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 >JO V
 
 PRACTICAL TREATISE 
 
 RAILWAY CURVES 
 
 AND 
 
 LOCATION, 
 
 FOR YOUNG ENGINEERS 
 
 MtTAINING A FTTLL DESCRIPTION OF THE INSTRUMENTS, THE MANNER OF ADJUSTING THEM, AND 
 THE METHODS OF PROCEEDING IN THE FIELD, NEW AND SIMPLE FORMULAE FOR COM- 
 POUND AND REVERSE CURVING, RULES FOR CALCULATING EXCAVATION AND 
 EMBANKMENT, STAKING OUT WORK, 4C., TOGETHER WITH TABLES OF 
 NATURAL SINES AND TANGENTS, RADII, CHORDS, ORDINATES, 
 AND OTHERS OF GENERAL USE IN THE PROFESSION. 
 
 WILLIAM F.^SHUNK, 
 
 crvn, ENCUCEER. 
 
 PHILADELPHIA : 
 ENRY CAREY BAIRD, 
 
 INDUSTRIAL PUBLISHER. 
 
 No. 406 WALNUT STREET. 
 
 18CG.
 
 Entered, according to the Act of Congress, in the year 1854, by 
 E. H. BUTLER A 00., 
 
 In the Clerk's Office of the District Court of the United States, in and for the Eastern 
 District of Pennsylvania,
 
 r 
 
 <* 
 
 PREFACE. 
 
 THE located line for railway is a series of curves and 
 straight lines, or tangents. These are first plotted to a 
 large scale from data gathered on preliminary survey. It 
 is therefore desirable that all explorations should be made 
 with extreme care, as upon their correctness depend, in no 
 small degree, the labour and time required in location. 
 It were better for accuracy that all angles should be made 
 and recorded from the plates, and the needle used only as 
 a test, or check. Good chaining is indispensable. Great 
 attention, too, should be given to the proper use of the 
 ; slope instrument. By these means a working map can be 
 ' constructed in the oflice upon which the proposed location, 
 ^ grade lines, &c., may be traced with tolerable resemblance 
 HEb fact. Still many errors attach to both data and map, 
 and these, together with the unexpected obstacles encoun- 
 tered in the field, require ready knowledge of the means 
 for overcoming them. 
 
 It has been my design to present this knowledge to my 
 younger fellows in the profession. I have endeavoured to 
 * % i-~*- ,..>. *-+. r~v \ )
 
 IV PREFACE. 
 
 do it lucidly and concisely without supposing unusual cases 
 without prolix proof or complex figuring. The problems 
 given are of frequent occurrence, and the tables appended 
 will be found useful and correct. 
 
 To STRICKLAND KNEASS, a gentleman whose professional 
 abilities are well known, I return thanks for valuable 
 assistance. I would likewise make my acknowledgments, 
 for useful suggestions, to CHARLES DELISLE, an engineer 
 of high mathematical attainment. 
 
 I am aware that much more might have been said 
 much more suggested on the subject of location ; but a 
 field book being the object, the compact plan precluded 
 any extensive essay. 
 
 If the work with brevity combines clearness, and ia 
 comprehensive withal, it is the work intended. 
 
 W. F. SHUNK.
 
 CONTENTS. 
 
 PACK 
 
 PREFACE ........ 3 
 
 Explanations ........ 7 
 
 ARTICLE I. Of the Instruments. The Level . . . ,11 
 
 Its adjustment . . . . . . 13 
 
 The Rod ....... 14 
 
 Levelling . . . . . . 15 
 
 The Transit . . . . . . .17 
 
 The Vernier ...... 18 
 
 Adjustment of Transit . . . . .19 
 
 II. Preliminary propositions .... 20 
 
 III. To avoid an obstacle in tangent . . . .21 
 
 IV. Triangulation ...... 22 
 
 V. Of calculating tangents to any degree of curvature . 24 
 
 VI. To trace a curve with transit and chain ... 25 
 
 VII. To triangulate on a curve .... 29 
 
 VIII. To change the origin of any curve, so that it shall termi- 
 nate in a tangent, parallel to a given tangent . . 31 
 IX. To change a P. C. C. with similar object. First. When the 
 
 second curve has the smaller radius . . .32 
 
 X. Second. "When the last curve has the larger radius . 34 
 
 Synopsis of formula; for compound curving . . 35 
 
 XI. Having located a compound curve terminating in any tan- 
 gent, to find the P. C. C. at which to commence another 
 curve of given radius which shall terminate in the same 
 tangent. First. When the latter curves have the smaller 
 radii ....... 3f 
 
 XII. Second. When the latter curves have the larger radii . 38
 
 VI CONTENTS. 
 
 PACI> 
 
 ART. XIII. To change a P. R. C. so that the second curve shall termi- 
 nate in a tangent parallel to a given tangent . . 39 
 XIV. How to proceed when the P. C. is inaccessible 41 
 XV. To avoid obstacles in the line of curve . . .43 
 XVI. To calculate reverse curves .... 45 
 
 XVII. Having given a located curve, terminating in any given 
 tangent, to find where a curve of different radius will ter- 
 minate in a parallel tangent . . . .40 
 
 XVIII. Having a curve located and terminating in a given tangent, 
 to find the P. C. C. whereat to begin another curve of 
 given radius which shall terminate in a parallel tangent 48 
 XIX. To locate a Y . . . . . 49 
 
 XX. To run a tangent to two curves . . . .51 
 
 XXI. Of ordinates ...... 52 
 
 XXII. To find the radius corresponding to any chord and deflexion 
 
 angle. Deflexion and tangential distances . . 54 
 
 XXIII. Of excavation and embankment . . . .56 
 
 XXIV. Side staking ...... 60 
 
 TABLES. 
 
 Natural Sines and Tangents ...... 63 
 
 Radii . . . 89 
 
 Long Chords . . . . . ..90 
 
 Ordinates ......... 91 
 
 Squares and Square Boots ...,. 94 
 
 Slopes and Distances for Topography > . 106
 
 EXPLANATIONS. 
 
 ALL railway curves are parts of circles. They are 
 designated generally from their character as simple, com- 
 pound, or reverse ; and specifically from the central angle 
 subtended by a chord of 100 feet at the circumference, 
 this being the length of the chain in common use. It is 
 found that the circle described with radius of 5730 feet 
 has a circumference of 36,000 feet. Since there are 360 
 in the circle, the central angle subtended by a chord of 
 100 feet is, in this case, equal to 1, and the curve is 
 named a one degree curve. So likewise in a circle with 
 radius of 2865 feet, half of 5730, the central angle cor- 
 responding to the chord 100 is 2; the curve is then called 
 a two degree curve. 
 
 The beginning of a curve is called the point of curva- 
 ture, or simply the P. C., and its termination the point of 
 tangent, marked P. T. 
 
 A compound curve is composed of two curves of different 
 radii, turning in the same direction, and having a common 
 tangent at their point of meeting. This point is called the 
 point of compound curvature, or P. C. C. 
 
