IB^^K ^ 
 
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 REESE LIBRARY 
 
 
 UNIVERSITY OF CALIFORNIA. 
 Deceived MAR ...1.5 1893 , /^ . 
 'Otr, Class No. 
 
PRACTICAL TREATISE 
 
 RAILWAY CURVES 
 
 LOCATION, 
 
 FOR YOUNG ENGINEERS 
 
 ^OBTAINING A FULL DESCRIPTION OP THE INSTRUMENTS, THE MANNER OP ADJUSTING THEM, AN* 
 
 tHE METHODS OF PROCEEDING IN THE FIELD, NEW AND SIMPLE FORMULA FOR <M- 
 
 POUND AND REVER8B CURVING, RULES FOR CALCULATING EXCAVATION A>T> 
 
 EMBANKMENT, STAKING OUT WORK, &C., TOGETHER WITH TABLES OF 
 
 NATURAL SINES AND TANGENTS, RADII, CHORDS, ORDINATES, 
 
 AMD OTHERS OF GENERAL USE IN THE PROFESSION. 
 
 BY 
 
 WILLIAM F. SHUNK, 
 UNIVERSITY 
 
 CIVIL ENGINEER. 
 
 PHILADELPHIA: 
 HENRY CAREY BAIRD & CO., 
 
 INDUSTRIAL PUBLISHERS, BOOKSELLERS AND IMPORTERS, 
 810 WALNUT STREET. 
 
Entered, according to the Act of Congress, in the year 1854, by 
 E. H. BUfLER ft CO., 
 
 In the Clerk's Office of the District Court of the United States, in and for the Extern 
 District of Pennsylvania. 
 
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 
 elope instrument. By these means a working map can be 
 constructed in the office upon which the proposed location, 
 grade lines, &c., may be traced with tolerable resemblance 
 to 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 
 
 (3) 
 
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 is 
 comprehensive withal, it is the work intended. 
 
 W. F. SHUNK. 
 
CONTENTS. 
 
 1'A.GK 
 
 PREFACE ........ 3 
 
 Explanations ........ 7 
 
 ARTICLE I. Of the Instruments. The Level . . . It 
 
 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 .... 2? 
 
 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 ihe 
 
 second curve has the smaller radius . . . 52 
 
 X. Second. When the last curve has the larger radius . 34 
 
 Synopsis of formulae for compound curving . . 35 
 XL 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 ....... 3C 
 
 XII. Second. When the latter curves have the larger radii . 38 
 
 1* (5) 
 
VI CONTENTS. 
 
 PA.GJ* 
 ART. XIII. To change a P. R. C. so that the second curve shall .ermi- 
 
 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 givon a located curve, terminating in any given 
 tangent, to find where a curve of different radius will ter- 
 minate in a parallel tangent . . . 46 
 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 . . . .61 
 
 XXL Ofordinates 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 ...... f>3 
 
 Radii ..... . 89 
 
 Long Chords ... . 90 
 
 Ordinates ..... 91 
 
 Squares and Square Roots . 94 
 
 Slopes and Distances for Topography . . , 108 
 
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) 
 
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. II. 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. 
 
 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 is 
 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 
 
 ON 
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE I. 
 
 OF THE INSTRUMENTS. 
 THE LEVEL. 
 
 THE level is an instrument used in ascertaining the 
 undulations of the ground 
 along the line of a survey, 
 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 
 6, the Y's c c, the rectan- 
 gular bar o d, the axis e, the plates and levelling screws 
 / m, and the tripod g. 
 
 In the tube of the telescope at /*, 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 isjield jjrposition by four 
 
 UNIVERSITY 
 
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 s, 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 Level. 
 
 First. To make the line of collimation coincide with the 
 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, i. 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 <?, ,/. The hori- 
 zontal line af is called the datum line. This is arbitrarily 
 
 assumed. It may be considered, for example, at any dis 
 tance above the point #, 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 datum. In* 
 land, the summer surface of the nearest stream, or, when 
 commencing on a ridge, the highest neighbouring knoll is 
 issumed. 
 
 Well ! suppose a to be zero, and the instrument, for 
 nstance, set and levelled at b. Stand the rod at , and 
 nilide 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 c above a, take the rod thither and lower tho 
 target until coincidence results as before. If the rod 
 reads 2 feet, of course c 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 -j- t>, 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 -{- 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 th<* 
 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-bock 
 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 
 
 
 + 3-00 
 
 i . . 
 
 
 
 " 
 
OF THE TRANSIT. 17 
 
 The advantages of this method of levelling over the 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 d, the compass- 
 box, &c., e g, and the axis 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 h, 
 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 7 . 
 
 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 1 J 
 
 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 b, at any convenient distance. Reverse the telescope 
 on its axis, and fix another point in the opposite direction, 
 
 ..... 
 
 of 
 
 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 5, 
 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 b, and the adjustment is complete. 
 
20 
 
 RAILWAY CURVES AND LOCATION. 
 
 After finishing this adjustment, the telescope may still 
 net revolve truly in the meridian. This inaccuracy there 
 is no method of removing in the field. It should be sent 
 tc 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 / 6, 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 fa b at the circumference. 
 . 3. The exterior angle dbf, formed at the circumference 
 by the two equal chords a 5, b /, is called a deflexion angle, 
 and is equal to the central angle b of, or double the tan- 
 gential angle e bf. df is called the deflexion distance, 
 and e /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 <?, and placed the instrument at that point, the angle 
 he d must be made equal to twice d be, and the distance 
 
 c d equal to the distance b c. Still another deflection at 
 d, 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 dbc exceeds 4, and the distance be is 
 greater than 200 feet, or even with an angle of 2|, 
 should the distance be greater than 300 feet, be will 
 differ sensibly in length from b k, and a calculation of the 
 latter becomes necessary. To effect this, multiply the 
 natural cosine of the angle k b c by b c. This result 
 doubled will give b k d, the length proper along tangent. 
 
 Thus : Suppose k b c, 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 b k. 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 
 
 AND it is desirable on any account not to run the line 
 from d y 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 c a d, 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 d a. The object now is 
 to find the point c and the angle dca. 
 The angle a d e subtracted from 180 will supply the 
 
AVOIDING OBSTACLES. 23 
 
 angle c d a, 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, dea = 42, 
 and the base d e = 200 feet. Then in the triangle dea 
 we have 
 
 Nat. sine d a e = (-5736) : nat. sine d e a = (-6691) : : 
 d e = (200) : d , 
 
 f)691 v 200 
 Wherefore, d a = &>-- = 233-3 feet. 
 
 *O ( o\) 
 
 Again, the angle cda = 77. The angle d a c being 
 6,acd is consequently = 97, and in the small triangle 
 we have, Nat. sin. a c d (-9925) : nat. sin. c a d = 
 (1045):: da = (233-3): c d. 
 
 1045 X 233-3 
 Therefore c d = - 79925 = 24-564 feet, and d cf 
 
 180 97 = 83. 
 
 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 <?, sight to c?, 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 Id = ~r 
 
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 
 <HVEN RADIUS. 
 
 Draw the bisecting line d c. Then the angle d c i = 
 
 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. 
 do a : Rad. ca : ad. There- 
 fore ad = Nat. Tang, d c a 
 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 *4452 ; and multiplying by Rad. 
 1460, we have 650 feet for the length of a d or d bj the 
 tangents. 
 
 2. If a die given and radius required, 
 
 ad 650 
 
 Kad. = c^ T A A /--> 1460. 
 
 Nat. Tang, a c d -4452 
 
 The following rule? 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 a c b (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. t 
 
 THE legree 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 1 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 h, see that the vernier has not been disturbed, 
 and sight back to d. The index now shows the angle 
 (d b e -j- h d </), and the object is to turn the angle d hf, 
 i. o. repeat the angle fd A, as was done before at d, and, 
 at the same time, have the whole angle (d b e -f- hfy) 
 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. d b e. 
 The vernier, turned to this resultant angle, will put the 
 telescope in tangent line to h. And so on. 
 
 ^Example. u 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 -j- 50 it is at once seen that sta. 33 + 30 
 is the P. T 
 
 Let a be the P. C., = 24 -f 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, = h. 
 Move up, and sight back to sta. 28. The index now reada 
 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 hundredth 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 hundredth* of 
 a degree, and for 30 feet it would be 60 hundredths. 
 
 Well ! when the P. T., = 33 -f 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. 
 
28 RAILWAY CURVES AND LOCATION. 
 
 The above is recorded in the field book as follows 
 
 6 
 
 03 . 
 
 J3 O 
 o> -M o 
 
 3 S oi2 
 
 S 
 
 g 
 
 o S 
 
 ctt co 
 
 r- I <M <M CO CO 
 
 O O O O O O O 
 O O O O O O CO 
 
 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. 0., 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 moreover 
 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 sut 
 
 the point c. Move to 6, measure the base b <?, the angle 
 a b c y 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 LOCAUON- 
 
 Should b 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, = 12; and to find the length of the chord a d y 
 we have, in the triangle efa, 
 
 Rad. : Sin. a ef : : ea : af, that is 
 
 Rad. of 1 : Nat. Sin. 12 : : 9554, 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 df/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 fk. 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 y parallel to d I. Measure <?/, 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. 
 
82 
 
 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 = k 6, describe the curve ?/, 
 terminating tangentially in / g, parallel to d h. 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 e m, 
 x, and the angle e If, or its equal, lcn,y. Let the dis- 
 tance if, or h g, be named D. Now the line c g is made 
 up of the lines c m -f- m h -{- kg, 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) + r. Therefore cosin. 
 x (R r ) _|_ r -f_ D = cosin. y (R r) + r, and reducing, 
 
TO CHANGE A ? P. C. C. 33 
 
 cosin. x (R r) 4- D 
 cosm. y == T^ ; so that the distance z/, 
 
 or Jig, measured rectangularly between the two tangents, 
 being added to the nat. cosin. x 9 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) 4- 64 
 
 - i '_ - 
 
 , -j \ r\r * rt ' ~~~- 
 
 (R r) 9o4-6 
 
 8101 = nat. cosin. 35 53'. Subtracting this from 42*, 
 we have 6 07', the angle to be advanced on the 8 curve ; 
 or, reducing minutes to hundredths, and dividing by 3, we 
 find 204 feet, the distance from b to the correct P. C. G 
 
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 vice 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, c n. Then, 
 nf = c h, and, consequently, h g = o n. Again, letting 
 k m fall, perpendicular to eg, we have c m = n I, and o I 
 = c m -j- o n', i. e., cosin. y (R r) = cosin. x (R r) 
 -f-D- 
 
 We observe that, with a curve of this nature, in order 
 tc throw the line farther out, it is necessary to go back, 
 toward b ; or, having located to y, if the object tangent lie 
 within, we must advance toward e. 
 
 Example. Suppose a b a 5 curve, b the P. C. C., and 
 bg a 2 curve. Setting the instrument at g, the P. T., 
 and turning into tangent, we find that we are a distance 
 Jig, = 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. 0. 35 
 
 be 28. Now we know, that, in order to strike farther to 
 the right, we must advance on the 5 curve. Consequently, 
 D 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 =. 9137, = 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 formulce. 
 
 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) J-J) 
 
 (R-rj" 
 
 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) P 
 (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. 
 
36 
 
 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE XL 
 
 HAVING LOCATED THE COMPOUND CURVE a b d, TERMINAT- 
 ING IN THE TANGENT df, IT IS REQUIRED TO FIND THE 
 P. C. C. 5, 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 , the lesser 
 centres, drop Jc m, i Z, perpendicular to c g. Call the great 
 radius R, the smaller radius r, and the intended radius of 
 
 the second curve /. Likewise name h k d, the angle of 
 the small curve located, x, and b i e 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 m g, i. e., c g 
 = (R r) cosin. x + r. It also consists of the lines c I 
 
COMPOUND CURVES 37 
 
 -f- lg, i. e., eg = (R /) cosin. y -}- /, and, reducing, 
 
 (R r) cosin. x -f- r / 
 nat. cosin. y = ^ ,r . This, there 
 
 fore, is the formula by means of which we can ascertain 
 the point #, as follows : 
 
 ^Example. Imagine a 2 curve, a A, compounding into 
 a 6 curve, h d, which terminates at c?, 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 A, by means of an intercalary 4 curve. 
 We wish, then, to know the distance, h b, back to the new 
 P. C. C. 
 
 The radius of a 2 curve = 2865 feet, of a 6 curve =r 
 955-4 feet, and their difference (R r) = 1909-6. The 
 radius of a 4 curve = 1433, and the difference (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) + 9554 1433 
 
 Then, * 1432~ ~ = ' ^ 
 
 nat. cosin. 47 43'. Subtracting 41, we have 6 43', the 
 angle h c b. Reducing minutes to hundredths, and divid- 
 ing by 2, we find 336 feet to bo the distance from h to 6. 
 A 4 curve of 47 43', traced from this latter point, will 
 terminate in the tangent ej. 
 
RAILWAY CURVES AND LOCATION. 
 
 ARTICLE XII. 
 
 IF THE LATTER CURVES HAVE LARGER RADII THAN THE 
 
 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 6, and tangent df, 
 being constant, the distance m g, or d w, is here constant 
 Call it A. Now A, in the first place, is equal to d i n i, 
 i. e., = r (r R) cosin. x\ and, in the second place, 
 
 it is equal to g c m <?, = / (/ R) cosin. y ; where 
 fore r (r R) cosin. x = / (/ R) cosin. ^, and 
 
 (r R) cosin. x -}- / r 
 consequently cosin. y = - > ' , , __ -p \ 
 
 Example. Suppose a b a T c curve, compounding, at , 
 into a 5 curve, b d, which latter subtends an angle of 38, 
 and terminates in the tangent df. We wish to substitute 
 a terminal 2 curve, Jig, and to know the position, A, 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. 
 
 mi k ^ r i ('788 X 327) + 2865 1146 
 Then, by the formula, * on/ia - ~ == 
 
 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. R. 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 
 
 in the tangent I h. Call the radius c J, R, and the radius 
 b e, r. Suppose ig the given tangent. At a distance 
 
40 RAILWAY CURVES AND LOCATION. 
 
 from it equal to i e, the radius of the second curve, draw 
 the parallel line, op. With c as a centre, and radius of, 
 = R -j- r, describe the integral curve, /e, cutting op in e. 
 d then, is the centre of the curve adjusted. 
 
 Application. 
 
 Place the transit at the P. T., Z, and turn into a tan- 
 gent, Im, parallel to d n, the common tangent of the two 
 curves at d. Unless some wide mistake has been made, 
 the distance Z&, measured along this line to ig, the tan- 
 gent proper, will be about equal to the distance ef, and 
 we shall have the proportion, cf :fe : : c d : db, i. e., R 
 
 + r : ef : : R : d b, which gives -^5 , as a simple 
 
 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 -f- 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 Z, turn into tangent 
 I m as before, and trace the curve i Z, until it strikes the 
 tangent ig. The angle which il subtends, being divided by 
 the degree of curvature of a 6, will give the distance, d 6, 
 to the P. R. C. proper. The curve retraced from 6, will 
 terminate tangentially in ig, and its angle, bei, will be 
 equal to df I deb. 
 
