± ---- s=2 ）， «： !== © © *（-- © ± •）: £ € }， ¿№. *** • ** * * • • • • • • ， . ſºlºiſillilull - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -a - - - - - - - - - - - - - - - w ºutnutiustituºusºulºulºuſºu. ſſ | | | | ! }} º. ſae XXIV*I th IT \\W\\ ºs ſuae， §§§ ſae Zº º ſí #ſº /2ºº2 | rity ºf *ae ſae ºl * . . . . .,…;x&raeºſ?” ‘’******|- ::：*（ſººſ ’”+ ', ！ -、、。ș&#*** „ægsæſºº’ſ LIBRARY ·|-* · * . --**· · \----* *:, ，- *---* --， -„• } ~^ ---- ~** ******! • į. -* |-~~ ~ ~„eſ“ * & -} | TRA WSPORT yº - / Transportation . Library . . " 'FI E L D-Boo w 4-7 mm; K - - - --mm * RAILROAD ENGINEERS. Fo R M U L E For LAYING OUT curves, DETERMINING FROG ANGLES, LEVELLING, CALCULATING EARTH-WORK, ETC., ETC., TOGETHER WITH T A B L E S of RADI1, ORDINATES, DEFLECTIONS, LONG CHORDS, LoGARITHMS, LOGA- RITHMIC AND NATURAL SINES, TANGENTS, ETC., ETC, .* º. \\º Jo HN B. HENGK, A. M., c 1 W 11, ENG IN E E R , REVISED EDITION. - NEW YORK : D. A P P L E TO N A N D C O M P A NY, 1, 3, AND 5 BOND STREET. LO N DO N : 1 6 LIT T L E B R ITA I N. 1885. * irsnsp0ttation { }}}rarſ - 93. § 2"| ".. , H 4 ºf ! zy P- ENTERED, according to Act of Congress, in the year 1854, By D. APPLETON & CO., In the Clerk's Office of the District Court of the United States for the Southern District of New York. ENTERED, according to Act of Congress, in the year 1881, By D. APPLETON & CO., In the Office of the Librarian of Congress, at Washington. o ; ; ; i © P R E F A C E . The object of the present work is to supply a want very generally felt by Assistant Engineers on Railroads. Books of convenient form for use in the field, containing the ordi- nary logarithmic tables, are common enough ; but a book combining with these tables others peculiar to railroad work, and especially the necessary formulae for laying out curves, turnouts, crossings, &c., is yet a desideratum. These formulae, after long disuse perhaps, the engineer is often called upon to apply at a moment's notice in the field, and he is, therefore, obliged to carry with him in manuscript such methods as he has been able to invent or collect, or resort to what has received the very appropriate name of “fudging.” This the intelligent engineer always considers a reproach; and he will, therefore, it is hoped, receive with favor any attempt to make a resort to it inex- cusable. Besides supplying the want just alluded to, it was thought that some improvements upon former methods might be made, and some entirely new methods introduced. Among the processes believed to be original may be specified those in §§ 41 –48, on Compound Curves, in Chapter II. on Parabolic Curves, in §§ 106 – 109, on Vertical Curves, and in the article on Excavation and Embankment. It is vi PREFACE. but just to add, that a great part of what is said on Reversed Curves, Turnouts, and Crossings, and most of the Miscel- laneous Problems, are the result of original investigations. In the remaining portions, also, many simplifications have been made. In all parts the object has been to reduce the operation necessary in the field to a single process, indi- cated by a formula standing on a line by itself, and distin- guished by a ſº. This could not be done in all cases, as will be readily seen on examination. Certain preliminary steps were sometimes necessary, and these, whenever it was practicable, have been indicated by words in italics. Of the methods given for Compound Curves, that in § 46 will be found particularly useful, from the great variety of applications of which it is susceptible. Methods of laying out Parabolic Curves are here given, that those so disposed may test their reputed advantages. Two things are certainly in their favor; they are adapted to unequal as well as equal tangents, and their curvature generally decreases towards both extremities, thus making the transition to and from a straight line easier. Some labor has been given to devising convenient ways of laying out these curves. The method of determining the radius of curvature at certain points is believed to be entirely new. Better processes, however, may already exist, par- ticularly in France, where these curves are said to be in general use. The mode of calculating Excavation and Embankment here presented, will, it is thought, be found at least as sim- ple and expeditious as those commonly used, with the ad- vantage over most of them in point of accuracy. The usual Tables of Excavation and Embankment have been omitted. To include all the varieties of slope, width of road-bed, and depth of cutting, they must be of great extent, and unfitted PREFACE. vii for a field-book. Even then they apply only to ground whose cross-section is level, though often used in a manner shown to be erroneous in § 128. When the cross-section of the ground is level, the place of the tables is supplied by the formula of § 119, and whén several sections are calcu- lated together, as is usually the case, and the work is ar- ranged in tabular form, as in § 120, the calculation is be- lieved to be at least as short as by the most extended tables. The correction in excavation on curves (§ 129) is not known to have been introduced elsewhere. In a work of this kind, brevity is an essential feature. The form of “Problem ’’ and “Solution” has, thereforo, been adopted, as presenting most concisely the thing to be done and the manner of doing it. Every solution, how- ever, carries with it a demonstration, which is deemed an equally essential feature. These demonstrations, with a few unavoidable exceptions, principally in Chapter II., pre- suppose a knowledge of nothing beyond Algebra, Geome- try, and Trigonometry. The result is in general expressed by an algebraic formula, and not in words. Those familiar with algebraic symbols need not be told how much more intelligible and quickly apprehended a process becomes when thus expressed. Those not familiar with these sym- bols should lose no time in acquiring the ready use of a language so direct and expressive. It may be remarked that it was no part of the author's design to furnish a col- lection of mere “rules,” professing to require only an abil, ity to read for their successful application. Rules can sel- dom be safely applied without a thorough understanding of the principles on which they rest, and such an understand- ing, in the present case, implies a knowledge of algebraic formula3. - The tables here presented will, it is hoped, prove relia- viii PREFACE. j V ble. Those specially prepared for this work have been computed with great care. The values have in some cases been carried out farther than ordinary practice requires, in order that interpolated values may be obtained from thern more accurately. The remaining tables have been care- fully examined by comparing them with others of approved reputation for accuracy. Many errors have in this way been detected in some of the tables of corresponding ex- tent in general use, particularly in the Tables of Squares, Cubes, etc., and the Tables of Logarithmic and Natural Sines, Cosines, etc. The number of tables might have been greatly increased, but for an unwillingness to insert any thing not falling strictly within the plan of the work or not resting on sufficient authority. J. B. H. BosTon, February, 1854. Up to the present time more than sixteen thousand copies of this work have been issued, without any change from the first edition, except the correction of a few typo- graphical errors. Certain changes and additions have now been made. Tables VIII and IX, which had lost some of their usefulness, have been replaced by tables for computing heights by the aneroid barometer, and for the comparison of English and French weights and meas- ures. Tables I and X have been slightly changed, and a new table, Table XVII, added. An Appendix has also been added, containing twelve articles, the titles of which may be seen on page xy. They treat of subjects sug- gested by later experience or by engineers of repute, and they will, it is hoped, give increased value to the book. J. B. H. August, 1881. TABLE OF CON TENTS. CIIAPTER I. CIRCULAR CURWES. ARTICLE I. — SIMPLE CURWEs . Definitions. Propositions relating to the circle Angle of intersection and radius given, to find the tangent . Angle of intersection and tangent given, to find the radius . Degree of a curve -> - º t Deflection angle of a curve . - o º A. Method by Deflection Angles. 9. Radius given, to find the deflection angle - e 10. Deflection angle given, to find the radius e º * ll. Angle of intersection and tangent given, to find the deflection - angle - - te º 4 e e º 12. Angle of intersection and deflection angle given, to find the tangent e e e • * * 13. Angle of intersection and deflection angle given, to find the - length of the curve . - - s e 14 Deflection angle given, to lay out a curve . © ſº w 16. To find a tangent at any station . tº - o º e B. Method by Tangent and Chord Deflections, 17. Definitions . - . ſº * º - e 18. Radius given, to find the tangent deflection and chord deflection 19. Deflection angle given, to find the chord deflection . 21. To find a tangent at any station tº G e 22. Chord deflection given, to lay out a curve • * * ! 4. {} 6 TA BLE OF CONTENTS. SECT 24. 25. 26. 27. 7 29. 30. 3] 4 l. 42. 44. 45. 46. 47. 48 C. Ordinates. Definition . & Deflection angle or radius given, to find ordinates Approximate value for middle ordinate . & Method of finding intermediate points on a curve approxi- mately D. Curving Rails. Deflection angle or radius given, to find the ordinate for curv- ing rails - - º º ARTICLE II. — REVERSED AND COMPOUND CURw cs. Definitions . © º • - e r Radii or deflection angles given, to lay out a reversed or com- pound curve & I- t g e & º A. Reversed Curves. . Reversing point when the tangents are parallel . º . To find the common radius when the tangents are parallel . One radius given, to find the other when the tangents are par- allel e & º º e º º e . Chords given, to find the radii when the tangents are parallel . Iładii given, to find the chords when the tangents are parallel . Common radius given, to run the curve when the tangents are not parallel , e e e º º . One radius given, to find the other when the tangents are not parallel . tº 0. º . To find the common radius when the tangents are not parallel . Second method of finding the common radius when the tan- gents are not parallel . e te B. Compound Curves. Common tangent point . º o To find a limit in one direction of each radius © One radius given, to find the other . º © º ºs Second method of finding one radius when the other is given To find the two radii . e & º To find the tangents of the two branches Q e Second method of finding the tangents of the two branches PA(\{’ 1 l | 1 I 3 | 6 16 17 18 1S 19 19 2] 22 23 2 | 25 26 27 29 30 TABLE OF CONTENTS. XI ARTICLE III. — TURNOUTS AND CROSSINGs. SECT. PA'iR 49. Definitions - & * * & tº . 3.} A. Turnout from Straight Lines. 50. Radius given, to find the frog angle and the position of the frog 32 51. Frog angle given, to find the radius and the position of the frog 33 52. To find mechanically the proper position of a given frog . 34 53. Turnouts that reverse and become parallel to the main track 34 54. To find the second radius of a turnout reversing opposite the rog * & wº - * & © S5 B. Crossings on Straight Lines. 55, References to proper problems te * 36 56. Radii given, to find the distance between switches s. $6 C. Turnout from Curves. 57. Frog angle given, to find the radius and the position of the frog 88 58. To find mechanically the proper position of a given frog . 4 i 59 Proper angle for frogs that they may come at the end of a rail 4 i 60 Radius given, to find the frog angle and the position of the frog 42 62 Turnout to reverse and become parallel to the main track , 44 D. Crossings on Curves. 63. References to proper problems ſe e & * - ~5 64. Common radius given, to find the central angles and chords 45 ARTICLE IV. — MISCELLANEOUS I’RobleMs. b5. To find the radius of a curve to pass through a given point 46 06. To find the tangent point of a curve to pass through a given point * º & * * * * 47 t;7. To find the distance to the curve from any point on the tan- gent. ſº & & * * e 47 ū8 Second method for passing a curve through a given point . 47 69. To find the proper chord for any angle of deflection . 48 70. To find the radius when the distance from the intersection point to the curve is given e sº e * * 48 71 To find the distance from the intersection point to the curve when the radius is given - * 49 kii TABLE OF CONTENTS SECT 72. 73. 74. 75. 76. . To change the tangent point of a curve, so that it may pass 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 89. 90. 95 To find the taugent point of a curve that shall pass through a given point tº e tº To find the radius of a curve without measuring angles To find the tangent points of a curve without measuring an- gles º To find the angle of intersection and the tangent points when the point of intersection is inaccessible . To lay out a curve when obstructions occur through a given point. o e © • © To change the radius of a curve, so that it may terminate in a tangent parallel to its present tangent . e To find the radius of a curve on a track already laid . To draw a tangent to a given curve from a given point . To flatten the extremities of a sharp curve . © g To locate a curve without setting the instrument at the ta gent point To measure the distance across a river C[IAPTER II. PARABOLIC CURWES. ARTICLE I. — Locating PARAbolio CURWEs, Propositions relating to the parabola . To lay out a parabola by tangent deflections To lay out a parabola by middle ordinates To draw a tangent to a parabola. To lay out a parabola by bisecting tangents To lay out a parabola by intersections ARTICLE II. — RADI Us of CURVATURE. - Definition 93. To find the radius of curvature at certain stations . Simplification when the tangents are equal TAGS, {U} 5] 52 52 55 56 57 5S 59 59 60 63 65 66 67 67 68 69 7] 7] 76 TABLE of contexts. X111 CILAPTER III. LEVELLING, ARTICls I. — IIEIGHTS AND SLOPE STARE8 SECT PAGE 96. Definitions & e e º tº º © & 7 97. To find the difference of level of two points . 78 98. Datum plane . & e tº e e º w º 79 99. To find the heights of the stations on a line tº º , 80 100. Sights denominated plus and minus . . © . Sl 101. Form of field notes * • * • e . 82 102. To set slope stakes & ſº t e e 82 ARTICLE II. — Correction For Tri E EART ii's CURVAT cre AND For REFRACTION. l()3. Earth’s ct:rvature , - - iº º ſº & º . 84 104. Refraction . º e e © ſº * 84 105. To find the correction for curvature and refraction & 85 ARTI cle III. — VERTICAL CURVEs. 106. Manner of designating grades - - 86 107. To find the grades for a vertical curve at whole stations S6 109. To find the grades for a vertical curve at sub-stations 88 ARTICLE IV. — FLEvATION OF THE OUTER RAIL on CU R v Es. ! 10. To find the proper elevation of the outer rail 89 111. Coning of the wheels . e - 89 * CIIAPTER IV. EART II - W. O. R. R. ARTICLE I. — l'Rission DA i. For M tº L.A. 112 Definition of a prismoid - t , 92 113. To find the solidity of a prisinoid 92 A RTICLE II – Borrow-Pits. 114 Manner of dividing the ground . . e 33 Riv TABLE OF CONTENTS. SECT. - PAGE 115. To find the solidity of a vertical prism whose horizontal sec- tion is a triangle . . © © e © tº - 93 | 16. To find the solidity of a vertical prism whose horizontal sec- tion is a parallclogram e & - o º e 94 117. To find the solidity of a number of adjacent prisms having the same horizontal section e tº tº º 95 ARTICLE III. — ExcAvATION AND EMBAN KMENT. A. Centre IIeights alone given. 119. To find the solidity of one section 97 120. To find the solidity of any number of successive sections 98 D. Centre and Side Heights given. 121. Mode of dividing the ground \}'} 122. To find the solidity of one section . e º e | 00 123. To find the solidity of any number of successive sections 104 125. To find the solidity when the section is partly in excavation and partly in embankment . , 105 126. Beginning and end of an excavation . 107 C. Ground very Irregular. 127. To find the solidity when the ground is very irregular 108 128. Usual modes of calculating excavation 109 - D. Correction in Ercavation on Curves. 129. Nature of the correction º º © o | 10 130. To find the correction in excavation on curves . . . 12 132. To find the correction when the section is partly in excava tion and partly in embankment | 13 A PP E N DI X. - - PAGR, I. Note on laying out Curves by Deflection Angles . , l l 5 II. Laying out Curves by Tangent Deflections . º . 116 III, Turnouts Tangent to Main Line . 4. , 117 TABLE OF CONTENTS, NO, IV. WI. VII, VIII. IX. X. XI. XII. N 0. II. III. IV. VI. VII. VIII. IX. IXa. X. XI. XII. XIII. XIV. XV. XVI. XVII. Double Turnouts Length of Parabolic Arcs Note on setting Slope Stakes . Expansion of Rails Easing Grades on Curves Transition Curves To Change a Tangent Point so that the Tangent may pass through a Given Point * © º e To connect Two Curves by a Common Tangent Note on the Computation of Earth-work . o T A B L E S. Radii, Ordinates, Tangent and Chord Deflections, and Or- dinates for Curving Rails Long Chords . • - Correction for the Earth's Curvature and for Refraction Elevation of the Outer Rail on Curves Frog Angles, Chords, and Ordinates for Turnouts . Length of Circular Arcs in Parts of Radius Expansion by Iſeat Heights by Aneroid Barometer . IIeights by Aneroid Barometer º º Comparison of French and English Weights and Measures Trigonometrical and Miscellaneous Formulae . Squares, Cubes, Square Roots, Cube Roots, and Recip- rocals • e Logarithms of Numbers º - - Logarithmic Sines, Cosines, Tangents, and Cotangents Natural Sines and Cosines Natural Tangents and Cotangents Rise per Mile of Various Grades º Tangents and Shortest Distanees from Intersection of a One-degree Curve Point PAGE F. X P L A N A TI O N () F S I GNS , TH E sign + indicates that the quantities between which it is prºced àle to be added together. - The sign — indicates that the quantity before which it iſ placed is to be subtracted. The sign x indicates that the quantities between which it is placed are to be multiplical together. The sign -- or indicates that the first of two quantities between which it is placed is to be divided by the second. The sign = indicates that the quantitics between which it is placed uſe equal. Tho sign co indicates that the difference of the two quantities be- tween which it is placed is to be taken The sign . . . stands for the word “hence" or “therefore.” The ratio of one quantity to another may be regarded as the quo. tient of the first divided by the second. Hence, the ratio of a to b is expressed by a b, and the ratio of c to d by c : d. A proportion ex presses the equality of two ratios. IIence, proportion is represented by placing the sign = between two ratios; as, a b = c : d. In the text and in the tables the foot has becn taken as the unit of aneasure when no other unit is specified. FI E L D-B O O K. CHAPTER I. CIRCULAR CURVES. ARTICLE I. — SIMPLE CURWEs 1. Tire railroad curves here considered are either Circulal or Para bolic. Circular curves are divided into Simple, Reversed, and Com pound Curves. We begin with Simple Curves. 2. Let the arc A D E FB (fig. 1) represent a railroad curve, unit K 2 CIRCULAR CU RV ES. ing the straight lines GA and B H. The length of such a curve is measured by chords, each 100 feet long.” Thus, if the chords A D, DE, E F, and F B are each 100 feet in length, the whole curve is said to be 400 feet long. The straight lines G A and B H are always tangent to the curve at its extremities, which are called tangent points. If G A and B H are produced, until they meet in C, A C and B C are called the tangents of the curve. If A C is produced a little beyond C to K, the angle K C B, formed by one tangent with the other pro- duced, is called the angle of intersection, and shows the change of direc. tion in passing from one tangent to the other. The following propositions relating to the circle are derived from Geometry. I. A tangent to a circle is perpendicular to the radius drawn through the tangent point. Thus, A C is perpendicular to A O, and B C to B O. II. Two tangents drawn to a circle from any point are equal, and iſ a chord be drawn between the two tangent points, the angles between this chord and the tangents are cqual. Thus A C = B C, and the angle B A C = A B C. III. An acute angle between a tangent and a chord is equal to half the central angle subtended by the same chord. Thus, C A B = # A O B. IV. An acute angle subtended by a chord, and having its vertex in the circumference of a circle, is equal to half the central angle sub- tended by the same chord. Thus, D A E = 3 D 0 E. V. Equal chords subtend equal angles at the centre of a circle, and also at the circumference, if the angles are inscribed in similar seg- ments. Thus, A O D = D O E, and D A E = E A F VI. The angle of intersection of two tangents is equal to the cen. tral angle subtended by the chord which unites the tangent points. Thus, K C B = A O B. 3. In order to unite two straight lines, as G A and B H, by a curve, the angle of interscetion is measured, and then a radius for the curve may be assumed, and the tangents calculated, or the tangents may be assumed of a certain length, and the radius calculated. * Some engineers prefer a chain 50 feet in length, and measure the length of a curve by chords of 50 instead of 100 feet. The chord of 100 fect has been adopted throughout this article; but the formulae deduced may be very readily unodiſied to suit chords of auv length. See also $13 SIMPLE CURW’ ES. - 3 4 Problem. Given the angle of intersection K C B = 1 (fig 1) and the radius A O = R, to find the tangent A C = T. K C Fig 1 Solution. Draw CO. Then in the right triangle A O C we have (Tab. X. 3) #. = tan. A 0 C, or, since A O C = \ 1 ($ 2, VI.) T R = tan. # I; [[F ... T = R tan. A 1. Erample. Given 1 = 22° 52', and R = 3000, to find T. Here R = 3000 3.477 121 & I = 1 1926, tan, 9.305S69 T = 606 72 2.782990 5. Problem. Given the angle of intersection K C B = 1 (fig 1), and the tangent A C = T, to find the radius A O == R 4 CIRCULAR C U R V ES. Solution. In the right triangle A O C we have (Tab. X 6) A O R A 6 - cot. A O C, or # = cot. A I; ſº ... R = T cot. § I. Erample. Given I = 31° 16' and T = 950, to find R. Here T = 950 2.97 7724 § 1 = 15° 38 cot. 0.553 I O2 IR = 339.4.89 3.530S26 6. The degree of a curve is determined by the angle subtended at its centre by a chord of 100 feet. Thus, if A O D = 6° (fig. 1). A D / F B is a 6° curve. 7. The deſiection angle of a curve is the acute angle formed at any point between a tangent and a chord of 100 feet. The deflection angle is, therefore (§ 2, III.), half the degree of the curve. Thus, CA D or C B F is the deflection angle of the curve A DE FB, and is half A O D or half F O B. A Method by Deflection Angles. 8. The usual method of laying out a curve on the ground is by means of deflection angles. 9. Problem. Given the radius A O = R (fig. 1), to find the de- flection angle C B F = D. Solution. Draw O L perpendicular to B F. Then the angle B O L = , B O F = D, and B L = } B F = 50. But in the right triangle B L O B L we have (Tab. X. 1) sin. B O L = Foº 50 ſº sin. D = * /* Erample. Given R = 57 29.65, to find D. Here - 50 1.698970 IR = 5729.65 3.758 128 D = 30! sin. 7.940842 Hence a curve of this radius is a 1° curve, and its deflection angle is 30!, 10. Problem. Given the deflection angle C B F = D (fig. 1), to jind the radius A O = R. METHOD BY lyk, FLECTION ANGL ES. 5 - - º º 50 Solution. By the preceding section we have sin. D = R’ whence f' sin. D = 50 ; 50 IGP ... R = .75. By this formula the radii in Table I. are calculated. Erample Given D = 1°, to find R IIere 50 1.69S970 D = 1° sin. 8 24 1855 R = 2S64.93 3.457 l l 5 li. Problemr. Given the angle of intersection KCB = I (fig. 1), and the tangent A C = T, to find the deflection angle CA D = D. 50 R and from $ 5, 18 = T' cot. # 1. Substituting this value of R in the first equation, we get 50 . T cot. I’ ſº . . . sin. D = Solution. From S 9 we have sin. D = Sin. D = 50 tan. # I T Erample. Given I = 21° and T = 424.8, to find D. Here 50 1,69S 970 # I = 10° 30 tan. 9,2679t,7 0.9660S / T = 424.8 2.628 l85 IJ = 1° 15' sin 8.338752 12. Problem. Given the angle of intersection K C B = 1 (fig 1) and the deflection angle C A D = D, to find the tangent A C = T. - - ... tº e 50 tº l Solution. From the preceding section wo have sin. L = wº ſIence, Tsin. D = 50 tan. I; ſº - •'. _50 tan. A 1. sin. D Erample. Given 1 = 28° and D = 1°, to find T. Here 0 tan. 14° = **** = i is al. sin lº 6 CI RC U LA R C U R W H. S. 13. Probleman. Given the angle of interse twon KCB = I (fig. 1), and the deflection angle CA D = D, to find the length of the curve. Solution. By $ 2 the length of a curve is measured by chords of 100 feet applied around the curve. Now the first chord A D makes with the tangent A C an angle C A D = D, and each succeeding chord D E, E F. &c. subtends at A an additional angle D A E, EA F. &c. each equal to D; since each of these angles (§ 2, IV.) is half of a central angle subtended by a chord of 100 feet. The angle C A B = # A O B = # I is, therefore, made up of as many times D, as there are chords around the curve. Then if n represcnts the number of chords, we have n D = } I; gº . n = H .. D If D is not contained an even number of times in A 1, the quotient above will still give the length of the curve. Thus, in ſig. 2, suppose D is contained 45 times in I. This shows that there will be four whole chords and § of a chord around the curve from A to B. The angle G A B, the fraction of D, is called a sub-deflection angle, and G B, the fraction of a chord, is called a sub-chord.* The length of the curve thus found is not the actual length of tho arc, but the length required in locating a curve. If the actual length of the arc is required, it may be found by means of Table VI. Erample. Given I = 16° 52' and D = 1° 20', to find the length of So 26, 506 * the curve. Here n = º = 15.7 = s.ſ. i 6.325, that is, the curve is 6325 feet long. To find the arc itself in this example, we take from Table VI. the length of an are of 16° 52', since the central angle of the whole curve is cqual to I (§ 2, VI), and multiply this length by the radius of the Cllr Ye, Arc 10° = .1745329 “ GP = .1047 198 * 50/ = .01 .45444 “ 2/ = .0005 Sl 8 “ 16O 52' = .294.3789 * This method of finding the length of a sub-chord is not mathematically accu- rate; for, by geometry, angles inscribed in a circle are proportional to the arcs ol. which they stand; whereas this method supposes them to be proportional to the chords of these arcs. In railroad curves, the error arising from this supposition i ; too small to be regart'ed MET1LC D BY DEFLECTION ANGLES. 7 The radius of the curve is found from Table I. to be 2148.79, and this multiplied by .2943789 gives 632.558 feet for the length of the arc. 14. Problem. Given the deflection angle D, to lay out a curve fiom a given tangent point. Ağ. Il Solution. Let A (fig. 2) be the given tangent point in the tangent H. C. Set the instrument at A, and lay off the given deflection angle D from A. C. This will give the direction A D, and 100 feet being measured from A in this direction, the point D will be determined Lay off in succession the additional angles D A E, E A F, &c., each equal to D, and make DE, E F. &c. each 100 feet, and the points E, F, &c. will be determined. The points D, E, F, &c., thus deter- mined, are points on the required curve (§ 7, and § 2, III., IV), and are called stations. - If there is a sub-chord at the cnd, as G B, the sub-deficetion angle (; A B must be the same part of D that G B is of a whole chord (§ 13). 15. It is often impossible to lay out the whole of a curve, without removing the instrument from its first position, either on account of the great length of the curve, or because some obstruction to the sight may be met with. In this case, after determining as many stations as possible, and removing the instrument to the last of these stations, we uught to be able to find the tangent to the curve at this station , for 8 . CIRCULAR CUIRVES. then the curve could be continued by deflections from the new tangent in precisely the same way as it was begun from the first tangent. 