 (7)
 
 8 RAILWAY CURVES AND LOCATION. 
 
 A reverse curve is composed of two curves turning in 
 different directions, and having a common tangent at their 
 point of meeting, which latter is named the point of re- 
 versed curvature, or the P. R. C. 
 
 All sines and tangents made use of in this work are 
 from the table at the end of the volume. For calculating 
 curves it is not necessary to use more than four decimals. 
 
 A Bench is a shoulder hewn with the axe on the but- 
 tressed base of a tree, and so shaped at the top as to afford 
 footing to the rod. The tree is blazed and the elevation 
 of the bench marked on it with red chalk. Benches serve 
 as permanent reference points to the level. They are 
 placed, where it is possible, about one thousand feet 
 apart. 
 
 Points. The operation termed pointing is the fact of 
 putting a peg firmly into the ground, and of driving in its 
 top a tack, or making thereon an indentation whose place 
 is indicated by cross keel marks, directly in the line of col- 
 limation of the transit. Thus true lines are traced on the 
 ground, and angles measured accurately. When the transit 
 is set over a point it is so posited that the plumb hangs 
 immediately above the tack head. If the head plate of 
 the tripod be much inclined the plumb should be examined 
 after levelling the instrument, as that operation disturbs it 
 to some extent. 
 
 Stations. The line of a survey is marked on the ground 
 at regular intervals, by stakes two feet in length, blazed, 
 and numbered from up in arithmetical progression. 
 These stakes are named stations. On exploration they 
 are commonly placed two hundred, and on location, one 
 hundred feet apart. 
 
 It is customary, when locating, to drive pegs even with 
 the surface along the true line, and to place the stations a 
 couple of feet to the right, numbers facing in, to show their
 
 EXPLANATIONS. 9 
 
 position. The pegs are less liable to disturbance from frost, 
 animals, &c. 
 
 In locating for construction stakes are driven on sharp 
 curves at intervals of 50, sometimes 25 feet. 
 
 The Chain in general use for railway surveys is made 
 of soft iron. It is 100 feet long, and divided into 100 
 links, each one foot in length. At every tenth link ia 
 attached a brass drop, toothed so as to indicate its distance 
 from the end. It presents the advantages of durability, 
 accuracy, and expedition.
 
 A PRACTICAL TREATISE 
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE I. 
 
 OF THE INSTRUMENTS. 
 THE LEVEL. 
 
 level is an instrument used in ascertaining the 
 undulations of the ground 
 along the line of a survey, rv, "- l 
 
 and of measuring these ir- ^ 
 regularities accurately in 
 reference to an assumed 
 base called the datum. It 
 consists mainly of the 
 telescope k i, the spirit- 
 level and its encasement 
 5, the Y's c c, the rectan- 
 gular bar o d, the axis e, the plates and levelling screws 
 / ?, and the tripod g, 
 
 In the tube of the telescope at 7i, and at right angles to 
 its axis, is placed a flat ring, called the diaphragm. To 
 this ring the cross-hairs are attached two delicate spider 
 lines stretched over it vertically and horizontally, and inter- 
 secting at the centre. It is held in position by four 
 
 (11)
 
 12 RAILWAY CURVES AND LOCATION. 
 
 slightly movable screws, which pierce the tube in the 
 direction of the " cross-hairs." i is a milled head for 
 adjusting the focus of the object glass, and k an inserted 
 tube, containing several lenses, which may be moved out 
 or in so as to make the spider-lines distinctly visible. 
 
 A straight line looked along from the eye glass at k 
 through the intersection of the cross-hairs is the line of 
 sight, technically named the line of collimation, 
 
 The immediate supports of the telescope are called the 
 Y's, from their resemblance to that letter. If a small arch 
 were sprung between the two legs of the Y it would give 
 a good idea of the clasping pieces which hold the telescope 
 in place. They are jointed to One leg and secured to the 
 other by pins which may be withdrawn and the pieces 
 turned back in order to remove the telescope, or change it 
 end for end. 
 
 The Y's are attached at right angles to the bar d, which, 
 again, is connected firmly at right angles with the hollow 
 axis e. This latter fits closely over and is revolvable hori- 
 zontally around a solid axis , which, passing through the 
 plate /, is secured to the head of the tripod by means of a 
 loose ball-and-socket joint. The plate /has four levelling 
 screws inserted in it ; with these the instrument may be 
 brought to a horizontal position even when the lower plate 
 is considerably inclined. 
 
 One of the Y's is movable for a short space up or down 
 by means of the capstan-head screw o. The spirit-level is 
 likewise movable both vertically and laterally by means of 
 screws at either end. 
 
 n is a clamp screw, and p a tangent-screw for slightly 
 turning the telescope in a horizontal direction.
 
 THE LEVEL, 13 
 
 To adjust the Lewi* 
 
 First. To make the line of collimation coincide with the 
 xxis of the telescope. 
 
 Set the instrument firmly, and direct the telescope 
 toward some distant, distinct object, such as a nail-head. 
 Clamp fast, and with tangent-screw fix the line of collima- 
 tion upon the object accurately. Revolve the telescope 
 half way round in the Y's, *'. e. until the bubble is above 
 it, and if the horizontal spider-line still covers the point, it 
 requires no adjustment. If it does not, reduce the error 
 one-half by means of the diaphragm screws, and complete 
 the reduction with the capstan-head screw. Revolve the 
 telescope round to its first position, and if the horizontal 
 line and point do not then coincide, repeat the operation 
 until they do, in any position of the telescope. In similar 
 wise the vertical hair may be adjusted, when the line of 
 collimation should cover the point through an entire revo- 
 lution of the telescope. 
 
 Great care should be taken in this as well as in all other 
 adjustments of cross-hairs, that the opposite screw of the 
 diaphragm be loosened before tightening its fellow, or 
 injury to the instrument must result. 
 
 Second. To make the axis of the spirit-level parallel to 
 the line of collimation. 
 
 With levelling screws bring the bubble to the middle of 
 its tube, reverse the telescope in its Y's, and if the bubble 
 does not then stand in the middle correct one-half the 
 deviation with the screw at the left end of the bubble-case, 
 and the other half with the capstan-head screw. Again 
 reverse the telescope in its Y's, and, if necessary, repeat 
 the operation. 
 
 Now revolve the telescope a short distance in its Y's, so 
 as to bring the spirit-level to one side of its lowest position. 
 If the bubble deviates from the middle, correct the error 
 
 2
 
 14 RAILWAY CURVES AND LOCATION. 
 
 with the lateral screws at the right end of the bubble-case, 
 and examine the previous adjustment before lifting the 
 instrument. 
 
 Third. To bring the line of collimation parallel to the 
 bar. 
 
 Turn the telescope until it stands directly over two of 
 the levelling screws, and with them bring the bubble to the 
 middle of the tube. Then revolve the telescope horizontally 
 until it stands over the same screws, changed end for end. 
 If the bubble does not still stand in the middle of the tube, 
 correct one-half the deviation with the capstan-head, and 
 one-half with the levelling screws. 
 