 Example. Let a d I be a reverse curve, composed of a 
 3 curve, a d, and a 6 curve, d 1. Let the angle df I 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 eke the curve. 
 Now the angle d c e, 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 c e or 
 c b, d, e, and 5, would be points injiajurcjimference, and 
 
42 RAILWAY CURVES AND LOCATION. 
 
 the angle dbe at once proven equal to half the angle dot. 
 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 
 
 Rad. of 1 : 9554 : : nat. sin. 9 = -1564 : ^, whereby 
 
 e G is shown equal to 298-8. Place the transit then at 6, 
 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 
 
 Rad. of 1 : nat. cosin. 4 30' = -9969 : : b c = 298-8 : b h 
 
 b e 
 , whereby b e is shown equal to 595-7 feet. Arriving 
 
 at e, the index reads 4 30'. Sight back to 6, 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 feg = 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 i, 
 we have, in the triangle efi, 
 
 Rad. of 1 : nat. sin. 5 15' = -0915 : : ef = 100 : fi 
 = 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 
 *vith the rod. At k, sighting back to /, and turning the 
 equal angle fke, 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 I, equal to it, at right angles to b e 
 Set the transit at Z, 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 LINE 0"F CURVE. 
 
 LET b Jc h be the curve. We can either follow the tan- 
 gents h d, 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 Jig, 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 9 i k, &c., along h b. 
 To find their length, h i, i k, &c., being equal to 100 
 feet, we have the proportion, c h : eg :: hi: x; i. e., 
 R, : 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 fea. 
 Rectangularly to the tangents at these points, and distant 
 h g, will be i, k, b of the curve proper. 
 
 Example. Let b 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 oft an equal distance to b, the point of tangency. 
 
FIND THE RADII OF REVERSE CURVE. 
 
 45 
 
 ARTICLE XVI. 
 
 HAVING GIVEN THE ANGLES d b k, mkl, AND THE DISTANCE 
 b &, IT IS REQUIRED TO FIND THE RADII C 6, ef OF THE 
 EASIEST REVERSE CURVE WHICH SHALL UNITE a d, ~k 
 
 THE angle d b e is equal to the angle a c e, half of which 
 is b c e. So likewise ef k is equal to half of I km. 
 
 Then, [nat. tang, b c e -f nat. tang, efk] : nat. tang, 
 
 b c e : : b k : b e, and b k b e = e k. Wherefore rad. c e 
 
 be ek 
 
 = v and rad. ef = 7 ?-. 
 
 nat. tang, bee nat. tang, kfe 
 
 Example. Suppose the angle d b e = 54 30', the 
 angle lkg = W 20', and the distance Ik = 832 feet. 
 
 Therefore the angle b c e = 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 = ^TH, and the radius ej = .2994? 
 
 = 1022 feet. 
 
46 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE XVII. 
 
 HAVING GIVEN THE CURVE /(/, LOCATED AND TERMINATING 
 IN THE TANGENT g m, IT IS REQUIRED TO FIND WHERE 
 A CURVE OF DIFFERENT RADIUS WILL TERMINATE IN A 
 TANGENT PARALLEL TO g m. 
 
 be the curve located, and fh the curve proposed, 
 commencing at the common point /, and terminating in 
 the parallel tangents, gm,hl. We wish to find the length 
 und direction of the line g h, connecting the points of tan- 
 
 Call the radii respectively R and r, and the central 
 angle, fdg orfch, x. Now g k = d e, = sine of x, c d 
 being radius ; i. e., = nat. sin. a; X (R r). Again, the 
 
 % 
 
 angle kg h = fh b = -, and g k = cosin. kg h, g h being 
 
 SY* 
 
 radius, i. e., g k = nat. cos. - X g h, and, consequently. 
 
 
 
 nat. cos. - X g h = nat. sin. x X (R r), wherefore 
 
 (R r) nat. sin. x 
 gh= . 
 
 nat. cos. <r 
 It will be observed that the angle leg h, included between 
 
TERMINATION OP A CURVE. 47 
 
 the tangent to a curve at any point g, and the line g h 
 connecting </ with an equivalent point h in any other curve 
 fh commencing at the same P. C.,/, and turning in the 
 same direction, is invariably equal to half the common 
 central angle, fdg orfch. 
 
 Example. Let fg be a 7 curve, subtending a central 
 angle, fdg, of 44 26'. Having arrived at g, the P. T., 
 and turned into tangent, g m, it is desired to fix the P. T., 
 A, of a 4 curve which shall likewise subtend an angle of 
 44 26'. Here the radii are, respectively, 819 and 1433 
 feet, and E, r = 614. The nat. sine of 44 26' = -700, 
 and the nat. cosine of half this angle, viz. : 22 13' = 
 
 614 y *7 
 9258. Then, by the formula, g h = --- = 464-2. 
 
 Deflecting, therefore, to the left, an angle of 22 13', and 
 laying off the distance 464-2 feet, we arrive at the point h. 
 Move to A, sight back to g, and a deflexion of 22 13' to 
 the right will direct the telescope along the tangent h L 
 
 If h were the P. T. located, and g the point required, 
 the same angle and distance would apply. 
 
48 RAILWAY CURVES AND LOCATION. 
 
 ARTICLE XVIII. 
 
 HAVING THE CURVE nfg LOCATED, AND TERMINATING IN 
 THE TANGENT g W, IT IS REQUIRED TO FIND THE POINT 
 /, WHEREAT TO COMPOUND WITH ANOTHER CURVE 01 
 GIVEN RADIUS, WHICH SHALL TERMINATE IN i I, PARAL- 
 LEL TO g m. 
 
 [See previous figure.] 
 
 NAME the radii and angle as before. Measure the dis- 
 tance g i, between the tangents, and call it D. Then c h 
 is equal to c e -j- e k -f- g i or k h ; i. e., R = (R r) 
 
 R_( r _D) 
 cosm. x -f- r -\- D, or, transposing, cosm. x = j^ \ 
 
 Thus discovering the angle f d g^ divide it by the curvature 
 nfg, and we have the distance, <//, to the P. C. C./. The 
 second curve, traced from this point, will terminate tan- 
 gentially in h I. 
 
 ^Example. Let the curvatures equal those of the last 
 problem, and suppose the distance g i to be 175-6 feet. 
 
 Then, by the formula, cosin. x = gTT~~ ~ == '7143 
 
 = cosin. 44 26'. Reducing minutes to hundredths, and 
 dividing by 7, we have 635 feet, the distance back to the 
 P. C. C. 
 
COMMENCEMENT OF A CLRVE 
 
 49 
 
 ARTICLE XIX. 
 
 HAVING GIVEN A TANGENT a b, AND A CURVE b k, LOCATED, 
 IT IS REQUIRED TO FIND THE POINT d, OR /, AT WHICH 
 TO COMMENCE A CURVE OF GIVEN RADIUS, WHICH SHALL 
 BE TANGENT TO BOTH. 
 
 DRAW the radius b c, and call it R. Name the radius 
 of the other curve r. Make b e equal to r, and, through 
 e. draw the line e I, parallel to the tangent a b. From c 
 as a centre, with eg, = (R -|- r), as radius, sweep the arc 
 of a circle ; which arc will intersect # I at g, and, from the 
 equidistances, prove g the centre of the other curve touch- 
 ing a b y b k, tangentially, at the points d and /. 
 
 Now, to find the point / for purposes of location, we 
 must know the angle b cf. Call it x. In the triangle 
 eg e, we have eg : c e : : radius : cosin. g c e, or b cf\ i, e*, 
 (R -f r) : (R r) : : 1 : nat. cosin #, wherefore nat. cosin. 
 
 x = TO" r { So that, having divided the difference of 
 
 ^rt -f- r) 
 
 the radii by their sum, we shall find opposite to the quo- 
 tient, in the table of nat. cosines, the angle b cf required. 
 This angle, divided by the degree of curvature of b k, will 
 
 5 
 
60 RAILWAY CURVES AND LOCATION 
 
 give the distance from b to the P. R. C., /; and 180 x 
 will equal the angle f g d, to be turned on the other curve. 
 
 Another plan, which may be preferred, for finding the 
 point d, is as follows : 
 
 Make c g the diameter of a semicircle, ctg. This semi- 
 circle is tangent to b d at t, and tf, perpendicular to eg, 
 is a common tangent to the curves b k, df. We have then, 
 
 / X /^ = tf\ i. e., R X r = tang. 2 1. Multiplying the 
 
 radii together, and extracting the square root of the product, 
 we find the distance tf, or its equal t b, which, doubled, is 
 the distance b d, from b to the point of curvature, d. 
 
 Example. Suppose k b a 3 curve, tangent to the line 
 a b. We wish to know the point /, at which to begin a 5 
 curve, which shall be tangent to both. Here R = 1910, 
 and r = 1146, wherefore (R + r) = 3056, (R r) = 
 
 764, and = = - , = -25, = nat. cosh, 75 31'. 
 
 Reducing minutes to hundredths, and dividing by 3, we 
 have 2517 feet, the distance from b to the P. R. C.,/. 
 
 If the point d be required, we have ^/1910 X 1146 = 
 1478-9. Doubling this result we find 2957'8 feet, the dis- 
 tance from b to the P. C., d. 
 
 If the radii are equal, of course the distance b d is equal 
 to their sum, and the angle # is a right angle. 
 
TO RUN A TANGENT TO TWO CURVES. 51 
 
 ARTICLE XX. 
 
 TO RUN A TANGENT TO TWO CURVES. 
 
 LET g , e /r, be the two curves. First plot them 
 carefully to a large scale, and, finding from this plot an 
 approximate P. T., say 6, run the tangent ef. Now the 
 chord in the curve g b, to which ef is parallel, is known, 
 and consequently the shortest distance, fg, between the 
 tangent and the curve may be measured. Then ef : f g 
 
 :: rad. : tang, feg, and nat. tang, feg = -^> Practi- 
 cally, this angle feg will be found equal to g a b or dc e, 
 so that, dividing it by the degree of curvature of k <*, we 
 shall have the distance, e d, to the proper P. T. 
 
 Strictly speaking, the angle g ef is too great by the 
 small angle bhg. There are two modes of calculating 
 this latter, but they are both complex, and in any but a 
 most unusual case, the above rule is sufficiently accurate. 
 
 Example. Let k e be a 5 curve. Suppose the tangent 
 ef = 1632 feet, and the distance gf = 42 feet, 
 
 Then 9 = -0257 = nat. tang, of 1 28'. Reducing 
 
52 
 
 RAILWAY CURVES AND LOCATION. 
 
 minutes to hundredths, and dividing by 5 
 feet, the distance back to the correct P. T. 
 
 we have 29 
 
 Should the curves turn in the same direction, or should 
 e be a fixed point from which to run a tangent to g b, the 
 above illustration is still applicable. 
 
 ARTICLE XXI. 
 
 OBDINATES. 
 
 TO FIND THE MIDDLE ORDINATE TO ANY GIVEN CHORD, IN 
 A CURVE OF ANY GIVEN RADIUS. 
 
 1st. (See figure in Art. XVL) Ic c = */ a e 2 a F, and 
 a c or be k e = the ordinate required. That is, from 
 the square of the radius subtract the square of half the 
 chord, and take the square root of the remainder from 
 radius, for the middle ordinate. 
 
 Example. The radius a c being 819 feet, and the chord 
 a/ 100 feet, to find the middle ordinate, Ik. 
 
 Here a <? a t? = 670761 2500, = 668261, the 
 square root of which is c k, = 817-5, which taken from 
 radius 819, leaves 1-5, the required middle ordinate. 
 
ORT>I NATES. 63 
 
 2d. Subtract the nat. cosine of the tangential ar/gle 
 from 1, and multiply the remainder by radius. 
 
 Example. Suppose a b a 7 curve. Here the nat. 
 cosine of 3 30', the tangential angle, is -9981, which, sub- 
 tracted from 1, leaves *0019. Multiplying this latter by 
 radius 819, we have 1*5, the middle ordinate as before 
 
 HAVING GIVEN THE MIDDLE ORDINATE, TO FIND ANY OTHER. 
 
 1st. eg = \/ c e 2 eg*, and ed = eg g d or clt. 
 
 Example. Suppose the distance Jed or eg to be 20 
 feet. Thence e e 2 c g 2 = 670761 400 = 670361. the 
 square root of which is eg, = 818*76. Taking from this 
 the distance gd = 817 '5, we have the ordinate ed = 
 1-26. 
 
 *2d. Multiply the ordinates of a 1 curve by the deflexion 
 angle of the curve whose ordinates are required. This is 
 an approximation sufficiently exact for railwa-y curves. 
 
54 
 
 RAILWAY CURVES AND LOCATION, 
 
 ARTICLE XXII. 
 
 TO FIND THE RADIUS CORRESPONDING TO ANY DEFLEXION 
 ANGLE, AND TO EQUAL CHORDS OF ANY GIVEN LENGTH. 
 
 HERE the deflexion angle, d b a, is of course equal to 
 the central angle d c b, subtended by the given chord d 6, 
 
 SIA-*'- i ,jiv 180 deb 
 and the triangle c d b being isosceles we have ~ 
 
 A 
 
 = c d b, or d b c. 
 
 Then nat. sin. deb : nat. sin. c b d :: db : c d 9 the 
 radius. 
 
 Example. Let the deflexion angle be 5, and the chord 
 100 feet. Required the radius. The nat. sine of 5 = 
 
 -I OAO CO 
 
 0872 and = 87 30', the nat. sine of which is 
 z 
 
 9990. 
 
 Then -0872 : -9990 : : 100 : radius = 1146 feet. 
 
 An approximate result, sufficiently accurate for all pur- 
 poses of railway location, may be found by simply dividing 
 5730 by the deflexion angle. 
 
 To find the deflexion angle corresponding to any given 
 
RADIUS CORRESPONDING TO DEFLEXION ANGLE. .> 
 
 radius and chord. In this case, we have c d : df : : rad 
 of 1 : nat. sin. of half the deflexion angle ; therefore, divide 
 half of the chord by the radius of the curve ; the quotient 
 will }yc the nat. sine of half the deflexion angle. 
 
 JExample. Radius as before 1146 feet, and chord 100 
 
 feet. Then j||g = -0436, the nat. sin. of 2 30'. Dou- 
 bling this result, we have the deflexion angle, viz. 5 00'. 
 
 To find the deflexion distance with chord of 100 feet and 
 any radius. Divide the constant number 10000 by the 
 radius in feet ; the quotient will be the deflexion distance : 
 for the deflexion distance with a radius of 10000 ft-et is 
 1 foot, and the deflexion distances for other radii increase 
 inversely as the radii. 
 
 Example. What is the deflexion distance for a 5 curve, 
 
 the chord being 100 feet? Here ^^ = 8-72 feet, the 
 
 deflexion distance. 
 
 To find the deflexion distance with any given chord and 
 radius. The deflexion distance is equal to twice the 
 natural sine of half the deflexion angle, multiplied by the 
 chord. Thus, the chord being 100 feet, and the deflexion 
 angle 5, we find the nat. sine of 2 30' equal to -0436, 
 which doubled, and multiplied by 100, gives 8*72 feet, as 
 above. 
 
 The tangential distance, with any radius and chord, is 
 in like manner equal to twice the nat. sine of half the tan- 
 gential angle multiplied by the chord. Thus, the tangential 
 angle being 2 30' and the chord 100 feet, we find the nat. 
 S'ne of 1 15' = -0218. Multiplying by 2, and 100, we 
 have 4*36 feet, the tangential distance. 
 
 For all curves under 11, the tangential distance is equal 
 to half the deflexion distance. 
 