16. Problem. After running a curve a certain number of stutions, to find a tangent to the curve at the last station. Solution. Suppose that the curve (fig. 2) has been run three stations to F, and that F L is the tangent required. Produce A F to K, and we have the angle K F L = A FC. But (§ 2, II.) A F C = FA C. Therefore K F L = FA C. Now I' A C is the sum of all the deflec- tion angles laid off from the tangent at A, that is, in this case, FA C = 3 D, and the tangent FI, is, therefore, obtained by laying off from A F produced an angle K F L equal to the total deflection from the preceding tangent. - If the curve is afterwards continued beyond F, as, for instance, to B a tangent B N at B is obtained by laying off from F B produced an angle M B N = L B F = L. F B, the total deflection from, the pre ceding tangent F L. - B. Method by Tangent and Chord Deflections. 17. Let A B C D (fig. 3) be a curve between the two tangents EA and D L, having the chords A B, B C, and C D of the same length 2^ II 2' Fig 3 I) *=} \ , > P G N A o E Produce the tangent E A, and from I; draw B G perpendicular to A G. Produce also the chords A B and B C, and make the produced METHOD BY TA NGENT AND CHIOR D DE FLECT ONS. 9 parts B II and C K of the same length as the chords. Draw CH and D. K. B G is called the tangent deflection, and C 11 or D R the chord deflection. 18. H” rob learn. Given the radius A O = R (fig. 3), to find the tangent deflection B G, and the chord deflection C. H. Solution. The triangle C B H is similar to B O C; for the angle B O C = 180° — (O B C + B CO), or, since B CO = A B O, B O C = 180° — (O B C + A B O) = C B H, and, as both the triangles are isosceles, the remaining angles are equal. The homologous sides are, therefore, proportional, that is, B O : B C = B C : C II, or, represent- ing the chord by c and the chord deflection by d, R.; c = c : d'; cº ſº d = T. To find the tangent deflection, draw B J1 to the middle of C 11, bisecting the angle C B II, and making B M C a right angle. Then the right triangles B M C and A G B are cqual; for B C = A B, and the angle. C B M = # C B II = } B O C = A A O B = B A G (§ 2, III.). Therefore B G = CM = # C L = }, d, that is, the tangent de flection is half the chord deflection. 19. Problem. Given the deflection angle D of a curve, to find the chord deflection d. Solution. By the preceding section we have d = cº R and by § 10, 50 tº e * * * e R = i. p. Substituting this value of R in the first equation, we find ſº l c” sin. D ſº (1 = – 50 This formula gives the chord deflection for a chord c of any length though D is the deflection angle for a chord of 100 feet (§ 7). When c = 100, the formula becomes d = 200 sin D, or for the tangent de- flection à d = 100 sin. D. By these formulae the tangent and chord deflections in Table I, may be easily obtained from the table of natural SlT1 (2S 20. The length of the curve may be found by first finding () (; 9 of § 1 l), and then proceeding as in § 13. 21. Problem. To draw a tangent to the curve at any station, is B (fig. 3). Solution. Bisect the cilord deflection II C of the next station in M 10 CIRCULAR CURVICS, A line drawn through B and M will be the tangent required; for it has been proved (§ 18) that the angle C B M is in this case equal to # B O C, and B M is consequently (§ 2, III.) a tangent at B. If B is at the end of the curve, the tangent at B may be found with- out first laying off H C. Thus, if a chain equal to the chord is extend. ed to H on A B produced, the point H marked, and the chain then swung round, keeping the end at B fixed, until H M = }; d, B M will be the direction of the required tangent.” 22. Problem. Given the chord deflection d, to lay out a curce from a given tangent point. Solution. Let A (fig. 3) be the given tangent point, and suppose a has been calculated for a chord of 100 feet. Stretch a chain of 100 feet from A to G on the tangent E A produced, and mark the point G. Swing the chain round towards A B, keeping the end at A fixed, until B G is cqual to the tangent deflection 3 d, and B will be the first station on the curve. Stretch the chain from B to II on A B pro duced, and having marked this point, swing the chain round, until H. C. is equal to the chord deflection d. C is the second station on the curve Continue to lay off the chord deflection from the preceding chord pro duced, until the curve is finished. Should a sub-chord D F occur at the end of the curve, find the tan gent D L at D ($ 21), lay off from it the proper tangent deflection L. F. for the given sub-chord, making D F of the given length, and F will be a point on the curve. The proper tangent deflection for the sub- chord may be found thus. Itcpresent the sub-chord by c', and the cor- 12 responding chord deflection by d", and we have (§ 18) # d' = # but in ra t = * •e have d' : , d = c/* : c”. Therefore 3, d' = d(£)” since # ( = 2.É. W C I) a VC : 3 (I = C' " : C*. erefore 3, d' = 4 (#) Example. Given the intersection angle I between two tangents equal to 16° 30', and R = 1250, to find T, d, and the length of the rurve in stations. Here (§ 4) T = R tan. I = 1250 tan. 82 15' = 181.24; cº 100? * The distance 13 MI is not exactly equal to the chord, but the error arising from tukiug it equal is too small to be regarded in any curves but those of very sunal] radius. If necessary, the true length of 13 AI may be calculated ; for B MI = /TTE-TETTI. ORDIN ATES. 11 e () * (§ 9) sin. D = ** = ... = .04 = nat, sin. 2° 17'; t] 4 I 80 15, 495 (§ 13) n = E == 2517; F 1375 - 3.60. These results show, that the tangent point A (fig. 3) on the first tan gent is 181.24 feet from the point of intersection, — that the tangent deflection G B = 3 d = 4 feet, — that the chord deflection H C or KD = 8 feet, — and that the curve is 360 feet long. The three whole sta- tions B, C, and D having been found, and the tangent D L drawn, the tangent deflection for the sub-chord of 60 feet will be, as shown above, 60 N2 º & d = 4 (i. = 4 × .6* = 4 × .36 = 1 44. L. F = 1.44 feet being laid off from D L, the point F will, if the work is correct, fall upon the second tangent point. A tangent at F may be found ($ 21) by producing D F to P, making FP = D F = 60 feet, and laying off F'N = 1.44 feet. FN will be the direction of the required tangent, which should, of course, coincide with the given tangent. 23. Curves may be laid out with accuracy by tangent and chord deflections, if an instrument is used in producing the lines. But if an instrument is not at hand, and accuracy is not important, the lines may be produced by the eye alone. The radius of a curve to unite two given straight lines may also be found without an instrument by $73, or, having assumed a radius, the tangent points may be found by § 74. C. Ordinates. 24. The preceding methods of laying out curves determine points i00 feet distant from each other. These points are usually sufficient for grading a road; but when the track is laid, it is desirable to have intermediate points on the curve accurately determined. For this pur. pose the chord of 100 feet is divided into a certain number of equal parts, and the perpendicular distances from the points of division to the curve are calculated. These distances are called ordinates. If the chord is divided into eight equal parts, we shall have points on the curve at every 12.5 feet, and this will be often enough, if the rails, which are sclélom shorter than 15 feet, have been properly curved (§ 28). 25. Problema. Given the deflection angle D or the radius R of a ºurve, to find the ordinates for any chord. Solution. I. To find the middle ordinate. I.et A E B (fig 4) be a portion of a curve, subtended by a chord A B, which may be de- S) 12 CIRCULAR UURWES. noted by c. Draw the middle ordinate E D, and denote it by in Pro. duce E D to the centre F, and join A F and A. E. Then (Tab. X. 3) E l 2-T ~s A & ſº | {} – B Fig. 4. F G E. #. = tan. E A D, or E D = A D tan. E A D. But, since the angle E A D is measured by half the arc BJE, or by half the equal arc A. E. we have E A D = } A FE. Therefore E D = A D tan. ; A FE, ol pº m = 3 c tan. ; A F E. When c = 100, A FE = D (§ 7), and m = 50 tan. 5 D, whence in may be obtained from the table of matural tangents, by dividing tan. & D by 2, and removing the decimal point two places to the right. The value of m may be obtained in another form thus. In the triangle A D F we have D F = V2ſ F*- Aſ D* = y Rº-4 cº. Then m = E F – D F = R — D F, or º m = It — VI: * – 3 cº. II. To find any other ordinate, as R N, at a distance D N = b from the centre of the chord. Produce R N until it meets the diametc. parallel to A B in G, and join R. F. Then R G = VR FTFG% = JR*-*, and R N = R G – N G = R G – D F substituting the value of R G and that of D F found above, we have - ſº RN == VF-W-- yī ETC. ORDIN AT ĐS. 13 By these formulae the ordinates in Table I. are calculated. The other ordinates may also be found from the middle ordinate by the following shorter, but not strictly exact method It is founded on the supposition, that, if the half-chord B D be divided into any number of equal parts, the ordinates at these points will divide the arc E B into the same number of equal parts, and upon the further supposition, that the tangents of small angles are proportional to the angles themselves. These suppositions give rise to no material error in finding the ordi- nates of railroad curves for chords not exceeding 100 feet. Making, for example, four divisions of the chord on each side of the centre, and joining A R, A S, and A T, we have the angle R A N = # E A D, since R B is considered equal to # E B. But E A D = 3 A FE. Therefore, R A N = } A FE. In the same way we should find SA O = } A FE, and TA P = } A FE. We have then for the ordinates, & N = A N tan. R A N = 3 c tan. ; A FE, SO = A O tan. SA O = 3 c tan. # A FE, and T P = A P tan. T A P = | c tan. ; A FE. But, by the second supposition, tan. ; A FE = 3 tan. ; A FE, tan. # A FE = } tan. ; A FE, and tan. ; A FE = 4 tan. ; A FE, Substituting these values, and recollecting that § c tan. ; A FE = in, we have 15 15 R N = i, x * c tan ; 4 FE = i. m, 8 8 [[F SO = + x 3 c tan. ; A FE = + m, TP – 4 x , c tan. A FE – 4 m (1 - 16 as 3 - “ . . . . 12 – 16 ºn. In general, if the number of divisions of the chord on each side of the centre is represented by n, we should find for the respective ordi- (n + 1) (n − 1) m (n + 2) (n — 2) m ------. --- º n? } 71.2 mates, beginning nearest the centre, it -- 3) (n — 3) on - Quº , &c. Erample Find the ordinates of an 8° curve to a chord of 100 feet. - 15 Here m = 50 tan. 20 = 1.746, R N = is m = 1.637, SO = m = 1.310, 7 and T P = is m = 0.764. 26. An approximate value of m also may be obtained from the for. mula in = R — MR” – 4 cº. This is done by adding to the quantity e g c4 e tº under the radical the very small fraction di Rs, making it a perfect 14 CIRCULAR CURVES. e c2 square, the root of which will be R — sis. We have, then, n = R e? \ . *s (R º-º à) y ſº . . m = -t-. 8 IR 27. From this value of m we see that the middle ordinates of any two chords in the same curve are to each other nearly as the squares of the chords. If, then, A E (fig. 4) be considered equal to 3. A. B. its middle ordinate C H = } E D. Intermediate points on a curve may, therefore, be very readily obtained, and generally with sufficient accu- racy, in the following manner. Stretch a cord from A to B, and by means of the middle ordinate determine the point E. Then stretch the cord from A to E, and lay off the middle ordinate C H = 4 E D, thus determining the point C, and so continue to lay off from the suc- 2 cessive half-chords one fourth the preceding ordinate, until a sufficient number of points is obtained. D. Curving Rails. 28. The rails of a curve are usually curved before they are laid. To do this properly, it is necessary to know the middle ordinate of the curve for a chord of the length of a rail. 29. Problem. Guven the rudius or deflection angle of a curve, to find the middle ordinate for curving a rail of given length. Solution. Denote the length of the rail by l, and we have (§ 25) the exact formula m = R — VI* — 4 l’, and (§ 26) the approximate formula 4 lº 7/2 = ..." --- tº * = .75 This formula is always near enough for chords of the length of A ruil 50 If we substitute for ſº its value (§ 10) R = gill B we have, sin. D =: } lº [[F m = 3 * x *i. Erámple. In a 19 curve find the ordinate for a rail of 18 feet in length. Here R is found by Table I. to bo 5729 65, and therefore, H EVERSED AND COMPOUND CURV ES. 15 py the first formula, m = nº = .00707. By the second formula, m = .81 sin. 30! = 00707. The exact formula would give the same result even to the fifth decimal. By keeping in mind, that the ordinate for a rail of 18 feet in a 1° curve is .007, the corresponding ordinate in a curve of any other de- gree may be found with sufficient accuracy, by multiplying this deci. mal by the number expressing the degree of the curve. Thus, for a curve of 5° 36' or 5.6°, the ordinate would be .007 × 5.6 = .039 ft. = 468 in. For a rail of 20 feet we have 4 lº = 100, and, consequently, m = sin. D. This gives for a 1° curve, m = 0087. The corresponding or dinate in a curve of any other degree may be found with sufficient accuracy, by multiplying this decimal by the number expressing the degree of the curve. By the above formula for m, the ordinates for curving rails in Table I, are calculated. ARTICLE II. — REVERSED AND Cox:Poux D CURves. 30. Two curves often succeed each othér having a common tangent at the point of junction. If the curves lie on opposite sides of the com- mon tangent, they form a reversed curve, and their radii may be the same or different. If they lie on the same side of the common tangent, A \ & R they have different radii, and form a compound curve. Thus A B C "fig. 5) is a reversed curve, and A B D a compound curve. 16 CIRCUI, A R CURWES. 31. Problemma. To lay out a reversed or a compound curve, when the radii or deflection angles and the tangent points are known. Solution. Lay out the first portion of the curve from A to B (fig. 5), by one of the usual methods. Find B F, the tangent to A B, at the point B (§ 16 or $ 21). Then B F will be the tangent also of the sec- ond portion B C of a reversed, or B D of a compound curve, and from this tangent either of these portions may be laid off in the usual man- Iler A. Reversed Curves 32 Theorersi. The reversing point of a reversed curve between varallel tangents w in the line joining the tangent points. Fig. 6. F H A 2 D B É E Demonstration. Ilet A C B (fig. 6) be a reversed curve, uniting the parallel tangents II A and B A, having its radii equal or unequal, and reversing at C. If now the chords A C and C B are drawn, we have to prove that these chords are in the same straight line. The radii E C and C F being perpendicular to the common tangent at C (§ 2, 1.), are in the same straight line, and the radii A E and B F, being per- pendicular to the parallel tangents II A and B K, are parallel. There- fore, the angle A E C = CFB, and, consequently, E CA, the half supplement of A E C, is equal to FC B, the half supplement of CFB; but these angles cannot be equal, unless A C and C B are in the same Straight line. 33. Problem. Given the perpendicular distance between two par- allel tangents B D = b (fig. 6), and the distance between the two tangent points A B = a, to determine the reversing point C and the common radius E C = C F = R of a reversed curve uniting the tangents HA and B K. Solution. Ilet A C B be the required curve. Since the radii are REVERSED CURVES. 17 equal, and the angle A E C = B FC, the triangles A E C and B FC are equal, and A C = C B = 3 a. The reversing point C is, therefore, the middle point of A B. To find R, draw E G perpendicular to A. C. Then the right tri- angles A E G and B A D are similar, since (§ 2, III.) the angle B A D = } A E C = A E G. Therefore A E . A G = A B : B D, or R : 3 a = a + b ; º ... R = *. 4 b Corollary. If R and b are given, to find a, the equation R = #. gives a” = 4 ſº b; [F ... a = 2 vſ. U. Jºramples. Given b = 12, and a = 200, to determine R. Here 2002 10000 Given R = 675, and b = 12, to find a. IIere a = 2.76.75 × 12 = 2 V8100 == 2 × 90 = 180. 34. Probkesma. Given the perpendicular distance between two par- allel tangents B D = b (ſig. 7), the distance between the two tangent points 21 B = a, and the first radius E C = R of a reversed curve uniting the tingents H A and B K, to find the chords A C = a' and C B = a!", and the second radius C F = R'. H A _% -- Z R / > BT-R / * Fig. 7. E * Solution. Draw the perpendiculars E G and FL. Then the right triangles A B D and E A G are similar, since the angle B A D = 18 CIRCULAR CURVES. * A E C = A E G Therefore A B : B D = E A : A G, or a b = R : a'; 2 IR b tº- . a' = Q. Since a' and a' are (§ 32) parts of a, we have ſº a' = a – a'. To find R the similar triangles A B D and F B L give A B : B D = F B : B L, or a b = R! 3 a! ; - a all ſº . . . R! = 2 Example. Given b = 8, a = 160, and I = 900, to find a', a!", and 2 X 900 × 8 R. IIere a' = **** = 90, a = 160–90 = 70, and R == 160 × 70 T2 x 8 = 700. 35. Corollary R. If b, a', and a” are given, to find a, IR, and 18, we have (§ 34) = a! a! : R = ** R = **i. º a = a' + a! ; * = g : 2 b Example. Given b = 8, a = 90, and a' = 70, to find a, R, and R. 160 × 00 160 × 70 Here a = 90 + 10 = 160, R = *...* = 900, and R = *** = 700. 2 x 5 = 36. Coroilary 2. If ſº, R', and b are given, to find a, a', and a ', f // I iſ 2 we have (§ 35), R + R' = a a º - sº - #. Therefore a” = 2 b (R -- R'); Af ſº ‘... a = J^2 b (R+ R') Having found a, we have (§ 34) - [º a – 2 R8, an 3 R'b. a ' Ol Erample. Given R = 900, It' = 700, and b = 8, to find a, a!, and '. Here a = x/2 × 8 (900 + 700) = x/16 x 1600 = 160, a = (). 2x 900 x 8 n 2 X 700 × 8 *i;- = 90, and a' = −iº = 70. REVERSED CURWES. 19 37. Problem. Given the angle A K B = K, which shows the change of direction of two tangents II A and B K (fig 8), to inite theſe tangents by a reversed curve of given common radius R, starting from a giv- en tangent pownt A. / __ / De / vº s *A* = ~~ }3 R E Fig 8 Solution. With the given radius run the curve to the point D, where the tangent D N becomes parallel to B K. The point D is found thus. Since the angle N G K, which is double the angle H A D (§ 2, II.), is to be made equal to A KB = K, lay off from HA the angle II ...! D = 3 A. Measure in the direction thus found the chord A D = 2 R sin. & K This will be shown ($69) to be the length of the chord for a deflection angle # K. Having found the point D, measure the perpendicular dis tance D M = b between the parallel tangents. The distance D B = 2 D C = a may then be obtained from the for. mula (§ 33, Cor.) lſº a = 2 v Rö. The second tangent point B and the reversing point C are now de termined. The direction of D B or the angle B L V unay also be ob $º, g - - D MI r tained; for sin B D N = sin. D B M = #, or tº sin B D N = } (1 38. Problem. Given the line A B = a (fig. 9) which joins the fired tangent points A and B, the angles HA. B = A and A B L = B, Ind the first radius A E = R, to find the second radius (; F = R. of a reversed curve to unite the tangents H" A and B K. First Solution. With the given radius run the euroe to the point D, where the tangent D N becomes parallel to 13 K. The point D is found 20 CIRCULAR CTJRVES. thus. Since the angle H G N, which is double H A D (§ 2, II.), is equal to A &o B, lay off from HA the angle H A D = } (A Co B), and measure in this direction the chord A D = 2 R sin. (Aco B) (§ 69) 1F Fig. 9. / - E. Setting the instrument at D, run the curve to the reversing point C ºn the line from D to B (§ 32), and measure D C and CB Then the similar triangles DE C and B F C give D C : DE = CB. B. F. or D C : R = C B . It! ; ſº ... R = 9.5 × R. DC Second Solution. By this method the second radius may be found by calculation alone. The figure being drawn as above, we have, in the triangle A B D, A B = a, A D = 2 R sin. # (A – B), and the included angle D A B = H A B – H A D = A — 3 (A – B) = # (A + B). Find in this triangle (Tab. X. 14 and 12) B D and the angle A B D. Find also the angle D B L = B -- A B D. Then the chord C B = 2 Ję' sin. A B F C = 2 R' sin. D B L, and the chord D C = 2 R sin. , D E C = 2 R sin. D B L ($ 69). But C B = B D – D C ; whence 2 R' sin. D B L = B D — 2 R sin D B L , - • ??' – B D — R tº . . " = gº ºf ". When the point D falls on the other side of A, that is, when the angle B is greater than A, the solution is the same, except that the angle D A B is then 180°– (A + B), and the angle D B L = B — A B D. REW ERSED CURVES. 21 39. Problesm. Given the length of the common tangent D G = a, and the angles of intersection I and I' (fig. 10), to determine the common radius C E = C F = R of a reversed curve to unite the tangents H 4 ana B L - - F Fig. 10. Fº Solution. By $ 4 we have D C = R tan. $ 1, and C G = R tan. ; I'; whence R (tan. ; I + tan. ; I’) = D C + C G = a, or ſº R = (I º tan. # 1 + tan. # 1' This formula may be adapted to calculation by logarithms; for we in. Il e have (Tab. X. 35) tan, 3 I + tan. # 17 = ; # #. Substituting this value, we get - os. # I cos. & I" tº- R a cos. # 1 cos. 34. \ sin. ; (I + 1") The tangent points A and B are obtained by measuring from D a distance A D = R tan. # I, and from G a distance B G = R tan, ; 17 Example Given a = 600, 1 = 12°, and I = Sº, to find R. Here a = 600 2.7781.51 3 1 = 0° COS, 9.9976l 4 * I' = 4° cos. 9.998.94 l 2.774706 ; (1 + 1) = 10° sin. 9.239670 R = 3427.96 . 3.535036 22 CIRCULAR CURVES, 40. Problem. Given the line A B = a (fig. 10), which joins the fired tangent points A and B, the angle D4 B = 4, and the angle A B G = B, to find the common radius E C = C F = R of a reverged curve to unite the tangents H 4 and B L. F. /* Fig. 10. Solution. Find first the auxiliary angle A K E = B K F, which may be denoted by K. For this purpose the triangle A E R gives A E: EA = gin. K : sim. E A K. Therefore E K sin. K = A E sin. E A K = R cog. A, since E A K = 90° — A. In like manner, the triangle B F K gives F. K sin K = B F sin. FB K = R cos. B. Adding these equations, we have (E K+ F K) sin. K = R (cos. A + cos. B), or, since E. K. -- F K = 2 R, 2 IR sin. K = R (cos. A + cos. B) Therefore, sin. K = } (cos. A + cos. B). For calculation by loga- rithms, this becomes (Tab. X 28) º sin. K = cos. 3 (A + D) cos. 3 (A – B). IIaving found K, we have the angle A E K = E = 180° — K — E A K = 180° — K — (90° — A) = 90°-H A — K, and the angle B FK = F = 1809 – K– FB K = 180° — K — (90° – B) = 90° -- B — K. Moreover, the triangle A E K gives A E : A K = sin. K. sin. E, or R sin. E = A K sin. K, and the triangle B F K gives B F : B K = sin. K: sin, F, or R sin. F = B K sin. K. Adding these equations, we have R (sin. E + sin. F) = (A K-H B K) sin. K == a sin. K. Substituting for sin, E -- sin. F its value 2 sin. § (E + F) compound cuRVEs. 23 cos. (E – F) (Tab. X. 26), we have 2 It sin. # (E -- F) coa, º # a sin. K e # (E--F) = a sin. K. Therefore R = sin. (E+ F) cos. 3 (E-F)* Fi- nally, substituting for E its value 90°-H A – K, and for F its value 90° -- B — K, we get # (E -- F) = 90° — [K– 3 (4 + B)], and # (E – F) = } (A – B); whence -> * = § a sin. K - * * ~ ºn TR-TV.iiijºs I (T-1) Erample. Given a = 1500, A = 18°, and B = 6°, to find R. IIere # (A + B) = 12° COS. 9.990.404 # (A – B) = 6° COS 9.9976 l 4 R = 7.6° 36' 10" Sin, 9.9SS018 # a = 750 2.S75061 2.S63079 A — (A + B) = 64° 36' 10 cos. 9.632347 } (A — B) = 6° cos. 9.997614 9,62990 1 R = 1 × 10,48 3.233 l l 8 B. Compound Curves. 41. Theorem. If one branch of a compound curve be produced, until the tangent at its extremity is parallel to the tangent at the extremity of the second branch, the common tangent point of the two arcs is in the straight line produced, which passes through the tangent points of these par- allel tangents. Demonstration. Let A CB (fig. 11) be a compound curve, uniting the tangents HA and B K. The radii C E and CF, being perpen- dicular to the common tangent at C (§ 2, I.), are in the same straight line. Continue the curve A C to D, where its tangent O D becomes parallel to BK, and consequently the radius DE parallel to B F. Then if the chords CD and C B be drawn, we have the angle CE D = CFB ; whence E C D, the half-supplement of C E D, is equal to FC B, the half-supplement of CF B. But E C D cannot be equal to FC B, unless C D coincides with CB. Therefore the line B D pro- ūnced passes through the common tangent point C. 24 cIRCULAR CURVES. 42. Problem. To find a limit in one direction of each radius of a compound curve. J. Fig. 11 k E. Solution. Let A 1 and B I (fig. 11) be the tangents of the curve Through the intersection point I, draw I M bisecting the angle 4 I B. Draw A L and B M perpendicular respectively to A. I and B I, meet- ing I M in L and M. Then the radius of the branch commencing on the shorter tangent A. I must be less than A L, and the radius of the branch commencing on the longer tangent B I must be greater than B.M. For suppose the shorter radius to be made equal to A L, and make IN = A I, and join L. N. Then the equal triangles A IL and NIL give A L = L N; so that the curve, if continued, will pass through N, where its tangent will coincide with IN. Then (§ 41) the common tangent point would be the intersection of the straight line through B and N with the first curve; but in this case there can be no intersection, and therefore no common tangent point. Suppose next, that this radius is greater than A L, and continue the curve, until its tangent becomes parallel to BI. In this case the extremity of the COMPOUND CURWES. 25 curve will fall outside the tangent BI in the line A N produced, and a straight line through B and this extremity will again ſail to intersect the curve already drawn As no common tangent point can be found when this radius is taken equal to A L or greater than A L, no com- pound curve is possible. This radius must, therefore, be less than A. L. In a similar manner it might be shown, that the radius of the other branch of the curve must be greater than B M. If we suppose the tan- gents A I and B I and the intersection angle I to be known, we have (§ 5) A L = A I cot. § I, and B M = B I cot. § 1. These values are. therefore, the limits of the radii in one direction, 43. If nothing were given but the position of the tangents and the tangent points, it is evident that an indefinite number of different com- pound curves might connect the tangent points; for the shorter radius might be taken of any length less than the limit found above, and a corresponding value for the greater could be found. Some other con- dition must, therefore, be introduced, as is done in the following problems 44. Problem. Given the line A B = a (fig. 11), which joins the /ired tangent points A and B, the angle B A R = A, the angle A B 1 = B, and the first radius A E = R, to find the second radius B F = R! of 2 compound curve to unite the tangents HA and B K. Solution Suppose the first curve to be run with the given radius from 4 to D, where its tangent DO becomes parallel to B I, and the angle I A D = } (A + B). Then (§ 41) the common tangent point C is in the line B D produced, and the chord CB = G D + B D. Now in the triangle A B D we have A B = a, A D = 2 R sin. (A + B) ($ 69), and the included angle D A B = I A B — I A D = A — 3 (A + B) = } (A – B). Find in this triangle (Tab. X. 14 and 12) the angle A B D and the side B D. Find also the angle C B I = B — A B D. Then (§ 69) the chord CB = 2 R' sin. CB I, and the chord CD = 2 R sin. CDO = 2 R sin. C B I. Substituting these values of CB and CD in the equation found above, C B = C D + B D, we have 2 R sin. C B I = 2 R sin. C B I + B D; B D tº ‘. It! = R + —# + 2 sin. C B I When the angle B is greater than A, that is, when the greater radius ls given, the solution is the same, except that the angle D A B = 26 CIRCULAR CRJ RVES. # (B-4), and C B I is found by subtracting the supplement of A B D from B. We §ll also find C B = C D – B D, and consequently R = R – 3 iſ a Fr. If more convenient, the point D may be determined in the field, by laying off the angle I A D = } (A + B), and measuring the distance A D = 2 IR sin. (A + B). B D and CB I may then be measured, instead of being calculated as above. Erample. Given a = 950, A = 8°, B = 7°, and R = 3000, to find R". Here A D = 2 x 3000 sin. ; (8° +7°) = 783.16, and D A G = # (8° — 7°) = 30'. Then to find A B D we have A B — A D = 166.84 2.22230U, # (A D B + A B D) = 89° 45' tan. 2.360 180 4.582.480 A B -- A D = 1733.16 3.238839 } (A D B — A B D) = 87°24' 17" tan. 1,343641 - . . . A B D = 2° 20' 43'ſ Next, to find B D, A D = 783.16 2.89.3849 D A B = 30! Sin, 7.940849 0.834691 A B D = 2° 20' 43'ſ sin. 8.6l 1948 B D = 167.01 2.2227.43 B – A B D = C B I = 4° 39' 17" sin. 8.909292 2 (R – R) = 2058.03 3.31.345) . It" — R = 1029.01 . It' = 3000 + 1029.01 = 4029.01 To find the central angle of each branch, we have CF B = 2 C B 4 = 9° 18' 34", which is the central angle of the second branch; aud A E C = A E D – C E D = A + D — 2 C B I = 5° 41' 26", which is the central angle of the first branch. 