 Place the telescope over the other levelling screws and 
 proceed in a similar manner, and continue the corrections 
 until the bubble stands without varying during an entire 
 revolution of the instrument upon its axis. 
 
 This completes the adjustment of the level. 
 
 THE ROD. 
 
 The rod used in levelling consists of a staff and a target, 
 which latter is so attached to the staff as to be movable 
 along it from end to end. The rod is commonly seven feet 
 long, but, being composed of two rectangular pieces fitted 
 together by means of a sliding groove, it can be extended 
 to nearly double that length. It is graduated to feet and 
 tenths of a foot. The target is a circle of wood or iron, 
 usually four-tenths in diameter, and divided into quadrantal 
 sectors by a horizontal and vertical line which intersect at 
 its centre. The sectors are painted alternately red and 
 white, so that their dividing lines are visible at a consider- 
 able distance. On the back of the target, where it meets 
 the graduated side of the rod, is fixed a chamfered brass 
 edging, whereon the space of one-tenth is graven from the 
 centre down. This is subdivided into ten spaces marking
 
 LEVELLING. 
 
 15 
 
 hundredths, and these latter divided into halves, so that the 
 height of the middle of the target above the base of the 
 rod may be accurately read to within -005 of a foot. 
 
 There is a similar graduated tenth on the standing part 
 of the rod, to be used for high sights when the sliding 
 groove comes into play. 
 
 Both target and rod are provided with clamp screws. 
 
 LEVELLING. 
 
 The operation technically called levelling is performed 
 thus : 
 
 Suppose a the starting point, or zero, in reference to 
 which all the inequalities of the surface along the line of 
 survey are measured, as at the points c, e, f. The hori- 
 zontal line a /is called the datum line. This is arbitrarily 
 
 assumed. It may be considered, for example, at any dis- 
 tance above the point a, and the irregularities of the ground 
 measured from an imaginary level line in ether ; but for 
 convenience of figuring, and other politic reasons, it is cus- 
 tomary in seaport towns to take high tide as datura. In- 
 land, the summer surface of the nearest stream, or, when 
 commencing on a ridge, the highest neighbouring knoll is 
 assumed. 
 
 Well ! suppose a to be zero, and the instrument, for 
 instance, set and levelled at b. Stand the rod at a, and 
 slide the target up until its cross-lines are covered by the 
 cross-hairs in the telescope ; i. e., until the line of collima- 
 tion coincides with the centre of the target. The leveller 
 directs the movements of the target by raising or lowering
 
 16 RAILWAY CURVES AND LOCATION. 
 
 his hand. A circular motion of the hand signifies " make 
 fast." The bubble should always be examined before the 
 rod is taken down, and the latter should be read twice, or, 
 if convenient, shown to the leveller, in order to guard 
 against mistake. If in this case it reads 8 feet, the height 
 of the instrument is then 8 feet above a. To find the ele- 
 vation of o above , take the rod thither and lower the 
 target until coincidence results as before. If the rod 
 reads 2 feet, of course e is 8 2 = 6 feet above a. 
 
 If it is necessary to lift the instrument here, a small peg 
 is driven at c before sighting to that point, to insure firm 
 footing for the rod. Sighting back from the new position, 
 d, the rod reads 5 feet ; then 5 + 6, the elevation of c 
 above a, = 11, the height of the telescope at d above a. 
 If at e the reading is 6, the elevation of that point is 11 
 6 = 5, and if at / the reading is 8 the elevation of 
 that point in reference to a is 11 8 = 3, marked -f- 3. 
 
 The rule, therefore, in levelling is, at each new stand of 
 the instrument, to add the reading of the rod sighted back 
 at, to the discovered elevation of the point at which the 
 rod stands, for the height of the instrument ; and to sub- 
 tract from this height the reading of the rod at any points 
 observed from the new position in order to find the eleva- 
 tion of those points. The above is noted in the field-book 
 as follows : 
 
 Station. 
 
 Rod. 
 
 Height of 
 Instrument. 
 
 Total, or 
 Elevation. 
 
 a 
 
 8-00 
 
 8-00 
 
 00 
 
 c 
 
 2-00 
 
 
 + 6.00 
 
 
 5-00 
 
 11-00 
 
 
 e 
 
 6-00 
 
 
 -f 5-00 
 
 f 
 
 8-00 
 
 
 -f 3-00
 
 OP THE TRANSIT. 17 
 
 The advantages of this method of levelling over tkf old 
 system of backsights and foresights are, that it affords 
 readier facilities for testing the correctness of the work, 
 and it may be carried on more rapidly. By the old plan 
 each sight at the rod was linked with that which preceded 
 it, and added one more to a continuous calculation in which 
 a single error affected all the following work. Here, how- 
 ever, if haste is required, the calculation of the interme- 
 diate sights or "cuttings" may be omitted entirely while 
 in the field, the reading of the rod only being set down ; 
 the "totals" may be worked from peg to peg, and the lia- 
 bilities to mistake thus decreased about eighty per cent. 
 
 OF THE TRANSIT. 
 
 The- transit is an instrument for measuring horizontal 
 angles. It consists of the tele- 
 scope a c, the Y's (i, the compass- . A ^C ^P$~" ^ 
 box, &c., e g, and the axia k. 
 The telescope is furnished like 
 that of the level, and the instru- 
 ment is similarly fitted to its tri- 
 pod. The telescope revolves in 
 a vertical circle, and is attached 
 to the Y's by means of a trans- 
 verse axis whose extremities turn in smooth journals at the 
 head of the Y's. The body of the instrument at/ contains 
 a magnetic needle, with its usual circular surrounding, 
 graduated to degrees and quarter degrees. The flooring of 
 this box has, on one side, an opening with chamfered edge 
 upon which the vernier is engraved. This latter, together 
 with the telescope, Y's, and all the upper part of the 
 instrument, is-made to revolve by means of the screw /*, 
 upon a solid plate beneath, which is likewise graduated 
 from to 180 each way. Thus angles may be measured 
 
 2*
 
 18 RAILWAY CURVES AND LOCATION. 
 
 accurately without using the needle at all. It need be 
 regarded merely as a check, g is a clamp screw for secur- 
 ing the plates together, and i a screw for fastening the 
 needle so as to prevent its vibrations while the instrument 
 is being carried from place to place. A plumb is suspended 
 from the axis of the transit, by means of which its centre 
 may be placed over a point on the ground. 
 
 THE VERNIER. 
 
 The vernier, in the transit, is a graduated index which 
 serves to subdivide the divisions of the graduated arc on 
 the lower plate. There are many varieties of the vernier, 
 but familiarity with one renders easy the acquaintance 
 with all, since the same general principle is pervading. 
 
 The figure represents a common form. Let a b be part 
 of any graduated arc, and c d the vernier. It will be 
 observed that the degrees on the limb are divided into 
 spaces of 15' each. Now if the vernier be made equal in 
 length to fourteen of those spaces, and be further divided 
 into fifteen equal parts, it is evident that each of these 
 parts will contain 14'. 
 