J>6 RAILWAY CURVER AND LOCATION 
 
 ARTICLE XXIII. 
 
 OF EXCAVATION AND EMBANKMENT. 
 
 A RAILWAY line having been located, the first office dutiea 
 are to map it down, to make a continuous profile, and an 
 approximate estimate of the cost of grading, &c. The " ele- 
 vation" of each stake above or below a certain assumed 
 base being fixed, the profile is drawn, on paper prepared 
 for the purpose, to a horizontal scale of 400 feet, and a 
 vertical scale of 40 feet, to the inch. This distorted pic- 
 ture presents at a glance, in compact shape, the undula- 
 tions of the surface, and from it grades are adopted, to 
 balance as nearly as possible the excavation and embank- 
 ment. By these grades, noted in the record, the cutting 
 or filling is ascertained at each station. Suppose, for 
 instance, that the elevation of station 40 is 12 feet, and 
 that of station 100 is 72 feet. The distance between 40 
 and 100 is 6000 feet, and the difference between 12 and 
 72 is 60 feet. Consequently, to connect those points, we 
 require an ascending grade of 1 in 100 feet. Grade at 
 station 54 is therefore 26 feet, and at station 60, 32 feet. 
 If the elevation at station 60 be 38 feet, of course we have 
 6 feet of excavation, and this is marked in our estimate 
 sheets -f 6. If the elevation at that point should be 27 
 feet, o feet of embankment is the consequence, which is 
 marked accordingly, 5 ; plus indicating excavation, and 
 minus embankment. 
 
 The usual slope for embankment is 1 feet horizontal 
 to 1 foot vertical, making an angle of about 34 with the 
 
OF EXCAVATION AND EMBANKMENT. 57 
 
 horizon ; that for earth cut, 1 to 1, or 45, and for rock 
 cut J to 1, or 76. The slopes of the ground surface at 
 each station being known, together with the breadth of the 
 intended roadway, we are prepared to calculate the cross- 
 sectional areas, and from them to determine the cubic yards 
 of excavation and embankment. 
 
 To facilitate these operations the following rules were 
 prepared. With Trautwine's common diagram, and the 
 table of squares and square roots appended to the volume, 
 they will be found an observable assistance. 
 
 Suppose li the road bed, and fh the depth of an ordi 
 nary clay cut. Produce the side slopes until they meet in 
 the point k. Then the angle being 45, ef=fk, and 
 ef 2 = twice the triangle efk, = the triangle e kg. For 
 like reasons I h = h k, and I h 2 = triangle I k i. But the 
 triangle Iki, taken from the triangle ekg, leaves the area 
 elig, to find which we have therefore the following 
 
 Rule. To half the breadth of the roadway, add the 
 depth of the cut. From the square of this sum, subtract 
 the square of half the breadth of the roadway for the 
 area. 
 
 Example. Suppose the breadth of the roadway to be 
 32 feet, and the depth hf, 7 feet. Half of 32 = 16, the 
 square of which, viz., 256, becomes a constant subtrahend. 
 16 -{- 7 = 23, and a reference to the table shows the 
 square of 23 equal to 529. Therefore 529 256 = 273, 
 the area required in square feet. 
 
 If there be a regular slope, as eg (p. 58), we can plot it on 
 the diagram, and read at once the side cuts, g k, en. 
 
58 RAILWAY CURVES AND LOCATION. 
 
 the area is equal to that of the trapezoid e n k g, minus 
 the two triangles e n h, mk g. 
 
 Calling e ft, or its equal n li, a, m k or g k, 5, and half 
 the breadth of the roadway, c, we have the triangle mk g 
 
 = b X 77 = - The triangle e n h is in like manner = , 
 
 22 2 
 
 and the area of the trapezoid = a-\-b-{-2cX J . 
 
 2 
 
 Then the area required = a-j-5-|-2tfX - ~ 
 
 2 
 2 ab -f 2 ac -f 2bc 
 
 = (a -f b) c + a.5. 
 
 Should the slopes be J to 1, as in rock cut, the area 
 might, in similar-wise, be shown equal to (a -f- b) c -\ ^-, 
 and that of embankment, where the slopes are 1J to 1, 
 equal to (a -f- #) c -j ^-. Wherefore, we arrive at the 
 
 following general rule for finding cross-sectional areas, 
 where the ground surface is a regular declivity : 
 
 Multiply the sum of the side cuttings by half the breadth 
 of the roadway, and mark the product. Multiply the pro- 
 duct of the side cuttings by the ratio of the side slopes to 
 1, and add this result to the previous product marked for 
 the area. 
 
 Example 1. Earth excavation road bed 32 feet, side 
 
OF EXCAVATION AND EMBANKMENT. 59 
 
 slopes 1 to 1, right cutting 12 feet, left cutting 3 feet. 
 Required the cross-sectional area. Here (a -f- b) c -\- a b 
 = (12 + 3) 16 -f 12 X 3 = 240 + 36 = 276 square 
 feet, the area required. 
 
 Example 2. Rock cut road bed 28 feet, side 
 slopes J to 1, cuttings as before. Required the area. 
 
 Here (a -f b) c + t-*J. = 210 +^ = 219 square 
 
 feet, the area. 
 
 Example 3. Embankment roadway 27 feet, side 
 slopes 1| to 1, cuttings as before. Here ( + #)<? + 
 
 3 ^ 2 X ^ = 202-5. + 54 = 256-5 square feet, the cross- 
 
 sectional area. 
 
 Other formulae might be given for varying surface slopes, 
 but they would involve a simple matter, and require more 
 time for calculation than a division of the diagram area 
 into triangles. 
 
 To find the cubic content of an excavation or embank- 
 ment : 
 
 Multiply half the sum of the two end areas by the 
 distance between them ; thus : supposing 282 and 310 
 to be the end areas, and 100 feet the distance, we have 
 (282 + 310) 
 
 yards. Or we may multiply the sum of the end areas by 
 100, and divide the product by 6 and 9, the factors of 54, 
 which is the number of cubic feet contained in two cubic 
 
 yards ; thus, 5?|?? = 9866-6, and this divided by 9 = 
 1096-3, as above. 
 
60 
 
 RAILWAY CURVES AND LOCATION. 
 
 PRISMOIDAL FORMULA. 
 
 To the two end areas add four times the middle area, 
 and multiply the sum by one-sixth of the length of the pris- 
 moid. Thus : from the foregoing example, the sum of the 
 end areas, 592, added to four times their mean, 1184, 
 gives 1776, and 1776 X 16-7 = 29659-2 cubic feet, = 
 1098'5 cubic yards. The former rule is approximate 
 sufficiently so for rough preliminary calculations, but where 
 strict correctness is required, as in a final field estimate, 
 the prismoidal formula should be usefl. It applies to all 
 solid bodies with plane faces and parallel ends. 
 
 ARTICLE XXIV. 
 
 SIDE STAKING. 
 
 THE object in side staking is to find the point where 
 the surface of the ground intersects the slope of the road 
 
 formation. For performing this work with level and rod, 
 place the instrument in such a position as to command as 
 many stations as possible, whether they be on the upper 
 
SIDE STAKING 61 
 
 or lower side, and ascertain by transfers from the bench 
 the height of your instrument with reference to grade. 
 This will facilitate operations by giving usually small num- 
 bers to work with. Knowing your rate of grade, subtract 
 or add, as the case may be, as you change from station to 
 station. 
 
 Having the level in place, we will suppose on an exca- 
 vation, our object is first to place the lower side stake. 
 Let the width of the road bed be 26 feet, and the side 
 slopes 1 to 1. Suppose the instrument to be 11 feet above 
 grade and the centre cut 5 feet, which latter is already 
 recorded in the field book for construction duties. Look- 
 ing at the fall of the hill we judge that at the distance of 
 17 feet from the centre stake the descent is 1'5 feet, which 
 would give us 3-5 feet cutting. We take an observation 
 at that point and find that the rod reads 10 feet, which, 
 subtracted from our height above grade, leaves us only one 
 foot cutting, and shows that our judgment has been at 
 fault. The distance measured is too much, for were we to 
 increase the distance we would reduce the cutting, and it 
 is therefore evident that we are too far from the centre 
 Let us then make the distance from the centre stake 15 
 feet, and take another sight, when the rod reads 9, show- 
 ing 2 feet cutting, which, added to half the road formation, 
 slope being 1 to 1, equals the distance measured. The 
 stake is therefore correct. The same process applies to 
 the upper side. 
 
 The only difference in staking out for banks is that you 
 add 1J times the height of the bank to the distance mea- 
 sured, or, as a general rule, the height of the bank multi- 
 plied by the ratio of the side slopes to unity. 
 
 Some judgment is required to be expeditious in this 
 work, which is obtained only by experience. 
 
 6 
 
62 
 
 RAILWAY CURVES AND LOCATION. 
 
 The following is a common form of field record for con 
 struction : 
 
 ,* 
 
 jji8t. 
 
 
 Left Side. 
 
 Centre. 
 
 i 
 Right Sido 
 
 Course. 
 
 Mag. 
 Course. 
 
 Elevation 
 
 Grade. 
 
 Dist. 
 
 AorB 
 
 A 
 
 B 
 
 Dist. 
 
 AorB 
 
 S-ll A 
 
 530 
 
 100 
 
 N20OOW 
 
 N 20-05 W 
 
 + 560-4 
 
 + 555-4 
 
 15-0 
 
 2-0 A 
 
 5-0 
 
 
 21-0 
 
VTUEAL SINES AND TANGENTS 
 
 (63) 
 
|l NAT. SINE. 65 ' 
 
 ' ' 
 
 O o [o 2 
 
 3 t 4 
 
 5 6 
 
 70 : / 
 
 , o 
 
 000 OOCO 017 4524 034 8995 
 
 052 3360 069 7565 
 
 087 1557 
 
 104 5285 
 
 121 8693! 60 
 
 i 
 
 29091 7432,035 1902 
 
 6264 070 0467 
 
 4455 
 
 R178 
 
 1 221 581 59 
 
 '2 
 
 581b 018 0341 
 
 4809 
 
 91691 3368 
 
 7353 105 1070 
 
 4468 58 
 
 3 
 
 8727 
 
 3249 
 
 7716 
 
 053 2074 6270 
 
 0880251 
 
 3963 
 
 7355i 57 
 
 4 
 
 001 1636 
 
 6158 
 
 036 0623 
 
 4979 9171 
 
 3148 
 
 6856 
 
 1230241 
 
 56 
 
 ; 5 
 
 4544 
 
 9066 
 
 3530 
 
 7883,071 2073 
 
 6046 
 
 9748 
 
 312- 
 
 55 
 
 6 
 
 7453019 1974 
 
 6437 
 
 054 0788J 4974 
 
 8943 
 
 106 2G41 
 
 6015 
 
 54 
 
 i 7 
 
 C02 0362 
 
 48S3 
 
 9344 
 
 3693| 7876 
 
 089 1840 
 
 5533 
 
 8901 
 
 53 
 
 I 8 
 
 3271 
 
 7791 
 
 037 2251 
 
 6597 
 
 072 0777 
 
 4738 
 
 8425 
 
 124 1768 
 
 52 
 
 9 
 
 6180 020 0699 
 
 5158 
 
 9502 
 
 3678 
 
 7635 
 
 107 1318 
 
 4674 
 
 51 
 
 10 
 
 9089 3608 
 
 8065 
 
 055 2406 
 
 6580 
 
 090 0532 
 
 4210 
 
 7560 
 
 
 
 11 
 
 003 1998 6516 
 
 0380971 
 
 5311 
 
 9481 
 
 3429 
 
 7102 
 
 125 0446 
 
 49 
 
 1 12 
 
 4907J 9424 
 
 3878 
 
 8215 
 
 073 2382 
 
 6326 
 
 9994 
 
 3332 
 
 48 
 
 13 
 
 7815 
 
 021 2332 
 
 6785 
 
 Oo6 1119 
 
 5283 
 
 9223 
 
 108 2885 
 
 6218 
 
 47 
 
 14 
 
 004 0724 
 
 5241 
 
 9692 
 
 4024 
 
 8184 
 
 091 2119 
 
 5777 
 
 9104 
 
 46 
 
 i 15 
 
 3633 
 
 8149 
 
 039 2598 
 
 6928 
 
 074 1085 
 
 5016 
 
 8669 
 
 126 1990 
 
 45 
 
 16 
 
 6542 
 
 022 1057 
 
 5505 
 
 9832 
 
 3986 
 
 7913 
 
 109 1560 
 
 4875 
 
 44 
 
 17 
 
 9451 
 
 3965 
 
 8411 
 
 057 2736 
 
 6887 
 
 392 0809 
 
 4452 
 
 7761 
 
 43 
 
 18 
 
 005 2360 
 
 6873 
 
 040 0318 
 
 5640 
 
 9787 
 
 3706 
 
 7343 
 
 127 0646 
 
 42 
 
 19 
 
 5268 
 
 9781 
 
 4224 
 
 8544 
 
 075 268S 
 
 6602 
 
 1100234 
 
 3531 
 
 41 
 
 20 
 
 8177 
 
 023 2690 
 
 7131 
 
 058 1448 
 
 5589 
 
 9499 
 
 3126 
 
 6416 
 
 40 
 
 21 
 
 006 1086 
 
 5598 
 
 041 0037 
 
 4352 
 
 8489 
 
 093 2395 
 
 6017 
 
 9302 
 
 39 
 
 22 
 
 3995 
 
 8506 
 
 2944 
 
 7256 
 
 076 1390 
 
 5291 
 
 8908 
 
 128 2186 
 
 ;$) 
 
 23 
 
 6904 
 
 024 1414 
 
 5850 
 
 059 0160 
 
 4290 
 
 8187 
 
 111 1799 
 
 5071 
 
 37 
 
 24 
 
 9813 
 
 4322 
 
 8757 
 
 3064 
 
 7190 
 
 094 1083 
 
 4689 
 
 7956 
 
 36 
 
 25 
 
 007 2721 
 
 7230 
 
 042 1663 
 
 5967 
 
 0770091 
 
 3979 
 
 7580 
 
 1290841 
 
 35 
 
 2G 
 
 5630 
 
 025 0138 
 
 4569 
 
 8871 
 
 2991 
 
 6875 
 
 1120471 
 
 3725 
 
 34 
 
 i 27 
 
 8539 
 
 3046 
 
 7475 
 
 060 1775 
 
 5891 
 
 9771 
 
 3361 
 
 6609 
 
 33 
 
 28 
 
 008 1448 
 
 5954 
 
 043 0382 
 
 4678 
 
 8791 
 
 0952&66 
 
 6252 
 
 9494 
 
 32 
 
 29 
 
 4357 
 
 8662 
 
 3288 
 
 7582 
 
 078 1691 
 
 5562 
 
 9142 
 
 130 2378 
 
 31 
 
 30 
 
 7265 
 
 026 1769 
 
 6194 
 
 061 0485 
 
 4591 
 
 8458 
 
 1132032 
 
 5262 
 
 30 
 
 31 
 
 009 0174 
 
 4677 
 
 9100 
 
 3389 
 
 7491 
 
 ^96 1353 
 
 4922 
 
 8146 
 
 20 
 
 32 
 
 3083 
 
 7585 
 
 044 2006 
 
 6292 
 
 079 0391 
 
 4248 
 
 7812 
 
 131 1030 
 
 28 
 
 33 
 
 5992 
 
 027 0493 
 
 4912 
 
 9196 
 
 3290 
 
 7144 
 
 1140702 
 
 3913 
 
 27 
 
 34 
 
 8900 
 
 3401 
 
 7818 
 
 062 2099 
 
 6190 
 
 097 0039 
 
 3592 
 
 6797 
 
 2G 
 
 35 
 
 010 1809 
 
 6309 
 
 045 0724 
 
 5002 
 
 9090 
 
 2034 
 
 6482 
 
 . 9681 
 
 25 
 
 36 
 
 4718 
 
 9216 
 
 3630 
 
 7905 
 
 080 1989 
 
 5829 
 
 9372 
 
 132 2564 
 
 24 
 
 37 
 
 7627 
 
 028 2124 
 
 6536 
 
 063 0808 
 
 4880 
 
 8724 
 
 1152261 
 
 5447 
 
 23 
 
 38 
 
 Oil 0535 
 
 5032 
 
 9442 
 
 3711 
 
 7788 
 
 098 1619 
 
 5151 
 
 8330 
 
 22 
 
 39 
 
 3444 
 
 7940 
 
 046 2347 
 
 6614 
 
 081 0687 
 
 4514 
 
 8040 
 
 133 1213 
 
 21 
 
 40 
 
 6353 
 
 029 0847 
 
 5253 
 
 9517 
 
 3587 
 
 7408 
 
 1160929 
 
 4096 
 
 20 
 
 41 
 
 9261 
 
 3755 
 
 8159 
 
 064 2420 
 
 6486 
 
 099 0303 
 
 3818 
 
 6979 
 
 HI 
 
 42 
 
 012 2170 
 
 6662 
 
 047 1065 
 
 5323 
 
 9385 
 
 3197 
 
 6707 
 
 9S62 
 
 Ifc 
 
 43 
 
 5079 
 
 9570 
 
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 TABLE OF RADII, &c.~- Chord 100 Feet. 
 
