DEM. 5 Concl. 6 P. 7. then on this sup., in As EDF, EGF, on the which is impossible. 3... since base BC coincides with base EF, the sides BA, CA, coincide with sides ED, FD: Wherefore, ▲ BAC must coincide with 7 D. 2, 8 D. 7. 9 Ax. 8. 10 Recap. Therefore, if two triangles have two sides, &c. QE.D. L SCH.-The Equality established is that of the angles,—but the sides being equal, the triangles also must be equal. This is the second criterion of the equality of triangles. USE.-1. By the aid of this proposition, and of Prop. 22, the angle at a given point C, made by st. lines from two objects, as A and B, may be determined without a theodolite. 2. When the instruments for angular magnitude cannot be employed, by reason of the inequalities of surface, or the difficulty of placing the instruments, this proposition is useful for measuring and cutting angles in a solid body, as in a block of stone, or for bevelling, i.e., for giving the desired shape to the D E G angular edges of timber, &c. For instance, a groove of the same triangular shape and size with the triangle ABC, is to be cut in a block of marble DEFGH. At the point in the edge DG of the block, where the groove is to commence, set off a line ac equal to AC; and on the plane surface DEFG, with ac for one side, construct a triangle abc, with sides equal to the sides in triangle ABC: abc will be the end of the groove, and if the guidance of abc be followed, the whole groove when finished will be of the same angular magnitude with ABC. Q K N P M M L On the same principles, a beam of timber, the end of which is represented by fig. KLMN, may be bevelled so that the bevelled edge shall be of the same angle with a given angle NPM; for, on the end of the beam draw a triangle, the sides of which shall equal those of the triangle NMP; then, by Prop. 8, the bevelled edge NQM is equal to the given angle NPM. PROP. 9.-PROB. To bisect a given rectilineal angle, that is, to divide it into two equal parts. SOL.-P. 3. From the greater st. line to cut off a part equal to the less. Pst. 1. A line may be drawn from one point to another. DEM.-P. 8. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one, shall be equal to the angle contained by the two sides equal to them of the other. DEM. 1 by C. 2 & P. 1| 2 P. 8. 3 Recap. Wherefore, the BAC is bisected by the st. line Q.E.F. SCH.-1. The bisection of the arc which measures an angle is also effected by the bisection of the angle. The arc DGF, on fig. to Prop. 9, the measure of the BAC, or DAE; the DAG is one half, and EAG the other half, of DAE; and halves of the same being equal, the arc DG is equal to the arc GE. 2. An isosceles triangle would serve equally well for the solution and demonstration. 3. By successive bisections an angle may be divided into any number of equal parts, indicated by a power of two, as into four, eight, sixteen, thirty-two, &c., equal parts. 4. Hitherto no method has been discovered of geometrically trisecting an angle, so that the division of the quadrant of 90° into single degrees is in part effected mechanically: by simple bisection, we divide 90° into two 45°; by setting off a semi-diameter from one extremity of the quadrant, we cut off an arc of 60°; 60° bisected gives 30°; and 30° bisected gives 15°: but for the division of the 15° we require to have the means of trisecting an angle, which means Geometry does not supply. But having mechanically divided an arc of 15° into three 5°, and from one arc of 5° set off an arc of 3°, the simple bisection of the remaining arc of 2° gives an arc of 1°-a unit in the measure of the circumference. Of course this is only a practical, not a theoretical proof. If, however, instead of taking an arbitrary quantity, 360, as the measure of the equal parts in the circumference of a circle, those who first made such division had followed the strictly geometrical process of this 9th Proposition, they would have arrived at a unit for the degrees in a given circumference with as much absolute certainty as they do now at the unit for a scale of equal parts. By making use of the powers of 2, and by their aid dividing the circle, the unit of the division is demonstrably accurate. Suppose the number of equal parts into which the circle had been divided, had been represented by the 9th power of 2, or 512, the bisection would have given 256 for the semicircle; 128 for the quadrant; and 64 for the octant and 64 by successive bisection, would have given 32, 16, 8, 4, 2, and 1 equal parts. Thus, every step in the division would have been strictly in accordance with geometrical verities. Again-the unit of such degrees represented by 64, equal parts, would in the same way have been divisible into 32, 16, 8, 4, 2, and 1 minutes, and so on, to whatever extent of minuteness we might wish to carry our bisections. Probably no fact in geometrical measurements more clearly shows the unscientific nature of the early geometry, than the division of the circle into 360 equal parts. It is now, however, too late to attempt an alteration on purely geometrical grounds; and, fortunately, there is no real inconvenience or inaccuracy in the received method; for an arc of the 360th part of a circle is in practice as readily obtained as the arc of the 512th part of the same circle would be. We have only to bear in mind that Plane Geometry does not supply the means for any division of a circle, except by the method of bisections, i. e., by using in regular series the powers of 2. USE AND APPLICATION.-1. Practically the angle BAC, in the figure to Prop. 9, p. 67, would be bisected by drawing the arc DGE, and with any radius from D and E drawing arcs intersecting in. F; AF is the bisecting line. 2. By Prop. 9, we show that the angles at the base of an isosceles triangle are equal; for bisecting ACB by CD, CA is equal to CB, CD common, and ACD equal to 4 BCD; therefore, by P. 4, < CAD equals < CBD: 3. Also that the line which bisects the vertical angle of an isosceles triangle, bisects the base perpendicularly; for AC equalling BC, DC being common, and ACD by construction equalling BCD, by Prop. 4, AD equals DB, and ▲ ADC equals A BDC, and ADC equals BDC; consequently, by Def. 10, st. line DC is perpendicular to A B. A B D 4. The Mariner's Compass is divided into its 32 parts or points by Props. 9 and 10. The bisection of the diameter by another diameter at rt. angles, gives the cardinal points N., E., S., W.; the quadrants, being bisected, give the points N.E., S.E., S.W., N.W.; and these bisections are continued until the number of equal divisions of the circle amounts to 32-the arc at each pair of points enclosing an angle of 11° 15'. This instrument is used for taking the bearings or directions of places from some central station of observation; and the correctness of the method depends on the physical law that a magnetised needle or index, working freely on a pivot, does, allowing for deviation, from the same place, point in the same direction. To be quite accurate in making the observation, it is requisite to know the amount of deviation from the true North, at any given place. For Great Britain, the deviation amounts to about 24° west of north; so that when the magnetised steel index, or needle, points 24° west, the pointer on the compass card marked N. indicates the true North. PROP. 10.-PROB. To bisect a given finite straight line. SOL.-P. 1. On a given st. line to construct an equilateral triangle. DEM.-P. 4. Two triangles are equal in every respect when two sides and the included angle of one are equal to two sides and the included angle of the other triangle. EXP. 1 Datum. 2 Quæs. CONS.1 by P. 1 2 P. 9 3 Sol. Given the st. line AB; On AB make an equil. A let st. line CD make LACD= = 4 BCD; then AD=BD, i.e, AB is DEM. 1 by C.1& 2. in As ACD, BCD, AC BC, CD is common, = and ACD = 4 BCD; B 2 P. 4. .. base AD = base DB. 3 Recap. Wherefore, AB is bisected in D. Q.E.F. SCH. By successive bisections, a line may thus be divided into any number of equal parts indicated by a power of 2; as 4, 8, 16, 32, 64, &c. USE.-1. In practice, the st. line AB would be divided into equal parts by drawing with equal radii arcs intersecting in . C and. E; the st. line CE will bisect the st. line AB. 2. Also, if the length of the line AB be ascertained by means of a scale of equal parts, the division into any required number of equal parts, as 2, |