 Then, if of the vernier coincides with any division of 
 the limb, the first line of the vernier to the left will be 
 just one minute behind the first line of the limb to the 
 left ; the second vernier line two minutes behind the second 
 limb line, and so on ; so that if the vernier be moved to 
 the left over the space of 15' on the limb, the lines from 
 to 15 of the vernier would coincide successively with lines
 
 TO ADJUST THE TRANSIT. 19 
 
 of the limb, and thus any angle may be read accurately to 
 minutes. 
 
 The vernier in the figure reads 48' to the left. A 
 vernier graduated decimally is much more convenient on 
 railway locations than those with the common graduation 
 to minutes. This is principally on account of its adapted- 
 ness to running in curves when the 100 feet chain is used. 
 The work can be done with more ease and rapidity. One 
 objection to it is that the tables in general use are calcu- 
 lated for degrees and minutes. 
 
 TO ADJUST THE TRANSIT. 
 
 Place the instrument firmly at a, level it, clamp all 
 fast, and with tangent-screw set the cross -hairs on the 
 point 6, at any convenient distance. Reverse the telescope 
 on its axis, and fix another point in the opposite direction, 
 
 as nearly as possible equidistant from a. Now loose the 
 lower clamp and revolve the entire upper part of the 
 instrument half way round on its axis. Clamp fast, and 
 having brought the cross-hairs again to coincide with 6, 
 reverse the telescope. If the sight strikes as before, the 
 instrument is in adjustment. If not, place another point, 
 d, where it does strike, and suppose c to be the point pre- 
 viously fixed : the point e, midway between d and c, is then 
 in the straight line. With the adjusting pin carefully 
 place the vertical cross-hair upon /, distant from d one- 
 quarter of the space d c with tangent-screw set it on e, 
 and reverse the telescope. If the points have been cor- 
 rectly placed, and the hair properly moved, the sight will 
 strike 5, and the adjustment is complete.
 
 20 
 
 RAILWAY CURVES AND LOCATION. 
 
 After finishing this adjustment, the telescope may still 
 not revolve truly in the meridian. This inaccuracy there 
 is no method of removing in the field. It should be sent 
 to an instrument-maker for repairs. 
 
 ARTICLE II. 
 
 \ 
 
 PRELIMINARY PROPOSITIONS. 
 
 1. In any circle the angle o cf at the centre, subtended 
 by the chord o /, is double the angle o af, at any part of 
 the circumference on the same side of the chord. 
 
 2. The angle fbe, formed by any chord fb, with a 
 tangent at either extremity, is called a tangential angle.
 
 AVOIDING OBSTACLES. 21 
 
 and is equal to half the angle / c b at the centre, or is 
 equal to the angle / a b at the circumference. 
 
 3. The exterior angle dbf, formed at the circumference 
 bj the two equal chords a b, b /, is called a deflexion angle, 
 and is equal to the central angle b c f, or double the tan- 
 gential angle e bf. df is called the deflexion distance, 
 and e f the tangential distance. 
 
 4. The exterior angle p o m of two unequal chords, is 
 equal to the sum of their tangential angles, or half the sum 
 of their central angles. 
 
 5. The exterior angle i k o, formed by tangents, is equal 
 to the central angle b c o, subtended by the chord which 
 connects their points of contact with the curve. 
 
 ARTICLE III. 
 
 TO AVOID AN OBSTACLE IN THE LINE OF TANGENT. 
 
 A GLANCE at the figure will show, that having deflected 
 to c, and placed the instrument at that point, the angle 
 hcd must be made equal to twice dbc, and the distance 
 
 c d equal to the distance b c. Still another deflection at 
 c?, equal to the original angle turned, is necessary in order 
 to sight again along the tangent. 
 
 Should the obstruction be continuous a parallel line 
 may be run, as from c to /, by deflecting at c an angle
 
 22 RAILWAY CURVES AND LOCATION. 
 
 equal to that at 5, and at /, repeating the deflection in 
 order to strike tangent. 
 
 If the angle die exceeds 4, and the distance b c is 
 greater than 200 feet, or even with an angle of 2|, 
 should the distance be greater than 300 feet, b c will 
 'differ sensibly in length from b &, and a calculation of the 
 latter becomes necessary. To effect this, multiply the 
 natural cosine of the angle kbc by be. This result 
 doubled will give bJcd, the length proper along tangent. 
 
 Thus : Suppose kbc, the angle deflected, to have been 
 5, and the distance b c 340 feet. Then -9962, the natural 
 cosine of 5, multiplied by 340, gives 338-7 for the dis- 
 tance bJc. Double this makes bkd = 677*4, and shows 
 a difference of 2'6 feet between b d and bed. 
 
 ARTICLE IV. 
 
 SHOULD THE OBSTRUCTION LIE ON THE OPPOSITE BANK 
 OF A STREAM, 
 
 AND it is desirable on any account not to run the line 
 from d, corresponding to a d, set the 
 instrument at a, in tangent, and deflect 
 clear of the obstacle to d. Point at d, 
 deflect to e, and point also there 
 marking the angles cad, d a e. Chain 
 the base de, and placing the transit at 
 e, measure the angle dea. Data are 
 thus obtained sufficient for the calcula- 
 tion of the line da. The object now is 
 to find the point c and the angle d c a. 
 The angle a d e subtracted from 180 will supply the
 
 
 AVOIDING OBSTACLES. 
 
 -"r, 
 
 23 
 
 angle c da, so that in the smaller triangle we have 
 obtained two angles and their included side. The dis- 
 tance cd, and angle dca readily follow. The transit 
 standing at e, c is placed, of course, in the prolongation 
 of the base d e, and the distance c d is carefully set off 
 with the rod. Moving the instrument to c, and turning 
 the angle ecf = 180 dca, we are again in tangent. 
 
 Example. Let cad = 6, dae = 35, d e a 42, 
 and the base d e 200 feet. Then in the triangle d e a 
 we have 
 
 Nat. sine d a e (-5736) : nat. sine d e a = (-6691) : : 
 d e = (200) : d a, 
 
 6691 x 200 
 Wherefore, d a = = 233-3 feet. 
 
 Again, the angle c d a 77. The angle d a c being 
 = 6, ac d is consequently = 97, and in the small triangle 
 we have, Nat. sin. a c d = (-9925) : nat. sin. c a d = 
 (-1045) : : d a = (233-3) : c d. 
 
 1045 X 933-3 
 Therefore cd = - ^r - = 24-564 feet, and d cf 
 
 9925 
 
 180 97 = 83 C 
 
 NOTE. A common and convenient plan for triangulat- 
 ing a creek is as per figure. Set the in- 
 strument at b, fix a point d on the oppo- 
 site shore, and making d b c a right angle, 
 place c at any convenient distance. Now 
 move to c, sight to d, and making d c a a 
 right angle also, fix a, in the same line 
 with b and d. a, c, and d are points in 
 the circumference of a circle whose dia- 
 meter is a d, a b : b c : : b c : b d, and therefore b d = 
 
 ab. 

 
 24 
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE V. 
 