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 34380-0 -291 
 
 30 
 
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 21-790 ! 
 
 15 
 
 22920-0 -436 
 
 35 
 
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 20 117190-01 -581 
 
 40 
 
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 620-2 
 
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 27 
 
 214-2 
 
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 40 
 
 21490 
 
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 573-7J17-430 
 
 28 
 
 206-7 
 
 48-380 ! 
 
 45 
 
 2084-0 
 
 4-799 
 
 15 
 
 559-7 17-870 
 
 29 
 
 199-7 
 
 50-070 
 
 50 
 
 2023-0! 4-9 14 
 
 30 
 
 546.4 
 
 18-300 
 
 30 
 
 193-2 
 
 51-760 
 
 ! o 55 
 
 1965-0 5-090 
 
 o 45 
 
 533-8J18-730 
 
 31 
 
 187-1 
 
 53-450 
 
 3 
 
 1910-0 
 
 5-235 
 
 11 
 
 521-719-170 
 
 32 
 
 181-4 
 
 55-130 
 
 5 
 
 1859-0 5-380 
 
 15 
 
 510-119-610 
 
 33 
 
 176-0 
 
 56-800 
 
 10 
 
 1810-0 5-526 
 
 30 
 
 499-120-050 
 
 34 
 
 171-0 
 
 58/470 
 
 15 
 
 1763-0! 5-671 
 
 o 45 
 
 488-520-500 
 
 35 
 
 166-3 
 
 60-140" 
 
 20 
 
 1719-0 
 
 5-817 
 
 12 
 
 478-3j20-940 
 
 36 
 
 .161-8 
 
 61-800 j 
 
 
 8* 
 
90 
 
 TABLE OF LONG CHORDS. 
 
 Raclics in feet. 
 
 Angle of 
 Deflection. 
 
 Length 
 1 Station. 
 
 of Chord in f< 
 2 Stations. 
 
 jet required tc 
 3 Stations. 
 
 subtend 
 4 Stations. 
 
 5730-0 
 
 1 
 
 100 
 
 200-0 
 
 300-0 
 
 400-0 
 
 4584-0 
 
 * 
 
 100 
 
 200-0 
 
 300-0 
 
 399-9 
 
 3820-0 
 
 | 
 
 100 
 
 200-0 
 
 300-0 
 
 399-9 
 
 3274-0 
 
 
 100 
 
 200-0 
 
 300-0 
 
 399-8 
 
 2865-0 
 
 2 f 
 
 100 
 
 200-0 
 
 299-9 
 
 399-7 
 
 I 2547-0 
 
 * 
 
 100 
 
 200-0 
 
 299-9 
 
 399-6 
 
 2292-0 
 
 
 100 
 
 200-0 
 
 299-8 
 
 399-5 
 
 2084-0 
 
 f. 
 
 100 
 
 200-0 
 
 299-8 
 
 399-4 
 
 1910-0 
 
 3 
 
 100 
 
 200-0 
 
 299-7 
 
 399-3 
 
 1763-0 
 
 i 
 
 100 
 
 200-0 
 
 299-7 
 
 399-2 
 
 . 1637-0 
 
 | 
 
 100 
 
 200-0 
 
 299-6 
 
 399-1 
 
 1528-0 
 
 
 100 
 
 200-0 
 
 299-6 
 
 399-0 
 
 1433-0 
 
 4 f 
 
 100 
 
 199-9 
 
 299-6 
 
 398-9 
 
 1348-0 
 
 \ 
 
 100 
 
 199-9 
 
 299-5 
 
 398-7 
 
 1274-0 
 
 
 100 
 
 199-9 
 
 299-4 
 
 398-5 
 
 1207-0 
 
 I 
 
 100 
 
 199-9 
 
 299-3 
 
 398-3 
 
 1146-0 
 
 5 
 
 100 
 
 199-9 
 
 299-2 
 
 398-0 
 
 1092-0 
 
 k 
 
 100 
 
 199-8 
 
 299-1 
 
 397-8 
 
 1042-0 
 
 
 100 
 
 199-8 
 
 299-0 
 
 397-6 
 
 1 996-8 
 
 ! 
 
 100 
 
 199-7 
 
 298-9 
 
 397-5 
 
 955.4 
 
 6 
 
 100 
 
 199-7 
 
 298-8 
 
 397-3 
 
 917-0 
 
 i 
 
 100 
 
 199-7 
 
 298-7 
 
 397-0 
 
 882-0 
 
 
 
 100 
 
 199-7 
 
 298-6 
 
 396-7 
 
 849-3 
 
 1 
 
 100 
 
 19^fc 
 
 298-5 
 
 396-5 
 
 819-0 
 
 7 
 
 100 
 
 199-6 
 
 298-4 
 
 396-2 
 
 790-8 
 
 i 
 
 100 
 
 199-6 
 
 298-3 
 
 396-0 
 
 764-5 
 
 * 
 
 100 
 
 199-6 
 
 298-2 
 
 395-7 
 
 739-9 
 
 1 
 
 100 
 
 199-6 
 
 298-1 
 
 395-4 
 
 1 716-8 
 
 8 
 
 100 
 
 199-6 
 
 298-0 
 
 395-1 
 
 695-1 
 
 i 
 
 100 
 
 199-5 
 
 297-9 
 
 394-8 
 
 674-6 
 
 
 100 
 
 199-5 
 
 297-8 
 
 394-5 
 
 655-5 
 
 3. 
 
 100 
 
 199-4 
 
 297-7 
 
 394-3 
 
 637-3 
 
 9 
 
 100 
 
 199-4 
 
 297-5 
 
 394-1 
 
 620-2 
 
 i 
 
 100 
 
 199-4 
 
 297-4 
 
 393-7 
 
 603-8 
 
 
 100 
 
 199-3 
 
 297-3 
 
 393-2 
 
 588-4 
 
 1 
 
 100 
 
 199-2 
 
 297-2 
 
 392-8 
 
 573-7 
 
 10 
 
 100 
 
 199-2 
 
 297-0 
 
 392-4 
 
 
: 
 
 TABLE OF ORDINATES. 
 
 Ordinates 10 feet apart. Chord 100 feet. 
 
 
 Distances of the Ordinates from the end of the 100 feet Chord. 
 
 Deflexion Angle 
 in Degrees and 
 
 50 feet. 
 
 40 feet. 
 
 30 feet. 
 
 20 feet. 
 
 10 feet. 
 
 
 Lengths of Ordinates in feet. 
 
 o / 
 
 
 
 
 
 
 5 
 
 018 
 
 017 
 
 015 
 
 012 
 
 006 
 
 10 
 
 036 
 
 035 
 
 031 
 
 023 ' -013 
 
 15 
 
 054 
 
 052 
 
 046 
 
 035 
 
 019 
 
 20 
 
 073 
 
 070 
 
 061 
 
 047 
 
 026 
 
 25 
 
 091 
 
 087 
 
 076 
 
 058 
 
 032 
 
 30 
 
 109 
 
 105 
 
 092 
 
 070 
 
 039 i 
 
 35 
 
 127 
 
 123 
 
 108 
 
 082 
 
 045 
 
 40 
 
 145 
 
 140 
 
 123 
 
 093 
 
 052 i 
 
 45 
 
 163 
 
 157 
 
 137 
 
 105 
 
 058 
 
 50 
 
 182 
 
 175 
 
 153 
 
 117 
 
 065 
 
 o 55 
 
 200 
 
 192 
 
 168 
 
 128 
 
 071 
 
 1 
 
 218 
 
 209 
 
 183 
 
 140 
 
 078 
 
 5 
 
 236 
 
 226 
 
 198 
 
 152 
 
 085 
 
 10 
 
 254 
 
 244 
 
 214 
 
 163 
 
 091 
 
 15 
 
 273 
 
 261 
 
 229 
 
 175 
 
 098 
 
 20 
 
 291 
 
 279 
 
 244 
 
 187 
 
 104 i 
 
 25 
 
 309 
 
 296 
 
 259 
 
 198 
 
 111 
 
 30 
 
 327 
 
 314 
 
 275 
 
 210 
 
 117 
 
 35 
 
 345 
 
 331 
 
 290 
 
 *221 
 
 124 
 
 40 
 
 364 
 
 349 
 
 305 
 
 233 
 
 130 
 
 45 
 
 382 
 
 366 
 
 321 
 
 245 
 
 137 
 
 50 
 
 400 
 
 '-384 
 
 336 
 
 256 
 
 144 
 
 o 55 
 
 418 
 
 401 
 
 351 
 
 268 
 
 150 
 
 2 
 
 436 
 
 419 
 
 366 
 
 280 
 
 157 
 
 5 
 
 454 
 
 436 
 
 382 
 
 291 
 
 163 
 
 10 
 
 473 
 
 454 
 
 397 
 
 303 . -170 
 
 15 
 
 491 
 
 471 
 
 412 
 
 315 -17' 
 
 20 
 
 509 
 
 489 
 
 428 
 
 326 1. 
 
 25 
 
 527 
 
 506 
 
 443 
 
 338 
 
 190 
 
 30 
 
 545 
 
 524 
 
 458 
 
 350 
 
 196 
 
 35 
 
 564 
 
 541 
 
 474 
 
 361 
 
 203 
 
 40 
 
 582 
 
 559 
 
 489 
 
 373 
 
 209 
 
 45 
 
 600 
 
 576 
 
 504 
 
 384 
 
 216 ; 
 
 .50 
 
 618 
 
 594 
 
 519 
 
 396 
 
 222 
 
 o 55 
 
 636 
 
 611 
 
 535 
 
 408 
 
 229 i 
 
 3 
 
 654 
 
 629 
 
 550 
 
 419 
 
 235 
 
 5 
 
 673 
 
 646 
 
 565 
 
 431 
 
 242 
 
 10 
 
 691 
 
 664 
 
 581 
 
 443 
 
 249 
 
 15 
 
 709 
 
 681 
 
 596 
 
 454 
 
 255 
 
 ! 20 
 
 727 
 
 699 
 
 611 
 
 466 
 
 262 
 
 ; 25 
 
 j - 
 
 745 
 
 716 
 
 627 
 
 478 
 
 268 
 
1 1 
 
 92 
 
 TABLE OF ORDINATES. CONTINUED. 
 
 Ordinates 10 feet apart. Chord 100 feet. 
 
 
 Distances of the Ordinates from the end of the 100 feet Chord. 
 
 Deflexion Angle 
 in Degrees and 
 Minutes. 
 
 60 feet. 
 
 40 feet. 
 
 30 feet. 
 
 20 feet. 
 
 10 feet 
 
 
 Lengths of Ordinates in feet. 
 
 o / 
 
 
 
 
 
 
 3 30 
 
 764 
 
 734 
 
 642 
 
 489 
 
 275 
 
 35 
 
 782 
 
 751 
 
 657 
 
 501 
 
 281 
 
 ! 40 
 
 800 
 
 769 
 
 673 
 
 512 
 
 288 
 
 45 
 
 818 
 
 786 
 
 688 
 
 524 
 
 294 
 
 50 
 
 836 
 
 804 
 
 703 
 
 536 
 
 301 
 
 o 55 
 
 854 
 
 821 
 
 718 
 
 547 
 
 308 
 
 4 
 
 873 
 
 839 
 
 734 
 
 559 
 
 314 
 
 15 
 
 927 
 
 891 
 
 780 
 
 594 
 
 334 
 
 30 
 
 981 
 
 944 
 
 825 
 
 629 
 
 354 
 
 o 45 
 
 1-036 
 
 996 
 
 871 
 
 664 
 
 373 
 
 5 
 
 1-091 
 
 1-048 
 
 917 
 
 699 
 
 393 
 
 15 
 
 1-146 
 
 1-100 
 
 963 
 
 734 
 
 413 
 
 30 
 
 1-200 
 
 1-153 
 
 1-009 
 
 769 
 
 432 
 
 o 45 
 
 1-255 
 
 1-205 
 
 1-055 
 
 804 
 
 452 
 
 6 
 
 1-309 
 
 1-258 
 
 1-100 
 
 839 
 
 472 
 
 15 
 
 1-364 
 
 1-310 
 
 1-146 
 
 874 
 
 492 
 
 30 
 
 1-419 
 
 1-362 
 
 1-192 
 
 909 
 
 511 
 
 o 45 
 
 1-473 
 
 1-415 
 
 1-238 
 
 944 
 
 531 
 
 7 
 
 1-528 
 
 1-467 
 
 1-284 
 
 979 
 
 551 
 
 15 
 
 1-582 
 
 1-520 
 
 1-330 
 
 1-014 
 
 570 
 
 30 
 
 1-637 
 
 1-572 
 
 1-375 
 
 1-048 
 
 590 
 
 o 45 
 
 1-692 
 
 1-624 
 
 1-421 
 
 1-083 . 
 
