DESCRIPTIVE GEOMETRY. terminating in T penetrates the cutting plane; if therefore we P". 620 A 4 -B C By a similar construction, we may determine any number of points in the curve Q'LXR', which will be the section required. The curve required may be obtained still more simply by merely finding the perpendiculars NS"H,and describing the curve through L, S", м, &c. without determining the corresponding points in 'LXR. It is evident that the plane meeting the base at right angles in r'Yq' must also meet the upper division of the conic surface, and produce another section equal and similar to q'zR. The curve determined by this construction is an hyperbola. • PROBLEM VI. To construct the intersection of a conic surface by a plane parallel to one of the slant sides of the cone. Let AB be the ground line; p' the centre of the cone's base, which is supposed to be coincident with the horizontal plane ; and let the base EFK touch the ground line in K: in P'x produced, take KP" equal to the altitude or axis of the cone, and Let p" is the vertical projection of the summit of the cone. the cutting plane be parallel to the ground line, and meet the j G Y" H" 621 1 Z DJA PORTAL base in the horizontal trace EF, which will consequently be parallel to AB : in KP" produced if necessary, take KY" a fourth DESCRIPTIVE GEOMETRY. proportional to the three straight lines KP', KX, KP", and the straight line G"Y"H" parallel to AB, will be the vertical trace of the cutting plane. Ꮐ 622 The angle which the slant side passing through z makes with the horizontal plane is evidently the acute angle at the base of a right-angled plane triangle of which the base is zr', and perpendicular equal to KP"; and the angle which the cutting plane makes with the horizontal plane is also the acute angle at the base of a right-angled triangle of which the base is xx and perpendicular KY"; and since these two triangles are in the same plane and have the bases and altitudes proportionals, it is plain that the acute angles at their bases are equal, and that the slant side passing through z is parallel to the plane of which the traces are EF, G"H". Q To construct the curve of intersection draw any radius r'q; from a draw am at right angles to AB, and join P", then 'Q, and r' are the horizontal and vertical traces of the slant side passing through a. Find by prob. 2. chap. Iv the horizontal and vertical projections s' and s" of the point in which this slant side meets the cutting plane; and by prob. 12. chap. II. find the position on the horizontal plane of the point of which s' and s" are the projections by the rotation of the cutting plane about the intersection EF, and I is a point in the required I curve. In a similar manner we may proceed in determining any number of points in the required curve FIVE. The ordinates No, uw are obtained by the construction given in prob 2, chap. IV. for the slant sides passing through the extremities of the diameter DR parallel to the ground line AB. The vertex v is found by taking KA equal to KY", and P'D equal to KP"; then drawing ax and KD, we have the position c'of the vertex of the curve on the horizontal plane; and therefore making xv equal to xc, the point v will be the vertex of the curve. It is obvious that the curve FIVE is a parabola. LOGARITHMS OF THE NUMBERS FROM 1 to 1000. N. Log. N. Log. N Log. Log. • 1 0.000000| 26 |1.414973|| 51|1.707570|| 76 |1.880814 2 0.301030|| 27 |1.431364|| 52|1.716003 77 1.886491 3 0.477121|| 28|1.447158|| 53 |1,724276|| 78 |1.892095 4 0.602060 29 1.462398 54 1.732394|| 79 1 897627 5 0.698970|| 30 |1.477121|| 55 |1.740363|| 80 |1.903090 6 0.778151 311.491362 56 1.748188 81 1,908485 7 0.845098 32 1.505150|| 57 1.755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 831,919078 9 0.954243 34 1.531479 59 1.770852|| 84 1.924279 10 1.000000 351,544068 60 1.778151 851.929419 11 1.041393 361,556303|| 61 |1.785330|| 86 1.934498 12 1.079181|| 371,568202|| 62 | 1.792392 871 939519 13|1.113943|| 38 1.579784 631.799341 881.944483 14 1.146128 39 1,591065 64 1.806180 89 1.949390 15 1.176091| 40 1.602069|| 65 1.812913 90 1.954243 16 1.204120 41 1.612784|| 66 1.819544 911.959041 17 1.230449 42 1.623249| 67 |1.826075|| 92 1.963788 181,255273 43 1.653468 68 1.832509 93 1.968483 19 1.278754|| 44 1.643453 69 1.838849|| 94 | 1.973128 20 1.301030 45 1.653213 70 1.845098 95 1.977724 214322219|| 46 |1.662758 71 1.851258 96 1.982271 22 1,342423 47 1.672098 72 1.857333 97 1.986772 231 361728|| 48 |1.681241|| 73 |1.863323 98 1.991226 24 1,380211 49 1.690196|| 74 1.869232 99 1.995635 25 1.397940 50 1.698970 75 1.875061| 100 | 2.000000 N. F : N. B. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are now introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the corresponding natural number in the first column stands in the next lower line, and its annexed first two figures of the Logarithms in the second column. N. 0 102 104 . 2 3 45 67 718 89 123 136 137 3539 8858 4177 4496 4814 5133 5451 5769 6086 6403 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 : |