 HAVING GIVEN THE ANGLE edb, FORMED BY THE INTERSEC- 
 TION OF TWO STRAIGHT LINES, IT IS REQUIRED TO FIND 
 THE POINT a OR 6, AT WHICH TO COMMENCE A CURVE OF 
 GIVEN RADIUS. 
 
 Draw the bisecting line d o. Then the angle d o a = 
 
 half the angle acb or its 
 equal e d b ; and in the tri- 
 angle d c a, the angle d a c 
 being a right angle, we have 
 Rad. of 1 : Nat. Tang. 
 dca : Rad. ca: ad. There- 
 fore a d = Nat. Tang, dca 
 X Rad. e a. 
 
 Example 1. Let e d b 
 = 48 and a c = 1460 
 feet. Here half the angle 
 edb or acb = 24, the 
 Nat. Tang, of which is P 4452 ; and multiplying by Rad. 
 1460, we have 650 feet for the length of a d or d b, the 
 tangents. 
 
 2. If ad be given and radius required, 
 ad 650 
 
 Rad. = 
 
 = 1460. 
 
 Nat. Tang, a c d -4452 
 
 The following rules are approximate, and sufficiently 
 correct for all purposes of location. 
 
 To find the degree of curvature of a b divide 5730 by 
 the radius in feet ; and to find the length of the curve in 
 feet divide the angle acb (after reducing minutes to hun- 
 dredths) by the degree of curvature the chord in each 
 case being 100 feet in length.
 
 _ TO TRACE A CURVE. 25 
 
 ARTICLE VI. 
 
 TO TRACE A CURVE WITH TRANSIT AND CHAIN. 
 
 THE degree of curvature and the angle to be turned are 
 known. If the latter is expressed in degrees and minutes, 
 reduce the minutes to hundredths, since the 100 feet chain 
 is used, and divide the whole angle by the degree of cur- 
 vature. The quotient will be the length of the curve in 
 feet, and the P. T. is at once ascertained. 
 
 Let m a be the tangent, and a the P. C. Place the 
 transit at a, index reading 0, and direct the sight along 
 the tangent m a e. The first deflection will be half the 
 central angle subtended by the chord used, and all the 
 stakes put in from a will be fixed by similar tangential 
 deflections. (Prelim. Prop. 1.) 
 
 3 -
 
 26 RAILWAY CURVES AND LOCATION.. 
 
 When the point d is reached, the angle dab, shown 
 on the index, will be half the angle dbe, or its equal 
 a c d, at the centre. Move the instrument to d, sight 
 back to a, and turn to double the index angle. The 
 telescope is now directed along the tangent b d g, and 
 the angle dbe = acd = dab-\-adb, reads on the 
 index. Note this angle in the column of tangents oppo- 
 site station d. Continue the curve from this new posi- 
 tion, precisely as was done at a, and set the point h. 
 Move to A, see that the vernier has not been disturbed, 
 and sight back to d. The index now shows the angle 
 (db e -\- k d g\ and the object is to turn the angle d hf, 
 i. e. repeat the angle fd h, as was done before at d, and, 
 at the same time, have the whole angle (d b e -\- hfg) 
 indicated on the plate. To effect this, merely add this 
 angle fhd to the present reading. It will be found sim- 
 pler, in practice, to double the entire angle thus far turned, 
 and subtract from the product the last tangent, viz. db e. 
 The vernier, turned to this resultant angle, will put the 
 telescope in tangent line to h. And so on. 
 
 Example. " At sta. 24 -f- 50 commence a 4 curve to 
 the left for 35 12'." Suppose this a required duty. First, 
 reducing minutes to hundredths, we have 35-20, which, 
 divided by 4, gives 880 feet for the length of the curve. 
 Adding 880 to 24 + 50 it is at once seen that sta. 33 -f 30 
 is the P. T. 
 
 Let a be the P. C., = 24 + 50. Now the deflexion 
 angle being 4, the tangential angle is 2, with a chord of 
 100 feet. With a chord of 50 feet, therefore, the tan- 
 gential angle is 1, and this deflexion from tangent m a e 
 fixes station 25. A deflexion from this latter point of 2, 
 the chord being 100 feet, fixes station 26. And so on. 
 
 When you have fixed the point d, = sta. 28, the index 
 reads 7. Move up to station 28, sight back to the P. C., 
 and turn the index to 14. This throws you on tangent
 
 TO TRACE A CURVE. 27 
 
 Proceed as before, with the 2 deflexions, to sta. 31, = A. 
 Move up, and sight back to sta. 28. The index now reads 
 20. Multiplying by 2, and subtracting the last tangent, 
 we have the reading of the tangent at h = 26 : we have 
 turned 26 of the curve. Continue as before. After 
 putting in sta. 33, to find the deflexion which shall fix the 
 P. T., 33 + 30, say, as 100 feet : 30 feet : : 2 : the re- 
 quired deflexion, = 36'. We may here remark the great 
 convenience of an instrument graduated to hundredths of* 
 a degree instead of sixtieths. In the present example it 
 would be seen immediately that the tangential angle for 
 100 feet being 2, for 1 foot it would be 2 hundredths of 
 a degree, and for 30 feet it would be 60 hundredths. 
 
 Well ! when the P. T., = 33 + 30, is fixed, the index 
 reads 30 36'. Move up, see that the vernier has not been 
 disturbed, and sight back to sta. 31. Now twice the index 
 reading, minus the last tangent, = 61 12' 26, = 
 35 12^ the present tangent, which is the final tangent, 
 which finishes the curve. 
 
 The advantage of this manner of running a curve is that 
 the instrument shows at a glance the work done, and there- 
 fore errors may be detected with greater facility. By 
 comparing at the P. T. the total index angle with the 
 distance run, the work is tested at once.
 
 } RAILWAY CURVES AND LOCATION. 
 
 The above is recorded in the field book as follows 
 
 CO CO CO CO 
 
 o 
 
 In running compound and reversed curves the operation 
 is quite as simple as the foregoing. A point is fixed at 
 the P. C. C., or P. R. C., and turning into tangent at 
 that point, the second curve is traced from this tangent, 
 without regard to what precedes. In reverse curving, it 
 is a good plan to adjust the index in such manner at the 
 P. R. C., that when we turn into tangent it will read 0.
 
 TO TRIANGULATE ON A CURVE. 
 
 29 
 
 This saves troublesome work, and it is advisable more/.ver 
 to show in the field-book the contained angle of each curve, 
 as well as the test of the two tangents with the magnetic 
 course. 
 
 ARTICLE VII. 
 
 TO TRIANGULATE ON A CURVE. 
 
 SET the transit at a, and, as usual, sight back, and turn 
 into tangent. Estimate the distance to the farther bank 
 do it liberally and make a deflexion around the curve, 
 corresponding to your estimated distance. Fix a point b 
 in this line. Measure any convenient angle, b a c, and set 
 
 the point c. Move to J, measure the base b <?, the angle 
 a b c, and, before lifting the instrument calculate the line 
 b a. If the angle turned from tangent to d exceeds 4, 
 and the distance is greater than 200 feet, the chord a d 
 must also be calculated, as per example, and the difference 
 between this and b a will be the distance b d to the point 
 d, in the curve, which can be fixed from b. 
 