 610 
 
 8 
 
 1-746 
 
 1-677 
 
 1-467 
 
 1-118 
 
 629 
 
 15 
 
 1-801 
 
 1-729 
 
 1-513 
 
 1-153 
 
 649 
 
 30 
 
 1-855 
 
 1-782 
 
 1-559 
 
 1-188 
 
 609 
 
 o 45 - 
 
 1-910 
 
 1-834 
 
 1-605 
 
 1-223 
 
 689 
 
 9 
 
 1-965 
 
 1-886 
 
 1-651 
 
 1-258 
 
 708 
 
 15 
 
 2-019 
 
 1-939 
 
 1-696 
 
 1-293 
 
 728 
 
 30 
 
 2-074 
 
 1-991 
 
 1-742 
 
 1-328 
 
 748 
 
 I o 45 
 
 2-128 
 
 2-044 
 
 1-788 
 
 1-363 
 
 767 
 
 10 
 
 2-183 
 
 2-096 
 
 1-834 
 
 1-398 
 
 787 [ 
 
 15 
 
 2-238 
 
 2-148 
 
 1-880 
 
 1-433 
 
 807 
 
 30 
 
 2-292 
 
 2-201 
 
 1-926 
 
 1-468 
 
 827 
 
 o 45 
 
 2-347 
 
 2-254 
 
 1-972 
 
 1-503 
 
 846 
 
 11 
 
 2-401 
 
 2-306 
 
 2-018 
 
 1-538 
 
 866 
 
 15 
 
 2-456 
 
 2-359 
 
 2-064 
 
 1-574 
 
 886 
 
 30 
 
 2-511 
 
 2-411 
 
 2-110 
 
 1-609 
 
 906 
 
 o 45 
 
 2-566 
 
 2-464 
 
 2-156 
 
 1-644 
 
 926 
 
 12 
 
 2-620 
 
 2-516 
 
 2-203 
 
 1-680 
 
 946 
 
 15 
 
 2-675 
 
 2-569 
 
 2-249 
 
 1-715 
 
 966 
 
 30 
 
 2-730 
 
 2-621 
 
 2.295 
 
 1-750 
 
 985 
 
 if 
 

 93 
 
 TABLE OF ORDINATES. CONTINUED. 
 
 Ordinates 10 feet apart. Chord 100 feet. 
 
 
 Distances of the Ordinates from the end of the 100 feet Chord. ; 
 
 Deflexion Angle 
 iu Degrees and 
 
 50 feet. 
 
 40 feet. 
 
 30 feet. 
 
 20 feet. 
 
 10 feet. 
 
 Minutes. 
 
 Lengths of Ordinates in feet. 
 
 o / 
 
 
 
 
 
 
 12 45 
 
 2785 
 
 2-674 
 
 2-341 
 
 1-785 
 
 1-005 
 
 13 
 
 2-839 
 
 2-726 
 
 2-387 
 
 1-820 
 
 1-025 1 
 
 15 
 
 2-894 
 
 2-779 
 
 2-433 
 
 1-855 
 
 1-045 
 
 30 
 
 2-949 
 
 2-832 
 
 2-479 
 
 1-891 
 
 1-065 
 
 o 45 
 
 3-000 
 
 2-884 
 
 2-525 
 
 1-926 
 
 1-085 
 
 14 
 
 3-058 
 
 2-937 
 
 2-571 
 
 1-961 
 
 1-105 
 
 15 
 
 3-113 
 
 2-989 
 
 2-618 
 
 1-996 
 
 1-124 
 
 30 
 
 3-168 
 
 3-042 
 
 2-664 
 
 2-031 
 
 1-144 
 
 o 45 
 
 3-222 
 
 3-094 
 
 2-710 
 
 2-067 
 
 1 164 
 
 15 
 
 3-277 
 
 3-147 
 
 2-756 
 
 2-102 
 
 1-184 
 
 15 
 
 3-332 
 
 3-200 
 
 2-802 
 
 2-137 
 
 1-204 
 
 30 
 
 3-387 
 
 3-252 
 
 2-848 
 
 2-172 
 
 1-224 
 
 o 45 
 
 3-442 
 
 3-305 
 
 2-895 
 
 2-208 
 
 1-244 
 
 16 
 
 3-496 
 
 3-358 
 
 2-941 
 
 2-243 
 
 1-264 
 
 o 30 
 
 3-606 
 
 3-463 
 
 3-033 
 
 2-314 
 
 1-304 
 
 17 
 
 3-716 
 
 3-569 
 
 3-125 
 
 2-384 
 
 1-344 
 
 o 30 
 
 3-826 
 
 3-674 
 
 3-218 
 
 2-455 
 
 1-384 
 
 i 18 
 
 3-935 
 
 3-779 
 
 3-310 
 
 2-525 
 
 1-424 
 
 i o 30 
 
 4-045 
 
 3-885 
 
 3-403 
 
 2-596 
 
 1-464 
 
 19 
 
 4-155 
 
 3-990 
 
 3-495 
 
 2-666 
 
 1-504 
 
 o' 30 
 
 4-265 
 
 4-096 
 
 3-588 
 
 2-737 1-544 
 
 20 
 
 4-375 
 
 4-201 
 
 3-680 
 
 2-808 
 
 1-583 
 
 
94 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 1 
 
 1 
 
 1-0000 
 
 44 
 
 1936 
 
 6-6332 
 
 2 
 
 4 
 
 1-4142 
 
 45 
 
 2025 
 
 6-7082 
 
 i 3 
 
 9 
 
 1-7320 
 
 46 
 
 2116 
 
 6-7823 
 
 4 
 
 16 
 
 2-0000 
 
 47 
 
 2209 
 
 6-8556 
 
 5 
 
 25 
 
 2-2360 
 
 48 
 
 2304 
 
 6-9282 ' 
 
 i 6 
 
 36 
 
 2-4495 
 
 49 
 
 2401 
 
 7-0000 
 
 7 
 
 49 
 
 2-6457 
 
 50 
 
 2500 
 
 7-0711 
 
 8 
 
 64 
 
 2-8284 
 
 51 
 
 2601 
 
 7-1414 
 
 9 
 
 81 
 
 3-0000 
 
 52 
 
 2704 
 
 7-2111 
 
 10 
 
 100 
 
 3-1623 
 
 53 
 
 2809 
 
 7-2801 
 
 11 
 
 121 
 
 3-3166 
 
 54 
 
 2916 
 
 7-3485 
 
 12 
 
 144 
 
 3-4641 
 
 55 
 
 3025 
 
 7-4162 
 
 13 
 
 169 
 
 3-6055 
 
 56 
 
 3136 
 
 7-4833 
 
 14 
 
 196 
 
 3*7416 
 
 57 
 
 3249 
 
 7-5498 
 
 15 
 
 225 
 
 3-8730 
 
 58 
 
 3364 
 
 7-6158 
 
 16 
 
 256 
 
 4-0000 
 
 59 
 
 3481 
 
 7-6811 
 
 17 
 
 289 
 
 4-1231 
 
 60 
 
 3600 
 
 7-7460 
 
 18 
 
 324 
 
 4-2426 
 
 61 
 
 3721 
 
 7-8102 
 
 19 
 
 3G1 
 
 4-3589 
 
 62 
 
 3844 
 
 7-8740 
 
 20 
 
 400 
 
 4-4721 
 
 63 
 
 3969 
 
 7-9372 
 
 21 
 
 441 
 
 4-5826 
 
 64 
 
 4096 
 
 8-0000 
 
 22 
 
 484 
 
 4-6904 
 
 65 
 
 4225 
 
 8-0628 
 
 , 23 
 
 529 
 
 4-7958 
 
 66 
 
 4356 
 
 8-1240 
 
 24 
 
 576 
 
 4-8990 
 
 67 
 
 4489 
 
 8-1853 
 
 25 
 
 625 
 
 5-0000 
 
 68 
 
 4624 
 
 8-2462 
 
 26 
 
 676 
 
 5-0990 
 
 69 
 
 4761 
 
 8-3066 
 
 27 
 
 729 
 
 5-196L 
 
 70 
 
 4900 
 
 8-3666 
 
 28 
 
 784 
 
 5-2915 
 
 71 
 
 5041 
 
 8-4261 
 
 29 
 
 841 
 
 5-3852 
 
 72 
 
 5184 
 
 8-4853 
 
 30 
 
 900 
 
 5-4772 
 
 73 
 
 5329 
 
 8-5440 
 
 31 
 
 961 
 
 5-5678 
 
 74 
 
 5476 
 
 8-6023 
 
 32 
 
 1024 
 
 5-6568 
 
 75 
 
 5625 
 
 8-6603 
 
 33 
 
 1089 
 
 5-7446 
 
 76 
 
 5776 
 
 8-7178 
 
 34 
 
 1156 
 
 5-8309 
 
 77 
 
 5929 
 
 8-7750 
 
 35 
 
 1225 
 
 5-9161 
 
 78 
 
 6084 
 
 8-8318 
 
 36 
 
 1296 
 
 6-0000 
 
 79 
 
 6241 
 
 8-8882 
 
 37 
 
 1369 
 
 6-0828 
 
 80 
 
 6400 
 
 8-9443 ; 
 
 38 
 
 1444 
 
 6-1644 
 
 81 
 
 6561 
 
 9-0000 
 
 39 
 
 1521 
 
 6-2450 
 
 82 
 
 6724 
 
 9-0554 i 
 
 40 
 
 1600 
 
 6-3246 
 
 83 
 
 6889 
 
 9-1104 
 
 41 
 
 1681 
 
 6-4031 
 
 84 
 
 7056 
 
 9-1651 
 
 42 
 
 1764 
 
 6-4807 
 
 85 
 
 7225 
 
 9-2195 
 
 43 
 
 1849 
 
 6-5574 
 
 86 
 
 7396 
 
 9-2736 i 
 
 
 
 
 i 
 
 
95 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 ! 87 
 
 7569 
 
 9-3274 
 
 130 
 
 16900 
 
 11-4017 
 
 88 
 
 7744 
 
 9-3808 
 
 131 
 
 17161 
 
 11-4455 
 
 89 
 
 7921 
 
 9-4340 
 
 132 
 
 17424 
 
 11-4891 
 
 90 
 
 8100 
 
 9-4868 
 
 133 
 
 17689 
 
 11-5326 
 
 91 
 
 8281 
 
 9-5394 
 
 134 
 
 17956 
 
 11-5758 
 
 92 
 
 8464 
 
 9-5917 
 
 135 
 
 18225 
 
 11-6189 
 
 93 
 
 8649 
 
 9-6436 
 
 136 
 
 18496 
 
 11-6619 
 
 94 
 
 8836 
 
 9-6954 
 
 137 
 
 18769 
 
 11-7047 
 
 95 
 
 9025 
 
 9-7468 
 
 138 
 
 19044 
 
 11-7473 
 
 96 i 9216 
 
 9-7979 
 
 139 
 
 19321 
 
 11-7898 
 
 97 
 
 9409 
 
 9-8488 
 
 140 
 
 19600 
 
 11-8322 
 
 98 
 
 9604 
 
 9-8995 
 
 141 
 
 19881 
 
 11-8743 
 
 99 
 
 9801 
 
 9-9499 
 
 142 
 
 20164 
 
 11-9164 
 
 100 
 
 10000 
 
 10-0000 
 
 143 
 
 20449 
 
 11-9583 
 
 101 
 
 10201 
 
 10-0499 
 
 144 
 
 20736 
 
 12-0000 
 
 102 
 
 10404 
 
 10-0995 
 
 145 
 
 21025 
 
 12-0416 
 
 103 
 
 10609 
 
 10-1489 
 
 146 
 
 21316 
 
 12-0830 
 
 104 
 
 10816 
 
 10-1980 
 
 147 
 
 21609 
 
 12-1244 
 
 105 
 
 11025 
 
 10-2469 
 
 148 
 
 21904 
 
 12-1655 
 
 106 
 
 11236 
 
 10-2956 
 
 149 
 
 22201 
 
 12-2065 
 
 107 
 
 11449 
 
 10-3440 
 
 150 
 
 22500 
 
 12-2474 
 
 108 
 
 11664 
 
 10-3923 
 
 151 
 
 22801 
 
 12-2882 
 
 109 
 
 11881 
 
 10-4403 
 
 152 
 
 23104 
 
 12-3288 
 
 110 
 
 12100 
 
 10-4881 
 
 153 
 
 23409 
 
 12-3693 
 
 111 
 
 12321 
 
 10-5356 
 
 154 
 
 23716 
 
 12-4097 
 
 112 
 
 12544 
 
 10-5830 
 
 155 
 
 24025 
 
 12-4499 
 
 113 
 
 12769 
 
 10-6301 
 
 156 
 
 24336 
 
 12-4900 
 
 114 
 
 12996 
 
 10-6771 
 
 157 
 
 24649 
 
 12-5300 
 
 115 
 
 13225 
 
 10-7238 
 
 158 
 
 24964 
 
 12-5700 
 
 116 
 
 13456 
 
 10-7703 
 
 159 
 
 25281 
 
 12-6095 
 
 117 
 
 13689 
 
 10-8166 
 
 160 
 
 25600 
 
 12-6491 
 
 118 
 
 13924 
 
 10-8628 
 
 161 
 
 25921 
 
 12-6886 ! 
 
 119 
 
 14161 
 
 10-9087 
 
 162 
 
 26244 
 
 12-7279 , 
 
 120 
 
 14400 
 
 10-9544 
 
 163 
 
 26569 
 
 12-7671 
 
 i 121 
 
 14641 
 
 11-0000 
 
 164 
 
 26896 
 
 12-8062 | 
 
 122 
 
 14884 
 
 11-0454 
 
 165 
 
 27225 
 
 12-8452 
 
 123 
 
 15129 
 
 11-0905 
 
 166 
 
 27556 
 
 12-8841 
 
 124 
 
 15376 
 
 11-1355 
 
 167 
 
 27889 
 
 12-9228 
 
 125 
 
 15625 
 
 11-1803 
 
 168 
 
 28224 
 
 129615 
 
 126 
 
 15876 
 
 11.2250 
 
 169 
 
 28561 
 
 13-0000 
 
 127 
 
 16129 
 
 11-2694 
 
 170 
 
 28900 
 
 13-0384 
 
 128 
 
 16384 
 
 11-3137 
 
 171 
 
 29241 
 
 13-0767 
 
 129 
 
 16641 
 
 11-3578 
 
 172 
 
 29584 
 
 13-1149 1 
 
96 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Koots. 
 
