In Fig. 379, AFC : ABC :: AD: AB; since these two triangles have the common base AC, and their altitudes are in the above ratio. So too, BFC: ABC:: BE: BA. Hence, the remaining triangle AFB: ABC :: DE: AB. (541) By Art. (65), Note, ABC AC × CB X sin. ACB. But the angle ACB=ACD+DCB=† (180°—ADC)+† (180°—CDB)=180°—† (ADC+CDB). Hence, ABC=AC × CB X sin. † (ADC + CDB) = 1 AC × CB × sin. † ADB. Let r= = DA =DB=DC. Since AB is the chord of ADB to the radius r, and therefore equal to twice the sine of half that angle, we have Also, since the area of each of the three small triangles equals half the product of one of the equal sides (r), by the sine of the included angle at D, these triangles will be to each other as the sines of those angles. These angles are found thus: AC 2 r AB BC 2 r sin. ADB= ; sin. BDC= ; sin. | ADC= (542) The formulas in this article are obtained by substituting, in those of Art. (523), for the triangle DBE, its equivalent BD thus becomes= m ་་་་ X AB X BC X sin. B. m + n m m + n m m + n XABXBC); XABXBC). (544) In Fig. 383, conceive the sides AB and DC, produced, to meet in some point P. Then, by reason of the similar triangles, ADP: BCP :: AD2 : BC2; whence, by "division," ADP-BCP ABCD: BCP :: AD2-BC2: BC2. AD2 In like manner, comparing EFP and BCP, we get EBCF: BCP:: EF2-BC2; BC2. Combining these two proportions, we have ABCD: EBCF :: AD2— BC2: EF2 — BC2; m+n:m:: AD2 - BC2: EF2 or, BC2. Also, from the similar triangles formed by drawing BL parallel to CD, we have Also, the aréa BEFC=a=}. BG (EF + BC) = } x (y+b); whence y Substituting this value of y in the expression for x, and reducing, we obtain Replacing the symbols by their lines, we get the formulas in the text. · X. (546) ABEF=ABCD. But ABRP = ABEF, because of the common part ABRF, and the triangles FRP and FRE, which make up the two figures, and which are equivalent because of the parallels FR and PE. So for the other parts. (547) The truth of the foot-note is evident, since the first line bisects the trapezoid, and any other line drawn through its middle, and meeting the parallel sides, adds one triangle to each half, and takes away an equal triangle; and thus does not disturb the equivalency. (548) In Fig. 385, since EF is parallel to AD, we have ADG: EGF :: GH2 : GK2. EGF is made up of the triangle BCG a', and the quadrilateral BEFC= (a — a′). Hence the above proportion becomes m m ABCD m + n m + n (m+n) a: ma+na' :: GH2: GK2; whence GK=GH 'ma+na' GE is given by the proportion GH: GK:: GA: GE=GA · Q GH In Fig. 386, the division into p parts is founded on the same principle. The triangle EFG —GBC+ EFCB=a'+ Now ADG: EFG :: AG2: EG2; p (552) In Fig. 390, join FC and GC. Because of the parallels CA and BF, the triangle FCD will be equivalent to the quadrilateral ABCD, of which GCD will therefore be one half; and because of the parallels GE and CH, EHDC will be equivalent to GCD. (553) In Fig. 391, by drawing certain lines, the quadrilateral can be divided into three equivalent parts, each composed of an equivalent trapezoid and an equivalent triangle. These three equivalent parts can then be transformed, by means of the parallels, into the three equivalent quadrilaterals shown in the figure. The full development of the proof is left as an exercise for the student. In Fig. 392, draw CG. Then CBG ABCD. But CKQCGQ. Therefore CKQB ABCD. So for the other division line. (556) The division of the base of the equivalent triangle, divides the polygon similarly. The point Q results from the equivalency of the triangles ZBP and ZBQ, PQ being parallel to BZ. APPENDIX C. INTRODUCTION TO LEVELLING. (1) The Principles. LEVELLING is the art of finding how much one point is higher or lower than another; i. e., how much one of the points is above or below a level line or surface which passes through the other point. A level or horizontal line is one which is perpendicular to the direction of gravity, as indicated by a plumb-line or similar means. It is therefore parallel to the surface of standing water. A level or horizontal surface is defined in the same way. It will be determined by two level lines which intersect each other.* Levelling may be named VERTICAL SURVEYING, or Up-and-down Surveying; the subject of the preceding pages being Horizontal Surveying, or Right-and-left and Fore-and-aft Surveying. All the methods of Horizontal Surveying may be used in Vertical Surveying. The one which will be briefly sketched here corresponds precisely to the method of "Surveying by offsets," founded on the Second Method, Art. (6), "Rectangular Co-ordinates," and fully explained in Arts. (114), &c. The operations of levelling by this method consist, firstly, in obtaining a level line or plane; and, secondly, in measuring how far below it or above it (usually the former) are the two points whose relative heights are required. (2) The Instruments. A level line may be obtained by the following simple instrument, called a "Plumb-line level." Fasten together two pieces of wood at right angles to each other, so as to make a T, and draw a line on the upright one so as to be exactly perpendicular to the top edge of the other. Suspend a plumb-line as in the figure. Fix the T against a staff stuck in the ground, by a screw through the middle of the crosspiece. Turn the T till the plumb-line exactly covers the line which was drawn. Fig. 415. Then will the upper edge of the cross-piece be a level line, and the eye can sight across it, and note how far above or below