 3*
 
 30 RAILWAY CURVES AND LOCATION. 
 
 Should J fall between d and a the operation is analogous. 
 
 Example. Let a be a point in a 6 curve. Having set 
 the transit, and turned into tangent, the distance to the 
 farther verge is estimated 400 feet. The tangential angle 
 for 100 feet is 3, and to fix d, 400 feet distant, is conse- 
 quently 12. Deflect this angle, fix a point in line, and 
 complete the triangulation, as previously illustrated in 
 Art. IV., p. 22. Suppose a b found equal to 472 feet. 
 Now the tangential angle to d = half the central angle, 
 = fea t = 12; and to find the length of the chord a d, 
 we have, in the triangle efa, 
 
 Kad. : Sin. a ef : : e a : a/, that is 
 
 Had. of 1 : Nat. Sin. 12 : : 955-4, the Rad. of the 6 
 curve : half the chord required. Wherefore a d = twice 
 the Nat. Sin. 12 X 955-4 = -2079 X 2 X 955-4 == 
 397-25 feet. Subtracting this from 472, we have the 
 distance, 74-75 feet, back to the point in the curve. Move 
 the instrument to d, set the index at 12, sight back to a, 
 and turning to 24, the telescope is in tangent. A deflexion 
 of 3 will fix the next station. 
 
 NOTE. In this case, if preferred, a third proportional 
 might be formed with the chord of crossing, as shown in 
 the note to Art. IV.
 
 TO CHANGE THE ORIGIN OF A CURVE. 
 
 31 
 
 ARTICLE VIII. 
 
 TO CHANGE THE ORIGIN OF ANY CURVE, SO THAT IT SHALL 
 TERMINATE IN A TANGENT PARALLEL TO A GIVEN TAN- 
 GENT. 
 
 LET eZ/be the located curve, terminating in a tangent/^, 
 and the nature of the ground requires that it should ter- 
 minate in the tangent e i, parallel to / k. At /, the tele- 
 scope being directed along the tangent fk, turn to the 
 
 right an angle equal to the central angle d bf, previously 
 turned to the left on the curve. This will direct the tele- 
 scope along ef, parallel to d I Measure ef, and go back 
 on the tangent, d I, a distance, c d, equal to it. The curve, 
 retraced from c and consuming the same angle, -will termi- 
 nate tangentially in e i. An example in this case is not 
 necessary.
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE IX. 
 
 TO CHANGE A P. C. C. SO THAT THE SECOND CURVE SHALL 
 TERMINATE IN A TANGENT PARALLEL TO A GIVEN TAN- 
 GENT. 
 
 LET a b d be the compound curve, located and terminat- 
 ing in the tangent d h. Continue the larger curve to e, 
 and from e, with radius e I = kb, describe the curve ef, 
 terminating tangentially in f g, parallel to dh. From c, 
 the centre of the larger curve, let fall upon fg the perpen- 
 
 dicular eg, and fill up the figure as above. Call the radii 
 respectively R and r, the angle b k d, or its equal, k c m, 
 x, and the angle e If, or its equal, lcn,y. Let the dis- 
 tance if, or h <7, be named D. Now the line c g is made 
 up of the lines cm -{- mh -\- hg, i. e., eg = cosin. x 
 ( R r) -f- r -f- D. c g is also made up of the lines c n 
 + ng, i. e., eg = cosin. y (R r) -f- r. Therefore cosin. 
 z(R r)-fr + D = cosin. y (R r) + r, an d reducing,
 
 TO CHANGE A P. C. C. 33 
 
 cosin. x (R r) -4- D 
 cosm. y = -- ^5 - - - ; so that the distance if. 
 
 - (-CV - T) 
 
 or Jig, measured rectangularly between the two tangents, 
 being added to the nat. cosin. x, will give the nat. cosin. of 
 the angle e If, to be turned on the smaller curve. The 
 angle y, subtracted from the angle x, gives of course the 
 angle b c e, to be advanced on the larger curve ; or, divid- 
 ing this angle by the degree of curvature of a b, we find 
 the distance from b to e the P. C. C. proper. 
 
 If ef be the second curve located, and the tangent to be 
 touched lies within, it is evident that we must retreat upon 
 the large curve, and, by subtracting D from the cosine of 
 the angle y, we obtain the cosine of the angle x. 
 
 Example. Suppose a b a 3 curve located, and com- 
 pounding, at b, into a 6 curve, which latter is continued 
 to the right through an angle of 42. At the P. T. we 
 discover that the proper tangent is 64 feet to the left. 
 We must throw our curve out, then we must advance on 
 the 3 curve a certain distance. How to find this distance : 
 The radius of a 3 curve = 1910 ; the radius of a 6 curve 
 = 955-4; R r, therefore, = 954-6. The nat. cosin. 
 42 = '7431. Now, by the formula just obtained, we 
 cos, x (R r} + D (-7431 X 954-6) + 64 _ 
 ~-~ ~~ 
 
 8101 = nat. cosin. 35 53'. Subtracting this from 42, 
 we have 6 07', the angle to be advanced on the 3 curve ; 
 or, reducing minutes to hundredths, and dividing by 3, we 
 find 204 feet, the distance from b to the correct P. C. C:
 
 34 
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE X. 
 
 SHOULD THE SECOND CURVE BE ONE OF LONGER RADIUS 
 THAN THE FIRST, 
 
 OUR illustration takes simpler form, and the application 
 of D varies vic versa. 
 
 See figure, analogous to that of the previous problem. 
 Here of, eg are equal and parallel radii; fh a perpen- 
 dicular connecting them. Draw its fellow, en. Then, 
 nf = c h, and, consequently, h g = 6 n. Again, letting 
 k m fall, perpendicular to c g, we have c m = n I, and o I 
 = c m -f- o n ; i. e., cosin. y (R r) = cosin. x (R r) 
 + D. 
 
 We observe that, with a curve of this nature, in order 
 to throw the line farther out, it is necessary to go back, 
 toward b ; or, having located to g, if the object tangent lie 
 within, we must advance toward e. 
 
 Example. Suppose a b a 5 curve, b the P. C. C,, and 
 b g a 2 curve. Setting the instrument at g, the P. T., 
 and turning into tangent, we find that we are a distance 
 hg, = 53 feet, too far to the left. The first question is, 
 what angle have we turned on the second curve. Let it
 
 TO CHANGE A P. C. C. 35 
 
 be 28. Now we know, that, in order to strike farther to 
 the right, we must advance on the 5 curve. Consequently, 
 I) must be added to the cosine of 28, to give us the cosine 
 of the proper angle for the 2 curve ; and the difference 
 between 28 and this newly found angle will be the angle 
 we are to advance on the 5 curve. Thus : the rad. of a 
 5 curve = 1146 feet, that of a 2 curve = 2865 feet, 
 and their difference = 1719 feet. The nat. cos. of 28 
 = -8829. Then ('8829 X 1719) +jg = . 9m| = nat 
 
 cos. 23 58*. This, subtracted from 28, leaves 4 02', = 
 80 feet, from b to the correct P. C. C. 
 