 173 
 
 29929 
 
 13-1529 
 
 216 
 
 46656 
 
 14-6969 
 
 174 
 
 30276 
 
 13-1909 
 
 217 
 
 47089 
 
 14-7309 
 
 175 
 
 30625 
 
 13-2287 
 
 218 
 
 47524 
 
 14-7648 
 
 I 176 
 
 30976 
 
 13-2665 
 
 219 
 
 47961 
 
 14-7986 
 
 177 
 
 31329 
 
 13-3041 
 
 220 
 
 48400 
 
 14-8324 
 
 178 
 
 31684 
 
 13-3417 
 
 221 
 
 48841 
 
 14-8661 
 
 179 
 
 32041 
 
 13-3791 
 
 222 
 
 49284 
 
 14-8997 
 
 180 
 
 32400 
 
 13-4164 
 
 223 
 
 49729 
 
 14-9332 
 
 181 
 
 32761 
 
 13-4536 
 
 224 
 
 50176 
 
 14-9666 
 
 182 
 
 33124 
 
 13-4907 
 
 225 
 
 50625 
 
 15-0000 
 
 183 
 
 33489 
 
 13-5277 
 
 226 
 
 51076 
 
 15-0333 
 
 184 
 
 33856 
 
 13-5647 
 
 227 
 
 51529 
 
 15-0665 
 
 185 
 
 34225 
 
 13-6015 
 
 228 
 
 51984 
 
 15-0997 
 
 186 
 
 34596 
 
 13-6382 
 
 229 
 
 52441 
 
 15-1327 
 
 187 
 
 34969 
 
 13-6748 
 
 230 
 
 52900 
 
 15-1657 
 
 188 
 
 35344 
 
 13-7113 
 
 231 
 
 53361 
 
 15-1987 
 
 189 
 
 35721 
 
 13-7477 
 
 232 
 
 53824 
 
 15-2315 
 
 190 
 
 36100 
 
 13-7840 
 
 233 
 
 54289 
 
 15-2643 
 
 191 
 
 36481 
 
 13-8203 
 
 234 
 
 54756 
 
 15-2970 
 
 192 
 
 36864 
 
 13-8564 
 
 235 
 
 55225 
 
 15-3297 
 
 193 
 
 37249 
 
 13-8924 
 
 236 
 
 55696 
 
 15-3623 
 
 194 
 
 37636 
 
 13-9284 
 
 237 
 
 56169 
 
 15-3948 
 
 195 
 
 38025 
 
 13-9642 
 
 238 
 
 56644 
 
 15-4272 
 
 196 
 
 38416 
 
 14-0000 
 
 239 
 
 57121 
 
 15-4596 
 
 197 
 
 38809 
 
 14-0357 
 
 240 
 
 57600 
 
 15-4919 
 
 198 
 
 39204 
 
 14-0712 
 
 241 
 
 58081 
 
 15-5242 
 
 199 
 
 39601 
 
 14-1067 
 
 242 
 
 58564 
 
 15-5563 
 
 200 
 
 40000 
 
 14-1421 
 
 243 
 
 59049 
 
 15-5885 
 
 201 
 
 40401 
 
 14-1774 
 
 244 
 
 59536 
 
 15-6205 
 
 202 
 
 40804 
 
 14-2127 
 
 245 
 
 60025 
 
 15-6525 
 
 203 
 
 41209 
 
 14-2478 
 
 246 
 
 60516 
 
 15-6844 
 
 204 
 
 41616 
 
 14-2828 
 
 247 
 
 61009 
 
 15-7162 
 
 205 
 
 42025 
 
 14-3178 
 
 248 
 
 61504 
 
 15-7480 
 
 206 
 
 42436 
 
 14-3527 
 
 249 
 
 62001 
 
 15-7797 
 
 207 
 
 42849 
 
 14-3874 
 
 250 
 
 62500 
 
 15-8114 
 
 208 
 
 43264 
 
 14-4222 
 
 251 
 
 63001 
 
 15-8430 
 
 209 
 
 43681 
 
 14-4568 
 
 252 
 
 63504 
 
 15-8745 
 
 210 
 
 44100 
 
 14-4914 
 
 253 
 
 64009 
 
 15-9060 
 
 211 
 
 44521 
 
 14-5258 
 
 254 
 
 64516 
 
 15-9374 
 
 212 
 
 44944 
 
 14-5602 
 
 255 
 
 65025 
 
 15-9687 
 
 213 
 
 45369 
 
 14-5945 
 
 256 
 
 65536 
 
 16-0000 
 
 214 
 
 45796 
 
 14-6287 
 
 257 
 
 66049 
 
 16-0312 
 
 215 
 
 46225 
 
 14-6629 
 
 258 
 
 66564 
 
 16-0624 
 
97 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 259 
 
 67081 
 
 16.0935 
 
 302 
 
 91204 
 
 17-3781 
 
 260 
 
 67600 
 
 16-1245 
 
 303 
 
 91809 
 
 17-4069 
 
 261 
 
 68121 
 
 16-1555 
 
 304 
 
 92416 
 
 17-4356 
 
 262 
 
 68644 
 
 16-1864 
 
 305 
 
 93025 
 
 17-4642 
 
 263 
 
 69169 
 
 16-2173 
 
 306 
 
 93636 
 
 17-4928 
 
 264 
 
 69696 
 
 16-2481 
 
 307 
 
 94249 
 
 17-5214 
 
 265 
 
 70225 
 
 16-2788 
 
 308 
 
 94864 
 
 17-5499 
 
 266 
 
 70756 
 
 16-3095 
 
 309 
 
 95481 
 
 17-5784 
 
 267 
 
 71289 
 
 16-3401 
 
 310 
 
 96100 
 
 17-6068 
 
 268 
 
 71824 ' 
 
 16-3707 
 
 311 
 
 96721 
 
 17-6352 
 
 269 
 
 72361 
 
 16-4012 
 
 312 
 
 97344 
 
 17-6635 
 
 270 
 
 72900 
 
 16-4317 
 
 313 
 
 97969 
 
 17-6918 
 
 271 
 
 73441 
 
 16-4621 
 
 314 
 
 98596 
 
 17-7200 
 
 272 
 
 73984 
 
 16-4924 
 
 315 
 
 99225 
 
 17-7482 
 
 273 
 
 74529 
 
 16-5227 
 
 316 
 
 99856 
 
 17-7764 
 
 274 
 
 75076 
 
 16-5529 
 
 317 
 
 100489 
 
 17-S045 
 
 275 
 
 75625 
 
 16-5831 
 
 318 
 
 101124 
 
 17-8325 
 
 276 
 
 76176 
 
 16-6132 
 
 319 
 
 101761 
 
 17-8606 
 
 277 
 
 76729 
 
 16-6433 
 
 320 
 
 102400 
 
 17-8885 
 
 278 
 
 77284 
 
 16-6733 
 
 321 
 
 103041 
 
 17-9165 
 
 279 
 
 77841 
 
 16-7033 
 
 322 
 
 103684 
 
 17-9444 
 
 280 
 
 78400 
 
 16-7332 
 
 323 
 
 104329 
 
 17-9722 
 
 281 
 
 78961 
 
 16-7630 
 
 324 
 
 104976 
 
 18*0000 
 
 282 
 
 79524 
 
 16-7928 
 
 325 
 
 105625 
 
 18-0277 
 
 283 
 
 80089 
 
 16-8226 
 
 326 
 
 106276 
 
 18-0555 
 
 284 
 
 80656 
 
 16-8523 
 
 327 
 
 106929 
 
 18-0831 
 
 285 
 
 81225 
 
 16-8819 
 
 328 
 
 107584 
 
 18-1108 
 
 286 
 
 81796 
 
 16-9115 
 
 329 
 
 108241 
 
 18-1384 
 
 287 
 
 82369 
 
 16-9411 
 
 330 
 
 108900 
 
 18-1659 
 
 288 
 
 82944 
 
 16-9706 
 
 331 
 
 109561 
 
 18-1934 
 
 289 
 
 83521 
 
 17-0000 
 
 332 
 
 110224 
 
 18-2209 
 
 290 
 
 84100 
 
 17-0294 
 
 333 
 
 110889 
 
 18-2483 
 
 291 
 
 84681 
 
 17-0587 
 
 334 
 
 111556 
 
 18-2757 
 
 292 
 
 85264 
 
 17-0880 
 
 aas 
 
 112225 
 
 18-3030 
 
 293 
 
 85849 
 
 17-1172 
 
 336 
 
 112896 
 
 18-3303 
 
 294 
 
 86436 
 
 17-1464 
 
 337 
 
 113569 
 
 18-3576 
 
 295 
 
 87025 
 
 17-1756 
 
 338 
 
 114244 
 
 18-3848 
 
 296 
 
 87616 
 
 17*2046 
 
 339 
 
 114921 
 
 18-4119 
 
 297 
 
 88209 
 
 17-2337 
 
 340 
 
 115600 
 
 18-4391 
 
 298 
 
 88804 
 
 17-2627 
 
 341 
 
 116281 
 
 18-4662 
 
 299 
 
 89401 
 
 17-2916 
 
 342 
 
 116964 
 
 18-4932 
 
 300 
 
 90000 
 
 17-3205 
 
 343 
 
 117649 
 
 18-5203 
 
 301 
 
 90601 
 
 17-3493 
 
 344 
 
 118336 
 
 18-5472 
 
98 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 345 
 
 119025 
 
 18-5742 
 
 388 
 
 150544 
 
 19-6977 
 
 346 
 
 119716 
 
 18-6011 
 
 389 
 
 151321 
 
 19-7231 
 
 347 
 
 120409 
 
 18-6279 
 
 390 
 
 152100 
 
 19-7484 
 
 348 
 
 121104 
 
 18-6548 
 
 391 
 
 152881 
 
 19-7737 
 
 349 
 
 121801 
 
 18-6815 
 
 392 
 
 153664 
 
 19-7990 
 
 350 
 
 122500 
 
 18-7083 
 
 393 
 
 154449 
 
 19-8242 
 
 351 
 
 123201 
 
 18-7350 
 
 394 
 
 155236 
 
 19-8494 
 
 352 
 
 123904 
 
 18-7617 
 
 395 
 
 156025 
 
 19-8746 
 
 353 
 
 124609 
 
 18-7883 
 
 396 
 
 156816 
 
 19-8997 
 
 354 
 
 125316 
 
 18-8149 
 
 397 
 
 157609 
 
 19-9248 
 
 355 
 
 126025 
 
 18-8414 
 
 398 
 
 158404 
 
 19-9499 
 
 356 
 
 126736 
 
 18-8680 
 
 399 
 
 159201 
 
 19-9750 
 
 357 
 
 127449 
 
 18-8944 
 
 400 
 
 160000 
 
 20-0000 
 
 358 
 
 128164 
 
 18-9209 
 
 401 
 
 160801 
 
 20-0250 
 
 359 
 
 128881 
 
 18-9473 
 
 402 
 
 161604 
 
 20-0499 
 
 360 
 
 129600 
 
 18-9737 
 
 403 
 
 162409 
 
 20-0749 
 
 361 
 
 130321 
 
 19-0000 
 
 404 
 
 163216 
 
 20-0997 
 
 362 
 
 131044 
 
 19-0263 
 
 405 
 
 164025 
 
 20-1246 
 
 363 
 
 131769 
 
 19-0526 
 
 406 
 
 164836 
 
 20-1494 
 
 364 
 
 132496 
 
 19-0788 
 
 407 
 
 165649 
 
 20-1742 
 
 365 
 
 133225 
 
 19-1050 
 
 408 
 
 166464 
 
 EO-1990 
 
 366 
 
 133956 
 
 19-1311 
 
 409 
 
 167281 
 
 20-2237 - 
 
 367 
 
 134689 
 
 19-1572 
 
 410 
 
 168100 
 
 20-2485 
 
 368 
 
 135424 
 
 19-1833 
 
 411 
 
 168921 
 
 20-2731 
 
 369 
 
 136161 
 
 19-2094 
 
 412 
 
 169744 
 
 20-2978 
 
 370 
 
 136900 
 
 19-2354 
 
 413 
 
 170569 
 
 20-3224 
 
 371 
 
 137641 
 
 19-2614 
 
 414 
 
 171396 
 
 20-3470 
 
 372 
 
 138384 
 
 19-2873 
 
 415 
 
 172225 
 
 20-3715 
 
 373 
 
 139129 
 
 19-3132 
 
 416 
 
 173056 
 
 20-3961 
 
 374 
 
 139876 
 
 19-3391 
 
 417 
 
 173889 
 
 20-4206 
 
 375 
 
 140625 
 
 19-3649 
 
 418 
 
 174724 
 
 20-4450 
 
 376 
 
 141376 
 
 19-3907 
 
 419 
 
 175561 
 
 20-4695 
 
 377 
 
 142129 
 
 19-4165 
 
 420 
 
 176400 
 
 20-4939 
 
 378 
 
 142884 
 
 19-4422 
 
 421 
 
 177241 
 
 20-5183 
 
 379 
 
 143641 
 
 19-4679 
 
 422 
 
 178084 
 
 20-5426 
 
 380 
 
 144400 
 
 19-4936 
 
 423 
 
 178929 
 
 20-5670 
 
 381 
 
 145161 
 
 19-5192 
 
 424 
 
 179776 
 
 20-5913 
 
 382 
 
 145924 
 
 19-5448 
 
 425 
 
 180625 
 
 20-6155 
 
 383 
 
 146689 
 
 19-5704 
 
 426 
 
 181476 
 
 20-6398 
 
 384 
 
 147456 
 
 19-5959 
 
 427 
 
 182329 
 
 20-6640 
 
 385 
 
 148225 
 
 19-6214 
 
 428 
 
 183184 
 
 20-6882 
 
 386 
 
 148996 
 
 19-6469 
 
 429 
 
 184041 
 
 20-7123 
 
 387 
 
 149769 
 
 19-6723 
 
 430 
 
 184900 
 
 20-7364 
 
 11 
 
 
 
 
 
 
99 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 431 
 
 185761 
 
 20-7605 
 
 474 
 
 224676 
 
 21-7715 
 
 432 
 
 186624 
 
 20-7846 
 
 475 
 
 225625 
 
 21-7945 
 
 433 
 
 187489 
 
 20-8086 
 
 476 
 
 226576 
 
 21-8174 
 
 434 
 
 188356 
 
 20-8327 
 
 477 
 
 227529 
 
 21-8403 
 
 435 
 
 189225 
 
 20-8566 
 
 478 
 
 228484 
 
 21-8632 
 
 43G 
 
 190096 
 
 20-8806 
 
 479 
 
 229441 
 
 21-8861 
 
 437 
 
 190969 
 
 20-9045 
 
 480 
 
 230400 
 
 21-9089 
 
 438 
 
 191844 
 
 20-9284 
 
 481 
 
 231361 
 
 21-9317 
 
 439 
 
 192721 
 
 20-9523 
 
 482 
 
 232324 
 
 21-9545 I 
 
 440 
 
 193600 
 
 20-9762 
 
 483 
 
 233289 
 
 21-9773 1 
 
 441 
 
 194481 
 
 21-0000 
 
 484 
 
 234256 
 
 22-0000 
 
 442 
 
 195364 
 
 21-0238 
 
 485 
 
 235225 
 
 22-0227 
 
 443 
 
 196249 
 
 21-0476 
 
 486 
 
 236196 
 
 22-0454 
 
 444 
 
 197136 
 
 21-0713 
 
 487 
 
 237169 
 
 22-0689 
 
 445 
 
 198025 
 
 21-0950 
 
 488 
 
 238144 
 
 22-0907 
 
 446 
 
 198916 
 
 21-1187 
 
 489 
 
 239121 
 
 22-1133 
 
 447 
 
 199809 
 
 21-1424 
 
 490 
 
 240100 
 
 22-1359 
 
 448 
 
 200704 
 
 21-1660 
 
 491 
 
 241081 
 
 22-1585 
 
 449 
 
 201601 
 
 21-1896 
 
 492 
 
 242064 
 
 22-1811 
 
 450 
 
 202500 
 
 21-2132 
 
 493 
 
 243049 
 
 22-2036 
 
 451 
 
 203401 
 
 21-2368 
 
 494 
 
 244036 
 
 22-2261 
 
 452 
 
 204304 
 
 21-2603 
 
 495 
 
 245025 
 
 22-2486 j 
 
 453 
 
 205209 
 
 21-2838 
 
 496 
 
 246016 
 
 22-2711 
 
 454 
 
 206116 
 
 21-3073 
 
 497 
 
 247009 
 
 22-2935 
 
 455 
 
 207025 
 
 21-3307 
 
 498 
 
 248004 
 
 22-3159 
 
 456 
 
 207936 
 
 21-3542 
 
 499 
 
 249001 
 
 22-3383 
 
 457 
 
 208849 
 
 21-3776 
 
 500 
 
 250000 
 
 22-3607 
 
 458 
 
 209764 
 
 21-4009 
 
 501 
 
 251001 
 
 22-3830 
 
 459 
 
 210681 
 
 21-4243 
 
 502 
 
 252004 
 
 22-4054 
 
 460 
 
 211600 
 
 21-4476 
 
 503 
 
 253009 
 
 22-4277 
 
 461 
 
 212521 
 
 21-4709 
 
 504 
 
 254016 
 
 22-4499 
 
 462 
 
 213444 
 
 21-4942 
 
 505 
 
 255025 
 
 22-4722 
 
 463 
 
 214369 
 
 21-5174 
 
 506 
 
 256036 
 
 22-4944 i 
 
 464 
 
 215296 
 
 21-5407 
 
 507 
 
 257049 
 
 22-5167 
 
 465 
 
 216225 
 
 21-5639 
 
 508 
 
 258064 
 
 22-5388 1 
 
 466 
 
 217156 
 
 21-5870 
 
 509 
 
 259081 
 
 22-5610 
 
 467 
 
 218089 
 
 21-6102 
 
 510 
 
 260100 
 
 22-5832 I 
 
 1 468 
 
 219024 
 
 21-6333 
 
 511 
 
 261121 
 
 22-6053 
 
 469 
 
 219961 
 
 21-6564 
 
 512 
 
 262144 
 
 22-6274 
 
 470 
 
 220900 
 
 21-6795 
 
 513 
 
 263169 
 
 22-6495 
 
 471 
 
 221841 
 
 21-7025 
 
 514 
 
 264196 
 
 22-6716 | 
 
 472 
 
 222784 
 
 21-7256 
 
 515 
 
 265225 
 
 22-6936 
 
 473 
 
 223729 
 
 21-7486 
 
 516 
 
 266256 
 
 22-7156 
 
100 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. ! 
 