any other point this level line, prolonged, would strike. It will be easier to look across sights fixed on each end of the cross-piece, making them of horsehair stretched across a piece of wire, bent into three sides of a square, and stuck into each end of the cross-piece; taking care that the hairs are at exactly equal heights above the upper edge of the cross-piece. * Certain small corrections, to be hereafter explained, will be ignored for the present, and we will consider level lines as straight lines, and level surfaces as planes. A modification of this is to fasten a common carpenter's square in a slit in the top of a staff, by means of a screw, and then tie a plumb-line at the angle so that it may hang beside one arm. When it has been brought to do so, by turning the square, then the other arm will be level. Another simple instrument depends upon the principle that "water always finds its level," corresponding to the second part of our definition of a level line. If a tube be bent up at each end, and nearly filled with water, the surface of the water in one end will always be at the same height as that in the other, however the position Fig. 417. Fig. 416. of the tube may vary. On this truth depends the "Water-level." It may be easily constructed with a tube of tin, lead, copper, &c., by bending up, at right angles, an inch or two of each end, and supporting the tube, if too flexible, on a wooden bar. In these ends cement (with putty, twine dipped in white-lead, &c.), thin phials, with their bottoms broken off, so as to leave a free communication between them. Fill the tube and the phials, nearly to their top, with colored water. Blue vitriol, or cochineal, may be used for coloring it. Cork their mouths, and fit the instrument, by a steady but flexible joint, to a tripod. Figures of joints are given on page 134, and of tripods on page 133. To use it, set it in the desired spot, place the tube by eye nearly level, remove the corks, and the surfaces of the water in the two phials will come to the same level. Stand about a yard behind the nearest phial, and let one eye, the other being closed, glance along the right-hand side of one phial and the left-hand side of the other. Raise or lower the head till the two surfaces seem to coincide, and this line of sight, prolonged, will give the level line desired. Sights of equal height, floating on the water, and rising above the tops of the phials, would give a better-defined line. The "Spirit-level" consists essentially of a curved glass tube nearly filled with alcohol, but with a bubble of air left within, which always seeks the highest spot in the tube, and will therefore by its movements indicate any change in Fig. 418. the position of the tube. Whenever the bubble, by raising or lowering one end, has been brought to stand between two marks on the tube, or, in case of expansion or contraction, to extend an equal distance on either side of them, the bottom of the block (if the tube be in one), or sights at each end of the tube, previously properly adjusted, will be on the same level line. It may be placed on a board fixed to the top of a staff or tripod. When, instead of the sights, a telescope is made parallel to the level, and various contrivances to increase its delicacy and accuracy are added, the instrument becomes the Engineer's spirit-level. Fig. 419. (3) The Practice. By whichever of these various means a level line has been obtained, the subsequent operations in making use of it are identical. Since the "water-level" is easily made and tolerably accurate, we will suppose it to be employed. Let A and B, Fig. 419, represent the two points, the difference of the heights of which is required. Set the instrument on any spot from which both the points can be seen, and at such a height that the level line will pass above the highest one. At A let an assistant hold a rod graduated into feet, tenths, &c. Turn the instrument towards the staff, sight along the level line, and note what division on the staff it strikes. Then send the staff 2 A B to B, direct the instrument to it, and note the height observed at that point. If the level line, prolonged by the eye, passes 2 feet above A and 6 feet above B, the difference of their heights is 4 feet. The absolute height of the level line itself is a matter of indifference. The rod may carry a target or plate of iron, clasped to it so as to slide up and down, and be fixed, at will. This target may be variously painted, most simply with its upper half red and its lower half white. The horizontal line dividing the colors is the line sighted to, the target being moved up or down till the line of sight strikes it. A hole in the middle of the target shows what division on the rod coincides with the horizontal line, when it has been brought to the right height. If the height of another point, C, Fig. 420, not visible from the first station, be required, set the instrument so as to see B and C, and proceed exactly as with A and B. If C be 1 foot below B, as in the figure, it will be 5 feet below A. If it were found to be 7 feet above B, it would be 3 feet above A. The comparative height of a series of any number of points, can thus be found in reference to any one of them. The beginner in the practice of levelling may advantageously make in his notebook a sketch of the heights noted, and of the distances, putting down each as it is observed, and imitating, as nearly as his accuracy of eye will permit, their pro |