 Synopsis of the preceding formula. 
 
 Call D the distance between tangents as before, a the 
 angle of the second curve located, and b the angle of the 
 same curve to be substituted for it. 
 
 FIRST, when the second curve has the smaller radius 
 Tangent falling within the point, cosine b = 
 cos. a (R r) -f- D 
 
 (R-r)~ 
 
 Tangent falling without the point, cosine b = 
 cos. a (R r) D 
 
 (*-') 
 
 SECOND, when the second curve has the larger radius 
 Tangent falling within the point, cosine b = 
 cos. a (R r) D 
 
 (R-r) ' 
 
 Tangent falling without the point, cosine b = 
 cos. a (R r) + D 
 
 Very little attention will familiarize these formulae, and 
 render the field practice easy.
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE XL 
 
 HAVING LOCATED THE COMPOUND CURVE a I d, TERMINAT- 
 ING IN THE TANGENT df, IT IS REQUIRED TO FIND THE 
 P. C. C. 6, AT WHICH TO COMMENCE ANOTHER CURVE OF 
 GIVEN RADIUS, WHICH SHALL ALSO TERMINATE TAN- 
 GENTIALLY IN df. 
 
 PLOT the curves as per figure. From c let fall c g, per- 
 pendicular to the tangent, df. From k and z, the lesser 
 centres, drop k m, i ?, perpendicular to c g. Call the great 
 radius B, the smaller radius r, and the intended radius of 
 
 the second curve /. Likewise name h k d, the angle of 
 the small curve located, #, and lie the angle to be found 
 for the proposed curve, y. Now, the tangent df, and the 
 curve a b, lying unstirred, the line c g is an unvarying dis- 
 tance, and it is made up of the lines c m -f mg, i. e., eg 
 = (B r) cosin. x + r. It also consists of the lines c I
 
 COMPOUND CURVES 37 
 
 -f- Iff, i. e., off = (R r'} cosin. y + /, and, reducing, 
 
 (R r) cosin. x -4- r / 
 
 nat. cosin. y = '^ -^ . This, there- 
 
 (K, r ) 
 
 fore, is the formula by means of which we can ascertain 
 the point 5, as follows : 
 
 Example. Imagine a 2 curve, a h, compounding into 
 a 6 curve, h d, which terminates at d, in the tangent df. 
 ^The tangent lies well ; the curve a h likewise ; but it is 
 desired to throw the line to the left, on better ground, 
 between d and h, by means of an intercalary 4 curve. 
 We wish, then, to know the distance, h 6, back to the new 
 P. C. C. 
 
 The radius of a 2 curve = 2865 feet, of a 6 curve = 
 955-4 feet, and their difference (R r) = 1909-6. The 
 radius of a 4 curve = 1433, and the difference (R r') 
 is, therefore, 1432 feet. Let h k d, the angle turned on 
 the 6 curve, be 41, the nat. cos. of which = -7547. 
 
 T , (-7547 X 1909-6) + 955-4 1433 RYOft 
 
 Inen, - ^32 * bTJ > = 
 
 nat. cosin. 47 43'. Subtracting 41, we have 6 43', the 
 angle h c 6. Reducing minutes to hundredths, and divid- 
 ing by 2, we find 336 feet to be the distance from h to b. 
 A 4 curve of 47 43', traced from this latter point, will 
 terminate in the tangent ef.
 
 38 
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE XII. 
 
 IF THE LATTER CURVES HAVE LARGER RADII THAN THE 
 FIRST, 
 
 THE solution retains its shape and simplicity. 
 
 Draw the figure as above, and, for the sake of uniformity, 
 name the radii as before. The curve a 5, and tangent df, 
 being constant, the distance m g, or d n, is here constant 
 Call it A. Now A, in the first place, is equal to d i n i 
 i. e., =; r (r -r R) cosin. x\ and, in the second place 
 
 it is equal to g c m c, = i 1 (/ R) cosin. y ; where- 
 fore r (r R) cosin. x = / (/ R) cosin. y, and 
 
 consequently cosin. y 
 
 _(r R) cosin. x + / r 
 
 (/-R) 
 
 Example. Suppose a b a 7 C curve, compounding, at b, 
 into a 5 curve, 5 d, which latter subtends an angle of 38, 
 and terminates in the tangent df. We wish to substitute 
 a terminal 2 curve, hg, and to know the position, h, of 
 the new P. C, C. 
 
 The radius of a 7 curve = 819 feet == R. r and r',
 
 TO CHANGE A P. R. C. 
 
 39 
 
 the radii respectively of 5 and 2 curves, are equal to 
 1146, and 2865 feet, r R, therefore, = 327, and 
 / R = 2046 feet. The nat. cosin. of 38 = -788. 
 
 mr, k ' A ' * 1 C 788 X 327 ) + 2865 1146 
 
 Then, by the formula, * 2fi46 
 
 9661, = the nat. cosin. of 14 57', the angle to be turned 
 on the 2 curve. Subtracting this from 38, we have 23 
 03', the angle to be continued on the 7 curve. Reducing 
 minutes to hundredths, and dividing by the degree of curva- 
 ture, 7, we find 329 feet, the distance from b to the new 
 P. C. C. h. 
 
 ARTICLE XIII. 
 
 TO CHANGE A P. 11. C., SO THAT THE SECOND CURVE SHALL 
 TERMINATE IN A TANGENT PARALLEL TO A GIVEN TAN- 
 GENT. 
 
 LET a d I be the reverse curve, located and terminating 
 
 1 K 
 
 in the tangent I L Call the radius c 5, R, and the radius 
 6 e, r. Suppose ig the given tangent. At a distance
 
 40 RAILWAY CURVES AND LOCATION. 
 
 from it equal to * e, the radius of the second curve, draw 
 the parallel line, op. With c as a centre, and radius cf, 
 == R + r, describe the integral curve, fe, cutting op in e. 
 e, then, is the centre of the curve adjusted. 
 
 Application. 
 
 Place the transit at the P. T., ?, and turn into a tan- 
 gent, Im, parallel to dn, the common tangent of the two 
 curves at d. Unless some wide mistake has been made, 
 the distance Ik, measured along this line to ig, the tan- 
 gent proper, will be about equal to the distance ef, and 
 we shall have the proportion, cfife : : c d : db, i. e., R 
 
 -f- r : ef : : R : d 5, which gives -^. , as a simple 
 
 XV J~ 7* 
 
 formula for finding the distance back from d to 5, the cor- 
 rect P. R. C. This rule, though sufficiently true for most 
 cases, is not mathematically justifiable. It will be seen 
 that ef, or its equal i I, the distance we wish to measure, 
 is a curving distance, part of the circumference of a circle 
 concentric with a b. Its radius is (R + r), therefore its 
 
 5730 
 degree of curvature = , ., or, more simply, equals 
 
 the product of the degrees of curvature of the curves com- 
 posing the reverse, divided by their sum. To be strictly 
 accurate, then, set the instrument at I, turn into tangent 
 I m as before, and trace the curve i I, until it strikes the 
 tangent ig. The angle which il subtends, being divided by 
 the degree of curvature of a b, will give the distance, d b, 
 to the P. R. C. proper. The curve retraced from b, will 
 terminate tangentially in ig, and its angle, b e i, will be 
 equal to dfl deb. 
 