 i 517 
 
 267289 
 
 22-7376 
 
 560 
 
 313600 
 
 23-6643 
 
 1 518 
 
 268324 
 
 22-7596 
 
 561 
 
 314721 
 
 23-6854 
 
 519 
 
 269361 
 
 22-7816 
 
 562 
 
 315844 
 
 23-7065 
 
 520 
 
 270400 
 
 22-8035 
 
 563 
 
 316969 
 
 23-7276 
 
 521 
 
 271441 
 
 22-8254 
 
 564 
 
 318096 
 
 23-7487 
 
 522 
 
 272484 
 
 22-8473 
 
 565 
 
 319225 
 
 23-7697 
 
 523 
 
 273529 
 
 22-8692 
 
 566 
 
 320356 
 
 23-7907 
 
 524 
 
 274576 
 
 22-8910 
 
 567 
 
 321489 
 
 23-8118 
 
 525 
 
 275625 
 
 22-9129 
 
 568 
 
 322624 
 
 23-8327 
 
 526 
 
 276676 
 
 22-9347 
 
 569 
 
 323761 
 
 23-8537 
 
 527 
 
 277729 
 
 22-9565 
 
 570 
 
 324900 
 
 23-8747 
 
 528 
 
 278784 
 
 22-9782 
 
 571 
 
 326041 
 
 23-8956 
 
 529 
 
 279841 
 
 23-0000 
 
 572 
 
 327184 
 
 23-9165 
 
 1 530 
 
 280900 
 
 23-0217 
 
 573 
 
 328329 
 
 23-9374 
 
 531 
 
 281961 
 
 23-0434 
 
 574 
 
 329476 
 
 23-9583 
 
 532 
 
 283024 
 
 2a-0651 
 
 575 
 
 330625 
 
 23-9792 
 
 533 
 
 284089 
 
 23-0868 
 
 576 
 
 331776 
 
 24-0000 
 
 534 
 
 285156 
 
 23-1084 
 
 577 
 
 332929 
 
 24-0208 
 
 535 
 
 286225 
 
 23-1301 
 
 578 
 
 334084 
 
 24-0416 
 
 536 
 
 287296 
 
 23-1517 
 
 579 
 
 335241 
 
 24-0624 
 
 537 288369 
 
 23-1733 
 
 580 
 
 336400 
 
 24-0832 
 
 538 289444 
 
 23-1948 
 
 581 
 
 337561 
 
 24-1039 
 
 539 290521 
 
 23-2164 
 
 582 
 
 338724 
 
 24-1247 
 
 540 
 
 291600 
 
 23-2379 
 
 583 
 
 339889 
 
 24-1454 
 
 541 
 
 292681 
 
 23-2594 
 
 584 
 
 341056 
 
 24-1661 
 
 542 
 
 293764 
 
 23-2809 
 
 585 
 
 342225 
 
 24-1868 
 
 543 
 
 294849 
 
 23-3021 
 
 586 
 
 343396 
 
 24-2074 
 
 544 
 
 295936 
 
 23-3238 
 
 587 
 
 344569 
 
 24-2281 
 
 545 
 
 297025 
 
 23-3452 
 
 588 
 
 345744 
 
 24-2487 
 
 546 
 
 298116 
 
 23-3666 
 
 589 
 
 346921 
 
 24-2693 
 
 547 
 
 299209 
 
 23-3880 
 
 590 
 
 348100 
 
 24-2899 
 
 548 
 
 300304 
 
 23-4094 
 
 591 
 
 349281 
 
 24-3105 
 
 549 
 
 301401 
 
 2, 4307 
 
 592 
 
 350464 
 
 24-3310 
 
 550 
 
 302500 
 
 i3-4521 
 
 593 
 
 351649 
 
 24-3516 
 
 551 
 
 303601 
 
 23-4734 
 
 594 
 
 352836 
 
 24-3721 
 
 552 
 
 304704 
 
 23-4947 
 
 595 
 
 354025 
 
 24-3926 
 
 553 
 
 305809 
 
 23-5159 
 
 596 
 
 355216 
 
 24-4131 
 
 554 
 
 306916 
 
 23-5372 
 
 597 
 
 356409 
 
 24-4336 
 
 555 
 
 308025 
 
 23-5584 
 
 598 
 
 357604 
 
 24-4540 
 
 556 
 
 309136 
 
 23-5796 
 
 599 
 
 358801 
 
 24-4745 
 
 557 
 
 310249 
 
 23-6008 
 
 600 
 
 360000 
 
 24-4949 
 
 558 
 
 311364 
 
 23-6220 
 
 601 
 
 361201 
 
 24-5153 
 
 559 
 
 312481 
 
 23-6432 
 
 602 
 
 362404 
 
 24-5357 
 
101 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 603 
 
 363609 
 
 24-5560 
 
 646 
 
 417316 
 
 25-4165 
 
 604 
 
 364816 
 
 24-5764 
 
 647 
 
 418609 
 
 25-4362 
 
 605 
 
 366025 
 
 24-5967 
 
 648 
 
 419904 
 
 25-4558 
 
 606 
 
 867236 
 
 24-6171 
 
 649 
 
 421201 
 
 25-4755 
 
 607 
 
 368449 
 
 24-6374 
 
 650 
 
 422500 
 
 25-4950 
 
 1 608 
 
 369664 
 
 24-6576 
 
 651 
 
 423801 
 
 25-5147 
 
 609 
 
 370881 
 
 24-6779 
 
 652 
 
 425104 
 
 25-5343 
 
 610 
 
 372100 
 
 24-6982 
 
 653 
 
 426409 
 
 25-5539 
 
 611 
 
 373321 
 
 24-7184 
 
 654 
 
 427716 
 
 25-5734 
 
 612 
 
 374544 
 
 24-7386 
 
 655 
 
 429025 
 
 25-5930 
 
 613 
 
 375769 
 
 24-7588 
 
 656 
 
 430336 
 
 25-6125 
 
 614 
 
 376996 
 
 24-7790 
 
 657 
 
 431649 
 
 25-6320 
 
 615 
 
 378225 
 
 24-7992 
 
 658 
 
 432964 
 
 25-6515 
 
 616 
 
 379456 
 
 24-8193 
 
 659 
 
 434281 
 
 25-6710 
 
 617 
 
 380689 
 
 24-8395 
 
 660 
 
 435600 
 
 25-6905 
 
 618 
 
 381924 
 
 24-8596 
 
 661 
 
 436921 
 
 25-7099 
 
 619 
 
 383161 
 
 24-8797 
 
 662 
 
 438244 
 
 25-7204 
 
 620 
 
 384400 
 
 24-8998 
 
 663 
 
 439569 
 
 25-7488 
 
 621 
 
 385641 
 
 24-9199 
 
 664 
 
 440896 
 
 25-7682 
 
 622 
 
 386884 
 
 24-9399 
 
 665 
 
 442225 
 
 25-7876 
 
 623 
 
 388129 
 
 24-9600 
 
 666 
 
 443556 
 
 25-8070 
 
 624 
 
 389376 
 
 ' 24-9800 
 
 667 
 
 444889 
 
 25-8263 
 
 625 
 
 390625 
 
 25-0000 
 
 668 
 
 446224 
 
 25-8457 
 
 626 
 
 391876 
 
 25-0200 
 
 669 
 
 447561 
 
 25-8650 
 
 627 
 
 393129 
 
 25-0400 
 
 670 
 
 448900 
 
 25-8844 
 
 628 
 
 394384 
 
 25-0600 
 
 671 
 
 450241 
 
 25-9037 
 
 629 
 
 395641 
 
 25-0799 
 
 672 
 
 451584 
 
 25-9230 
 
 630 
 
 396900 
 
 25-0998 
 
 673 
 
 452929 
 
 25-9422 
 
 631 
 
 398161 
 
 25-1197 
 
 674 
 
 454276 
 
 25-9615 
 
 632 
 
 399424 
 
 25-1396 
 
 675 
 
 455625 
 
 25-9808 
 
 633 
 
 400689 
 
 25-1595 
 
 676 
 
 456976 
 
 26-0000 
 
 634 
 
 401956 
 
 25-1794 
 
 677 
 
 458329 
 
 26-0192 
 
 635 
 
 403225 
 
 25-1992 
 
 678 
 
 459684 
 
 26-0384 
 
 636 
 
 404496 
 
 25-2190 
 
 679 
 
 461041 
 
 26-0576 
 
 637 
 
 405769 
 
 25-2389 
 
 680 
 
 462400 
 
 26-0768 
 
 638 
 
 407044 
 
 25-2587 
 
 681 
 
 463761 
 
 26-0960 
 
 639 
 
 408321 
 
 25-2785 
 
 682 
 
 465124 
 
 26-1151 
 
 640 
 
 409600 
 
 25-2982 
 
 683 
 
 466489 
 
 26-1343 
 
 641 
 
 410881 
 
 25-3180 
 
 684 
 
 467856 
 
 26-1534 
 
 642 
 
 412164 
 
 25-3377 
 
 685 
 
 469225 
 
 26-1725 
 
 643 
 
 413449 
 
 25-3574 
 
 686 
 
 470596 
 
 26-1916 
 
 644 
 
 414736 
 
 25-3772 
 
 687 
 
 471969 
 
 26-2107 
 
 645 
 
 416025 
 
 25-3968 
 
 688 
 
 473344 
 
 26-2297 
 
 ! ! 
 
 
 
 i ! 
 
102 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. Squares. 
 
 Square Roots. 
 
 689 
 
 474721 
 
 26-2488 
 
 7S2 
 
 535824 
 
 27-0555 
 
 690 
 
 476100 
 
 26-2678 
 
 733 
 
 537289 
 
 27-0740 
 
 691 
 
 477481 
 
 26-2869 
 
 734 
 
 538756 
 
 27-0924 i 
 
 692 
 
 478864 
 
 26-3059 
 
 735 
 
 540225 
 
 27-1109 
 
 693 
 
 480249 
 
 26-3249 
 
 736 
 
 541696 
 
 27-1293 
 
 j 694 
 
 481636 
 
 26-3439 
 
 737 
 
 543169 
 
 27-1477 
 
 695 
 
 483025 
 
 26-3629 
 
 738 
 
 544644 
 
 27-1662 
 
 696 
 
 484416 
 
 26-3818 
 
 739 
 
 546121 
 
 27-1846 
 
 697 
 
 485809 
 
 26-4008 
 
 740 
 
 547600 
 
 27-2029 
 
 698 
 
 487204 
 
 26-4197 
 
 741 
 
 549081 
 
 27-2213 
 
 699 
 
 488601 
 
 26-4386 
 
 742 
 
 550564 
 
 27-2397 
 
 700 
 
 490000 
 
 26-4575 
 
 743 
 
 552049 
 
 27-2580 
 
 701 
 
 491401 
 
 26-4764 
 
 744 
 
 553536 
 
 27-2764 
 
 702 
 
 492804 
 
 26-4953 
 
 745 
 
 555025 
 
 27-2947 
 
 703 
 
 494209 
 
 26-5141 
 
 746 
 
 556516 
 
 27-3130 
 
 704 
 
 495616 
 
 26-533C 
 
 747 
 
 558009 
 
 27-3313 
 
 705 
 
 497025 
 
 26-5518 
 
 748 I 559504 
 
 27-3496 
 
 706 
 
 498436 
 
 26-5707 
 
 749 
 
 561001 
 
 27-3679 
 
 707 
 
 499849 
 
 26-5895 
 
 750 
 
 562500 
 
 27-3861 
 
 708 
 
 501264 
 
 26-6083 
 
 751 
 
 564001 
 
 27-4044 
 
 709 
 
 502681 
 
 26-6271 
 
 752 
 
 565504 
 
 27-4226 
 
 710 
 
 504100 
 
 26-6458 
 
 753 
 
 567009 
 
 27-4408 
 
 711 
 
 505521 
 
 26-6646 
 
 754 
 
 568516 
 
 27-4591 
 
 712 
 
 506944 
 
 26-6833 
 
 755 
 
 570025 
 
 27-4773 
 
 713 
 
 508369 
 
 26-7021 
 
 756 
 
 571536 
 
 27-4955 
 
 714 
 
 509796 
 
 26-7208 
 
 757 i 573049 
 
 27-5136 
 
 715 
 
 511225 
 
 26-7395 758 i 574564 
 
 27-5318 
 
 716 
 
 512656 
 
 26-7582 759 ! 576081 
 
 27-5500 
 
 717 
 
 514089 
 
 26-7769 760 577600 
 
 27-5681 
 
 718 
 
 515524 
 
 26-7955 761 579121 
 
 27-5862 
 
 719 
 
 516961 
 
 26-8142 ! 762 580644 
 
 27-6043 
 
 720 
 
 518400 
 
 26-8328 
 
 763 582169 
 
 27-6225 
 
 721 
 
 519841 
 
 26-8514 
 
 764 
 
 583696 
 
 27-6405 
 
 722 
 
 521284 
 
 26-8701 
 
 765 
 
 585225 
 
 27-6586 
 
 723 
 
 522729 
 
 26-8887 
 
 766 
 
 586756 
 
 27-6767 
 
 724 
 
 524176 
 
 26-9072 
 
 767 588289 
 
 27-6948 
 
 725 
 
 525625 
 
 26-9258 
 
 768 589824 
 
 27-7128 
 
 726 
 
 527076 
 
 26-9444 
 
 769 591361 
 
 27-7308 i 
 
 727 
 
 528529 
 
 26-9629 
 
 770 592900 
 
 27-7489 j 
 
 728 
 
 529984 
 
 26-9815 
 
 771 594441 
 
 27-7669 
 
 729 
 
 531441 
 
 27-0000 
 
 772 ! 595984 
 
 27-7849 
 
 730 
 
 532900 
 
 27-0185 
 
 773 
 
 597529 
 
 27-8029 
 
 731 
 
 534361 
 
 27-0370 
 
 774 
 
 599076 
 
 27-820S 
 

 103 
 
 A TABLE OF THE SQUARES AND SQUARE ROO'i'S 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 775 
 