45. Problem. Given (fig. 11) the tangents A I = T, BI = T', the angle of intersection = 1, and the first radius A E = R, to find the second radius B F = R. Solution. Suppose the first curve to be run with the given radius from A to 19, where its tangent DO becomes parallel to L I. Through COMPOUND CURVES. 27 D draw D. P. parallel to A I, and we have I P = D 0 = A O = R tan. ; I (; 4). Then in the triangle D P B we have DP = I O = A 1 – A O = T – R tan. 3 I, B P = B I — I P = T' – R tan. $ 1, and the included angle D PB = A I B = 180°– 1. Find in this tri- angle the angle C B 1, and the side B D. The remainder of the solution is the same as in § 44. The determination of the point D in the field is also the same, the angle I A D being here = } I. When B is greater than A, that is, when the greater radius is given, the solution is the same, except that DP = R tan. # I — T, and B P = R tan. # 1 — T'. Erample. Given T = 447.32, T' = 510.84, 1 = 15°, and R = 3000, to find IR. Here R tan. 3. I = 3000 tan. 73° = 394.96, D P = 447.32 — 394.96 = 52.36, B P = 510.84 — 394.96 = 1 15.88, and D P L == 1802 – 15° = 165°. Then (Tab. X. 14 and 12) B P – D P = 63.52 1.8029 . () § (B DP + P B D) = 7° 30' tan, 9.1 ! 94.29 ().92233.9 B P + D P = 16S 24 2,225929 3 (B DP — P B D) = 2° 50' 44" tan, 8 696.410 • P B D = C B I = 4° 39' 16" Next, to find B D, A) P = 52.36 1.71 9000 D P D = 15° sin. 9.412996 1.13 1996 P B D = 4° 39' 16" sin, 8,909266 I3 D = 167.005 2.2227.30 The tangents in this example were calculated from the example in § 44. The values of CBI and B D here found differ slightly from those obtained before. In general, the triangle D B P is of better form for accurate calculation than the triangle A D B. 46. If no circumstance determines either of the radii, the condition may be introduced, that the common tangent shall be parallel to the line joining the tangent points. Problem. Given the line A B = a (fig. 12), which untes the Jired tangent points A and B, the angle 1A B = A, and the angle A BI = B, to find the radii A E = R and B F = R! of a compound curve, having the common tangent D G parallel to A. B. 28 CIRCULAR C U R W ES, Solution. Let A C and B C be the two branches of the required curve, and draw the chords A C and B C. These chords bisect the I Fig. 12. D C is angles A and B; for the angle D A C = # 1 D G = } I A B, and the angle G B C = } D G 1 = } A B 1. Then in the triangle A C B we have A C : A B = sin. A B C : sin. A C B. But A C B = 180° — (CA B + C B A) = 180° — 3 (A + B), and as the sine of the sup- plement of an angle is the same as the sine of the angle itself, sin. A C B = sin. (A + B). Therefore A C : a = sin. B : sin. # (A + B), or A C = iº. In a similar manner we should Y a sin. A find B .# TFE o R! = sin. E 9", substituting the values of A C and B C just found. .4 C Now we have (§ 68) R = -## , and # a sin. B I # a sin. ; A [F R Tsin. A sin. (A + Bj' 2 Tsin. Hsin (ATB): Eranſple. Given a = 950, A = 8°, and B = 7°, to find R and R' Here compoun D CURVES. 29 # a = 475 2,676ſ;94 * B = 3° 30' sin, 8.785675 1.462369 # A = 4° sin. 8.84.3585 § (A + B) = 7° 30' sin. 9.115698 - 7.959.283 R = 31.84.83 3,503086 Transposing these same logarithms according to the formula for I" we have # a = 475 - 2.676694 # A = 4° sin. 8.84.3585 1.520279 * B = 3° 30 sin. 8.785675 # (A -- B)= 7° 30 sin. 9.1 15698 7.90.1373 R = 4.158.21 3.61 8906 47. Probleman. Given the line A B = a (fig. 12), which unites the fired tangent points A and B, and the tangents A I = T and BI = T', to find the tangents A D = r and B G = y of the two branches of a come pound curve, having its common tangent D G parallel to A B. Solution. Since D C = A D = a, and C G = B G = y, we have D G = a + y, Then the similar triangles I D G and I A B give I D : I.A = D G : A B, or T — a T = a + y : a. Therefore a T — a r = Tz + Ty (1). Also 4 D : A I = B G : B I, or r: T = y : T. Therefore Ty = T r (2). Substituting in (1) the value of Tyin (2), we have a T'— a r = Tr + T'a, or a r + Tr + Tº r = a T; º - —“*- fº . If a-FT-HT' ſ and, since from (2), y = # } T. º = —“ — * * - HTTTT, The intersection points D and G and the common tangent point C are now easily obtained on the ground, and the radii may be found by the usual methods. Or, if the angles I A B = A and A B I = B 30 CIRCULAR CURVES. have been measured or calculated, we have (§ 5) R = r cot. # A, and R = y cot. § B. Substituting the values of r and y found above, we y _ a T cot + 4 a T' cot. B have R = a FT-HT'ſT+ T and IR' = a + T-F Tſ .* Erample. Given a = 500; T = 250, and T’ = 290, to find r and y Here a + T + T = 500 + 250 + 290 = 1040; whence r = 500 x 250 + 1040 = 120.19, and y = 500 × 290 + 1040 = 139.42. 48. Problem. Given the tangents A I = T, B1 = T', and the angle of intersection I, to unite the tangent points A and B (fig. 13) oy a compound curve, on condition that the two branches shall have their angles of intersection A D G and I G D equal. l Fig 13. Solution. Since 1 D G = 1 G D = } 1, we have I D = 1 G, ſtep. resent the line I ly,– I G by r. Then if the perpendicular I H be le * The radii of an oval of given length and breadth, or of a three-centre arch of given Span and rise, may also be found from these formulae. In these cases A + B = 90°, and the valuce of R and R! may be reduced to R -Hº-f and R' = T - H++ . These values admit of an easy construction, or they may be readily Oalculated. - TU RNOU TS ANI) CROSSINGS. 3] (all from I, we have (Tab. X. 11) D H = 1 D cos, I D G = r cos. # 4, and D G = 2 r cos. # I. But D G = D C + C G = A D + B G = T — r + T — r = T + T — 2 r. Therefore 2 r cos. # 1 = T + T – 2 r, or 2 r + 2 r cos. I = T + T'; whence r = (T + Tr) {{#}, or (Tab. X. 25) _ 3 (T + T ) gº T cos.” # 1 The tangents A D = T — r and B G = T' — r are now readily found. With these and the known angles of intersection, the radii or deflection angles may be found (§ 5 or § 11). This method answers very well, when the given tangents are nearly equal; but in general the preceding method is preferable. Erample. Given T = 480, T = 500, and I = lSº, to find r. Here 4 (T + T') = 245 2.3S9166 # I = 4° 30' 2 cos. 9.997.318 a = 246.52 2.391848 Then A D = 480 – 246.52 = 233.48, and B G = 500 – 246.52 = 253.48. The angle of intersection for both branches of the curve being 9°, we find the radii A. E = 233.48 cot. 4° 30' = 2966.65, and B F == 253, 48 cot. 4° 30' = 3220.77. ARTICLE III. — TURNOUTS AND CROSSINGs. 49. THE usual mode of turning off from a main track is by switch- ing a pair of rails in the main track, and putting in a turnout curve tangent to the switched rails, with a frog placed where the outer rail of the turnout crosses the rail of the main track. A B (fig. 14) repre- sents one of the rails of the main track switched, B F represents the outer rail of the turnout curve, tangent to A B, and F shows the posi- tion of the frog. The switch angle, denoted by S, is the angle D A B, formed by the switched rail A B with A D, its former position in the main track. The frog angle, denoted by F, is the angle G FM made by the crossing rails, the direction of the turnout rail at F being the tangent FM at that point. In the problems of this article the gauge of the track D C denoted by g, and the distance D B, denoted by d are supposed to be known. The switch angle S is also supposed to be known, since its sine (Tab. X. l) is equal to d divided by the length 32 CIRCUI, A R CURV ES, of the switched rail. If, for example, the rail is 18 feet in length and ſt = .42, we have S == 1° 20' A. Turnout from Straight Lines. 50. Problem. Given the radius R of the centre line of a turnout (fig. 14), to find the frog angle G F M = F and the chord B. F. Fig 14. “L E H R. Solution. Through the centre E draw E K parallel to the main track. J)raw B H and FA perpendicular to E K, and join Ł F. Then, since E F is perpendicular to FM and FK is perpendicular to FG, the angle E FR == G FM = F; and since E B and B H are respectively perpendicular to A B and A D, the angle E B H = D A B - FK = S. Now the triangle E FE gives (Tab. X. 2) cos E F K = # , But E F, the radius of the outer rail, is equal to IR + 3 g, and & R = C H = B H – B C = B E cos. E B H — 13 C = \R + $g) cos. S – (g — d). Substituting these values, we have eos. E. F. K = R + 3 g : , O - — d ſº cos, F = cos. S – 2 T. * R + 3 g From this formula F may be found by the table of natural cosines To adapt it to calculation by logarithms, we may consider g – d to be equal to (g -- d) cos S, which will lead to no material error since TU RN OUT FROM STRAIGHT LINSS. 83 y – d is very small, and cos. Salmost equal to unity The value of coº. F then becomes t;" cos, F - (R – 3 g + d) cos. 8. R + 3 y To find B F, the right triangle B C F gives (Tab. X. 9) B F == ###. But B C = g – d and the angle B F C = BFE – C F E = (900 — 3 B E F) — (90° — F) = F — ; B E F. But B E F = B L F – E B L = F — S. Therefore B F C = A' — # (F — S) = } (F-H S). Substituting these values in the formula for B F, we have — d B F = 9 T *— . º sin. (F-H S) By the above formulae the columns headed F and B Fin Table V are calculated. - Example. Given g = 4.7, d = .42, S = 1° 20', and R = 50 to find F and B F. Iſere nat. cos. S = .9997.29, g – d = 4.28, R + š q = 502.35, and 4.28 -i- 502 35 = .008520. Therefore mat. cos. R = .9997.29 — .008520 = .99.1209, which gives F = 7° 36' 10". Next to finil B. F. g — d = 4.28 0.631.4 $4 § (F + S) = 4°28' 5" sin. S.89.1555 B F = 54.94 1.739883 51. Problesia. Giren the frcg angle G F M = F (fig. 14), u, find the radius R of the centre line of a turnout, and the chord B. F. Solution. From the preceding solution we have cos. F == (R + 3 g) cos, S – (g — d) + # R -H # g (5 – d). Therefore (R + 3 g) cos, F = (R + # 9) cos, S – (g — d.), or tº Fº { * = * cos, S – cos. Fº R + # 9 = For calculation by logarithms this becomes (Tab. X. 20) -: § (g — d.) ſº R + $g sin. ; (F-FS) sin. (FTS) ' Having thus found R + 3 g, we find R by subtracting ; g. Bº is found, as in the preceding problem, by the formula tº- RF=-gi."—r. sin. A (F -- S) 34 CIRCULAR CU. R.W. E.S. Erample Given g = 4.7, d = .42, S = 1° 20', and F = 79, to find fº. Here # (g — d) = 2.14 0.3304 l 4 # (F-H S) = 4° 10' sin. 8.86 1283 º # (F– S) = 2° 50' sin. 8.693998 7.555281 It -- 3 g = 595.85 2 775 1.33 • A = 593.5 52. Problem. To find mechanically the proper position of a given frog. Solution. Denote the length of the switch rail by l, the length of the frog by f, and its width by w. From B as a centre with a radius B H = 2 l, describe on the ground an arc G H K (fig. 15), and from the inside of the rail at G measure G LI = -2 d, and from II measure H K such that H K : B L = } w; f, or II K: 2 l = } w; f; that is, H A = º Then a straight line through B and the point K will strike the inside of the other rail at F, the place for the point of the frog. For the angle II B K has been made equal to $ F, and if B M be drawn parallel to the main track, the angle M1 B H is seen to be equal to 3 S. Therefore, M B K = B F C = } (F + S), and this was shown ($ 50) to be the true value of B FC. 53. If the turnout is to reverse, and become parallel to the main track, the problems on reversed curves already given will in general be sufficient. Thus, if the tangent points of the required curve are fixed, the common radius may be found by § 40. If the tangent point at the switch is fixed, and the common radius given, the reversing point and the other tangent point may be found by § 37, the change »ſ direction of the two tangents being here equal to S. But when the TURNOUT FROM STRAIGIIT LIN ES. 3 F- J frog angle is given, or determined from a given first radius, and the point of the frog is taken as the reversing point, the radius of the sce- ond portion may be found by the following method. 54. Problem. Given the frog angle F and the distance H B = b (fig. 16) between the main track and a turnout, to find the radius R' of the second branch of the turnout, the reversing point being taken opposite F, the Loint of the frog. Fig. 16. / 2’ wº-sº- • *2. … ºs / \s IV. Solution, Let the arc FB be the inner rail of the second branch, F' G = R — 3 g its radius, and B the tangent point where the turneut becomes parallel to the main track. Now since the tangent FK is one side of the frog produced, the angle H F K = F, and since the angle of intersection at K is also equal to F, B F K = } F (§ 2, II); whence B F H = 3 F. Then (; 68) F G = ####, or R — g = Aſ / ly E * B F -y FH F § b jºr. But B F = Tºsh (Tab. X. 9), or : B F=#IF. Sub stituting this value of B F, we have ſº R – a = }" . R! — g sin.” A F : In measuring the distance H B = b, it is to be observed, that the widths of both rails must be included. 3 36 CIRCULAR CURWES. £rample. Given b = 6.2 and F = 8°, to find R. Here # b = 3.1 0.49 1362 ; F = 4° sin. 8.84.3585 # B F = 44.44 1.647 777 # F = 4° Sin. 8.S4.3585 A — 3 g = 637.08 2.S0-4 192 , '. R' = 633.43 B. Crossings on Straight Lines. 55. When a turnout enters a parallel main track by a second switch, it becomes a crossing. As the switch angle is the same on both tracks, a crossing on a straight line is a reversed curve between parallel tan- gents. Let H D and NK (fig. 17) be the centre lines of two parallel tracks, and HA and B K the direction of the switched rails. If now the tangent points A and B are fixed, the distance A B = a may be measured, and also the perpendicular distance B P = b between the tangents II P and B K. Then the common radius of the crossing . A C B may be found by § 33; or if the radius of one part of the cross ing is fixed, the second radius may be found by § 34. But if both frog angles are given, we have the two radii or the common radius of a crossing given, and it will then be necessary to determine the distance A B between the two tangent points. 56. Problem. Given the perpendicular distance G N = b (fig. 17) between the centre lines of two parallel tracks, and the radii E C == IR ana C F = R! of a crossing, to find the chords A C and B C. Solution. Draw E G perpendicular to the main track, and A L, CM, and B L' parallel to it. Denote the angle A E C by E. Then, since the angle A E L = A H G = S, we have C E L = E + S, and in the right triangle C E M (Tab. X. 2), C E cos. C E M = R cos. (E -- S) = E M = E L – L. M. But E L = A E cos. A E I = R cos. S, and L. M. : L' M = A C : B C. Now A C : B C = E C : C F = R : R!. Therefore, L. M.: Ll M == R : R', or L M ; L }} + L' M = R : R + R'; that is, L. M.: b – 2 d = R : R + R', whence L M = *F #! . Substituting these values of E L and LM in the Dºnation for R cos. (E -- S), we have R cos. (E -- S) = R cos. S – H (b — 2 d) fiT Rf CROSSINGS ON STRAIGHT LIN ES. 37 b — 2 d R – R' Having thus found E + S, we have the angle E and also its equal CF B. Then ($ 69) ºf A C = 2 R sin. E, B C = 2 R' sin. E. We have also A B = A C + B C, since A C and B C are in the 9aine straight line (§ 32), or A B = 2 (R + R) sin E. º ... cos. (E -H S) = cos. S – :F * –—% 21 D L F=\s-- _^ | N. \ * i P * M!— :^C "F- ºs L- 21 >'' T- Tº- 2 º - JK E Fig. 17 When the two radii are equal, the came formulae apply by making H' = f'. In this case, we have tº- cos (E+ S) = cos. S-3-34, 2 R º A C = D C = 2 R sin. E. #: - Erample. Given d = 42, g = 4.7, S = 1° 20', b = 11, and the an- gles of the two frogs each 79, to find A C = B C = 3 A. B. The common radius R, corresponding to F = 7°, is found (§ 51) to be 593.5. Then 2 IR = 1187, b – 2 d = 10.16, and 10.16 -i- 1187 = ,00856. Therefore, nat, cos. (E + S) = .99973 — .00856 = .991.17; whence E + S = 7°37' 15". Subtracting S, we have E = 69 17' 15" Next 2 R = l l 87 3.07445l # E = 398' 37}” sin. 8.739106 A C = 65.1 - | 8 || 3557 3S CIRCULAR CURVES. C. Turnout from Curves 57. Problem. Given the radius R of the centre inne of the main track and the frog angle F, to determine the position of the frog by moans of the chord B F (fig3, 18 and 19), and to find the radius R' of the cent the line of the turnout. .--—-º- _T | - / El-T & Fig 18. l, Solution. I. When the turnout is from the unside of the curve (fig. 18). Let A G and C F be the rails of the main track, A B the switch rail, and the arc B F the outer rail of the turnout, crossing the inside rail of the main track at F. Then, since the angle E FIV has its sides perpendicular to the tangents of the two curves at F, it is equal to the acute angle made by the crossing rails, that is, E F K = F. Also E B L = S. The first step is to find the angle B K F denoted by K. To find this angle, we have in the triangle B F K (Tab. X. 14), B K-- KF: B K — K F = tan. 3 (B FK-- FBK): tan, (B FK– F B K). But B K = It + 3 g – d, and K F = IR – 3 J. Therefore, B K -- KF = 2 R — d, and B K — K F = g – d. Moreover, B F A = B F E + E F K = B F E + F, and FB A = E B F – E B K = B FE – S. Therefore, B F K — FB K = F + S. Lastly, B F K + F B K = 1809 — K. Substituting these values in the preceding proportion, we have 2 R -- d . g – d = tan. (90°–3 K): tan. 3 (F-H S), TU RN OUT FROM CURVES. 39 (2 R — d) tan. # (F + S) or tan. (90° — K) = g — d But tan (909 – 3 K) l º - = cot. A K = an. A R ſº . . . tan, 3 K = g — d (3 R — d) tan, (F-FS) ' Next, to find the chord B F, we have, in the triangle B FC in. B C F. (Tub. x 12), B F =##"... But B c-g–d, and B C F = 1809 – F C K = 1809 – (900 — 3 K) = 90° -- # K, or sin. B C F = cos. 3 K. Moreover, B F C = } (F + S); for B F K = K FC + B FC, and F B K = K C F – B FC = K FC – B FC. There- fore, B F K – F B K = 2 B FC. But, as shown above, B F A — F B K = F + S. Therefore, 2 B FC = F-- S, or B FC = } (F-H S). Substituting these values in the expression for B F, we have w — d) cos. 3 K I, E: (9 – d) cos. & K. ſº- sin. A (F -- S) ~. * B I/ Lastly, to find R, we have (; 6s) R' + 3 g = E F = in HFF . But B E F = B L F – E D L, and B L F = L F K -- L R F = F + K. Therefore, B E F = F + K — S, and f - 3 BF IF R + $9 – III*r-s, II. When the turnout is from the outside of the curve, the preceding solution requires a few modifications. In the present case, the angle E F K = F (fig. 19) and E B L = S. To find K, we have in the triangle B F K, K F + B K : R F — B K = tan. 3 (FB K -- B F K): tan. 3 (F B K — B F A). But KF = R + 3 g, and B K = R — 3 g + d. Therefore, K F + B K = 2 R + d, and KF — B K = g – d. Moreover, F B K = 1809 – F B L = 1809 – (E B F – E B L) = 1809 – (E B F — S), and B F K = 1s00 – B FK = 1802 – (B F E + E FRT) = 1809 – (E B F + F). Therefore, F B K — B FK = F + S. Lastly, F B K -- B F K = 180° — K. Substituting these values in the preceding proportion, we have 2 R + d . g – d = tan. (90° — 3 K) : tan. (F + S), or tan. (90° – 1 K) = ** fººt *. 1. g — d. cot, 3 K = tan. A R } s = g — d tº- . . . tan. K (2 R + d) tan. (F + S) ' But tan. (900 — 3 K) = 40 CIRCULAR CURVES. Next to find B F. we have, in the triangle B F C B F = B O sin. B C F jº. But B-c = g – d. and B CF = 90° – 1 K, of E Fig. 19. / lº ‘in. B C F = cos. 3 K. Moreover, 13 F C = } (F + S); for B P K = K FC – B FC, and F B K = K C F-H B F C = KFC -- B F C. Therefore, F B K – B FK = 2 B F C. But, as shown above, F B K– B FK = F-- S. Therefore, 2 B FC = F + S, or B F C = 3 (F-H S). Substituting these values in the expression for B F, we have, as before, - - (g – d) cos. & R. " rº- D F sin. (F-ELS) ' B F Lastly, to find R', we have ($68) R' + 3 g = E F = mº EF * Since # K is generally very small, an approximate value of B F may be obtained * - y - - __# - d - - by making cos. 3 K = 1. This gives B F sin. A (F + S) ' which is identical with the formula for B F in § 50. Table V, will, therefore, give a close approxima- 'll n to the value of B F on curves also, for any value of F contained in the table TURNOUT FROM CU RWES. 4l But B E F = B L F – E B L, and B L F = L FK — L R F = F'— K. Therefore, B E F = F — K — S, and 3 BF sin. # (F– K — S) [EP R! -- 4 g = Erample. Given g = 4.7, d = .42, S = 1° 20', R = 4583.75, and F = 79, to find the chord B F and the radius IR' of a turnout from the outside of the curve. Here - g — d = 4.28 0.631.444 0.631.444 2 R + d = 9167.92 3,962.271 # (F'+ S) = 4° 10' tan. 8.862433 Sin. 8.86 12S3 2.824704 1.77 0161 3 K = 22 1s" tan. 7.806740 coS 9.99999.1 B F = 58,905 1.770152 2 0.30.1030 § (F– K — S) = 2° 27' 5S.2" sin. 8.633766 8.934,796 R" + 3 y = 684.47 2.835.556 . . . R = 682.12 58. Problem. To find mechanically the proper position of a given frog. Solution. The method here is similar to that already given, when the turnout is from a straight line (§ 52). Draw B M (figs. 18 and 19) parallel to FC, and we have FB M = B F C = } (F -- S), as just shown (§ 57). This angle is to be laid off from B M ; but as F is the point to be found, the chord F C can be only estimated at first, and B M taken parallel to it, from which the angle # (F-H S) may be aid off by the method of $ 52. In this case, however, the first meas- are on the arc is d, and not 2 d: since we have here to start from BIM, and not from the rail. Having thus determined the point F approxi- mately, B M may be laid off more accurately, and F found anew. 59. When frogs are cast to be kept on hand, it is desirable to have them of such a pattern that they will fall at the beginning or end of a certain rail; that is, the chord B F is known, and the angle F is re- Juired. 42 CIRCULAR CURVES. Problem. Given the position of a frog by means of the chord BF (figs. 14, 18, and 19), to determine the frog angle F. Solution. The formula B F = miºs, which is exact on straight lines ($ 50), and near enough on ordinary curves (§ 57, note), gives ſº sin. (F4 S) = 9.7% B F" By this formula 3 (F-H S) may be found, and consequently F. 60. Problem. Given the radius R of the centre line of the main track, and the radius IV of the centre line of a turnout, to find the frog angle F, and the chord B F (figs. 18 and 19). Solution. I. When the turnout is from the inside of the curve (fig. 18). In the triangle B E K.find the angle B E A and the side E. A. For this purpose we have B E = R' + 4 g, B K = R + 3 g — d, and the included angle E B K = S. Then in the triangle E FIV we have E K, as just found, E F = R + 3 g, and FK = R — 3 g. The frog anglo E FK = F may, therefore, be found by formula 15, Tab. X., which gives tº- tan. ; F = W (s = b) (s = c), s (s — a) where s is the half sum of the three sides, a the side E A, and b and e the remaining sides. Find also in the triangle E FR the angle FE K, and we have the angle B E F = B E A – FE K. Then in the triangle B E F we have (§ 69) 35° D F = 2 (R, + 3 g) sin. A B E F." II. When the turnout is from the outside of the curve (fig. 19). In the triangle B E K find the angle B E K and the side E. K. For this purpose we have B E = R + 3 g, B K = R — 3 g + d, and the in- cluded angle E B K = 180° — S. Then in the triangle E FK we have E K, as just found, E F = R + 3 g, and FK = R + 3 g. The angle E FR may, therefore, be found by formula 15, Tab. X., which G-5) (s–c) gives tan. # E F K = But the angle E FK = F s (s — a) ' • The value of B F may be more easily found by the approximate formula B F = § - d.__ * sin. (F-H S) mark applies also to B F in the second part of this solution. , and generally with sufficient accuracy. See note to $ 57. This ro- TURNOUT FROM CURVES. 43 = 1809 – E F K. Therefore # F = 90° — # E F K, and cot. F = lan. # E FIC; s = 5) (s− c) tº ...co. F-S" s (s — a) where s is the half sum of the three sides, a the side E K, and b and c the remaining sides. Find also in the triangle E FK the angle FE K, and we have the angle B E F = F E K – B E K. Then in the triangle B E F we have (§ 69) º B F = 2 (R + 3 g) sin. B E F. Example. Given g = 4.7, d = 42, S = 1° 20', R = 4583.75, and R = 682.12, to find F and the chord B F of a turnout from the outside of the curve. Here in the triangle B E K (fig. 19) we have B E = R, + 3 g = 684.47, B K = R — 3 g + d = 4581.82, and the angles B E K -- B K E = S = 1° 20'. Then B K — B E = 3807.35 3.590,769 # (B E K+ B KE) = 40' tan. 8.065806 1.656575 B K -- B E = 5266.29 3.721505 = 3 (B E K – B KE) * = 29.6029" tan, 7.935070 . . . B E K = 1° 9,6029/ - B K sin. E. B. R. E K is now found by the formula E K = Fiji Ekº , or log. E R = log. 4581.82 + log. sin. 178° 40' -- log. sin. 129.6029' = 3.721491, whence E K = 5266.12. Then to find F, we have, in the triangle E F K, s = } (5266.12 + 684.47 -- 4586.10) = 5268.34, s — a = 2.22, s — b = 4583.87, and 0 – c = 682.24. s — b = 4583.87 3.66 1233 s — c = 682.24 2.S33937 6.495170 s = 5268.34 3.721674 s — a = 2.22 0.346353 4,06802? 2)2.427143 ; F = 3° 30' cot. 1,21357l • . F = 79 * This augle and the sine of 109 6029 below, are found by the method given in OUnnection with Table XIII. If the ordinary interpolations had been uscil, we Ghould have found F = 79 7", whereas it should be 79, since this example is the oonverse of that in $ 57. 44 CIRCULAR curves. To find FE K, we have s as before, but as a is here the side FK opposite the angle sought, we have s — a = 682.24, s — b = 4583.87, and s – c = 2.22. Then by means of the logaritnms just used, we find § FE K = 3°2'45". Subtracting 3 B E H = 34' 48", we have # B E F = 2° 27' 57". Lastly, B F = 1368.94 sin. 2° 27' 57" == 58.897. The formula B F = Iriºs, (§ 57, note) would give BF 58.306, and this value is even nearer the truth than that just found. owing, however, to no error in the formulae, but to inaccuracies inci- dent to the calculation. - 61. If the turnout is to reverse, in order to join a track parallel to the main track, as A CB (fig. 20), it will be necessary to determine the reversing points C and B. These points will be determined, if we find the angles A E C and B FC, and the chords A C and C.B. 62. Probleman. Given the radius D K = R (fig 20) of the centre line of the main track, the common radius E C = C F = R! of the centre line of a turnout, and the distance B G = b between the centre lines of the parallel tracks, to find the central angles A E C and B FC and the chords A J and B C. - D 2/ F G Fig. 20 { Solution. In the triangle A E R find the angle A E K and the did, CROSSINGS ON CURVES, 45 E K For this purpose we have A E = R', A K = R — d, and the included angle E A K = S. Or, if the frog angle has been previously calculated by § 60, the values of A E R and E K are already known." Find in the triangle E FK the angles E FIV and FE K. For this purpose we have E K, as just ſound, E F = 2 R', and FR = R + Iº! — b. Then A E C = A E A — FE K, and B F C = EFA. Lastly, (§ 69) - {ZF A C = 2 R sin & A. E. C., C B = 2 R' sin. 3 B F C. This solution, with a few obvious modifications, will apply, when the turnout is from the outside of a curve. D. Crossings on Curves. 63. When a turnout enters a parallel main track by a second switch, it becomes a crossing. Then if the tangent points A and B (fig. 21) are fixed, the distance A B must be measured, and also the angles which A B makes with the tangents at A and B. The common ra- dius of the crossing may then be found by § 40; or if one radius of the crossing is given, the other may be found by § 38. But if one tangent point A is fixed, and the common radius of the crossing is given, it will be necessary to determine the reversing point C and the tangent point B. These points will be determined, if we find the angles A E C and B FC, and the chords ºf C and C B. 64. Problemar. Given the radius D R = R (fig. 21) of the centre line of the main track, the common radius E C = C F = R' of the centre line of a crossing, and the distance D G = b between the centre lines of the parallel tracks, to find the central angles A E C and B F C and the chords A C and C B. - Solution In the triangle A E R find the angle A E R and the side E. K. For this pirpose we have A E = R', A R = R — d, and the included angle E A K = S. - - Find in the triangle B FK the angle B F K and the side FK. For this purpose we have B F = R', B K = R — b + d. and the included angle F B K = 180° — S. Find in the triangle E FIV the angles FER and E F K. For this * The triangle A E R does not correspond precisely with B E R in § 60, A being 3.n the centre line and B on the outer rail ; but the difference is too slight to affeot the calculations. 46 CIRCULAR CURVEs. purpose we have E A and FK as just found, and E F = 2 R'. Ihen A E C = A E K — FE K, and B F C = E F K – B FK. Lastly ($ 69,) ºr A C = 2 R' sin. ; A E C; CB = 2 R' sin. BFC. D ARTICLE IV — MISCELLANEous PROBLEMs. 