of CD = 60 links, and links are there between the s B and A ? PROB. VIII-By means of a line, DE, of which the length is known, to find the length of its parallel, AB, one end of which, B, only can be approached. Set out the line BE, and on it ascertain the point C, where a line joining A and D would cross BE; measure BC, CE; the As ACB and DCE are eq. ang., .. CE: CB = DE: AB; Ex. The parallel DE measures 800 links; the side EC, 900 links, and CB, 1800 links; required the links in AB. 1800 X 800 Here 900 1800 800: AB; .. AB 900 = 1600 links. OBS. An instrument in common use,-the Proportional Compasses, is an example of the last Problem; in this instrument, the common centre C, about which the legs turn, is changed at pleasure, yet so as to preserve any given proportion between EC and CB, and consequently between ED and AB. These compasses give great facility in enlarging or diminishing a plan or map; and with sufficient accuracy for many purposes,-the principle being perfect, but the application liable to fail for want of the requisite care. The Pentagraph and Eidograph are also instruments, of which the principle is, that the arms move parallel to each other, and consequently that the triangles formed are always similar, being exemplifications of Euclid's Prop. 4, Bk. VI. For accuracy and precision the Eidograph is far superior to the Pentagraph. See BRADLEY'S Pract. Geom. p. 59. PROP. 6.-THEOR. If two triangles have one angle of the one equal to the one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides. CON. 23, I.-32, I, DEM. 4, VI. 11, V. 9, V. 4, I. Ax. 1, I. 32, I. or FDG = BAC, EDF; & DFG;= ACB; A DGF is eq. ang. with ▲ ABC; .. BA: AC = GD: DF; but BA: AC = ED: DF; .. ED: DF = GD : DF; & ... ED = DG. .. EF = FG, ▲ EDF = ▲ GDF, / DFG= 6 C. 2. Ax. 1, I. but COR. 1.-It may be added, that the sides also about each pair of equal angles shall be proportional; i. e., by 4, VI, AB : BC = DE : EF; & BC: CA = EF: FD. COR. 2. If through any points, b, c, d, &c., of a straight line NM, parallels b B, c C, d D &c., be drawn, which are proportional to the distances A b, bc, cd &c., from any point A on the 1st line, then their extremities B. C, D &c., will be on the rt. line DA passing through A. B = AbbВ Ac: cC; & AbB A c C; ..▲ A B is similar to ▲ Ac C; & .. ▲ BA b = ¿ CA c ; and Ab coincides with A c; .. AB & AC, being on the same side of NM, also coincide. N. B.-The equation of a rt. line in Analytical Geometry depends on this principle. See LARDNER'S Euclid, p. 184. USE & APP. 1. Since all rectilineal figures may be divided into triangles, if Join AC, AD, AE; ac, ad, ae. = As ABC, abc, have B=Lb, & AB : BC=ab: bc; LACB a cb, and AC: CB = ac: cb; and: BCD = bcd, and ▲ ACB = La cb; ../ ACD = Lacd. Also AC CB ac: cb; and CB : CD = cb: cd; : ex. æquo. AC: CD ac: cd; AS ACD and a c d are similar. And thus As ADE, AEF, are sim. to As a de, a ef; :: L F = Lf; and s at A = s at a. = Also. AF AE af: a e; and AE : AD = ae: ad; and AD: AC ad: a c, and AC : AB = ac:ab: 11 22, V. 16, V... ex. æq. AF: AB = aƒ : a b; and alt. AF: aƒ =AB: ab. 2. Similar rectilineal figures may also be divided into the same number of similar triangles; and their homologous or corresponding sides, are to one another in the same ratio, each to each. PROP. 7.-THEOR. If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals; then, if each of the remaining angles be either less, or not less, than a rt. angle, or if one of them be a rt. angle; the triangles shall be equiangular, and shall have those angles equal about which the sides are proportionals. Or, "If two triangles have one angle equal to one angle, and the sides about the other angle proportional, and the remaining angles either both less, or both not less than a right angle, the triangles will be equiangular, and will have those angles equal about which the sides are proportional."-EUCLID. DEM. 23, I. 32, I. 4, VI. 11, V. 9, V. 5, I. 13 I. The s which one st. line makes with another upon one side of it, are either 17, I. Any two s of a ▲ are together less than two rt. s. E. 1 Hyp. 1 Let As ABC, DEF have A = ▲ D, and of the one respectively s DEF and EFD, of the other, and AB: BC = DE: EF, and BC: CA EF: FD. Let each of the rem. s C and F be a rt. Li |