 600625 
 
 27-8388 
 
 818 
 
 669124 
 
 28-6007 
 
 776 
 
 602176 
 
 27-8568 
 
 819 
 
 670761 
 
 28-6182 
 
 777 
 
 603729 
 
 27-8747 
 
 820 
 
 672400 
 
 28-6356 
 
 778 
 
 605284 
 
 27-8926 
 
 821 
 
 674041 
 
 28-6531 
 
 ! 779 
 
 606841 
 
 27-9106 
 
 822 
 
 675684 
 
 28-6705 
 
 780 
 
 608400 
 
 27-9285 
 
 823 
 
 677329 
 
 28-6880 
 
 781 
 
 609961 
 
 27-9464 
 
 824 
 
 678976 
 
 28-7054 
 
 782 
 
 611524 
 
 27-9643 
 
 825 
 
 680625 
 
 28-7228 
 
 783 
 
 613089 
 
 27-9821 
 
 826 
 
 682276 
 
 28-7402 
 
 784 
 
 614656 
 
 28-0000 
 
 827 
 
 683929 
 
 28-7576 
 
 785 
 
 616225 
 
 28-0179 
 
 828 
 
 685584 
 
 28-7750 
 
 786 
 
 617796 
 
 28-0357 
 
 829 
 
 687241 
 
 28-7924 
 
 787 
 
 619369 
 
 28-0535 
 
 830 
 
 688900 
 
 28-8097 
 
 788 
 
 620944 
 
 28-0713 
 
 831 
 
 690561 
 
 28-8271 
 
 789 
 
 622521 
 
 28-0891 
 
 832 
 
 692224 
 
 28-8444 
 
 i 790 
 
 624100 
 
 28-1069 
 
 833 
 
 693889 
 
 28-8617 
 
 791 
 
 625681 
 
 28-1247 
 
 834 
 
 695556 
 
 28-8791 
 
 792 
 
 627264 
 
 28-1425 
 
 835 
 
 697225 
 
 28-8964 
 
 793 
 
 628849 
 
 28-1603 
 
 836 
 
 698896 
 
 28-9137 
 
 794 
 
 630436 
 
 28-1780 
 
 837 
 
 700569 
 
 28-9310 i 
 
 795 
 
 632025 
 
 28-1957 
 
 838 
 
 702244 
 
 28-9482 
 
 ! 796 
 
 633616 
 
 28-2135 
 
 839 
 
 703921 
 
 28-9655 
 
 797 
 
 635209 
 
 28-2312 
 
 840 
 
 705600 
 
 28-9828 
 
 798 
 
 636804 
 
 28-2489 
 
 841 
 
 707281 
 
 29-0000 
 
 799 
 
 638401 
 
 28-2666 
 
 842 
 
 708964 
 
 29-0172 
 
 800 
 
 640000 
 
 28-2843 
 
 843 
 
 710649 
 
 29-0345 
 
 801 
 
 641601 
 
 28-3019 
 
 844 
 
 712336 
 
 29-0517 
 
 802 
 
 643204 
 
 28-3196 
 
 845 
 
 714025 
 
 29-0689 
 
 803 
 
 644809 
 
 28-3373 
 
 846 
 
 715716 
 
 29-0861 
 
 804 
 
 646416 
 
 28-3549 
 
 847 
 
 717409 
 
 29-1033 
 
 1 805 
 
 648025 
 
 28-3726 
 
 848 
 
 719104 
 
 29-1204 
 
 806 
 
 649636 
 
 28-3901 
 
 849 
 
 720801 
 
 29-1376 
 
 807 
 
 651249 
 
 28-4077 
 
 850 
 
 722500 
 
 29-1548 
 
 808 
 
 652864 
 
 28-4253 
 
 851 
 
 724201 
 
 29-1719 
 
 809 
 
 654481 
 
 28-4429 
 
 852 
 
 725904 
 
 29-1890 
 
 810 
 
 656100 
 
 28-4605 
 
 853 
 
 727609 
 
 29-2062 
 
 811 
 
 657721 
 
 28-4781 
 
 854 
 
 729316 
 
 29-2233 
 
 812 
 
 659344 
 
 28-4956 
 
 855 
 
 731025 
 
 29-2404 
 
 813 
 
 660969 
 
 28-5132 
 
 856 
 
 732736 
 
 29-2575 
 
 814 
 
 662596 
 
 28-5307 
 
 857 
 
 734449 
 
 29-2746 
 
 815 
 
 664225 
 
 28-5482 
 
 858 
 
 736164 
 
 29-2916 
 
 816 
 
 665856 
 
 28-5657 
 
 859 
 
 737881 
 
 29-3087 
 
 817 
 
 667489 
 
 28-5832 
 
 860 
 
 739600 
 
 29-3258 
 
r=-- _ - 
 
 104 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS : 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 ! 861 
 
 741321 
 
 29-3428 
 
 904 
 
 817216 
 
 30-0666 
 
 862 
 
 743044 
 
 29-3598 
 
 905 
 
 819025 
 
 30-0832 
 
 863 
 
 744769 
 
 29-3769 
 
 906 
 
 820836 
 
 30-0998 
 
 864 
 
 746496 
 
 29-3939 
 
 907 
 
 822649 
 
 30-1164 
 
 865 
 
 748225 
 
 29-4109 
 
 908 
 
 824464 
 
 30-1330 
 
 866 
 
 749956 
 
 29-4279 
 
 909 
 
 826281 
 
 30-1496 
 
 867 
 
 751689 
 
 29-4449 
 
 910 
 
 828100 
 
 30-1662 
 
 868 
 
 753424 
 
 29-4618 
 
 911 
 
 829921 
 
 30-1828 
 
 869 
 
 755161 
 
 29-4788 
 
 912 
 
 831744 
 
 30- 1993 
 
 870 
 
 756900 
 
 29-4958 
 
 913 
 
 833569 
 
 30 2159 
 
 871 
 
 758641 
 
 29-5127 
 
 914 
 
 835396 
 
 30-2324 
 
 872 
 
 760384 
 
 29-5296 
 
 915 
 
 837225 
 
 30-2490 
 
 873 
 
 762129 
 
 29-5466 
 
 916 
 
 839056 
 
 30-2655 
 
 874 
 
 763876 
 
 29-5635 
 
 917 
 
 840889 
 
 30-2820 
 
 875 
 
 765625 
 
 29-5804 
 
 918 
 
 842724 
 
 30-2985 
 
 876 
 
 767376 
 
 29-5973 
 
 919 
 
 844561 
 
 30-3150 
 
 877 
 
 769129 
 
 29-6142 
 
 920 
 
 846400 
 
 30-3315 
 
 878 
 
 770884 
 
 29-6311 
 
 921 
 
 848241 
 
 30-3480 
 
 879 
 
 772641 
 
 29-6479 
 
 922 
 
 850084 
 
 30-3645 
 
 880 
 
 774400 
 
 29-6648 
 
 923 
 
 851929 
 
 30-3809 
 
 881 
 
 776161 
 
 29-6816 
 
 924 
 
 853776 
 
 30-3974 
 
 882 
 
 777924 
 
 29-6985 
 
 925 
 
 855625 
 
 30-4138 
 
 883 
 
 779689 
 
 29-7153 
 
 926 
 
 857476 
 
 30-4302 
 
 884 
 
 781456 
 
 29-7321 
 
 927 
 
 859329 
 
 30-4467 
 
 885 
 
 783225 
 
 29-7489 
 
 928 
 
 861184 
 
 30-4631 
 
 886 
 
 784996 
 
 29-7658 
 
 929 
 
 863041 
 
 30-4795 
 
 887 
 
 786769 
 
 29-7825 
 
 930 
 
 864900 
 
 30-4959 
 
 888 
 
 788544 
 
 29-7993 
 
 931 
 
 866761 
 
 30-5123 
 
 889 
 
 790321 
 
 29-8161 
 
 932 
 
 868624 
 
 30-5287 
 
 890 
 
 792100 
 
 29-8329 
 
 933 
 
 870489 
 
 30-5450 
 
 891 
 
 793881 
 
 29-8496 
 
 934 
 
 872356 
 
 30-5614 
 
 892 
 
 795664 
 
 29-8664 
 
 935 
 
 874225 
 
 30-5778 j 
 
 893 
 
 797449 
 
 29-8831 
 
 936 
 
 876096 
 
 30-5941 
 
 894 
 
 799236 
 
 29-8998 
 
 937 
 
 877969 
 
 30-6105 
 
 895 
 
 801025 
 
 29-9166 
 
 938 
 
 879844 
 
 30-6268 
 
 896 
 
 802816 
 
 29-9333 
 
 939 
 
 881721 
 
 30-6431 
 
 897 
 
 804609 
 
 29-9500 
 
 940 
 
 883600 
 
 30-6594 
 
 1 898 
 
 806404 
 
 29-9666 
 
 941 
 
 885481 
 
 30"i757 
 
 999 
 
 808201 
 
 29-9833 
 
 942 
 
 887364 
 
 30-6920 
 
 900 
 
 810000 
 
 30-0000 
 
 943 
 
 889249 
 
 30-7083 
 
 901 
 
 811801 
 
 30-0167 
 
 944 
 
 891136 
 
 30-7246 
 
 ' 902 
 
 813604 
 
 30-0333 
 
 945 
 
 893025 
 
 30-7409 
 
 903 
 
 815409 
 
 30-0500 
 
 946 
 
 894916 
 
 30-7571 
 
105 
 
 A TABLE OF THE SQUARES AND SQUARE ROOTS 
 
 OF NUMBERS. CONTINUED. 
 
 From 1 to 1000. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 No. 
 
 Squares. 
 
 Square Roots. 
 
 947 
 
 896809 
 
 30-7734 
 
 974 
 
 948676 
 
 31-2090 
 
 948 
 
 898704 
 
 30-7896 
 
 975 
 
 950625 
 
 31-2250 
 
 949 
 
 900601 
 
 30-8058 ' 
 
 976 
 
 952576 
 
 31-2410 
 
 950 
 
 902500 
 
 30-8221 
 
 977 
 
 954529 
 
 31-2570 
 
 951 
 
 904401 
 
 30-8383 
 
 978 
 
 956484 
 
 31-2730 
 
 952 
 
 906304 
 
 30-8545 
 
 979 
 
 958441 
 
 31-2890 
 
 953 
 
 908209 
 
 30-8707 
 
 980 
 
 960400 
 
 31-3050 
 
 954 
 
 910116 
 
 30-8869 
 
 981 
 
 962361 
 
 31-3209 
 
 955 
 
 912025 
 
 30-9031 
 
 982 
 
 964324 
 
 31-3369 
 
 956 
 
 913936 
 
 30-9192 
 
 983 
 
 966289 
 
 31-3528 
 
 957 
 
 915849 
 
 30-9354 
 
 984 
 
 968256 
 
 31-3688 
 
 958 
 
 917764 
 
 30-9516 
 
 985 
 
 970225 
 
 31-3847 
 
 959 
 
 919681 
 
 30-9677 
 
 986 
 
 972196 
 
 31-4006 
 
 960 
 
 921600 
 
 30-9839 
 
 987 
 
 974169 
 
 31-4166 
 
 961 
 
 923521 
 
 31-0000 
 
 988 
 
 976144 
 
 31-4325 
 
 962 
 
 925444 
 
 31-0161 
 
 989 
 
 978121 
 
 31-4484 
 
 963 
 
 927369 
 
 31-0322 
 
 990 
 
 980100 
 
 31-4643 
 
 964 
 
 929296 
 
 31-0483 
 
 991 
 
 982081 
 
 31-4802 
 
 965 
 
 931225 
 
 31-0644 
 
 992 
 
 984064 
 
 31-4960 
 
 966 
 
 933156 
 
 31-0805 
 
 993 
 
 986049 
 
 31-5119 ! 
 
 967 
 
 935089 
 
 31-0966 
 
 994 
 
 988036 
 
 31-5278 , 
 
 968 
 
 937024 
 
 31-1127 
 
 995 
 
 990025 
 
 31-5436 
 
 969 
 
 938961 
 
 31-1288 
 
 996 
 
 992016 
 
 31-5595 
 
 970 
 
 940900 
 
 31-1448 
 
 997 
 
 994009 
 
 31-5753 i 
 
 971 
 
 942841 
 
 31-1609 
 
 998 
 
 996004 
 
 31-5911 
 
 972 
 
 944784 
 
 31-1769 
 
 999 
 
 998001 
 
 31-6070 
 
 973 
 
 946729 
 
 31-1929 
 
 1000 
 
 1000000 
 
 31-6228 
 
 
106 
 
 TABLE OF SLOPES, &c. For TOPOGRAPHY. 
 
 
 Vertical Rise 
 
 Horizontal 
 
 
 Vertical Rise 
 
 Horizontal 
 
 Degrees. 
 
 in 100 feet 
 horizontal. 
 
 Distance to a 
 rise of 10 feet. 
 
 Degrees. 
 
 in 100 feet 
 horizontal. 
 
 Distance to a 
 rise of 10 feet. 
 
 
 
 1 
 
 1-75 
 
 572-9 
 
 o 
 
 19 
 
 34-43 
 
 29-0 
 
 2 
 
 3-49 
 
 286 : 4 
 
 20 
 
 36-40 
 
 27-5 
 
 3 
 
 5-24 
 
 190-8 
 
 21 
 
 38-40 
 
 26-0 
 
 4 
 
 6-99 
 
 143-0 
 
 22 
 
 40-40 
 
 24-7 
 
 5 
 
 8-75 
 
 114-3 
 
 23 
 
 42-45 
 
 23-5 
 
 6 
 
 10-51 
 
 95-1 
 
 24 
 
 44-52 
 
 22-4 
 
 7 
 
 12-28 
 
 81-4 
 
 25 
 
 46-63 
 
 21-4 
 
 8 
 
 14-05 
 
 71-2 
 
 26 
 
 48-77 
 
 20-5 
 
 9 
 
 15-83 
 
 63-1 
 
 27 
 
 50-95 
 
 19-6 
 
 10 
 
 17-63 
 
 56-7 
 
 28 
 
 53-17 
 
 18-8 
 
 11 
 
 19-44 
 
 51-4 
 
 29 
 
 55-43 
 
 18-0 
 
 12 
 
 21-25 
 
 47-0 
 
 30 
 
 57-73 
 
 17-3 
 
 13 
 
 23-09 
 
 43-3 
 
 35 
 
 70-02 
 
 14-2 
 
 14 
 
 24-93 
 
 40-1 
 
 40 
 
 83-91 
 
 11-9 
 
 15 
 
 26-79 
 
 37-3 
 
 45 
 
 100-00 
 
 10-0 
 
 16 
 
 28-67 
 
 34-9 
 
 50 
 
 119-17 
 
 8-4 
 
 17 
 
 30-57 
 
 32-7 
 
 55 
 
 142-81 
 
 7-0 
 
 ]8 
 
 32-49 
 
 30-7 
 
 60 
 
 173-20 
 
 5-7 
 
 ; 1^ ! 
 
 R. B. WEARS, STEREOTYPER. 
 
 
 
AN INITIAL FINE OF 25 CENTS 
 
 OVERDUE. 
 
p 
 
 YB 5360! 
 
 2/y 
 
 W~ - 
 
 _