65 Problem. Given A B = a (fig. 22) and the perpendicular B C = b, to find the radius of a curve that shall pass through C and the tangent point A. Solution. Let O be the centre of the curve, and draw the radii A 0 and CO and the line CD parallel to A. B. Then in the right triangle C O D we have O C* = C D* + O D*. But O C = R, C D = a, and O D = A O — A D = R — b. Therefore, Rº = a” + (R – b)* = a" + 1" — 2 R b + bºº, or 2 R b = a” + b%; % R = *. b, ſº ... + 3 Erample. Given a = 204 and b = 24, to find R. IIere R = 2042 . 24 3. 3ſ + j = 867 + 12 = 879. MISCELLANEOUS PRO BLEMS. 47 .* 66. Corollary 1. If R and b are given to find A B = a, that ls, to determine the tangent point from which a curve of given radius A TR s `-- \ 'i: I) / C _^ Fig. 22. /* /~ must start to pass through a given point, we have (§ 65) 2 R b = a" + b%, or a* = 2 R b — b”; tº . . . a = \Zb (2 R — b). 0 Erample. Given b = 24 and R = 879, to find a. Here u = J24 (1758–24) = y AT616 = 204. 67. Corollary 2. If I and a are given, and b is required, we have (§ 65) 2 R b = a” + bº, or bº — 2 R b = — a”. Solving this equation, we find for the value of b here required, tºº b = R — VR3 – a . 68. Problema. Given the distance A C = c (fig. 22) and the an- gle B A C = A, to find the radius R or deflection angle D of a curve, that shall pass through C and the tangent point A. Solution. Draw 0 E perpendicular to A. C. Then the angle A O E = }A O C = B.A. C = A (§ 2, III.), and the right triangle A OE gives A E (Tab. X. 9). A 0 = Ginº F : gº * = }º º 50 - - - * To find D, we have (; 9) sin. D = . Substituting for R its value # c Jast found wo have sin. D = 50 - sin. A 48 CIRCULAR CURWES. º . sin. D = 100 sin. A C Example. Given c = 285.4 and A = 5°, to find R and D. Here 142.7 º 100 sin. 50 sin. 50 tº R = sin. 50 F 1637.3; and sin. D = T2:...T = 2.854 = Sin. 1O 45 or D = 1° 45'. 69. Problem. Given the radius IR or the deflection angle D of a curve, and the angle B A C = A (fig. 22), made by any chord with the tangent at A, to find the length of the chord A C = c. 3 I Solution. If It is given, we have (§ 68) R = º: ; º . . . c = 2 R sin. A. ſº º º 100 sin. A If D is given, we have (§ 68) sin. D = wºn a ; tº C = 100 sin, 4. sin. D This formula is useful for finding the length of chords, when a cut ve is laid out by points two, three, or more stations apart. Thus, suppose that the curve A C is four stations long, and that we wish to find ºne length of the chord A. C. In this case the angle A = 4 D and , == lºgº . By this method Table II. is calculated. Example. Given R = 2455.7 or D = 1° 10', and A = 4° 40', to find c. Here, by the first formula, c = 4911.4 sin. 4° 40' = 399.59 100 sin. 40 40" By the second formula, c = Tsin. ISIOT = 399.59. 70. Problems. Given the angle of intersection KCB = I (fig. 23), and the distance C D = b from the intersection point to the curve in the direction of the centre, to find the tangent A C = T, and the radius A O. = R. - Solution. In the triangle A D C we have sin, CA D : sin. A D C = CD: A C. But CA D = } A O D = 3 I (§ 2, III, and VI.), and as the sine of an angle is the same as the sine of its supplement, sin. A D C = sin. A D E = cos. D A E = cos. # 1. Moreover, C D = b and A C = T. Substituting these values in the preccding pro- º • b cos, 4 I portion, we have sin. # I: cos, 4 I = b : T. or T = # ; whence (Tab. X. 33) MISCELLANEOUS PROBLEMS. 49 ºff" T = b cot. # 1. To find R, we have (; 5) R = T cot. I. Substituting for T its value just found, we have º R = b cot. 4 I cot. § 1 E '0 Example. Given 1 = 30°, b = 130, to find T and R. Here b = 130 2.1 13943 # I = 7o 30' cot. 0.880571 T = 987.45 2.99.4514 # I = 150 COL. 0.57 1948 R = 3685,21 3.566.402 71. Problem. Given the angle of intersection KCB = I (fig. 23). and the tangent A C = T, or the radius A O = R, to find C D = b, Solution. If T is given, we have ($ 70) T = b cot. 4 I, or b = T Rot TT 5 [5° ... b = T tan. # 1. If |. is given, we have (§ 70) R = b cot. 4 I cot. § 1, or b = COt. II cot. I } ſº , b = R tan. 4 I tan. $ 1. 50 CIRCULAR CURVES, Example, Given I = 279, T = 600 or R = 2499.18, to find b Here b = 600 tan. 6° 45' = 71.01, or b = 2499.18 tan. Go 45 tan. 13° 30' = 71.01 72. Problem. Given the angle of intersection I of two tangents A C and B C (fig. 24), to find the tangent point A of a curve, that shall pass through a point E, given by CD = a, DE = b, and the angle CD B = } I. C N F [. *6 Solution. Produce D E to the curve at G, and draw C O to the con- tie O. Denote D F by c. Then in the right triangle C D F we have (Tab. X. 11) D F = C D cos. C D F, or ſº * c = a cos. 3. I Denote the distance A D from D to the tangent point by r. Then, by Geometry, r* = D E X D G. But D G = D F + F G = D F + E F = 2 D F – D E = 2 c — b. Therefore, a” = b (2 c -— b), and Jºãº z = x/ö (2 c – b). Having thus found A D, we have the tangent A C = A D + DC = x + a. Hence, R or D may be found (§ 5 or § 11). - - If the point E is given by E H and CH perpendicular to each other, a and b may be found from these lines. For a = CH + D H = cH + E II cot I (Tab. x.9), and b = DE = #. MISCELLANEOUS PROBLEMS. 5] Example. Given I = 20° 16', a = 600, and b = 80, to find r and R. Here c = 600 cos. 100 8' = 590,64, 2 c – b = 1101.28, and r = V80 × 1101.28 = 296.82. Then T = 600 + 296.82 = 896.82, and R = 896.82 cot. 109 8' = 501 7.82. 73. Problem. Given the tangent A C (fig. 25), and the chora A B, uniting the tangent points A and B, to find the radius A 0 = R Solution. Measure or calculate the perpendicular CD. Then if C D be produced to the centre O, the right triangles A D C and CA O, having the angle at C common, are similar, and give C D : A D = A C : A O, or º R = ** * ~ * > . l O' D If it is inconvenient to measure the chord A B, a line E F, parallel to it, may be obtained by laying off from C equal distances C E and C F. Then measuring E G and G C, we have, from the similar tri- angles E G C and CA O, C G : G E = A C: A O, or R – ºfteC Eramp'e. Given A C = 246 and A D = 240, to find R. Here 240 × 24 CD = 54, and R = *** = 1093,33, 52 C1RCULAR CURWES. 74. Probienn. Given the radius AO = R (fig. 25), to find the tangent A C = T of a curve to unite two straight lines given on the ground Solution. Lay off from the intersection C of the given straight lines any equal distances CE and C.F. Draw the perpendicular CG to the mid- dle of E F, and measure G E and C. G. Then the right triangles E G C and CA O, having the angle at C common, are similar, and give G E : C G = A O : A C, or T. C G X 4 0. [º C. E. T Dy this problem and the preceding one, the radius or tangent points of a curve may be found without an instrument for measuring angles. Erample. Given R = 1093A, G E = 80, and C G = 18, to find T. IIere T = lsº = 246. 75, Problemsz. To find the angle of intersection 1 of two straight lines, when the point of intersection is inaccessible, and to determine the tan- gent points, when the length of the tangents is given. Solution. I. To find the angle of intersection I. Let A C and C V (fig. 26) be the given lines. Sight from some point A on one line to a point B on the other, and measure the angles C A B and T B V. These angles make up the change of direction in passing from one tangent to the other. . But the angle of intersection (§ 2) shows the change of di- rection between two tangents, and it must, therefore, be equal to the sum of C A B and T B | , that is, [ºr I = C A B + T B V. But if obstacles of any kind render it necessary to pass from A C to B V by a broken line, as A DE FB, measure the angles C A D, ND E, PE F. R F B, and SB V, observing to note those angles as minus which are laid off contrary to the general direction of these angles. Thus the general direction of the angles in this case is to the right; but the angle P E Flics to the left of DE produced, and is therefore to be marked minus. The angles to be measured show the successive changes of direction in passing from one tangent to the other. Thus C A D shows the change of direction between the first tangent and A D, N D E shows the change between A D produced and DE, P E F the change between D E produced and E F, RFB the change between E F produced and F B, and, lastly, SB P the change between B F pro- MISCELLANEOUS PROBLEMIS. 53 duced and the second tangent. But the angle of intersection (§ 2) shows the change of direction in passing from one tangent to another, and it must, therefore, be equal to the sum of the partial changes measured, that is, Fºr 1 – c. D + ND E – PE F4 R F D + SB v. Il. To determine the tangent points. This will be done if we find the distances A C and B C; for then any other distances from C may be found. It is supposed that the distance A B, or the distances A D, DE, E F, and FB have been measured. If one line A B connects A and B, find A C and B C in the triangle A B C. For this purpose we have one side A B and all the angles. If a broken line A DE FB connects A and B, let fall a perpendicular B G from B upon A C, produced if necessary, and find A G and B G by the usual method of working a traverse. Thus, if A C is taken as a meridian line, and D K, E L, and FM are drawn parallel to A C, and D. H. E. K., and F L are drawn parallel to B G, the difference of lati- tude A G is equal to the sum of the partial differences of latitude A H, D K, EL, and FM, and the departure B G is equal to the sum of the partial departures D. H. E. K., FL, and B M. To find these partial differences of latitude and departures, we have the distances A D, DE, E F, and F B, and the bearings may be obtained from the angles already measured. Thus the bearing of A D is C A D, the bearing of DE is KDE = KD N + N DE = C A D + N DE, the bearing of E F is I, E F = L E P – P E F = KD E — P E F, and the 54 CIRCULAR CURVES. bearing of F B is M. FB = MFR + R FB = L E F + R F B; that is, the bearing of each line is equal to the algebraic sum of the preced ing bearing and its own change of direction. The differences of lati. tude and the departures may now be obtained from a traverse table, or more correctly by the formulae: Diff of lat. = dist. × cos. of bearing ; dep. = dist. X sin. of bearing Thus, A LI = A D cos. CA D, and D H = A D sin. CA. D. Having found A G and B G, we have, in the right triangle B G C, (Tab. X. 9) g c = B G cot. BC G, and B c = Hºg But B C G = 1809 — I. Therefore, cot. B C G = — cot. I, and sin. B C G = sin. I. Hence G c = – B G cot. I, and B C = #%. Then, since A C = A G + G C, we have *~. fºr A c = A G – B G cot. I, B C = **. sin. I When I is between 90° and 180°, as in the figure, cot. I is negative, and —B G cot. I is, therefore, positive. When I is less than 909, G will fall on the other side of I; but the same formula for A C wil still apply ; for cot. I is now positive, and consequently, — B G cot. I is negative, as it should be, since, in this case, A C would equal A G in nus G C. Example. Given A D = 1200, DE = 350, E F = 300, FB = 310, C A D = 20°, ND E = 449, P E F = — 250, R F B = 310, and S B V = 30°, to find the angle of intersection I, and the distances A C and B O. Here I = 200 + 440 — 250 + 319 + 300 = 1000. To find A G and B G, the work may be arranged as in the following table : — *"... Bearings. Distances N ſº. O O --- 20 N. 20 E. 1200 | 127.63 4 10.42 44 64 350 l 53.43 3 l 4,58 —25 39 300 233.14 188.80 31 70 3 l () 106.03 291.30 | 1620.23 | 205.1 () The first column contains the observed angles. The second contain: the bearings, which are found from the angles of the first column, in MIISCELLANEOUS PROBLEMIS. 55 the manner already explained. A C is considered as running north from A, and the bearings are, therefore, marked N. E. The other col- umns require no explanation. We find A G = 1620.23, and B G = *205.10. Then G C = — B G cot. I = — 1205.1 X cot. 100° = 212.49. This value is positive, because it is the product of two nega- tive factors, cot. 100° being the same as —cot. 80°, a negative quanti- ty. Then A C = A G + G C = 1620.23 + 212.49 = 1832.72, and 1205. I º g B C = Hijºs = 1223.69. Having thus found the distances of A and B from the point of intersection, we can easily fix the tangent points for tangents of any given length. 76. Problem. To lay out a curve, when an obstruction of any kind pºrvents the use of the ordinary methods. Solution. First Method. Suppose the instrument to be placed at 4 (fig. 27), and that a house, for instance, covers the station at B, and also obstructs the view from A to the stations at D and E. Lay off from A C, the tangent at A, such a multiple of the deflection angle D, as will be sufficient to make the sight clear the obstruction. In the figure it is supposed that 4 D is the proper angle. The sight will then pass through F, the fourth station from A, and this station will be de. termined by measuring from A the length of the chord A F found by 56 CIRCULAR CURWES. § 69 or by Table II. From the station at F the stations at D and B may afterwards be fixed, by laying off the proper deflections from the tangent at F. Second Method. This consists in running an auxiliary curve paral lel to the true curve, either inside or outside of it. For this purpose lay off perpendicular to A C, the tangent at A, a line A A' of any con venient length, and from Aſ a line A' C/ parallel to A. C. Then A/C is the tangent from which the auxiliary curve A' E! is to be laid off. The stations on this curve are made to correspond to stations of 100 feet on the true curve, that is, a radius through B' passes through B, a radius through D' passes through D, &c. The chord A' B' is, there fore, parallel to A B, and the angle C'A' B' = CA B; that is, the de flection angle of the auxiliary curve is equal to that of the truc curve It remains to find the length of the auxiliary chords Al 13, B'D', &c Call the distance A A' = b. Then the similar triangles A B O and A! B O give A O : A' C = A B : A' B', or IR : R — b = 100 : A' 13'. 100 ( R — b 100 b & “º & Therefore, A'B' = ſº-n = 100 — =#. If the auxiliary curve were on the outside of the true curve, we should find in the same way A'B' = 100 + lº . It is well to make b an aliquot part of R ; for the auxiliary chord is then more easily found. Thus, if n is any whole number, and we make b = f , we have A'B' = 100 + lº 100 R = 100 + *... If, for example, b = 100 WC have n = 100, and A' B = 100 + 1 = 101 or 99. When the auxiliary curve has been run, the corresponding stations on the true curve are found, by laying off in the proper direction the distances B B', D D, &c., each equal to b. 77. Probleman. JIaving run a curve A B (fig. 28), to change the tangent point from A to C, in such a way that a curve of the same radiua may strike a given point D. Solution. Measure the distance B D from the curve to D in a direction parallel to the tangent CE. This direction may be sometimes judged of by the eye, or found by the compass. A still more accurate way is to make the angle D B E equal to the intersection angle at E, or to twice B A E, the total deflection angle from A to B; or if A can be seen from B, the angle D B A may be made equal to BA E. Measure on the tangent (backward or forward, as the case may be) a dis tance A C = BD, and C will be the new tangent point required. For, if t. H be drawn equal and esrallel to A F, we have FIH equal and par MiscELLANEous FROBLEMs. 57 allel to A C, and therefore equal and parallel to B. D. Hence D H = E F = A F = CH, and D H being equal to C H, a curve of radius U E from the tangent point C must pass through D IT Fº S. N \ ^. Fig. 28. /~7 O A. 12 78. Problern. Having run a curve A B (fig. 29) of radius R (a deflection angle D, terminating in a tangent B D, to find the radius R' or deflection angle D' of a curve A C, that shall terminate in a given parallel tangent C E. G is Solution Since the radii B F and C G are perpendicular to the par. fillel tangents C E and B D, they are parallel, and the angle A G C = A F B Therefore, A CG, the half-supplement of A G C, is equal to 58 CIRCULA (? CURVES. A B F, the half-supplement of A F B. Hence A B and B C are in the same straight line, and the new tangent point C is the intersection of A B produced with C. E. - Represent A B by c, and A C = c + B C by c'. Measure B C, or, if more convenient, measure D C and find B C by calculation. To calculate B C from D C, we have B C = #3 (Tab. X. 9), and the augle D B C = A B K = B.A. K., the total deflection from A to B. Then the triangles A F B and A G C give A B : A C = B F : C G, or c : c’ = R : It'; gº R = ? R. C 50 50 • To find D', we have ($ 10) R' = H.B. , and I: = H. H. Sub stituting these values in the equation for R', we have Hij = C! 50 X ºn D : C º . . . sin. D' = a sin. D. 79. Probleman. Given the length of two equal chords A C and B C (fig. 30), and the perpendicular CD, to find the radius R of the curve. __ Fig 30. /~ 2^ () Solution. From 0, the centre of the curve, draw the perpendiculaſ U E. Then the similar triangles O B E and B C D give B O : BE = B C : CD, or R : * B C = P C : CD. Hence B C* tºy- ^ = , º, MISCELLANEOUS PROBLEMIS. 59 This problem serves to find the radius of a curve on a track already laid. For if from any point C on the curve we measure two equal chords A C and B C, and also the perpendicular CD from C upon the whole chord A. L., we have the data of this problem. 80. Problem. To draw a tangent FG (fig. 30) to a given curve from a given point F. Solution. On any straight line FA, which cuts the curve in two points, measure FC and FA, the distances to the curve. Then, by Geometry, fº F G = y FC X FA. This length being measured from F, will give the point G. When FG exceeds the length of the chain, the direction in which to measure it, so that it will just touch the curve, may be found by one or two trials. 81. Problem. Having found the radius A 0 = R of a curve (fig. 31), to substitute for it two radii A E = IR, and D F = R, , the longer of which A E or B E! is to be used for a certain distance only at zach end of the curve. Fig. 31. E f Solution. Assume the longer radius of any length which may be thought 4 60 CIRCULAR CURVE.S. proper, and find ($ 9) the corresponding deflection angle D, . Suppogo that each of the curves A D and B D is 100 feet long. Then drawing CO, we have, in the triangle F O E, O E: FE = sin. OF E: sin. FOE. But the side O E = A E — A O = {1 — ſº, FE = D E — D F = R – R2, the angle F O E = 180° — A O C = 1809 – $ I, and the angle OF E = A OF — O E F = } I – 2 D1, since O E F = 2 D, (§ 7). Substituting these values, and recollecting that sin. (1809 – $ 1) = sin. # I, we have R1 — It : It, — R2 = sin. (; I – 2 D1) : sin. I IIence (R – R) sin. # 1. sin. (; I – 2 D1) R, is then easily found, and this will be the radius from D to D', or until the central angle D F D' = I — 4 Dr. The object of this problem is to furnish a method of flattening the extremities of a sharp curve. It is not necessary that the first curve should be just 100 feet long; in a long curve it may be longer, and in a short curve shorter. The value of the angle at E will of course change with the length of A D, and this angle must take the place of 2 D, in the formula. The longer the first curve is made, the shorter the Second radius will be. It must also be borne in mind, in choosing the first radius, that the longer the first radius is taken, the shorter will bo the second radius. º R1 — IR3 = Example. Given R = 1146.28 and I = 450, to find R, , if R, is as sumed = 1910.08, and A D and B D' cach 100. Ilere, by Table I., D1 = 1° 30'. Then R. — R = 763.8 2.88.2980 § 1 = 22° 30' sin. 9.582840 2.4658.20 * I – 2 D1 = 19930' Sin. 9,523495 R – R. = 875.64 2.94.93.25 ‘. R., e- R1 – 875.64 = 1034.44 82, Problem. To locate the second branch of a compound or re- tersed curve from a station on the first branch. Solution. Let A B (fig. 32) bo the first branch of a compound curve, and D its deflection angle, and let it be required to locate the second branch A B', whose deflection angle is D', from some station B On A B. MISCELLANEOUS PROBLEMS. Gl Det n be the number of stations from A to B, and n' the number of sta- tions from A to any station B on the second branch. Represent by V the angle A B B', which it is necessary to lay off from the chord B4 to strike B'. Let the corresponding angle A B' B on the other curve be repre: T A T sented by Vſ. Then we have V + V = S0° — BA B'. But if T' Tº be the common tangent at A, we have TA B + Tº A B' = n D + n’ D = 1800 — B A B'. Therefore, V + V' = n D + n’ D'. Next in the triangle A B B we have sin. V" : sin, W = A B : A B'. But A B : A B = n : n', nearly, and sin. V’: sin. W = }'': V, near- ly. Therefore we have approximately V'; W = n : n', or V = . Jy Substituting this value of V in the equation for V-- V', we have V + # W = n D + n, D. Therefore, n' P + n V = n' (n D + n’ D'), or [[F I, _ n' (n D + n’ D') The same reasoning will apply to reversed curves, the only change being that in this case V-H V = n D — n' D', and consequently n (n D — n' D') n + n’ When in this formula n' D becomes greater than n D, V becomes minus, which signifies that the angle V is to be laid off above BA in- stead of below. This problem is particularly useful, when the tangent point of a curve is so situated, that the instrument cannot be set over it. The same method is applicable, when the curve A B' starts from a straight line; for then we may consider A B' as the second branch of a com- pound curve, of which the straight line is the first branch, having its radius equal to infinity, and its deflection angle D = 0, Making D = 0, the formula for V becomes . tº- V = 62 CIRCULAR CURV H.S. y n'* D . n + n! When n and n' are each 1, the formula for V is in all cases exact for then the supposition that V' : V = n : n! is strictly true, since A B will equal A B', and V and V', being angles at the base of an isosceles triangle, will also be equal. Making n and n' equal to 1, we have W = 3 (D + D'). When the curve starts from a straight line, this formula becomes, by Inaking D = 0, W = } D. We have secn that when n or n! is more than 1, the value of V is only approximate. It is, however, so near the truth, that when nei. ther n nor n' exceeds 3, the error in curves up to 50 or 60 varies from a fraction of a second to less than half a minute. The exact value of V might of course be obtained by solving the triangle 'A B B', in which the sides A B and A B may be found from Table II., and the included angle at A is known. The extent to which these formula may be safely used may be seen by the following table, which gives the approximate values of V for several different values of n, n', D, and D', and also the error in each case Compound Curves, l{eversed Curves | m. | D. n!. | D', W. Error. n. D. it'. D'. W. Error. O O O º A l O O O | || | || 0 || 5 || 1 4 1 () 0.9 | || 3 || 4 || 3 || 7 || 2 || 27.2 | || 0 || 5 || 3 || 1 2 30 || 25.3 2| 3 || 4 || 3 || 4 0 || 23.5 2|| 0 || 3 || 3 5 24 22.] 3| 3 || 4 || 3 || 42% 8.3 3| 0 || 3: 3 4 30 20.7 3| || 3 || 3 || 3 45 24.0 | | | | 5 || 3 | l 3 20 | 18.6 2| l | 1 || 4 || 0 40 0.1 2| #| || 3 l 20 || 0.7 2|| 1 || 4 || 2 || 4 0 | l l .0 2| 2 || 3 || 3 7 4S | 15.0 1 || 6 || 2 || 6 || 4 0 || 23.5 2| 2 || 4 || 3 || 10 40| 24.7 1 || 5 || 3 || 5 || 7 30 51.8 3| 3 || 3 || 4 || 10 30| 54.0 || 2 3' 5| 3 || 6 25;| 52.8 As the given quantities are here arranged, the approximate values of V are all too great; but if the columns n and n' and the columns D and D were interchanged, and V calculated, the approximate value) of W would be just as much too small, the column of errors remaining the game. - MISCELLANEOUS PROBLEMS. 63 S3. Problemui. To measure the distance across a river on 3 9ttº traight line. Fig. 33 Solution. First Method. Let A B (fig. 33) be the required distance. Measure a line A C along the bank, and take the angles B A C and A C B. Then in the triangle A B C we have one side and two angles to find A. B. If A C is of such a length that an angle A C B = } D A C can be laid off to a point on the farther side, we have A B C = 3 D A C = A ("B. Therefore, without calculation, A B = A C. l; T) A fill | | # | º º | º A t t | N Fig. 34, Second Method. Lay off A C (fig. 34) perpendicular to A. B. Meas- are A C, and at Clay off CD perpendicular to the direction CB, and meeting the line of A B in D. Measure A. D. Then the triangles A CD and A B C are similar, and give A D : A C = A C : A B. A Cº. - Therefore, A B = HB . If from C, determined as before, the angle A C B' be laid off equal to A C B, we have, without calculation, A B = A B’. Third Method. Measure a line A D (fig. 35) in an oblique direction from the bank, and fix its middle point C. From any convenient point E in the line of Al B, measure the distance E C, and produce 64 MISCELLAN E O ÚS PROBLEM S. E C until C F = E C. Then, since the triangles A C E and D C F ure similar by construction, we see that D F is parallel to E B. Find now a point G, that shall be at the same time in the line of C B and of D F, and measure G D. Then the triangles A B C and D G C are equal, and G D is equal to the required distance A B. As the object of drawing E F is to obtain a line parallel to A B, this line may be dispensed with, if by any other means a line G F be drawn through D parallel to A. B. A point G being found on this parallel in the line of C B, we have, as before, G D = 4 B. PARABOLIC CURW ES. 65 CHAPTER II. PARABOLIC CURVES. ARTICLE I. — LOCATING PARABOLIC CURWEs. 84, LET A E B (fig. 36) be a parabola, A C and B C its tangents, and 4 B the chord uniting the tangent points. Bisect A B in D, and join CD. Then, according to Analytical Geometry, - Fig. 36. A. D I3 I. C. D is a diameter of the parabola, and the curve bisects C D in E. II. If from any points T, T', T'', &c., on a tangent A F, lines be ºl:awn to the curve parallel to the diameter, these lines T.M., T M T' 'M', &c., called tangent deflections, will be to each other as the squares of the distances A T, A T', A Tº", &c. from the tangent point A. - III. A line E D (fig. 37), drawn from the middle of a chord A B to the curve, and parallel to the diameter, may be called the middle ordi nate of that chord; and if the secondary chords A E and B E be drawn, the middle ordinates of these chords, K G and L H, are each equal to # E D. In like manner, if the chords A K, KE, E L, and L. B be drawn, their middle ordinates will be equal to 3 K G or 3 L H. 1V. A tangent to the curve at the extremity of a middle ordinate, is parallel to the chord of that ordinate. Thus MF, tangent to the . curve at E, is parallel to A B. 66 PARABOLIC CU HWES. V. If any two tangents, as A C and B C, be bisected in 31 and F. the line M1 F. joining the points of bisection, will be a new tangent, it middle point E being the point of tangency. 85. Problem. Given the tangents A C and B C, equal or unequal (Jig. 36,) and the chord A B, to lay out a parabola by tangent deflections. Fig. 36. - B Solution Bisect A B in D, and measure C D and the angle A CD ; or calculate C D* and A CD from the original data. Divide the tan. gent A C into any number n of equal parts, and call the deflection T'M for the first point a. Then (§ 84, II.) the deflection for the sec. ond point will be T'M' = 4 a, for the third point Tº M' = 9a, and so on to the nth point or C, where it will be n° a. Dut the deflection at this last point is C E = } CD ($ 84, I.). Therefore, n° a = CE, and Having thus found a, we have also the succeeding deflections 4 a, 9 a. 16 a, &c. Then laying off at T, T1, &c. the angles A TMſ, A T M &c. each equal to A CD, and measuring down the proper deflections, just found, the points M1, Mſ., &c. of the curve will be determined. The curve may be finished by laying off on A C produced n parts equal to those on A C, and the proper deflections will be, as before, a multiplied by the square of the number of parts from A. Dut all sº * Siuce C D is drawn to the middle of the base of the triangle A B C, we baſe, by (Heoruotry, C D2 = 3 (A C2 + B C2) — A D2. LOCATING PARABOLIC CURWES.. " 67 easier way generally of finding points beyond E is to divide the sec- ond tangent B C into equal parts, and proceed as in the case of A. C. If the number of parts on B C be made the same as on A C, it is obvi- ous that the deflections from both tangents will be of the same length for corresponding points. The angles to be laid off from B C must, of course, be equal to B C D. The points or stations thus found, though corresponding to equal distances cn the tangents, are not themselves equidistant. The length of the curve is obtained by actual measurement. 