 Example. Let a d I be a reverse curve, composed of a 
 3 curve, a d, and a 6 curve, dL Let the angle dfl be 
 equal to 52, and suppose the distance If to have been
 
 WHEN P. C. IS INACCESSIBLE. 
 
 41 
 
 found 34 feet. Being part of a 2 curve, it therefore sub- 
 tends a central angle of 41'. This corresponds to a dis- 
 tance of 23 feet, to be gone back on the 3 curve, and 
 52-00 41' = 51 19', the angle to be turned from h 
 on the 6 curve, in order to strike the tangent ig. 
 
 ARTICLE XIV. 
 
 HOW TO PROCEED WHEN THE P. C. IS INACCESSIBLE. 
 
 IN the figure, drawn to illustrate this case, let c be the 
 point of curvature, c a the tangent, and c k e the curve. 
 Now the angle dee, included between the tangent and 
 
 any chord, as c e, fixing the point e, is known. Make c b 
 along tangent, equal to c e, and connect be. If a circle 
 were now described from c .as a centre, with radius e e or 
 c b, d, e, and b, would be points in its circumference, and
 
 42 RAILWAY CURVES AND LOCATION. 
 
 the angle d b e at once proven equal to half the angle dee. 
 With proof precisely similar, d c e = half of d g e, and, 
 consequently, d b e is equal to one-fourth of the central 
 angle subtended by the chord c e. 
 
 Example. Suppose c to be the inaccessible point of 
 curvature of a 6 curve, eke. It is concluded to run to 
 the third station, e. First we must calculate the length 
 of the chord c e. The angle d c e = 9, and from Art. 
 VII. we have 
 
 Had. of 1 : 9554 : : nat. sin. 9 = -1564 : e -, whereby 
 
 m 
 
 ec is shown equal to 298-8. Place the transit then at 5, 
 298-8 feet distant from the P. C., and deflect to the left 
 an angle of 4 30', equal to half the angle dee. This is 
 in line to e, and b e must likewise be calculated as follows : 
 
 In the triangle b c h we have 
 
 Bad. of 1 : nat. cosin. 4 30' == -9969 : : b c = 298-8 : b h 
 
 be 
 -^-, whereby b e is shown equal to 595-7 feet. Arriving 
 
 28 
 
 at e, the index reads 4 30'. Sight back to 5, turn to 18, 
 and the telescope will be in tangent. Suppose, however, 
 that having reached /, 100 feet from e, this latter point is 
 also found inaccessible. We find k a different point in the 
 curve, thus: The angle/e^ = 18 4 30' = 13 30', 
 and the tangential .angle g e k = 3. Consequently the 
 angle fek = 10 30', and, drawing the bisecting line e z, 
 we have, in the triangle efi, 
 
 Had. of 1 : nat. sin. 5 15' = -0915 : : */ = 100 : / 
 = 9-159 feet. Therefore fk = 18-318 feet, and the 
 angle efk = 90 5 15' = 84 45'. At / deflect this 
 angle to the right, and measure the distance fk carefully 
 with the rod. At k, sighting back to /, and turning the 
 equal angle fk e, the telescope will be directed to e, and 
 the curve may be continued. 
 
 If it is inconvenient to run the line b e, the point e may
 
 TO AVOID OBSTACLES IN THE LINE OF CURVE. 43 
 
 be reached thus: Fix the P. C. Find the tangential 
 distance d e, corresponding to the angle dee. Carefully 
 with the rod lay off b Z, equal to it, at right angles to b e 
 Set the transit at I, and, in line with c, put in e. 
 The distance b I should not exceed 10 or 12 feet. 
 
 NOTE. The foregoing illustrations will apply when the 
 P. T. is likewise inaccessible. 
 
 ARTICLE XV. 
 
 TO AVOID OBSTACLES IN THE LINK OF CURVE. 
 
 LET b k h be the curve. We can either follow the tan- 
 gents hd, db, or trace a parallel curve, g a, within the 
 first ; which tracing is effected thus : Set the instrument 
 
 at A, the P. C., and offset any distance h g, at right angles 
 to the tangent h d. It will be observed that as the dis- 
 tance h g increases, the distance g a decreases, whilst the 
 angle subtended by g a remains equal to that subtended
 
 44 RAILWAY CURVES AND LOCATION. 
 
 by h b ; i. e., our deflexions on the offset curve stand 
 unchanged, but the corresponding chords, g /, f e, &c., 
 are less than their equivalents, h i, i k, &c., along h b. 
 To find their length, h z, i k, &c., being equal to 100 
 feet, we have the proportion, c h : eg : : h i : x ; i. e., 
 K : rad. h g : : 100 feet : x, where x symbols the un- 
 known chord. Now set the transit at g, turn into tangent 
 parallel to h d, and with the shortened chord, fix/ e a. 
 Rectangularly to the tangents at these points, and distant 
 h g, will be , k, b of the curve proper. 
 
 Example. Let 5 h be a 4 curve, and the offset distance 
 85 feet. The radius then is 1433, and 
 
 1433 : 1348 : : 100 : 94, the short chord. 
 
 To follow the tangents, suppose the angle b c h = 42. 
 Then by Art. V. we find the tangent h d = 550 feet, 
 which distance we duly measure, and, at d, deflecting 42, 
 lay off an equal distance to 5, the point of tangency.
 
 fIND THE RADII OP REVERSE CURVE. 
 
 45 
 
 ARTICLE XVI. 
 
 HAVING GIVEN THE ANGLES d b k, m k I, AND THE DISTANCE 
 b k, IT IS REQUIRED TO FIND THE RADII C e, ef OF THE 
 EASIEST REVERSE CURVE WHICH SHALL UNITE ad, km. 
 
 THE angle d b e is equal to the angle ace, half of which 
 is b o e. So likewise ef k is equal to half of I km. 
 
 Then, [nat. tang, b c e + nat. tang, eflc] : nat. tang. 
 o c e : : b k : b e, and b k b e = e k. Wherefore rad. c e 
 
 be ek 
 
 = , and rad. ef = ,-7-. 
 
 nat. tang, bee nat. tang, kj e 
 
 Example. Suppose the angle d b e = 54 30', the 
 angle Ikg = 33 20', and the distance b k = 832 feet. 
 
 Therefore the angle bee = 27 15', the nat. tang, of 
 which is -5150, and the angle efk = 16 40', the nat. 
 tang, of which is -2994. The sum of the tangents = -8144. 
 Then, to find b e, we have 
 
 As -8144 : -5150 : : 832 : 526, and subtracting this from 
 b k, we have e k = 306 feet. 
 
 526 306 
 
 Again, the radius c e = TFTH? and the radius ef -2994' 
 
 = 1022 feet.