86. Problem. Given the tangents A C and B C equal or unequal, (jiy. 37,) and the chord A B, to lay out a parabola by middle ordinates. C 2^ Fig. 37. Ní E R" 2 ~~~~ w 2- " º _*z- A Solution. Bisect A B in D, draw CD, and its middle point E will be a point on the curve ($ 84, I.). DE is the first middle ordinate, and its length may be measured or calculated. To the point E draw the chords A E and B E, lay off the second middle ordinates G K and H L, each equal to 3 D E (§ 84, III.), and K and L are points on the curve. IDraw the chords A K. R. E., E. L., and L B, and lay off third middle ordinates, each equal to one fourth the second middle ordi- nates, and four additional points on the curve will be determined. Continue this process, until a sufficient number of points is obtained i) 87. Problem. To draw a tangent to a parabola at any station. Solution. I. If the curve has been laid out by tangent deflections ($ 85), let M!" (fig, 36) be the station, at which the tangent is to be drawn. From the preceding or succeeding station, lay off, parallel to CD, a distance M' N or E L equal to a, the first tangent deflection ($ 85), and JT''' N or M" L will be the required tangent. The same thing may be done by laying off from the second station a distance M'T' = 4 a, or at the third station a distance G P = 9 a ; for tho 68 * PARABOLIC CURVES. required tangent will then pass through T' or G. It will be seen, also, that the tangent at M1''' passes through a point on the tangent at A corresponding to half the number of stations from A to M!!! ; that is, M1" is four stations from A, and the tangent passes through T', the second point on the tangent A C. In like manner, M!!! is six sta- tions from B, and the tangent passes through G, the third point on the tangent B C. II. If the curve has been laid out by middle ordinates ($ 86), the tan- gent deflection for one station is equal to the last middle ordinate made use of in laying out the curve. For if the tangent A C (fig. 37) were divided into four equal parts corresponding to the number of stations from A to E, the method of tangent deflections would give the same points on the curve, as were obtained by the method of $ 86. In this case, the tangent deflection for one station would be a = # C E = # DE; but the last middle ordinate was made equal to 3 GK or * D E. Therefore, a is equal to the last middle ordinate, and a tan- gent may be drawn at any station by the first method of this section. A tangent may also be drawn at the extremity of any middle ordi- nate, by drawing a line through this extremity, parallel to the chord of that ordinate (§ 84, IV.). 88. In laying out a parabola by the method in § 85, it may some- times be impossible or inconvenient to lay off all the points from the original tangents. A new tangent may then be drawn by § 87 to any station already found, as at M''' (fig. 36), and the tangent deflections a, 4 a, 9 a., &c. may be laid off from this tangent, precisely as from the first tangent. These deflections must be parallel to CD, and the dis- tances on the new tangent must be equal to T' N or NM'", which may be measured. 89. Problema. Given the tangents A C and B C, equal or unequal, (fig 38,) to lay out a parabola by bisecting tangents. Solution. Bisect A C and B C in D and F, join D F, and find E, the middle point of D F E will be a point on the curve ($ 84, V.). We have now two pairs of what may be called second tangents, Zi D and I) E, and E F and F.B. Bisect A D in G and D E in H, join G II, and its middle point M will be a point on the curve. Bisect E F and F B in K and L, join K L, and its middle point N will be a point on the curve. We have now four pairs of third tangents, A G and G M, M BI and H E, E K and KN, and NL and L. B. Bisect each pair in aurn, join the points of bisection, and the middle points of the joining Locating PARABOLIC CURVES. 69 lines will be four new points, M', Mº, Nº", and N'. The same method may be continued, until a sufficient number of points is obtained Fig. 38. 90. Problem. Given the tangents A C and B C, equal or uncgual, (fig. 89,) and the chord 21 B, to lay out a parabola by intersections. C Fig. 39. A. G R I} Solution. Bisect A B in D, draw CD, and bisect it in E. Divido the tangents A C and B C, the half-chords A D and D B, and the line C E, into the same number of equal parts, five, for example. Then the intersection M of A a and FG will be a point on the curve. For FM = # Ca, and Ca = # CE. Therefore, FM = 's CE, which is the proper deflection from the tangent at F to the curve ($85). In like manner, the intersection N of A b and H K may be shown to be a point on the curve, and the same is true of all the similar intersections indicated in the figure. If the line D E were also divided into five equal parts, the line A a would be intersected in M on the curve by a line drawn from B through a', the line A b would be intersected in N on the curve by a line drawn 7{) - PARABOLIC CURWES. from B through b', and in general any two lines, drawn from A and 3 through two points on CD equally distant from the extremities C and D, will intersect on the curve. To show this for any point, as JI, it is sufficient to show, that Ba' produced cuts F. G on the curve ; for it - has already becn proved, that A a cuts FG on the curve. Now Du': M G = B D : B G = 5:9, or M G = 3 Da'. But Da' = } C. E. Therefore, M G = , C E. Again, F G : CD = A G : A D = 1 : 5, Therefore, F G = } C D = É C E. We have then FM = F G — A1 G = 3 C E — ;, C E = ?, CE. As this is the proper deflection from the tangent at F to the curve ($ 85), the intersection of B aſ with F G is on the curve. This furnishes another method of laying out a parabola by intersections. 91. The following example is given in illustration of several of the preceding methods. Example. Given A C = B C = 832 (fig. 40), and A B = 1536, to lay out a parabola A E B. We here find C D = 320. To begin with the method by tangent deflections ($85), divide the tangent A C into eight equal parts. Then a = ... = , = 2.5. Lay off from the divisions on the tangent F l = 2.5, G 2 = 4 × 2.5 = 10, II 3 = 9 × 2.5 = 22.5, and A 4 = 16 × 2.5 = 40. Suppose now that it is inconvenient to continue this method beyond IV. In this case we may /C Fig. 40. / N I M. N. () E. J. g-#T fly Iſ 3 S R y^* G. 3 2% I- WY Fº V 2. S A. I} lº, find a new tangent at E, by bisecting A C and B C ($ 89), and draw: ing K L through the points of bisection. Divide the new tangent K E = 3 A D = 384 into four equal parts, and lay off from KE the {ADIUS OF CURVATURE. 7] same tangent deflections as were laid off from A K, namely, M5 = 22.5, N6 = 10, and O7 = 2.5. To lay off the second half of the curve by middle ordinates ($ 86), measure E B = 784.49. Bisect E B in P, and lay off the middle ordinate P R = 4 D E = 40. Measure E R = 386,08, and B R = 402.31, and lay off the middle or- dinates S T and V FV, each equal to 3 P R = 10. By measuring the chords ET, TIR, R IV, and WB, and laying off an ordinate from each, equal to 2.5, four additional points might be found ARTICLE II. — TAD1Us of CURVATURE. 92. THE curvature of circular arcs is always the same for the same arc, and in different arcs varies inversely as the radii of the arcs. Thus, the curvature of an arc of 1,000 feet radius is double that of an arc of 2,000 feet radius. The curvature of a parabola is continually changing. In fig. 39, for example, it is least at the tangent point A, the extremity of the longest tangent, and increases by a fixed law, un- til it becomes greatest at a point, called the vertex, where a tangent to the curve would be perpendicular to the diameter. From this point to B it decreases again by the same law. We may, therefore, con- sider a parabola to be made up of a succession of infinitely small cir- cular arcs, the radii of which continually increase in going from the vertex to the extremities. The radius of the circular arc, correspond- Íng to any part of a parabola, is called the radius of curvature at that point. If a parabola forms part of the line of a railroad, it will be necessa- ry, in order that the rails may be properly curved (§ 2S), to know how the radius of curvature may be found. It will, in general, be necessary to find the radius of curvature at a few points only. In Ghort curves it Inay be found at the two tangent points and at the mid- dle station, and in longer curves at two or more intermediate points besides. The rails curved according to the radius at any point should be sufficient in number to reach, on each side of that point, half-way to the next point. 93. Problem. To find the radius of curvature at certain stations on a parabola. - Solution. I.et A E B (fig. 41) be any parabola, and let it be re- quired to find the radii of curvature at a certain number of stations 72 PARAE OLIC CURVES. from 4 to E. These stations must be selected at regular intervals from those determined by any of the preceding methods, Let n de note the number of parts into which A E is divided, and divide CD into the same number of equal parts. Draw lines from A to the points Fig. 41. A. - D B of division. Thus, if n = 4, as in the figure, divide CD into four cqual parts, and draw A. F., A E, and A. G. Let A D = c, A F = ci A E = ca, A G = ca, and A C = T. Denote, moreover, CD by a and the area of the triangle A C B by A. Then the respective radii for the points E, 1, 2, 3, and A will be cº ciº coš ca” T's R = i, R = i, R = i , R, + = The area A may be found by form. 18, Tab. X.; c and T are known and c1, C2, ca may be found approximately by measurement on a figure carefully constructed, or exactly by these general formulae : — T* – c’ (n − 1) dº 2 — 22 C1 c” + 7, n” y T* – c' (n – 3) dº c.” – c.” –H *- 2 l Wł I, } Y: , , T – c' (n − 5) dº ca” = cz” + 73 *- n? 2 2 T2 — cº (n – 7) cº" = ca” –H ?! - m? &c. &c. It will be seen, that each of these values is formed from the preceding, º ... T2 – c.2 ... d2 * * * * * * by adding the same quantity **** , and subtracting a multiplied in Buccession by n – ?, n – 3, n - 5, &c, Making n = 4, we have RADIUS OF Cluſ RWATURE. 73 * c,” = c + 3 (Tº – c') — # d", c.” = c,” + 4 (Tº – c’) — ſº d”, c.” + 3 (T" — cº, + ſº d”. All the quantities, which enter into the expressions for the radii, are now known, and the radii may, therefore, be determined. The same method will apply to the other half of the parabola. The manner of obtaining the preceding formulae is as follows. The radius of curvature at any given point on a parabola is, by the Differ- ca” F ontial Calculus, It = gº gr, in which p represents the parameter of the parabola for rectangular coördinates, and E the angle made with a diameter by a tangent to the curve at the given point. First, let the middle station E (fig. 42) be the given point. Then the angle E is the g Fig. 42. __ E2 | . H sº |K __ A-L. —t angle made with E D by a tangent at E, or since A B is parallel to the tangent at E (§ 84, IV.), sin. E = sin. A DE = sin. B D E. Let p! be the parameter for the diameter E. D. Then, by Analytical Ge º & e p ometry, p = p' sin.” E. Therefore, at this point R = 2 sin.3 E F p' sin.2 E p' A D2 C2 c2 ††† = grin. E. But p' = EB = i. Therefore, R = HitF g= aire == º ; since A = c d sin. E (Tab. X. 17). Next, to find R, , or the radius of curvature at H, the first station from E. Through H draw FG parallel to CD, and from F draw the tangent FK. Join A K, cutting CD in L. Then from what has just been proved for the radius of curvature at E, we have for the radius G of curvature at H, R1 = **. Now A G : A L = * 3 - sº * -- - - - - ---- R G. B | | | < i. FT NI K N the centre A to the slope stakes at D and E would be A D = 21 E = M A + A N = } b + š c. But as the ground rises from A to C through a height C G = g, the slope stake must be scº farther out a distance E G = # g; and as the ground falls from A to B through a height B F = g, the slope stake must be set farther in a distance D F = $g. To find B and C, set the level, if possible, in a convenient position for sighting on the points A, B, and C. From the known cut at the centre find the value of A E = } b + š c. Estimate by the eye the rise from the centre to where the slope stake is to be set, and take this as the probable value of g. To A E add $g, as thus estimated, and measure from the centre a distance out, equal to the sum. Obtain now by the level the rise from the centre to this point, and if it agrees with the estimated rise, the distance out is correct. Iłut if the esti- mated rise prove too great or too small, assume a new value for g, measure a corresponding distance out, and test the accuracy of the estimate by the level, as before. These trials must be continued, until the estimated rise agrees sufficiently well with the rise found by the level at the corresponding distance out. The distance out will then be } b + š c + g. The same course is to be pursued, when the ground falls from the centre, as at B; but as g here becomes minus, the dis- tance out, when the true value of g is found, will be A F = A D — D F = } b + šc — ; g. For embankment, the process of setting slope stakes is the same as for excavation, except that a rise in the ground from the centre on embankments corresponds to a fall on excavations, and vice versá. This will be evident by inverting figure 45, which will then represent 5 84 LEVELLING. an embankment. What was before a fall to B, becomes now a rise. and what was before a rise to C, becomes now a fall. When the section is partly in excavation and partly in embankment, the method above applies directly only to the side which is in excava- tion at the same time that the centre of the road-bed is in excavation, or in embankment at the same time that the centre is in embank- ment. On the opposite side, however, it is only necessary to make c in the expressions above minus, because its effect here is to diminish the distance out. The formula for this distance out will, therefore, be- come & b – ; c + š g. ARTICLE II. — CoRRECTION FOR TIIE EART II's CURVATURE AND FOR REFRACTION. 103. IET A C (fig. 46) represent a portion of the carth's surface. Then, if a level be set at A, the line of sight of the level will be the tan- gent A D, while the true level will be A. C. The difference D C be. tween the line of sight and the true level is the correction for the earth's curvature for the distance A D. 104. A correction in the opposite direction ariscs from refraction. Refraction is the change of direction which light undergoes in passing from one medium into another of different density. As the atmos- pilere increases in density the nearer it lies to the earth's surface, light, passing from a point B to a lower point A, enters continually air of greater and greater density, and its path is in consequence a curve concave towards the earth. Near the earth's surface this path may be taken as the arc of a circle whose radius is seven times the radius of the earth.” Now a level at A, having its line of sight in the direction A D, tangent to the curve A B, is in the proper position to receive the light from an object at B; so that this object appears to the observer to be at D. The effect of refraction, therefore, is to make an object appear higher than its true position. Then, since the correction for the earth's curvature D C and the correction for refraction D B are in opposite directions, the correction for both will be B C = D C – D.B. * Peirce's Spherical Astronomy, Chap. X., § 125. It should be observed, how- ever, that the effect of refraction is very uncertain, varying with the state of the atmosphere. Sometimes the path of a ray is even made convex towards the earth. nnd sometimes the rays are refracted horizontally as well as vertically EARTH's CURVATURE AND REFRACTION. 85 This correction must be added to the height of any object as deter. mined by the level. 105. Problem. Given the distance A D = D (fig. 46), the radius of the earth A E = R, and the radius of the arc of refracted light = 7 R, to find the correction B C = d for the earth's curvature and for refraction. *— D / / Fig. 46 % Solution. To find the correction for the earth's curvature D C, we have, by Geometry, D C (DC + 2 E C) = A D*, or D C (D C+ 2 R, = D*. But as D C is always very small compared with the diameter of the earth, it may be dropped from the parenthesis, and we have º D2 º a D C X 2 R = D*, or D C = ; R. The correction for refraction D B may be found by the method just used for finding DC, merely chang º º D2 r ing R into 7 R. IIence D B = i. F. We have then d = B C – I)? I)2 D C – D B = #F – III, , or gº a 3 1?” . By this formula Table III. is calculated, taking R = 20,911,790 ft, as given by Bowditch. The necessity for this correction may be avoided, whenever it is possible to set the level midway between the points whose height is required. In this case, as the distance on each side of the level is the same, the corrections will be equal, and will destroy each other. 86 LEVELLING. A RTICLE III. — VERTICAL CURWEs. 106. VERTICAL curves are used to round off the angles formed by the meeting of two grades. Let A C and C B (fig. 47) be two grades meeting at C. These grades are supposed to be given by the rise per sta. tion in going in some particular direction. Thus, starting from A, the grades of A C and C B may be denoted respectively by 9 and g’; that is, g denotes what is added to the height at every station on A C, and g' denotes what is added to the height at every station on CB; but since C B is a descending grade, the quantity added is a minus quan- tity, and g! will therefore be negative. The parabola furnishes a very simple method of putting in a vertical curve. 107. Problem. Given the grade g of A C (fig. 47), the grade g of C B, and the number of stations n on each side of C to the tangent points 4 and B, to unite these points by u parabolic vertical curve. P - Fig. 47 <= wº-T -- – {r | --- | | ~. 2 ºf A. I’ In 2 ..] I R Solution. Let A E B be the required parabola. Through B and U draw the vertical lines F K and C II, and produce A C to meet FK in F. Through A draw the horizontal line A K, and join A B, cut- timg C II in D. Then, since the distance from C to A and B is meas. ured horizontally, we have A H = II K, and consequently A D = D B. The vertical line CD is, therefore, a diameter of the parabola (§ 84, I.), and the distances of the curve in a vertical direction from the stations on the tangent A F are to each other as the squares of the number of stations from A (§ 84, II.). Thus, if a represent this dis- tance at the first station from A, the distance at the sceond station would be 4 a, at the third station 9 a., and at B, which is 2 n stations e * F B - from A, it would be 4 nº a; that is, FB = 4 n°a, or a = H. To find a, it will then be necessary to find FB first. Through C draw the horizontal line C G and we have, from the equal triangles ('F G and VERTICAL CUB VES. 87 4 CH, F G = C H. But C H is the rise of the first grade 9 in the n stations from A to C; that is, C H = n g, or F G = n g. G B is also the rise of the second grade g' in n stations, but since g' is negative (§ 106), we must put G B = —ng'. Therefore, FB = F G + GB = n g – ng'. Substituting this value of F B in the equation for a, we have a = *H** ; Or [[F a = 2 = 9. - 4 n. The value of a being thus determined, all the distances of the curve from the tangent A F, viz. a, 4 a, 9 a., 16 a, &c., are known. Now if T and Tº be the first and sceond stations on the tangent, and verti. cal lines T P and T’ſ P’ be drawn to the horizontal line A K, the height TP of the first station above A will be g, the height T' P' of the second station above A will be 2 g, and in like manner for suc. ceeding stations we should find the heights 3 g, 4 g, &c. As we have already found T M = a, T'M' = 4 a, &c., we shall have for the heights of the curve above the level of A, MP = T P – T M = g — a, M! P = T'P' — Tº M' = 2 g – 4 a, and in like manner for the succeeding heights 3 g – 9 a., 4 g – 16 a, &c. Then to find the grades for the curve at the successive stations from A, that is, the rise of each height over the preceding height, we must subtract each height from the next following height, thus: (g — a) — 0 = g — a, (29 – 4 a) — (y – a) = g – 3 a, (3 g – 9 a) — (2 g – 4 a) = g – 5 a., (4 g – 16 a) — (3 g – 9 a) = g – 7 a., &c. The successive grades for the vertical curve are, therefore, ſº g — a, g – 3 a, g – 5 a., g – 7 a, &c. In finding these grades, strict regard must be paid to the algebraic signs. The results are then general ; though the figure represents but one of the six cases that may arise from various combinations of ascending and descending grades. If proper figures were drawn to represent the remaining cases, the above solution, with due attention to the signs, would apply to them all, and lead to precisely the same formulae. 108. Eramples. Let the number of stations on each side of C be 3, and let A C ascend .9 per station, and C B descend.6 per station. Here - — g! .9 — (–.6 1.5 * = 3, g = 9, and g = –0. Then, a = *...* = (−6) 1. 4 m = 4 × 3 = i. * = .125, and the grades from A to B will be 88 LEVELLING. g — a = .9 — .125 = .775, g — 3 a = .9 — .375 = .525, g — 5 a = .9 — .625 = .275, g — 7 a = .9 — .875 = .025, g – 9 a = .9 — 1.125 = — .225, g – l l a = .9 — 1.375 = — .475. As a second example, let the first of two grades descend .8 per sta. tion, and the second ascend 4 per station, and assume two stations on each side of C as the extent of the curve. Here g = —.8, g! = 4, —.8 — .4 — 1.2 and n = 2. Then a = -15. a = −3 = – 15, and the four grades required will be g – a = −.8 – (– 15) = – 8 + 15 = — 65, 9 – 3 a = — .8 — (– :45) = — 8 + .45 = — .35, 9 – 5 a = – 8 — (— .75) = — 8 + .75 = — .05, 9 – 7 a = — .8 — (– 1.05) = — 8 + 1.05 = + .25. It will be seen, that, after finding the first grade, the remaining grades may be found by the continual subtraction of 2 a. Thus, in the first example, each grade after the first is 25 less than the preceding grade, and in the second example, a being here negative, each grade after the first is .3 greater than the preceding grade. 109. The grades calculated for the whole stations, as in the fore- going examples, are sufficient for all purposes except for laying the track. The grade stakes being then usually only 20 feet apart, it will be necessary to ascertain the proper grades on a vertical curve for these sub-stations. To do this, nothing more is necessary than to let g and g' represent the given grades for a sub-station of 20 feet, and n the number of sub-stations on cach side of the intersection, and to apply the preceding formulae. In the last example, for instance, the first grade descends .8 per station, or .16 every 20 fect, the second grade ascends .4 per station, or .08 every 20 feet, and the number of sub-stations in 200 feet is 10. We have then g = — .16, g! = .08, and n = 10. — .16 — .08 — .24 * º IIence a = −izijº = -1 = — .006. The first grade is, there. fore, g — a = — .16 + .006 = — 154, and as each subsequent grade increases .012 (§ 108), the whole may be written down without farther trouble, thus: -— .154, −.142, — .130, −.118, - .106, - .094, −.082, —.070, −.058, -.046, -.034, −,022, —.010, + .002, +.014, --,026 +.038, -} .050, + .062, -i-.074. EL E VAT I O N OF THE OUTER RAIL ON CURV ES. 89 ARTICLE IV — ELEvATIox of THE OUTER RAIL on CURWEs. 110. Probleman. Given the radius of a curve R, the gauge of the track g, and the velocity of a car per Second v, to determine the proper ele- vation e of the outer rail of the curve. Solution. A car moving on a curve of radius R, with a velocity per sec- e º tº 2 ond = v, has, by Mechanics, a centrifugal force = TR To counteract this force, the outcr rail on a curve is raised above the level of the inner rail, so that the car may rest on an inclined plane. This eleva- tion must be such, that the action of gravity in forcing the car down the inclined plane shall be just equal to the centrifugal force, which impels it in the opposite direction. Now the action of gravity on a body resting on an inclined plane is equal to 32.2 multiplied by the ratio of the height to the length of the plane. But the height of the plane is the elevation e, and its length the gauge of the track g. This action of gravity, which is to counteract the centrifugal force, is, there- 3.22 e - s e fore, = gT Putting this equal to the centrifugal force, we have 32.2 g tº -ā- = R IIence fº e = .9" . º * 50 If we substitute for R its value (§ 10) R = si. B , we have e = g tº sin. D * c : º * º * * 30 x : x = 00062112 g v" sin. D. If the velocity is given in miles M x 5280 * = Tö0 × 60- Substituting this value of v, we find e = .0013361 g M* sin. D. When g = 4.7, this becomes e = .00627966 J1° sin. D. By this formula Table IV. is calculated. In determining the proper elevation in any given case, the usual practice is to adopt the highest customary speed of passenger trains as the value of J1. 111. Still the outer rail of a curve, though elevated according to the preceding formula, is generally found to be much more worn than the inner rail. On this account some are led to distrust the formula, and to give an increased elevation to the rail. So far, however, as the centrifugal force is concerned, the formula is undoubtedly correct, and the evil in question must arise from other causes, – causes which are not counteracted by an additional elevation of the outer rail. The principal of these causes is probably improper “coming” of the wheels. Two wheels, immovable on an axle, and of the same radius, must, iſ per hour, represent this velocity by iſ, and we have 90 LEVELLING. no slip is allowed, pass over equal spaces in a given number of ſevo. lutions. Now as the outer rail of a curve is longer than the inner rail, the outer wheel of such a pair must on a curve fall behind the inner wheel. The first effect of this is to bring the flange of the outer wheel against the rail, and to keep it there. The second is a strain on the axle consequent upon a slip of the wheels equal in amount to the dif. ference in length of the two rails of the curve. To remedy this, con- ing of the wheels was introduced, by means of which the radius of the Outer wheel is in effect increased, the nearer its flange approaches the rail, and this wheel is thus cnabled to traverse a greater distance than the inner wheel. To find the amount of coning for a play of the wheels of one inch, let r and r" represent the proper radii of the inner and outer wheels respectively, when the flange of the outer wheel touches the rail. Then r' — r will be the coning for one inch in breadth of the tire. To ena- ble the wheels to keep pace with each other in traversing a curve, their radii must be proportional to the lengths of the two rails of the curve, or, which is the same thing, proportional to the radii of these rails. If R be taken as the radius of the inner rail, the radius of the outer rail will be R + g, and we shall have r : r' = R : R + g. Therefore, r R + r g = r. ſº, or r — r = ′2. 18 As an example, let R = 600, r = 1.4, and g = 4.7. Then we have 1.4 × 4,7 - e º º © r' — r = **ś- = 011 ft. For a tire 3.5 in. wide, the coning would be 3.5 × .01 l = .0385 ft., or nearly half an inch. Wheels comed to this amount would accommodate themselves to any curves of not less than 600 feet radius. On a straight line the flanges of the two whecls would be equally distant from the rails, making both wheels of the same diameter. On a curve of say, 2400 feet radius, the flange of the outer wheel would assume a position one fourth of an inch nearer to the rail than the flange of the inner wheel, which would increase the radius of the outer wheel just one fourth of the necessary increase on a curve of 600 fect. Should the flange of the outer wheel get too near the rail, the disproportionate increase of the radius of this wheel would make it get the start of the inner wheel, and cause the flange to recede from the rail again. If the shortcst radius were taken tº o 1.4 × 4.7 as 900 feet, r and g remaining the same, we should have r"—r = Hº- ELEVATION OF THE OUTER RAIL ON CURV ES. 9 : = .0073, and for the coming of the whole tire 3.5 × .0073 = .0256 ft., or about three tenths of an inch. Wheels coned to this amount would accommodate themselves to any curve of not less than 900 feet radius. If the wheels are larger, the coning must be greater, or if the gauge of the track is wider, the coning must be greater. If the play of the wheels is greater, the coning may be diminished. Hence it might be advisable to increase the play of the wheels on short curves, by a slight increase of the gauge of the track. Two distinct things, therefore, claim attention in regard to the mo- tion of cars on a curve. The first is the centrifugal force, which is generated in all cases, when a body is constrained to move in a cur- wilinear path, and which may be effectually counteracted for any given velocity by elevating the outer rail. The second is the unequal length of the two rails of a curve, in consequence of which two wheels fixed on an axle cannot traverse a curve properly, unless some provision is made for increasing the diameter of the outer wheel. Coning of the wheels seems to be the only thing yet devised for obtaining this in- crease of diameter. At present, however, there is little regularity either in the coning itself, or in the distance between the flanges of wheels for tracks of the same gauge. The tendency has been to di- minish the coning,” without substituting any thing in its place. If the wheels could be made to turn independently of each other, the whole difficulty would vanish ; but if this is thought to be impracticable, the present method ought at least to be reduced to some system. * Bush and Lobdell, extensive wheel-makers, say, in a note published in Apple- tons' Mechanic's Magazine for August, 1852, that wheels made by them for the New York and Erie road have a coning of but one sixteenth of an inch. This coning ou A track of six feet gauge with the Cîher data us given above, would suit no curve Ji loss than a mile radius. 92 ſº ARTH- WORK, CHAPTER TV. I. A RTH-WWORK. ARTICLE I. — PRISMOIDAL ForMUL.A. 112. EARTH-work includes the regular excavation and embank ment on the line of a road, borrow-pits, or such additional excavations as are made necessary when the embankment exceeds the regular ex cavation, and, in general, any transfers of earth that require calcula. tion. We begin with the prismoidal formula, as this formula is fre- quently used in calculating cubical contents both of earth and masonry. A prismoid is a solid having two parallel faces, and composed of prisms, wedges, and pyramids, whose common altitude is the perpen- dicular distance between the parallel faces. ! 13. H* rob Heann. Given the areas of the parallel faces B and B', the middle area M, and the altitude a of a prismoid, to find its solidity S. Solution. The middle area of a prismoid is the area of a section midway between the parallel faces and parallel to them, and the alti- tude is the perpendicular distance between the parallel faces. If now b represents the base of any prism of altitude a, its solidity is a b. If b represents the base of a regular wedge or half-parallelopipedon of alti- tude a, its solidity is ; a b. If b represents the base of a pyramid of altitude a, its solidity is ; a b. The solidity of these three bodies ad mits of a common expression, which may be found thus. Let m rep resent the middle area of either of these bodies, that is, the area of a section parallel to the base and midway between the base and top. In the prism, m = b, in the regular wedge, m = , b, and in the pyramid, m = } b. Moreover, the upper base of the prism = b, and the upper base of the wedge or pyramid = 0. Then the expressions a b, a b, and a b may be thus transformed. Solidity of x * = 0 + 0 + 4*) = 0 + 3 + 4*). prism = a b = 6 wedge = } a b = x 30–40 + 1 + 2) = 0 + b + 4 m) . pyramid = { a } = × 2 = 0 + tº - (0 + b + 4 m) BO RROW-PITS. 93 Hence, the solidity of either of these bodies is found by adding togeth: er the area of the upper base, the area of the lower base, and four times the middle area, and multiplying the sum by one sixth of the altitude. Irregular wedges, or those not half-parallelopipedons, may be measured by the same rule, since they are the sum or difference of a regular wedge and a pyramid of common altitude, and as the rule applies to both these bodies, it applies to their sum or difference. Now a prismoid, being made up of prisms, wedges, and pyramids of common altitude with itself, will have for its solidity the sum of the solidities of the combined solids. But the sum of the areas of the upper and lower bases of the combined solids is equal to B -- B', the sum of the areas of the parallel faces of the prismoid; and the sum of the middle arcas of the combined solids is equal to JI, the middle area of the prismoid. Therefore ſº s–, (b+ b + M). ARTICLE II. — Borº Row-PITs. 114. For the measurement of small excavations, such as borrow. pits, &c., the usual method of preparing the ground is to divide the surface into parallelograms “ or triangles, small enough to be consid- ered planes, laid off from a base line, that will remain untouched by the excavation. A convenient bench-mark is then sclected, and levels taken at all the angles of the subdivisions. After the excavation is inade, the same subdivisions are laid off from the base line upon the bottom of the excavation, and levels referred to the same bench-mark are taken at all the angles. - This method divides the excavation into a series of vertical prisms, generally truncated at top and bottom. The vertical edges of these prisms are known, since they are the differences of the levels at the top and botto ºn of the excavation. The horizontal section of the prisms is also known, because the parallelograms or triangles, into which the surface is divided, are always measured horizontally. 115. Problenn. Given the cages h, h, , and ha, to find the solidit; * If the ground is divided into rectangles, as is generally done, and one side be made 27 feet, or some multiple of 27 feet, the contents may be obtained at once in oubic yards, by merely o-mitting the factor 27 in the calculation. 94. EARTH- WO R K. S of a vertical prism, whether truncated or not, whose lorizontal section is a triangle of given area A. Fig. 48. º Solution. When the prism is not truncated, we have h = h = hg. The ordinary rule for the solidity of a prism gives, therefore, S = A h = 4 X # (h –H h; + ha). When the prism is truncated, let A B C FG II (fig. 48) represent such a prism, truncated at the top. Through the lowest point A of the upper face draw a horizontal plane A D E cutting off a pyramid, of which the base is the trapezoid B D E C, and the altitude a perpendicular let fall from A on D E. Represent this perpendicular by p, and we have (Tab. X. 52) the solidity of the pyra- mid = } p X B D E C = } p x D E x 3 (B D + CE) = } p x DE x * (B D + C E) = Z[ x , (B D + CE), since , p x DE = A DE = A. But y (B D + C E) is the mean height of the verti. cal edges of the truncated portion, the height at A being 0 IIenco the formula already found for a prism not truncated, will apply to tho portion above the plane A D E, as well as to that below. The same reasoning would apply, if the lower end also were truncated. Hence, for the solidity of the whole prism, whether truncated or not, we have [EF" S = A X (h -- h, + ha). 116. Problem. Given the edges h, h, , h, , and ha, to find the solidity S of a vertical prism, whether truncated or not, whose horizontal Section is a parallelogram of given area A. BORROW-PITS, 95 Solution. Let B H (fig. 49) represent such a prism, whether tran- cated or not, and let the plane B F HD divide it into two triangular C h jº h ! l, h *— º \, T----- G. ‘S., ------" " -- are * - 1.------T, S, 'H. prisms A F BI and C F H. The horizontal section of each of these prisms will be A, and if h, h, , h, , and ha represent the edges to which they are attached in the figure, we have for their solidity (§ 115) A FII = } A x 3 (h+hi + ha), and C F H = } A X # (h, + k2 -- ha). Therefore, the whole prism will have for its solidity S = } A X § (h-H 2 h, + ha + 2 ha). Let the whole prism be again divided by the plane A E G C into two triangular prisms B E G and D E G Then we have for these prisms, B E G = } A X # (h + hi + ha), and D E G = } A × 3 (h -- ha + h), and for the whole prism, S = # A × 3 (2 h -- h, + 2 ha + lis). Adding the two expressions found for S, we have 2 S = } A (h -- h, + ha + ha), or º S = At X # (h -- hi + he + h9). It will be seen by the figure, that § (h+ ha) = R L = } (h, + ha), or h -- ha = hl + ha. The expression for S might, therefore, be re- duced to S = 4 × 3 (h -- ha), or S = A X 3 (hi + ha). But as the ground surfaces A B C D and E F G H are seldom perfect planes, it is considered better to use the mean of the four heights, instead of the mean of two diagonally opposite. 117. Corollary. When all the prisms of an excavation have the same horizontal section A, the calculation of any number of them 96 EART II - WORK. may be performed by one operation. Let figure 50 be a plun of Such in excavation, the heights at the angles being denoted by a, a, , 02, b, CZ. &/ cºg Z, 25, Ör & Ö 2 £5 0 6. / (* P * 2 (*A. 6's (Z d’. (ſe a’s Ž. Fig. 50. b, , &c. Then the solidity of the whole will be equal to 3 A multi- plied by the sum of the heights of the several prisms (§ 116). Into this sum the corner heights a, as , b, b, , cs, d, and d4 will enter but once, each being found in but one prism ; the heights at , bi , c, d, , d. , and dº will enter twice, each being common to two prisms; the heights b1, ba, aud ca will enter three times, each being common to three prisms; and the heights bs, C1, ca, and ca will enter four times, each being common to four prisms. If, therefore, the sum of the first set of heights is represented by s1, the sum of the second by ss, of the third by sa, and of the fourth by sa, we shall have for the solidity of all the prisms fº S = 4 A (si + 2 s, + 3 sq + 4 sa). ARTICLE III. – EXCAVATION AND EMBANKMENT. 118. As embankments have the same general shape as excavations, it will be necessary to consider excavations only. The simplest case is when the ground is considered level on each side of the centre line. Figure 51 represents the mass of earth between two stations in an ex- cavation of this kind. The trapezoid G B F H is a section of the imass at the first station, and G. B. F. H. a section at the second sta- tion; A E is the centre height at the first station, and A, E, the centre height at the second station; H II, F, F is the road-bed, G G, B, B the CENTRE HEIGHTS ALONE GIVEN . 97 surface of the ground, and G G, H, H and B B, F, F the planes form- ing the side slopes. This solid is a prismoid, and might be calculated by the prismoidal formula (§ 113). The following method gives the same result. A. Centre IIeights alone given. 119. Problem. Given the centre heights c and c. , the width of the road-bed b, the slope of the sides s, and the length of the scetion l, to find the solidity S of the ercavation. \ t w ! • * / Fig. 51. * * ‘. . \ , - w * * w • A 1 v *, * A v \, . \\, / N.' . A “. . Solution. Let c be the centre height at A (fig. 51) and c, the height at A1. The slope s is the ratio of the base of the slope to its perpen- ūicular height (§ 102). We have then the distance out A B = } b + sc, and the distance out Al B1 = } b + sci (§ 102). Divide the whole mass into two equal parts by a vertical plane A A. E. E. drawn through the centre line, and let us find first the solidity of the right- hand half. Through B draw the planes B E E1, B 41 E1, and B E, F , dividing the half-section into three quadrangular pyramids, having for their common vertex the point B, and for their bases the planes A A, E, E, E. E. F. F., and Al B. F. E. For the areas of these bases we have Area of A 4, E, E = } E E. × (4 E + A, E1) = }l (c -i- c1), “ “ E E, F, F == E F x E E, = } b l, “ “ All B. F. El = # 4, El X (El F +4, B1) = }(b c, -i- sc,”), and for the perpendiculars from the vertex B on these bases, produced when necessary, - 98 EART II - WORK. Perpendicular on A A. E., E = A B = } b + b c, (ſ “ E. E. F. F = A E = c, ({ {{ A, B, F. El = E E1 = l. Then (Tab. X. 52) the soliditics of the three pyramids are B - A 41 E, E = ? (; b + sc) × 3 l (c + c,) = { l (; b c + 3 & c, -} sc" -- S cel) B – E E, F, F = } c × 3 bl = }l b c, B – A, B, F. El = } l × (b c, -i- sc,”) = #l (b.ci + sc,”). Their sum, or the solidity of the half-section, is ; S = 3 l (; b (c + c,) + s (c" + c,” + c c,)]. Therefore the solidity of the whole section is S = } l (; b (c + c,) + s (c’ + c,” + c ci)], Or gº S = 3 l (b (c + ci) + 3 s (c” + c,” + c c1)] When the slope is 1} to 1, s = }, and the factor 3 s = 1 may be dropped. 120. Problem. To find the solidity S of any number n of succee- sive sections of equal length. Solution. Let c, c, , ca, ca, &c. denote the centre heights at the suc- cessive stations. Then we have (§ 1 19) Solidity of first section = 3 l (b (c + c,) + 3 s (cº + c,” + c cy)], “ “ second section = } l (b (c1 + c2) + 3 s (c.” -- c.” -- c1 c2)|, “ “ third section = } l (b (c., + ca) + 3 s (cº + c2” -- ca ca)|, &c &c. For the solidity of any number n of sections, we should have 3 l mul- tiplied by the sum of the quantities in n parentheses formed as those just given. The last centre height, according to the notation adopted, will be represented by ca, and the next to the last by cn — 1. Collect- ing the terms multiplied by b into one line, the squares multiplied by § s into a second line, and the remaining terms into a third line, we have for the solidity of n sections #3; S = }l b (c + 2 cl + 2 co -H 2 ca. . . . -- 2 cm – 1 + c,) + § s (c” + 2 c.1% + 2 c.” -- 2 ca”. ... + 2 cºn – 1 + cº,) + 3 s (c ci + c, co -- coca + caca .... + cn – 1 cm). When s = }, the factor 3 s = 1 may be dropped. - - - t C ENTRE AN D SIDE IIEIGIHTS GIVEN . 89 Erample. Given l = 100, b = 28, s = }, and the stations and cen: tre heights as set down in the first and second columns of the annexed table. The calculation is thus performed. Square the heights, and set the squares in the third column. Form the successive products cci, c1 c2, &c., and place them in the fourth column. Add up the last three columns. To the sum of the second column add the sum itself, minus the first and the last height, and to the sum of the third column add the sum itself, minus the first and the last square. Then 86 is the multiplier of b in the first line of the formula, 592 is the second line, since 3 s is here 1, and 274 is the third line. The product of 86 by b = 28 is 2408, and the sum of 274, 592, and 2408 is 3274. This mul- tiplied by 3 l = 50 gives for the solidity 163,700 cubic feet. Station. c. c?. C c 1 . 0 2 4 l 4 | 6 8 2 7 49 28 3 6 36 42 4 10 1 O0 60 5 7 49 70 6 6 36 42 7 4 | 6 24 46 -?. 306 - 27 | -- 40 2S6 592 86 - 592 2408 28 2)3274 2408 163700 B. Cºntre and Side Heights given. 121. When greater accuracy is required than can be attained by the preceding method, the side heights and the distances out ($ 102) are introduced. Let figure 52 represent the right-hand side of an excava tion between two stations. A A, B, B is the ground surface ; A E = c and 41 E1 = c are the centre heights; B G = h and B. G = hl, the side heights; and d and d. , the distances out, or the horizontal distan- ces of B and B, from the centre line. The whole ground surface may sometimes be taken as a plane, and sometimes the part on each side of the centre line may be so taken;" but neither of these Suppo- *- * * It is easy in any given case to ascertain whether a surface like A A 1 Bº B "g 100 EART II - WORK, sitions is sufficiently accurate to serve as the basis of a general method In most cases, however, we may consider the surface on each side of the centre line to be divided into two triangular planes by a diagonal passing from one of the centre heights to one of the side heights. A ridge or depression will, in general, determine which diagonal Ought to be taken as the dividing line, and this diagonal must be noted in the field. Thus, in the figure a ridge is supposed to run from B to A 1, from which the ground slopes downward on each side to A and I31. Instead of this, a depression might run from A to B, , and the ground rise each way to A1 and B. If the ridge or depression is very marked, and does not cross the centre or side lines at the regular sta- tions, intermediate stations must be introduced to make the triangular planes conform better to the nature of the ground. If the surface happens to be a plane, or nearly so, the diagonal may be taken in either direction. It will be seen, therefore, that the following method is applicable to all ordinary ground. When, however, the ground is very irregular, the method of $ 127 is to be used. 122. Problems. Given the centre heights c and cu, the side heights on the right h and h, , on the left h" and h', , the distances out on the right d and dº, on the left d' and d'I, the width of the road-bed b, the length of the section l, and the direction of the diagonals, to find the solidity S of the ercavation. Solution. Let figure 52 represent the right-hand side of the excava- tion, and let us suppose first, that the diagonal runs, as shown in the figure, from B to A 1. Through B draw the planes B E E1, BA, E1, and B El F , dividing the half-section into three quadrangular pyra- mids, having for their common vertex the point B, and for their bases the planes A A, E, E, E 12, F, F, and Al B. F. E.I . For the areas of these bases we have Area of A. A1 El E = } E E. × (4 E + A, E1) = } l (c + c,), “ “ E E, F, F = E FX E E, == } b l, “ “A, B, F, E, - 3 Al J21 × di + $ E, F: X hi = } di ci + 3 bhi, - and for the perpendiculars from the vertex B on these bases, produced when necessary, plane; for if it is a plane, the descent from A to B will be to the descent from A1 to B1, as the distance out at the first station is to the distance out at the second sta- tion, that is, c — h : Ci – ht = d : d1. If we had c = 9, h = 6, c1 = 12, h i = 8, == 24, and d 1 = 27, the formula would give 3: 4 = 24 : 27 which shows that the l saiface is not a plane. cENTRE AND slDE HEIGIITs GIVEN. 10] Perpendicular on A A. E., E = E G = d, {{ “ E E, F, F = B G = h, {{ {{ A, B, F, E1 = E E = i. A Fig 52 NT-- C `-- *. h. *— F, : S. * * * * * *---. \ ". \ * | j- Iº. T- Tuen (Tab X. 52) the solidities of the three pyramids are B – A A, E, E = } d x 3 l (c + c,) = 3 l (d. c + del), B – E E, F, F = } h X bl # lb h, B – A, B, F, E, - 3 l × ) (d. c. -i- 3 bhi) = # 1 (d. c + 3 bhi), - Their sum, or the solidity of the half-section, is § l (d. c + d, c, -i- d c, -i- bh + 3 bhi). (l) Next, suppose that the diagonal rims from A to B, . In this case, through B, draw the planes B, E, E, B, A E, and B, E F (not rep- resented in the figure), dividing the half-section again into three quadrangular pyramids, having for their common vertex the point B1, and for their bases the planes A 4, E, E, E E, F, F, and A B FE For the areas of these bases we have * Area of A A 1 E, E = } E E. × (4 E + Al Ei) = 3 l ( -- c1), “ “E E, F, F = E F x E E, = } b l, “ “A B FE = } A E x d -- # E F X h = 3 de + 3 h; and for the perpendiculars from B, on these bases, produced when Decessary, 102 I. A RTIſ - WO R.H. Perpendicular on A A 1 E, E = E, G, - u, , {{ “ E E, F, F = B; G = n, 4.( “ A B F E = E E = 4. Then (Tab. X. 52) the solidities of the three pyramids are 1, - 44, E, E = 3 d, × 3 l (c + c,) = ′ i (d. c + d, ...) 13, - E E, F, F = } h; × 3 bl # lb h, , 131 - 4 B FE = } l × 3 (d. c + 3 b h) = # l (d. c + , t h). E ~ Their sum, or the solidity of the half-section, is #! (d. c + d, c, -i- d. c + bh, -- bh). (2) We have thus found the solidity of the half-section for both direc tions of the diagonal. Let us now compare the results (1) and (2), and express them, if possible, by one formula. For this purpose let (l) be put under the form § 1 [d c + d c, -i- d c + 3 b (h 4 h, + h), and (2) under the form # l [d c + d, c, -i- d. c + 3 b (h -- h, + h)]. The only difference in these two expressions is, that d c, and the last h in the first, become di c and hi in the second. But in the first case, ci and h are the heights at the extremities of the diagonal, and d is the distance out corresponding to h; and in the second case, c and h, are the heights at the extremities of the diagonal, and di is the distance out corresponding to hi. Denote the centre height touched by the diagonal by C, the side height touched by the diagonal by H, and the distance out cor- responding to the side height H by D. We may then express both d c, and d. c by DC, and both h and hi.by II; so that the solidity of the half-section on the right of the centre line, whichever way the diago. nal runs, may be expressed by § 1 [d c + d, c, -i- D C + 3 b (h -- h, + II)}. (3) To obtain the contents of the portion on the left of the centre line, we designate the quantities on the left by the same letters used for cor- responding quantities on the right, merely attaching a (') to them to distinguish them. Thus the side heights are h' and k'i, and the dis. tances out d' and d'i, while D, C, and II become D', C7, and H/. The solidity of the half-section on the left may therefore he taken di. rectly from (3), which will become CENTRE AND SIDE HEIGHTS GIVEN. 103 # |d c + d", c, -i- D'C' + 3 b (h' + h", + H')}. (4) Finally, by uniting (3) and (4), we obtain the following formula for the solidity of the whole section between two stations tº S = }l (d+ d") c + (d. + d") c, -i- D C + D'C' + # 5 (h+ hi + H+ h' + h' + II')]. Example. Given l = 100, b = 18, and the remaining data, as ar ranged in the first six columns of the following table. The first col- amn gives the stations; the fourth gives the centre heights, namely, c = 13.6 and c1 = 8; the two columns on the left of the centre heights give the side heights and distances out on the left of the centre line of the road, and the two columns on the right of the centre heights give the side heights and distances out on the right. The direction of the diagonals is marked by the oblique lines drawn from h! = 8 to c1 = 8 and from c = 13.6 to h, = 12. Sta. | dºl. h". C. A. d. ||d +d. (a +d). D. c.| D c. | O| 2l 8 S. 13.6 S. l 0 || 24 45 || 6 || 2 1 | 15| 4 |> 8.0 |\ 12| 27 42|| 336 168] 367.2 lº | 2 1 GS 20 367.2 54 × 9 = 4 S6 6) 1969.20 32S20. To apply the formula, the distances out at each station are added together, and their sum placed in the seventh column; these sums, multiplied by the respective centre heights, are placed in the eighth column ; the product of d = 21 (which is the distance out correspond. ing to the side height touched by the left-hand diagonal) by c, = 8 (which is the centre height touched by the same diagonal) is placed In the ninth column, and the similar product of d, = 27 by c = 13.6 is placed in the last column. The terms in the formula multiplied by # b are all the side heights, and in addition all the side heights touched by diagonals, or 8 + 4 + 10 + l2 + 8 + 12 = 54. Then by sub- stitution in the formula, we have S = | x 100 (612 + 336 -- 168 + 367.2 + 9 X 54) = 32,820 cubic feet." * The examplo hore given is the same as that calculated in Mr. Borden's “8ys 104 EAHTH-WORK By applying the rule given in the note to § 121, we see that the Hur. face on the left of the centre line in the preceding example is a plane; since 13.6 – 8: 8 – 4 = 21 : 15. The diagonal on that side might, therefore, be taken either way, and the same solidity would be ob. tained. This may be easily seen by reversing the diagonal in this ex- ample, and calculating the solidity anew. The only parts of the for mula affected by the change are D'C' and 3 b Iſ". In the one case the sum of these terms is 21 × 8 + 9 × 8, and in the other 15 × 13.6 + 9 × 4, both of which are equal to 240. 123. Problein. To find the solidity S of any number n of succes Sive sections of equal length. Solution. Let c, c, , ca, ca, &c. be the centre heights at the succes 5ive stations; h, hi, h, , ha, &c. the right-hand side heights; h", h', , h'2, h's , &c. the left-hand side heights; d, di, d. , da , &c. the distances out on the right; and d', d'i, d'2, d's, &c. the distances out on the left. Then the formula for the solidity of one section (§ 122) gives for the Sclidities of the successive sections ël [(d+ d") c + (d. + d") c, -i- D C + D'C' + 3 b (h+h, + H+ h' + h", + EI")], tl (d. + d") cl + (d. + d'.) cz + D, C, + D', C.", + § b (hi + ha + Hi + h' + h^2 + H',)], § 1 [(d. + d'2) co -H (da + d'a) ca + D. C., + D'. C'2 + 3 b (he + h9 H., + h^2 + h's + H'2)], and so on, for any number of scetions. For the solidity of any num, ber n of sections, we should have l multiplied by the sum of n paren. theses formed as those just given IIence - iſº S = } (d+ d") c + 2 (d1+ d'I) ci-H 2 (d. + d'a) c. ...+ (d. + d'A) ch + DC + D'C' -- D, C, -i- D', C", + D. C., +- D', C/2 + &c. + $ b|h + 2 ha + 2 he . . . . . + ha + H+ Hi + H2 + &c + h!-H 2h'1+ 2 h’s ... + h", -- II'-H II'1+II's + &c. term of Useful Formulae, &c.,” page 187. It will be seen, that his calculation makes Chu solidity 32,400 cubic feet, which is 360 cubic feet less than the result above. This diſſerence is owing to the omission, by Mr. Borden's method, of a pyramid in- closed by the four pyramids, into which the upper portion of the right-hand half Section is by that method diviled. C ENTRE AND SIDE HEIGHTS GIVEN. 105 Erample. Given l = 100, b = 28, and the remaining data as gived in the first six columns of the following table. * sia; d. A' | < || |_d. a + d. (a + d') c. D. C. D c. * -º-º- i O || 17 | 2 s 2 || 2 | 17 34 6S - l 18.5| 3 > 4 ~T-5; 21.5 || 40 | 60 68; 43 2 | 20 || 4 ~i_5 T6: 23 43 215 S0 92 3 23 6 - || 5 || 8 || 26 49 29.4 I l 5 || 130 4 || 21.5| 5 - || 6 |>7| 24.5 || 46 276 129 147 5 20 || 4 || 6 - || 4 20 40 240 | 120 | 1.47 6 15.5 iTT 4-T3 is 5. 34 136 || 93 80 25 35 1389 605 6.39 22 30 1 1 S5 22 7 605 69 102 630 1 O2 2894 l? I X 14 = 2394 6)02:12 103533 cubie feet. The data in this table are arranged precisely as in the example for eal. culating one section (§ 122), and the remaining columns are calculated as there shown. Then, to obtain the first line of the formula, add all the numbers in the column headed (d+ a ') c, making 1389, and after- wards all the numbers except the first and the last, making 1185. The next line of the formula is the sum of the columns D' C1 and DC, which give respectively 605 and 639. To obtain the first line of the quantities multiplied by § b, add all the numbers in column h, making 35, next all the numbers except the first and the last, making 30, and lastly all the numbers touched by diagonals (doubling any one touched by two diagonals), making 37. The second line of the quan- tities multiplied by § b is obtained in the same way from the column marked h . The sum of these numbers is 171, and this multiplied by # h = 14 gives 2394. We have now for the first line of the formula 1389 + 1185, for the second 605 + 639, and for the remainder 2394. By adding these together, and multiplying the sum by § l = lº , we get the contents of the six sections in feet. 124. When the section is partly in excavation and partly in embank. ment, the preceding formulae are still applicable; but as this applica. tion introduces minus quantities into the caleulation, the following method, similar in principle, is preferable. 125. Problemin. Given the widths of an ercavation at the rootd-bed 106 EARTH- WORK. 4 F = w and Al F = wit fig. 53), the side heights h and h, , the wagth of the section l, and the direction of the diagonal, to find the solidity S of the ercavation, when the section is partly in creavation and partly in ent bankment. Fig. 53. B ; ==LT Airs | ſº D-1 I' Solution. Suppose, first, that the surface is divided into two trian. gles by the diagonal BA1. Through B draw the plane B A F, , dividing that part of the section which is in excavation into two pyra Inids B-A A, F, F and B – A, B, F, , the solidities of which are B-A A, F, F = } h x 3 l (w 4-w,) = l (wh + wi h), B-A, B, F = x , whº = l w, h, , The whole solidity is, therefore, S = }l (wh + wi h, + wi h) Next, suppose the dividing diagonal to run from A to B1. Through B, draw a plane B, A F (not represented in the figure), dividing the excavation again into two pyramids, of which the solidities are B. - 4 A, F, F = h, × 3 l (w -- wh) = i ! (whº + wi h), B, - A B F = l x , whº = . 1 wh. - The whole solidity is, therefore, S = l (w/ -- w h, + whi). The only difference in these two expressions is, that wi h in the first becomes whi in the second. But in the first case the diagonal touch- cs wrand h, and in the second case it touches w and hi. If then, we designate the width touched by the diagonal by W, and the height touched by the diagonal by H, we may express both w, h and wh, by W II; so that the solidity in either case may be expressed by CENTRE AND SIDE REIGHTS GIVEN . 107 pºgº" S = l (w h + wº, hi + W II). Coroliatry. When several sections of equal length succeed one another, the whole may be calculated together. For this purpose, the preceding formula gives for the solidities of the successive sections # 1 (wh + w, b, + W. H.), # l (wi h, + we ha + JV, IIA), # l (wo ho + wº ha + W. Hz), and so on for any number of sections. IIence for the solidity of any number n of sections we should have ſº S = }l (wh + 2 win, -ī- 2 we ha . . -- whhn + WEI-F W LI, +- W. II, + &c.) Erample. Given l = 100, and the remaining data as given in the flrst three columns of the following table. station. w. h tº wº. 0 2 L | | 2 l 8 × 6 4 S 8 2 10 JS 7 70 56 {} 13 - |\ 7 9 l 7() | 4 9 || 4 36 52 - 2.47 186 209 186 6}642 10700. The fourth column contains the products of the several widths by the corresponding heights, and the next column the products of those widths and heights touched by diagonals. The sum of the products in the fourth column is 247, the sum of all but the first and the last is 209, and the sum of the products in the fifth column is 186. These three sums are added together, multiplied by 100, and divided by 6, according to the formula. This gives the solidity of the four sections = 10700 cubic feet. 126. When the excavation does not begin on a line at right angle, to tho centre line, intermediate stations are taken where the excava- tion begins on each side of the road-bed, and the section may be calcu. 6 108 EARTH - WORK. lated as a pyramid, having its vertex at the first of these points, and for its base the cross-section at the second. The preceding method gives the same result, since w and h in this case become 0, and reduce the formula to S = {l w, h, The same remarks apply to the end of an excavation, C. Ground very Irregular. 127. Problem. To find the solidity of a scetion, when the ground is very irregular. A - \c Fig. 64. D E. l 31 ...A F. `S. - H > 0, _2^ 2- _2^ F > * 2’ ..” `S 2^ Solution. Let A H B FE – A 1 CD B, F, E, (fig. 54) represent one side of a section, the surface of which is too irregular to be divided into two planes. Suppose, for instance, that the ground changes at H, C, and D, making it necessary to divide the surface into five trian- gles running from station to station.” Let heights be taken at H, C, and D, and let the distances out of these points be measured. If now we suppose the earth to be excavated vertically downward through the side line B B, to the plane of the road-bed, we may form as many vertical triangular prisms as there are triangles on the surface. This will be made evident by drawing vertical planes through the sides * It will often be necessary to introduce intermediate stations, in order to make the subdivision into triangles more conveniently and accurately, G ROUN L, V FRY I R REGULAR. I09 A C, BI C, HD, and HB, . Then the solidity of the half-section will be 24val to the sum of these prisms, minus the triangular mass B F G - B, F, G, . The horizontal section of the prisms may be found from the distan- ces out and the length of the section, and the vertical edges or heights are all known. Hence the solidities of these prisms may be calculated by § 115. To find the solidity of the portion B F G - B, F, G, , which is to be deducted, represent the slope of the sides by s (§ 102), the heights at B and B, by h and h, , and the length of the section by l. Then we have F G = sh, and F, GI = sh, . Moreover, the area of B F G = $s h”, and that of B, F, G, = } sh!”. Now as the triangles B F G and B, F, G, are similar, the mass required is the frustum of a pyra- mid, and the mean area is V} s h” × sº = } s h h, . Then (Tab. X. 53) the solidity is B F G – B, F, G, - # 1 s (h’ + h,” + h hi). Example. Given l = 50, b =18, s = }, the heights at A, E1, and B respectively 4, 7, and 6, the distances A LI = 9 and II B = 9, the heights at A1, C, D, and B, respectively 6, 7, 9, and 8, and the distan- ces A, C = 4, C D = 5, and D Bi = 12. Then the horizontal see- tion of the first prism adjoining the centre line is $ 1 × A, C, since the distance A 1 C is measured horizontally; and the mean of the three heights is (4 + 6 + 7) = } X 17. The solidity of this prism is therefore 3 l X At C. × 3 × 17 = }l × 4 × 17, that is, equal to Él multiplied by the base of the triangle and by the sum of the beights. Iu this way we should find for the solidity of the five prisms !! (4 x 17+ 9 × 18 + 5 x 23 + 2 × 2 + 9 x 21) = . 1 × 822. For the frustum to be deducted, we have # 1 × 3 (6* + 8* + 6 × 8) = + 1 x 222. Hence the solidity of the half-section is #! (822 — 222) = { X 50 × 600 = 5000 cubic feet. 128. Let us now examine the usual method of calculating excavo, tion, when the cross-section of the ground is not level. This method consists, first, in finding the area of a cross-section at each end of the mass ; secondly, in finding the height of a section, level at the top, equivalent in area to each of these end sections; thirdly, in finding from the average of these two heights the middle area of the mass 110 EARTH- WORK. and, lastly, in applying the prismoidal formula to find the contents. The heights of the equivalent sections level at the top may be found approximately by Trautwine's Diagrams,” or exactly by the follow- ing method. Let A represent the area of an irregular cross-section, Ö the width of the road-bed, and s the slope of the sides. Let ~ be the required height of an equivalent section level at the top. The bottom of the equivalent section will be b, the top b + 2 s 3, and the area will be the sum of the top and bottom lines multiplied by half the height or # 3 (2 b + 2 sw) = s. 3." -- b 2. But this area is to be equal to 4. Therefore, S a.” + b x = A, and from this equation the value of a may be found in any given case. According to this method, the contents of the section already cal- culated in § 122 will be found thus. Calculating the end areas, we find the first end area to be 387 and the second to be 240. Then as s is here 3 and b = 18, the equations for finding the heights of the equivalent end sections will be 4 x2 + 18 x = 387, and 3 x2 + 18 a. = 240. Solving these equations, we have for the height at the first station & = 11.146, and at the second, a = 8. The middle area will, therefore, have the height ] (11.146 + S) = 9.573, and from this height the middle area is found to be 309.78. Then by the prismoi. dal formula (§ 113) the solidity will be S = { X 100 (387 -- 240 + 4 × 309.78) = 31102 cubic feet. But the true solidity of this section was found to be 32820 cubic feet, a difference of 1718 feet. The error, of course, is not in the pris. moidal formula, but in assuming that, if the earth were levelled at the ends to the height of the equivalent end sections, the intervening earth might be so disposed as to form a plane between these level ends, thus reducing the mass to a prismoid. This supposition, however, may sometimes be very far from correct, as has just been shown. If the diagonal on the right-hand side in this example were reversed, that is, if the dividing line were formed by a depression, the true solidity found by § 122 would be 29600 feet; whereas the method by equiva- lent sections would give the same contents as before, or 1502 feet too much. D. Correction in Ercavation on Curves 129. In excavations on curves the ends of a section are not parallel * A New Method of Calculating the Cubic Contents of Excavations and Embank ments by the aid of Diagrams. By John C. Trautwine ' COR RECTION IN EXCAVATION ON CURVES. 1 11 to each other, but converge towards the centre of the curve. A section between two stations 100 feet apart on the centre line will, therefore, measure less than 100 feet on the side nearest to the centre of the curve, and more than 100 feet on the side farthest from that centre. Now in calculating the contents of an excavation, it is assumed that the ends of a section are parallcl, both being perpendicular to the chord of the curvo, Thus, let ſigure 55 represent the plan of two sections of JB IB II _--~~ A. N all excavation, E. F. G being the centre line, A L and CMſ the extreme Side lines, and 0 the centre of the curve. Then the calculation of the (irst section would include all between the lines A, C, and B, D, ; while the true section lies between A C and B D. In like manner, the calculation of the second section would include all between H K and NP, while the true section lies between B D and L. M. It is evident, therefore, that at each station on the curve, as at F, the calculation is too great by the wedge-shaped mass represented by KFD, , and too * > N. * . C. <-- ~... . . •º - — — — —-...------ - - * *-* **T * Small by the mass represented by B. F. H. These masses balance | 12 EA RTH - WOR tº . each other, when the distances out on each side of the centre line are equal, that is, when the cross-section may be represented by A D FRE (fig. 56). But if the excavation is on the side of a hill, so that the distances out differ very much, and the cross-section is of the shape A DFB E, the difference of the wedge-shaped masses may require consideration. 130. Problem. Given the centre height c, the greatest side height h, 'he least side height h", the greatest distance out d, the least distance out d", and the width of the road-bed b, to find the correction in excavation C, at any station on a curve of radius R or deflection angle D. Solution. The correction, from what has been said above, is a trian- gular prism of which B FR (fig. 56) is a cross-section. The height of this prism at B (fig. 55) is B, H, the height at R is R, S, and the height at F is 0. B. H. and R, S, being very short, are here considered straight lines. Now we have the cross-section B FR = F B E G — F R E G = (3 c d -- 4 bh) — (; c d" + 3 b h') = * c (d. — d") + 4 b (h — h'). To find the height B, H, we have the angle B F H = B FBI = D, and therefore B, II = 2 H F sin. D = 2 d sin. D. In like manner, IR, S = K D1 = 2 KF sin. D = 2 d" sin. D. Then since the height at F is 0, one third of the sum of the heights of the prism will be 3 (d. -- d") sin. D, and the correction, or the solidity of the prism, will be (§ 115) - tº c = }• (d-d') +400–40 x 3 (a + a sin D. When R is given, and not D, substitute for sin. D its value (§ J) sin; D - º . The correction then becomes 100 (d. -- d') 3 IR This correction is to be added, when the highest ground is on the convex side of the curve, and subtracted, when the highest ground is on the concave side. At a tangent point, it is evident, from figure 55, that the correction will be just half of that given above. tº C = |} c (d — d") + 3 b (h — h')] × Erample. Given c= 28, h = 40, h' = 16, d = 74, d' = 38, b = 28, and R = 1400, to find C. IIere the area of the cross-section B F R -= 2 - - º (74 — 38) + # (40 – 16) = 672, and one third of the sum of the . . 100 (74 + 88) 8 * 8 heights of the prism is lºgº = . . Hence C = 672 × 5 - 1792 cubic feet. CORRECTION IN EXCAVATION ON CURVES. I 13 131. When the section is partly in excavation and partly in em. hankment, the cross-section of the excavation is a triangle lying wholly on one side of the centre line, or partly on one side and partly on the other. The surface of the ground, instead of extending from B to D (fig. 56), will extend from B to a point between G and E, or to a point between A and G. In the first case, the correction will be a triangular prism lying between the lines B, F and HF (fig. 55), but not extending below the point F. "In the second case, the excavation extends below F, and the correction, as in § 129, is the difference be- tween the masses above and below F. This difference may be ob- tained in a very simple manner, by regarding the mass on both sides of F as one triangular prism the bases of which intersect on the line G F (fig. 56), in which case the height of the prism at the edge be- low F must be considered to be minus, since the direction of this edge, referred to either of the bases, is contrary to that of the two others. The solidity of this prism will then be the difference required. 132. Problem. Given the width of the excavation at the road-bed w, the width of the road-bed b, the distance out d, and the side height h, to find the correction in excavation C, at any station on a curve of radius R or deflection angle D, when the section is partly in excavation and partly in embankment. Solution. When the excavation lies wholly on one side of the centre line, the correction is a triangular prism having for its cross-section the cross-section of the excavation. Its area is, therefore, $ wh. The beight of this prism at B (fig. 56) is (§ 130) B, H = 2 H F sin. D = 2 d sin. D. In a similar manner, the height at E will be 2 G E sin. D = b sin. D, and at the point intermediate between G and E, the dis- tance of which from the centre line is $ b – w, the height will be 2 (§ b–w) sin. D = (b – 2 w! sin. D. Hence, the correction, or the solid- ity of the prism, will be (§ 115) C = } whº (2 d-i-b-i-b–2 w sin. D = } wh;K # (d + b – wy sin. D. When the excavation lies on both sides of the centre line, the cor- rection, from what has been said above, is a triangular prism having also for its cross-section the cross-section of the excavation. Its area will, therefore, be wh. The height of this prism at B is also 2d sin. D, and the height at E, b sin. D; but at the point intermediate between A and G, the distance of which from the centre line is w — 3 b, the height will be 2 (w — ; b) sin. D = (2 w — b) sin. D. As this height is to be considered minus, it must be subtracted from the others, and the correction required will be C = § w fi X (2d -- b – 2 w -- b) sin. D 114 EARTH-WORK. = } to h X 3 (d. -- 5 — wº sin. D. Hence, in all cases, when the see , tion is partly in excavation and partly in embankment, we have the formula, [35° C = 3 wo h3. 3 (d -H b — to) sin. D. When R is given, and not D, substitute for sin. D its value (; 9) . 50 e sin. D = T - The correction then becomes {} ſº c = wh; 199 (***-*2. 3 IR This correction is to be added, when the highest ground is on the convex side of the curve, and subtracted when the highest ground is on the concave side. At a tangent point the correction will be just half of that given above. - Erample. Given w = 17, b = 30, d = 51, h = 24, and R = 1600, to find C. Here the area of the cross-section is $ wha 17 × 12 = tº - . . 100 (d+ b — 204, and one third of the sum of the heights of the prism is wº-R lo. 100 (51 + 80 – 17 4 - := *::::: ) := i. Hence C = 20.4 × 3 = 272 cubic feet. 133. The preceding corrections (§ 130 and § 132) suppose the length of the sections to be 100 feet. If the sections are shorter, the angle B FH (fig. 55) may be regarded as the same part of D that FG is of 100 feet, and B, FB as the same part of D that E Fis of 100 feet. The true correction may then be taken as the same part of C that the Sum of the lengths of the two adjoining sections is of 200 feet. APPENDIX. ARTICLE I.—NoTE To $12, P. 5. WHEN, as generally happens, the beginning of a curve does not fall at a full station, the first stake on the curve will be at the end of a sub-chord, and the sub-deflection angle will be the same part of P that the sub-chord is of a whole chord. In laying out a curve, there is an obvious advantage in having the several deflection angles whole minutes. When the deflection angle is assumed, whole minutes would naturally be chosen. But when D is found from I and T by § 11, it generally happens that D does not come out even minutes. In such cases, unless it is absolutely neees- sary that the curve should commence at the assumed tangent point, it will be better to take D to the nearest minute, and calculate T for I and this new value of D by § 12. If, however, there is a sub-chord at the beginning of the curve, the sub-deflection angle will generally contain seconds, and if we start from the tangent with the instrument set at zero, all the deflection angles will contain seconds, although D contains none. To avoid this, set the vernier back the amount of the sub-deflection angle, so that, when this angle is turned off, the instru- ment will read zero. All the subsequent angles will then be whole minutes. The usual mode of designating curves by their “degree,” as given in § 6, is objected to by some, because when curves are laid out by chords shorter than 100 feet, as is usual on sharp curves, the degree of the curve is slightly increased, though its designation remains the same. If the arc of 100 feet is substituted for the chord of 100 feet in the definition, this difficulty vanishes; but so many greater difficul- ties are introduced that the general adoption of this method is not 116 APPENDIX. probable. Moreover, when American engineers use the metric sys- tem, as possibly they are now doing on Mexican roads, both these methods are inapplicable. We might designate a curve by the length of its radius, for this fixes the curve, however laid out, and any units of length may be used; but when D is even, R is generally frac- tional, which makes it inconvenient for exact definition. The length of the radius is also an indirect designation, when curves are laid out by deflection angles. If the curve were designated by its deflection angle for a certain length of chord, any length of chord and any units of length might be used, and the curve be still definitely described. Thus we might say: “Curve to the right, deflection angle for chords of 50 feet, 2° 10',” or, “Curve to the left, deflection angle for chords of 20 metres, 1° 35'.” ARTICLE II.-LAYING OUT CURVES BY TANGENT DEFLECTIONs. This method of laying out curves is sometimes advantageous, and, when due care is used, very accurate. To keep the work close to the line of the curve, and to prevent the tangent deflections becoming in- conveniently long, new tangents may have to be found by the method shown below. Let D be the deflection angle of the curve for a chord c. Let C (fig. 58) be any point on the curve distant from the tangent point Fig. 58. A T R N O A, a certain number of chords, whole or fractional, denoted by n. Then the angle B A C = n D, and the central angle 4 O C = 2nl). APPENDIX. II 7 We wish to find the distance on the tangent A B = a, and the tan- gent deflection B C = b, in order to fix the point C. We have & c sin. 2n D a = C D = R sin. 2n D = ***, sin. D #c sin. 2n D tan. n D c sin.”n D b = a tan. n D = º: E —- . sin. D sin. D In finding these values for successive points, the logarithm of C — or of -. sin. D sin. D tions. The position of the stakes is best fixed by measuring the suc- cessive chords, instead of depending on the right angle at B. A tangent T C at any point of the curve is readily obtained by #c tan. n D, TO sin. D remains constant, which facilitates the computa- measuring from A a distance A T = R tan, n D = should, of course, prove equal to A. T. ARTICLE III.-TURNOUTS TANGENT TO MAIN LINE. The switch-rail in this case is to be considered a part of the turn- out curve, and the relation of its length to the radius of the turnout must be determined. Denote the length of the switch-rail by l, the throw of the switch by d, and the radius of the turnout by R. Then d is the tangent deflection of a curve of radius R + 3 g, and, by § 18, a tangent deflection is equal to the square of the chord divided by J. * twice the radius. ... d = 2R + y ; whence if R is given to find l, ! = 4/(2R-Eg) di or, if l is given to find the radius of the outer rail, o R+ g =#| For turnouts tangent to the main track a switch-rail of the fol- \owing kind has been used, and appears to have much in its favor. Suppose the length of the switch-rail as calculated above to be 20 feet. A rail of 30 feet in length is for 10 feet of its length spiked down or otherwise securely fastened on the main track, back from the tan- gent point, leaving 20 feet free for the switch-rail. The free end being thrown in the usual way, a curve is formed, which, however, is not a circular curve, but an elastic curve. The inclination at the free end, in the case supposed, would be about three-fourths of that of the cir- cular curve that meets it. If it be desired to make the two inclina- 118 APPENDIX. tions equal, so that the two curves shall be tangent to each other, the free part of the switch-rail should be only three-fourths of the calculated length of l. Thus, in the case Supposed, 15 feet instead of 20 feet would be left free, beginning as before at the tangent point. The switch-rail may, however, be made to take a circular form by suitable stops attached to the sleepers; and the full length, as calculated above, will then, of course, remain free. To obtain the necessary formulae for turnouts when tangent to the main line, we have only to modify the formulae already given (pages . 32 to 46), by making S = 0 and d = 0. The switch-angle evidently vanishes, and though d still exists as the throw of the switch-rail, it vanishes at the new tangent point. The chord, called BF, will now give the distance from the tangent point on the main track to the point of the frog. The modified formulae will now be given. For $ 50, R given to find F and B F, we have Jº — # (ſ º JB F = g cos, F=#H. sin. 3. Fº For $ 51, F given to find R and B F, we have # 9 . _ 9 Jº —- + 9 = sin.”3 F' T sin. Fº For $ 52, to find mechanically the position of a given frog, de- scribe the arc of 2 l from the tangent point on the main track, and ! .. measure the chord, HR = º, , directly from the inside of the main rail. § 54 requires no change. § 56. This problem is to be now solved by § 36. For § 57, I., R of main track and F given, to find B F and R' of turnout from inside of curve (fig. 59), we have B F — g cos.; K. tan. 4 K = - - 4./ sin. #F ' –4– 2R tan. Fº 4B F sin. (F-E-R)' For § 57, II., R of main track and F given, to find B F and R' of turnout from outside of curve (fig. 60), we have F' + 3 g = — —”— . , 9 cos. 4K. tan. 3 K = 2R tan. #F'' F'— sin. F ' f 4B F R" + 3 g = H sin. (F’— K) APF ENDIX, 119 For $ 58, same modification as for § 32. For $ 59, B F given to find F, we have º — 9. sin. #F = BF * For $60, I., R of main track and R' of turnout from inside of curve given, to find F and B F (fig. 59), we have all the sides of the triangle E F K, namely, E K = R — R', E F = R + #9, FK = R – 3 g. Therefore, by 15, Tab. X., (R – R' – 3 g) g R × R' *. tan. #F= 4 For § 60, II., R of main track and R' of turnout from outside of curve given, to find F and B F (fig. 60), we have all the sides of the Fig. 59. Fig. 60. - R __ – --— - _- B _- E. K K! triangle E FR, namely, ER = R + R', EF= R'+ , g, F K = R + #9. Therefore, by 15, Tab. X., / R × R' 4. (R+RE i º) iſ * (R + R' + 4 g) # g JR × R' ſº For § 62, R of main track, R' common radius of the turnout, and b = B G, the distance between centre lines of the parallel tracks; given, to find the angles A B C and B FC and the chords A C and tan, 3 E FR = tan. #F = 120 APPENDIX. B C (fig. 61), we have all the sides of the triangle E FK, namely, E K = R – R', EF = 2 R', FK = R + R'— b. The angles F.E. K. Fig. 61. K and E. F. K. may therefore be found by 15, Tab. X. Then A E C = 180° – FE K and B F C = EF K, and A C = 2 R' sin. ; A E C : B C = 2 R' sin. BFC. § 64. This problem is now the same as that of § 62, just solved. ARTICLE IV.--Double TURNOUTs. When two turnouts start from the same point of the main track, a third frog is needed where one turnout crosses the other. If the turnouts turn opposite ways, one of them may be treated as a turnout from the outside of the other, and then the case falls under the prob- lems already given for such turnouts. Or, the third frog may be placed with its point in the centre line of the main track, and its an- © Fig. 62. 2. Az ~. Fe A 2 > º - gle may be taken as made up of two angles, F, and F, one on each side of said centre line, as in figure 62. On a straight main track the ! APPENDIX. 121 two turnouts would, in gencral, be symmetrical, and F be equal to F. On a curved main track these partial angles may be made equal or unequal. In this way all the relations between the frog angles and the radii concerned may be determined by previous problems, sub- stituting 3 g for g as the distance of the line B C from either rail. Thus in the figure the radius of A, B and the partial frog angle F. depend on each other; so also do the radius of A2 B and the partial frog angle F. When the curve B F is not a continuation of the curve A1 B, the relation between its radius and the frog angle F is to be determined by considering F, to be a switch angle and the curve B F to commence at the but-end of the frog. If both turnouts turn the same way, as in figure 63, the third frog Fig. 63. —s. at F, may be considered to be on a turnout A F, F, from the inside of the curve A F, and the case falls under the problems already given for such turnouts. ARTICLE V.—LENGTH OF PARABoLIC ARCs. Fig. 64. A The length s of the parabolic arc A B (fig. 64) from the vertex 4 to a point B whose rectangular coördinates are z and y is 122 APPENDIX. * * (,, , z) •=1/(' —H 3:2 + ac? hyp. log 2y + *(, +.) . #) Ay ºr “e —---, or, introducing the angle i which the tangent at B makes with the axis of 2, 22 . . . S E 4y [tan. sec. i + hyp. log, (tan, i + sec. ?)]; or, by series, 2 /2 2. , a.6 •=(1++ º-, 4+4.4- *). QC When y is small relatively to 2, two terms of this series are often sufficient. Whence 2 2 y S - === -s. - 3 -i- 3 a. nearly. The length s of the parabolic arc A B (fig. 65) from the origin of oblique coördinates A to a point B whose oblique coördinates are z Fig. 65. and y, is given by the following formula, in which i is the angle made by the tangent at B with a line perpendicular to the axis of the para- bola, and j is the angle made by y with a perpendicular to the axis 4 Å. _ 2% *(an i sec. 7 – tan. 7 sec. 7 -- hyp. log tan. : + º) T 4y º e an. 7 Sec. 7 -i- nyp. 10g. tan.j + sec.j/. In many cases a near approximation is . . . 2 9 cos. 27 s = x + y sin.j + . . -º-. 3 2 + y sin.j ARTICLE VI.-NotE on SETTING SLOPE STAKES, The following process is often of advantage in setting slope stakes, Figure 65a represents the operation at three stations: APPENDIX. 123 Fig. 65 a. Let C C C represent the datum plane, “ B C = height of instrument = H, “ C D = height of road-bed = h, “ A B = sight on the ground at the Supposed place of side-stake = S, “ A D = the side cut (minus cuts are fills) = c'; Then in all three of the cases represented A D = B C – C D – A B, or c' = H – h — s. Having thus the side-cut or fill at the supposed place for a slope stake, we have for the distance out (slope 1.5 to 1) d = } b + 3 c'. For the same setting of the instrument H — h is constant for any One cross-section, and varies with h from one station to another. It is obvious that the cut or fill at any point between the side Stakes can be obtained in the same manner. ARTICLE VII.—ExPANSIon of RAILs. The rails of a track exposed to a summer sun may rise to a tem- perature of 180° Fahrenheit. When, therefore, a track is laid at a much lower temperature, as is usual, provision for the expansion of the rails must be made by leaving a proper space between successive rails. The expansion of a bar of iron or steel may be taken as ,000 007 of its length for every degree of rise in temperature. The space to be left between the rails will vary with the length of the rails and with the number of degrees below 130° of the temperature When the track is laid. Suppose 30-feet rails are laid at a tempera- ture of 50°. Then the number of degrees of possible rise of tempera- 124 APPENDIX. ture is 130°– 50° = 80°, and the space to be left between the rails is ,000 007 x 80 x 30 = .0168 of a foot. In general, let s be the space to be left, n the number of degrees that the temperature is below 130°, and l the length of the rails in fect, and we have s = .000 007 72, l. ARTICLE VIII.-EASING GRADEs on CURVES. When a curve occurs on a steep grade, it is desirable to ease the grade on the curve, so as to make the joint resistance of the grade and curve equal to that of the grade alone on straight lines. The re- sistance on a grade is proportional to the rise of the grade per sta- tion, and the resistance due to a curve can be represented as equiva- lent to that of a grade having a certain rise per station. The rise per station of the eased grade will be simply the original rise dimin- ished by the rise that represents the curve resistance. The resistance caused by curves varies greatly with the state of the track and the kind of rolling stock, and is variously estimated as equivalent on a 1° curve to that of a grade of .025 to .06 of a foot per station. For a curve of any other degree the resistance increases with the de- gree ; so that a 6° curve has six times the resistance of a 1° curve. As an example, let a rise of .04 per station be taken as the average resistance on a 1° curve, and suppose a 6° curve to occur on a grade of 1.6 per station. Then the reduced grade will be 1.6 – .24 = 1.36 per station. ARTICLE IX. —TRANSITION CURWEs. The object of a transition curve is to make the change easy from a straight line to a circular curve. The proper super-elevation of the outer rail on the circular curve is also arrived at by a gradual rise from the straight line. If this rise is to be uniform, the radius of curvature of the transition curve must be infinite at its beginning on the straight line, decrease in such a way that at any point of the curve it shall be inversely as the distance of that point from the straight line, and finally become equal to the radius of the circular curve where it joins that curve. Two kinds of transition curves have been employed—the cubic parabola and a compound circular curve. 1. The Cubic Parabola. The cubic parabola fulfils all the essential requisites of a transi- tion curve. Its first application for this purpose is ascribed by Ran- APPENDIX. ' - 125 kine (Civil Engineering, p. 651) to William Froude, who is said to have used it as early as 1842, although his first publication in regard to it was in the year 1860 or 1861. Subsequently Nordling published a full discussion of the curve in the Annales des Ponts et Chaussées for 1867, but without mentioning Froude. Sarrazin and Oberbeck, in a small volume published in Berlin in 1874, give the German practice in the use of the cubic parabola, and ascribe its first use to Nordling. Let C D (fig. 66) be part of the circular curve of radius R. Let A B C be the transition curve, connecting the straight line at 4 with the circular curve, and let x and y be the rectangular coördi- nates of the transition curve with origin at A. Let the rise of the outer rail be taken as uniform for distances from A along the axis of 1. . a; instead of along the curve, and let iſ be the rate of rise. Then the rise at any distance z from A will be and if p is the radius of curvature at the corresponding point of the curve, we shall have by a gu” _ gvºi - - } § 110, p. 89, # - 32.25° or P = 32.3. When the velocity v has 1 º y?; been fixed, and also the rate of rise the quantity #. Q O X. & becomes a constant, and may be represented by P.; so that we shall have P = 0." y”; 32.2' and consequently f (1) p - 126 AIPPENDIX. - ds? dºc? dº? By the differential calculus p = dºdºy - dºdy (nearly) = dy' 2 d”y QC e - h Whence d. F P : and, integrating once, we have dy z* d; T 2P’ (2) and, integrating again, - 3.3 '?/ - – e. 3 The value of P is now to be determined. It depends, as we have seen, upon the values assumed for the velocity of trains in fixing the elevation of the outer rail, and on the rate of rise in attaining that elevation, that is, on the values of v and i. Now, when the circular curve is of large radius, v would generally be taken greater than When the radius is small. The rate of rise also might be taken less in the first case than in the second; so that both v and i, and consequently P, would be greater, the larger the radius of the circular curve. Ac- cording to Froude, as given by Rankine, i may be taken as 300; 300gy” 32.2 miles per hour, this would give P = 60 000, and for double this ve- locity P = 240 000. Nordling assigns values to P, according to the general character of a road—whether it is one having generally curves of short radius, or one having curves of large radius. For the first he gives P = 129 000, and for the second P= 430 000; taking g = 1.5m. = 4.92 feet. For the same value of g, Sarrazin and Ober- beck adopt for the first case P = 129 000; but in the second case they prefer P= 65.6 R, making P vary directly with R. All these values are for measures in feet. These examples may serve as a guide in assigning to P values suited to a particular road or to dif- ferent parts of it. The value of P having been fixed, equation (3) will enable us to find points on the curve corresponding to any as- sumed values of 2. Certain properties of the cubic parabola are now to be established. Let A E = a be the abscissa, and C E = b be the ordinate of the end C of the curve. We wish to find a when the radius R of the circular curve is given. Since the radius of curvature of the parabola at C becomes equal to R, we have by equation (1), putting z = a, whence P = For g = 4.7 and for a velocity slightly over 25 R = 0 = ; whence a =#. R Q. APPENDIX. 127 Next to find the tangent C F at the end of the curve, call the an- gle C FE = a, and we have by (2), putting a = a, tan. a = % - # 3 3 From (3) we have b = # Or P = º . Substituting this value of P, tan. a = †a and as tan. a = # , we have FE = ? a , so that this tangent is readily drawn by making A F = #a. If the circular curve is produced to G, where its tangent becomes parallel to A E, we shall find that the abscissa A H of the point G is 3a, and that the curve at B bisects the line G. H. Since C F is also tangent to the circular curve at C, the angle C O G = C F E = a, and tan. a = #. Therefore C K = H E = O Ktan. a = (R— K. G.) 1. tan. a = R tan. a, nearly, since K G is always small relatively to R. o 2 º - But we had above R = £and tan, a = #: Substituting these (, 2 2 values we have H E = P x * Q. 2 P Sects a. To show that G H is bisected at B, we have only to show that G H = 45, since we know by (3) that the ordinate B H = *ē. Now G H = C E – R G : but C E = b, and by § 26, p. 14, we may = }a = A H, and the point H bi- To lay out the curve we may have the intersection angle I given, and the radius R of the circular curve assumed, to find A. I = T. LIaving found a = #and also b, we have T = A H + H J = }a + O H tan. I. But O H = R + G H = R + 4b. '. T = }a + (R + +b) tan. 4I. If I is given and the tangent T assumed to find R, assume a probable value of ¥a, neglect #5, as small compared with R, and compute R by the formula, R = (7–34) cot. I. From this value of R compute the corresponding values of a and b, and the length of T, as abovo. If thereby the tangent point is changed too much, a nearer approximation to R may be found by the formula, R + +5 = (7'-3a) cot. #1, using the values of a and 5 just found. - Until the track is laid, it will generally be sufficient to fix the points A, B, and C. A is the tangent point, the coördinates of B 128 4: - APPENDIX. are 4a and #5, and of C they are a and b. If more points are after. wards needed, values of y are to be found from the equation of the 3 QC & curve y = 6P for any assumed values of 2. To run the circular curve, find the point F by measuring A F = 3a. Then F C is the tangent from which the circular curve beginning at C is to be run to meet the transition curve at the farther end. Curving the Rails.-The first rail on the transition curve may be curved by finding the deflection from the tangent by the formula 3 - - $/ = f , taking a equal to so much of the length of the rail as comes on the curve. If this does not exceed 8 or 10 feet, the curving will 2 not be appreciable. For other rails the ordinary formula m = in may be used, taking for R the mean of the radii of curvature at the two ends of the rail. These radii are easily computed by formula (1). The following method is more exact. Let A B (fig. 67) be the rail Fig. 67. to be curved. Let C on the axis of a be distant n rail-lengths from the origin, and D, also on the axis of r, be distant n + 1 rail-lengths, n being a whole or fractional number. Then if we represent by d the deflection from the tangent at the end of one rail-length from the origin, we have A C = nºd and B D = (n + 1)*d. We then have m = G F = E F – E G. But E F = } (A C + B D) = º + n° + 3n2 + 3n -- 1) = d (n3 + 3n2 + 3n - ?), and E 6 = d. (n + 3)*= d(n3 + 3n2 + 3n -- ). Therefore m = #d (2n + 1). In a similar way the ordinates HI and KL at the quarter points may be shown to be, HI = d (ſº n - #) = #m — & d, K L = d (§ n + #4) = \m + š, d. The angle made by the chord A B with the axis of z is in general so small that these ordinates may be APPENDIX. 129 measured at right angles to the chord instead of at right angles to the axis of z. If required, the angle in question can be found, since B D — A C d(3m.” + 3n -- 1) a B-- C D just found, being multiplied by the cosine of this angle, will give or- dinates at right angles to the chord. its tangent is , and the ordinates 2. Compound Transition Curve. The transition curve is here formed of successive circular arcs in- creasing in curvature a certain amount for each chord. Tables for curves of this kind have been computed by Mr. A. M. Wellington (Railroad Gazette for March 11, 1881), and by Mr. Augustus Tor- rey, of Burlington, Vermont, printed in a small pamphlet. A curve of this kind, A B C D (fig. 68) may be readily laid out Fig. 68. A T, | T B 2 C Ts D by tangent deflections, measuring at the same time the successive chords. Let c represent the length of each chord, n their number, and let D be the deflection angle for the first chord, 2D that for the second chord, 3D that for the third chord, and so on to the deflection angle for the last chord, which will be n D. Then it is easily seen that the angles T', A B, T., B C, T, CD, &c., will be successively D, 4D, 9 D, 16 D, &c., up to n*D. Calling the required deflections from the tangent A I, di, dº, ds, &c., and recollecting that, since these an- gles are all small, we may put sin. 4D = 4 sin. D, sin. 9 D = 9 sin. D, &c., we have di = c sin. D, da = di + 4c sin. D = di + 4d, = 5d., da = da + 9c sin. D = 5d. -- 9d, = 14d, &c., the successive deflec- tions being formed by multiplying the first by the terms of the series 1, 5, 14, 30, 55, 91, &c., formed by the successive additions of the Squares of the natural numbers. The projections A T, BT, CT's, &c., of the chords may be found thus. A T = c cos. D, B.T. = c cos. 4D, C T = c cos. 9.D, &c., and the differences between the chords and their projections will be c (1 — cos. D), c (1 — cos. 4D), c (1 — cos. 9 D), &c. For small an- 130 APPENDIX. gles these corrections become º sin.” D, º sin.” 4 D, . sin.” 9 D, &c., or sin.” D. 16 x sinº D, 81 × sin. D, .... ". . sin.” D. The sum of all the projections represented by A E (fig. 69) would therefore be 720 - . sin.” D (1* + 2* + 3 + . . . . nº). When the intersection angle I is given and the radius or deflection angle of the main or central curve is assumed, the deflection angle D and the tangent A I = T (fig. 69) are to be found. In the figure Fig. 69 O CK. R M K C G H. –T /~~ A H F E | the central curve M C is supposed to be run back to G, where its tangent becomes parallel to A I, the radius O Gº to be drawn and pro- duced to H, and the common tangent F C of the central curve and the transition curve A C to be drawn. Assume a suitable value for n, the number of chords in A. C. Then, as we have seen, the deflection angle for the last chord on A C will be n D. For the first chord of the same length c on the central curve the deflection angle should be (n + 1) D. Since the central curve is given, its deflection angle for a chord of the length c is known, and may be called D'. We have then (n + 1) D = D', or - D' in T 1 To find Tit will be necessary first to find C K = H E and H. G. C K = R sin, C. O G. The angle C O G = E FC is the sum of all the central angles of the transition curve. Call this sum g, and we shall have a = 2D (1 + 2 + 3 . . . . n) = n (n + 1) D. Therefore APPENDIX. 131 O K = H E = R sin. (n2+ n) D. Next H G = E C – G. K. But E C = d. - d. (1 + 4 + 9 + . . . .”) = d. (ºn” + 3n+++n) and G. K. – the chord G. C. × sin. 3 a. The chord G. C may be taken = #ng sº ... + a since the number of chords of length c in the curve G C is iy - **** = *n. Therefore G R = }ne sin. An (r. 4- 1) D = (n + 1) 1 (n3 + nº)e sin. D = 3 (tº + n) di. We have then H G = E C — ! sº & K = d. (kn? + 3n2 + n – #m° – #n”) = #. (n3 + 3n2 + 2n). We now have T = A H + H I = A E – H E + H I. 4 E and H E have been found, and HI = (R + H G) tan. #1 = [R + # (n3 + 3n2 + 2n)] tan. 31. sin. 2D (1 + 2 + . . . . n°) – R sin. (n°4 m) D + C .. T = me -— 2. [R + #(º + 3n2 + 2n)] tan. 31. When I is given and T assumed to find R, assume n, and in the expression for T' neglect the second term and that part of the fourth term that represents H G, both of which are small, and consider R sin. (mº + n) D = }nc (nearly). We then have T= }ne + R tan. #1 (nearly) ... R = (T- ºnc) cot. #1 (nearly). From this approximate value of R compute the corresponding val- ues of D and di, and find T by the first formula for T. If thereby the tangent point is changed too much, a nearer approximation to R may be found by substituting in the formula for T the values of D and di just found. All its terms may then be considered known ex- cept R. To run the central curve we must be able to fix the tangent F. C. This will be done if we find F. E. Now F E = E C cot. E. F. C. We have already found the angle E F C = a = (n” + n) D, and E C = d. ... FE =d, cot. (n” + n) D. The central angle of the central curve will be 2 G O M – 2 a = I — 2n (n + 1) D, and as (n + 1) Dis the deflection angle of this curve for chords equal to those on the transition curve, the number of these chords on the central curve will be #1 – n (n. + 1) D - #1 — ?? }~ (n + 1) L (n + 1) D ſ 132 APPENDIX. 3. Remarks. In regard to both these transition curves it may be remarked that there would be certain advantages in having the chords c of the length of one or two rails, and, when the track is laid, in placing a joint at both ends of the curve. The ends of each rail would then be definitely fixed, and each rail could be more satisfactorily curved. It would also be easier to maintain the track in its proper position, if : the trackmen knew that the tangent points were at joints, and, when rails were renewed, the new rails would be more likely to be properly curved and placed in their true position. It has been lately said (see Railroad Gazette for December 24, 1880), that German engineers find that the rails on their transition curves (cubic parabolas) are displaced near the tangent points by the shock of the wheels striking the outer rails, and, in order to counter- act this, they are said to recommend putting half the super-elevation of the outer rail on the straight line and half on the parabola. May not the difficulty arise, at least in part, from too much play in the gauge of the track near the tangent points 2 There would seem to be no reason why the gauge should not be kept close on the straight line up to the tangent point, since on the transition curve the “widening of the gauge on curves * can be made gradually as the curvature increases. - ARTICLE X.—To CHANGE A TANGENT PoinT so THAT THE TANGENT MAY PASS THROUGH A GIVEN POINT. If the given point is at a considerable distance, let D (fig. 70) be the required tangent point, and C the given distant point. At a sta- ZŽºr 70. A & T} APPENDIX. 133 tion A, back of D, moasure the angle B A C made by A C with the tangent at A. Then we shall have the angle B A D = }B A C, very nearly. Or, by § 69, compute the length of the chord A E, and we shall have the chord A D = }A E, very nearly. If the distance A C is known, we have E C = A C – A E, and D C may then be found by § 80. - If the point C is given by A B = a (fig. 71 or 72), measured on a tangent at A, and B C = b, at right angles to A B, draw C E parallel to A B to meet O A, produced if necessary. Then in the first case Az Z # * * D C 1. l O (fig. 71) we have the required angle A O D = 4 O C — D O C. But O D R tan. A 0 c=*= * , and cos. D O C = ** = −. O C. Woº-H(R—b)? E. 6 T R-5 In the second case (fig. 72) we have the required angle A O D = - – 0 P ––" D O C – A O C. But cos. D O C = - 1 = −, and O C T Wºl (R-F by ' E C a EOT FIL 5’ angle A O D is determined. tan. A O C = Hence, in both cases, the required ARTICLE XI.--To CONNECT. Two CURVES BY A CoMMON TANGENT. When both curves turn the same way (fig. 73), run a line A B cut- ting both curves in such a way as to make the middle ordinates E. G. and FH as nearly equal as can conveniently be done. Measure A B = a and the tangential angles C A B = A and D B A = B. Let E" F" be the required common tangent, and draw O E and P F per- pendicular to A B, and F. K parallel to A. B. Let A O = R and B P = R'. Then the required angle C A E' = }A O E' = #4 + 134 APPENDIX. 7%. 73. O T # E O E' = }A + 3 E"F" K, nearly. Now tan. E' F" K -*** 2R sin.”4A — 2 R' sin.”4B a – R sin. A – R' sin. B We have also the angle P B F = }B — ; E' F" K. When the curves turn opposite ways (fig. 74), A H = a should be run outside the Second curve, making FH as nearly equal to E G as can Žy, ZZ. nearly = Hence C A E" is determined. conveniently be done. FH must be measured. Then the required an- gle C A E = }A O E' = }A + 3 E O E' = }A + 4 E"F" K, nearly. Now tan. E. F. K– # "TºF H. nearly = 2 R sin. * * 4 – FH G. H. a – R sin. A Hence C. A. E." is determined. In both these cases E G has been supposed larger than F. H. If E G is smaller than FH, the point E" will fall on the other side of E, and the angle C A E' = }A — 3 E' F" K. It is obvious that, in both cases, if E G is exactly equal to FH, the angle E' F" Iſ van- ishes, and C A E = #4. APPENDIX. 135 ARTICLE XII.-NOTE ON THE COMPUTATION OF EARTH-work. The mode of computing earth-work on railroads by first finding equivalent level-top sections has already been examined in § 128, and the assumption made in applying the prismoidal formula is shown to lead to possibly serious errors. Another assumption that forms the basis of many formulae, tables, and diagrams, is that the natural Sur- face of the ground of such a section as that calculated in § 122 is a warped surface or hyperbolic paraboloid. The solidity is then com- puted by the prismoidal formula. Computing the Section just referred to on this assumption, we find the solidity 31 210 feet. Now we have seen in § 128 that, with the diagonal running in one direction, the solidity is 32 820 feet, and, with the diagonal running in the other direction, the solidity is 29 600 feet. The assumption of a warped surface gives, therefore, an exact mean between these two re- sults, being 1,610 feet too much or too little, according to the direc- tion of the diagonal. Errors so great would not perhaps be common; but they are at least possible. The objection to these methods is that they involve general assump- tions as to the natural Surface of the ground—assumptions that the engineer cannot readily test in the field for each section, or allow for, if seen to be wrong. No method would seem to be reasonably cor- Tect that does not require all the data used in the computation to be obtained directly in the field. Now the division of the ground into triangular planes, whether four as in § 122, or more as in § 127, sat- isfies this condition. Since three points determine a plane, it is comparatively easy to decide on the ground what heights should be adopted at the vertices, so that a triangular plane shall be a fair aver- age of the ground. Suppose the ground cross-sectioned in the usual Way, and the actual cuts marked on the stakes and recorded. These cuts remain to guide the contractor in his work; but the engineer is to examine each triangle, and see whether these cuts require any cor- rection in order to obtain a fair average of the surface. As he goes from section to section, two of the heights or cuts would in general be already fixed, and, standing at the third vertex, he readily de- termines whether the actual cut there should Stand, or have one, two, three, or more tenths added or subtracted. The correction, if any, may be noted in small figures over the aetual cut, and applied when the heights are taken off for the computations. Some additional labor is doubtless involved in thus obtaining directly all the data required, and dispensing with all general as- 136 APPENDIX. sumptions; but if justice to the contractor and to the company require such additional labor, the conscientious engineer will not hesitate on that account. The computations, as arranged in § 123, will be found, after a little practice, to admit of very rapid work. Of course, only final estimates require so much care. Preliuminary estimates and the usual monthly estimates admit of rougher methods, such as simply averaging end areas. TA B L E I. T. A DII, OR DIN AT ES, D E FL ECTIONS, .AN D ORDINATES FOR CURVING RAII,S. Formula for Rudii, $ 10; for Ordinates, § 25; for Deflectives, § 19 for Curving Rails, $ 29. 138 TABLE I. RADII, ORDINATEs, DEFLEctions, Ordinatos for Ordinates. Tangent| Chord Rails. I'egree. Radii. Deflec- || Deflec- 12}. 25. 873. 50. tion. tion. 18. 30. } 5| 6S754.94 | .008 .014) .017| .01S .073| .145 | .001 | .002 10| 34377.48 || .016| .027 .034 .036 ..] 45] .291 .001 | .003 15| 22918.33 .024 .04 || .051 | .055 .218| .436 | .002 . (105 20, 17188.76 .032 .055 .06S .073 .291 | .5S2 | .002 .007 25| 13751.02 . .040 .068| .0S5] .091 .364; .727 | .003] .008 30| 11459. 19 .048] .0S2] .102 100 .436| .873 . .004 .010 35| 9822.18 .056| .095 . I 19 . 127 .509| 1.018 .004 .011 40 8594.41 | .064| . 109| 136|| . 145 ,582| I. 164 .005 .013 45| 7639.49 .072 . 123 - 153| .164 .654|| 1.309 .005 .015 50, 6875.55 | .080 .136|| .170 . IS2 .727| 1.454 .006 .016 55| 6250.5l .0S7| .150 .187| 200 .800. 1.600 . . .006 .018 1 0 5729.65 .095 - 164] .205] .21S .873 1.745 .007| .020 5| 5288.92 || 103} , 177 .222| .236 .945] 1.891 | .008 .021 10| 491 I. 15 ..] ] I | . 191] .239| .255 | 1.018, 2.036 .008 .023 15| 4583.75 ..] 19| 205 .256] .273 | 1.091 2.182 .009 .025 20| 4297.28 | . 127] .218] .273 .291 || 1.164] 2.327 | .009 .026 25, 4044.51 | . 135 .232 .200 .309 || 1.236| 2.472 .010 .028 30 3819.83 | .143 245|, .307| .327 | 1.309| 2.618 .01 || 029 35 3618.80 | . 151 259 .324 .345 || 1 382 2.763| .01 I .031 40|| 3437.87| . 159; .273 .341 .364 || 1.454| 2.909 .012 .033 45) 3274.17 .167] 2S6 .35S] .3S2 | 1.527| 3.054 .012 .03 50 3125.36 .175 300 .375 400 | 1.600 3.200 | .013 .036 55] 2989.48 .183 .314| .392 .418 || 1.673| 3.345 .014 .038 2 0| 2864.93 . 191| 327 .400 .43G | 1.745; 3.490 t .014 .039 5 2750.35 .199| 341|| 426 .455 1.818; 3.636 .015 .041 10| 2644.58 .207 355| 443 .473 || 1.891 || 3.781 || 0.15} .043 15|| 2546,64 || .215| 368 460, .491 || 1.963. 3.927 || 016| (44 20] 2455.70 | .223 3S2 .477 .509 || 2.036 4.072 .016 .046 25 2371.04 || .231|| 395 .494 .527 2.109 4.218 .017| .047 30 2292.01 . .239 409 .5l I .545 2.181| 4.363 | .018 .049 35 22:18.09 | .247. 423 .528 .564 || 2.254. 4.50S .018 051 40 2148.79| .255] .436] .545 5S2 2.327| 4.654 .019 .052 45. 2083.68 .263 .450 562 600 2.400. 4,799 || .019 .054 50 2022.4] | .270 .464 .5S0 .61S 2.472| 4.945 .020" .056 55, 1964.64 .278 .477 .597. 636 || 2.545 5.090 .021 .057 3 O 1910.08 .2S6 .491 .614|| 655 2.61S 5.235 | .02|| .059 5| 1858.47 || 294 .505] .631| .673 || 2.690) 5.3S1 | .022 .061 10 1809.57 | .302 .518 .64S .691 || 2.763| 5,526 || .022| ,052 15| 1763 18 .310 .532 .665| .709 || 2.836 5.672 .023} ,064 20|| 1719. 12 | .318 .545 .682 727 || 2.90S 5.817 | .024] .065 25, 1677.20 | .326 .559 .699. ,745 2.9S1 5.962 .024 .067 30| 1637.28 .334|| .573 .716 .764 || 3.054| 6.108 .025 .069 35| 1599.21 .342 .5S6 .733 .782 || 3.127| 6.253 .025 ,070 40 1562.88 . .350 .600 .750) .800 || 3.199| 6.398 .026 072 45] 1528.16 .358 .614 .767 .818 || 3.272; 6.544 .027 .074 50 1494.95 .366 .627 .784 .836 || 3.345 6.6S9 .027 .075 55] 1463.16|| .374 .641. .801| .855 || 3,417| 6.835 | .028 .077 4 0 1432.69 | .3S2 .655; .818 .873 || 3,490| 6.9S0 | .028 .079 5| 1403.46 .390 .66S .835| 891 || 3.563 7.125 | .029 .080 10| 1375.40 .398 .682 .852 .909 || 3.635| 7.271 .029 .082 15| 1348.45 .406 .695 .869 .927 3.70S 7,416 | .030 .0S3 20|| 1322.53 || 414 .709 .886. .945 || 3.781| 7.561 | .031|| 085 25, 1297.58|| .422 .723 .903 .964 || 3.853| 7.707 | .031|| 087 30| 1273.57 .430] .736 .921 .9S2 3.926 7.852 | .032 0SS 35| 1250.42 | .438 .750 .938|| 1.000 || 3.999 7.997 . .032 080 40| 1228, l l .446 .764| .955| 1.018 4.07 || 8, 143 .033, .092 45 i266.57 .45|| 777 ;3| i.d36|| 4. iii. 8.285| 034 j 50. 1185.78 .462 .791| .989| 1.055 || 4.217 8.433 | .034 .095 55 1165.70 || 469 .805 1.006 1.073| 4,289 8.579) 035, 007 |l- 5 0 1 146,28 .477] .818| 1.023 1.091 4,362 8.724 . .035 098 AND OR DINATES FOR CURVING RAIL S. 139 - Ordinates for Ordinates. Tangent Chord Rails. Degree.| Radii. Deflec- Deflec- 2}. 25. 374. 50. tion. tion. 18. 30 | - . 5| | 127.50; .4S5| S32| 1.040 l. 109 4.435 S.S69 .036|| .100 10; 1100.33 || 493 .S.16|| 1.057| 1.127 4.507 9.014 .037| . 1()1 15, 1091.73 || 50 || .859| 1.074 1.146 4.5S0. 9. 160 | .037| . 103 20|| 1074.6S | .509| .873 1.091 | 1.164 4.653. , 9.305 | .03S. , 105 25| 105S. 16 | .517| SS7| 1.10S 1. 182 4.725] 9.450 .03S) .106 30| 1042. 14 .525] .900| 1.125] 1 200 4.798, 9.596 || .039 , 10S 5| 1026.60 | .533. .914 | 1. 142|| 1.21S 4.S70| 9.74 I .039 . 110 40|| 101 1.5 ! . .541 .92S| 1.159| 1.237 4.943| 9.SS6 | .040] .111 45| 996.87 | .549] .94 || 1.l76; 1.255 5.016; 10.031 .041: ... 113 5ſ) 9S2.64 .557| 955| 1, 193| 1.27 5.0SS 10.177 | .04] | .114 55: 96S.S1 | .565; .96S 1.210 l.291 5. 161 | 10.322 .042 .116 6 0 955 37 || 573 .9S2 1.22S 1.309 5.234|| 10.467 | .042 .118 5| 942.29 | .5S] . .996 | 1.245; 1.327 5.3(;6| 10.612 .043| . 119 10| 929.57 | .5S9| 1.009| 1.262| 1.346 5.379| 10.75S .044) . 121 15| 917. 19 .597 | 1.023| 1.279| 1.364 5.45|l 10.903 .044 .123 20. 905. 13 | .605. 1.037| 1.296 1.3S2 5.524| | 1.04S .045] . 124 2.5| S93.39 .613 1.050| 1.313| 1.400 5.597 1 1. I93 .045 .136 30| SS1.95 | .621 | 1.064| 1.330| 1.41S 5.669|| 1 | .339 .046 .12S 35| 870.79 .629| 1.078| 1.347| 1.437 5.742; 1 1.484 .047| .129 40| 859.92 | .637| 1.091 | 1.364 1.455 5.S14 l l .629 .047| .131 45| 849.32 .645. I. 105| 1.381 | 1.473 5. SS7 | | 1.774 .048 . 132 50| 838.97 .653| 1.1 |S| 1.39S 1.491 5.960|| | 1.919 . .04S) . 134 55: 828.88 . .661 | 1.132| 1.415 1.510 6.032 l?.U65 . .049| .136 7 0|| 819.02 . .669| 1.146|| 1.432. 1.52S 6. IU5| 12.210 | .049) . 137 5| S09.40 .677| i. 159| 1.449| 1.546 6, 177| 12.355 .050; .139 10| 800.00 .6S5| 1.173| 1.466; 1.564 6.25()| 12.5()() .051] .141 15| 790.81 .693| 1.lS7| 1.4S3| 1.5S2 6 323| 12.645 .051 | . 142 20|| 7S1.84 || .70 || 1.200| 1.50 || 1.600 6,395| 12.790 .052 .144 25| 773.07 | .709| 1.214 | 1.517. 1,619 t; 468 12.936 .052] .146 30| 764.49 || 7 || 7 | 1.228; 1.535. 1,637 6.5-1()} | 3.0Sl .053| .147 35 756. 10 . .725] 1.242|| 1.552| 1.655 6.6 | 3 || 3.226 .054! .149 4()| 747,S9 .733| 1.255| 1.569| 1.673 6.6S5| 13.37 | .054 .150 45| 739.S6 .740 l.269| 1.5S6] 1.691 6.75S 13.516 ,055 . 152 50| 732.0l .748; 1.2S3| 1.603| 1.710 6.S3 l 13.66] .055; . 154 55| 724,31 .756] 1.296 | 1.620 1.72S 6.9(3| 13.S06 .056| . 155 8 0 716.78 . .764 1.310|| 1.637| 1.746 6,976. 13.951 .057 .15% 5| 7(19.40 .772; 1.324| 1.654|| 1.764 7.04S 14.096 | .057 . 159 10| 702, IS .780| 1.337| 1.67 || 1.782 7.121 || || 4.24 l .058 .160 15 605.09 | .7SS| 1.351 | 1.6SS| 1.801 7. 193] 14.3S7 .058 .162 20|| 6SS. 16 . .796|| 1.365. 1.705| 1.S.19 7.266; 14.532 .059 .163 25 6Sl. 35 | .804| 1.378| 1.722, 1.S$7 7.338|| 14.677 | .059 . 165 30; 674.69 || S12 1.392| 1.739| 1,855 7.4 l l 14.S22 | .060 .167 35' 66S. 15 .820, 1.406 1.757| 1.S73 7.4S3| 14.967 . .061 .168 4ſ) 661.74 .828 1.419' 1.774 I.S92 7.556] 15, 112 | .061 | . 170 45 655.45 ,836|| 1.433 1.79] | 1.910 7.62S 15.257 .062] , 172 50 649.27 | .844|| 1.447| 1.80S 1.92S 7,701 || 15.402 .062 .173 55| 6:13.22 .852 1.460. I.S25; 1.94/3 7.773| 15.547 | .063} . 175 9 O 637.27 .860; 1.474 1.S42| 1.965 || 7.S46. 15.692 | .064 .177 5| 631.44 .868| 1.4SS| I.S59; 1.9S3 7 918 15,837 | .064 .178 10 625.71 ,876. 1.501 || |.S76| 2,00] 7.991. 15.9S2 .065i . 180 15 620,09 || SS4, 1.515 1.893 2.019 || 8.063. 16.127 .065 - 181 20; 614.56 | .892] 1.529| 1.910 2.037 || 8.136|| 16.272 .066 .1S3 25| 609. 14 | .9(iſ) 1.542 1.927| 2,056 8.20S 16.4 17 .066 .185 30, 603.S() { .90S 1.556| 1.944, 2.074 S.2S] | 16.562 .067 . 186 35| 598.57 | .916| 1.570) 1.961 2.092 S.353 16.707 .06S .188 40|| 593.42 .924, 1.5S3| 1.979| 2.11() 8.426|| 16.S52 || 068| . 190 45| 588.36 .932 1.597.| 1.996 2.12S 8.498 16.996 .069 .191 50| 583.3S .940| 1.6] 1 || 2,013. 2.147 8,571 17.141 .069 . 193 55| 578.49 .948 º 2,030) 2.165 8.643| l?.2S6 .070. .195 10 0} 573.69 .956] 1.638 2,017| 2. IS3 8.716, 17.4.31 l ,071 -196 140 TABLE I. RA DII, OR DINATES, DEFLECTIONS, &c. Degree. Radii. O | | () |ſ) 564.3| 2.) 555.23 30 546.44 40 537.92 50 529.67 1 1 0 521.67 10 513.91 20 506.3S 30 499.06 40 491.96 50 4S5.05 12 0 47S.34 J () 47 I.S.I 20 465.46 30 459.2S 40 453.26 50 447.4() 13 0 44 I.6S 10 436. 12 2ſ) 430.69 3() 425,40 40 420.23 50 415. 19 14 0 4} 0.2S 10 405.47 20|| 400.78 3ſ) 396.20 40 391.72 50 387.34 15 0 3S3.06 | 0 78.8S 20 374.79 30 370.78 40 366.86 50 363.02 16 0 359,26 10 355.59 20 351.9S 30 348.45 40 344.99 50 341.60 17 O 33S.27 10 3.35.01 20 3.31.82 30 32S,6S 40 325.60 50 322.59 18 () 3 (9.62 10 3|6.7l 20 313. S6 3ſ) 3| 1,06 4() 30S.30 50 305.60 19 () 302.94 10 300.33 20 297.77 30 295.25 40 292.77 50 290.33 20 0 287.94 - - rulinates H. Ordinates. Th ngent ("hord O Rails. for Leflec- , Deflec- - 12}. 25. 873. 50, tion. tion. 18. | 30. .972; 1.665| 2.0SI 2.219 8.860, 17.72] 072| .199 .9SS 1.693| 2. I 15| 2.256 9.005] 18.0|| 073 .203 1.004. 1.720 2. 149; 2.292 9. 150 18.300 074 .205 1.020; 1.74S 2. IS4| 2.329 9.295| 1S.500 075|| .209 1.036 1.775. 2.21s 2.365 9.440 18.8SO 07G] .212 1.0.72| 1.802| 2.252| 2.402 9.5S5| 19.160 l .078] .216 1.0153. 1.830; 2.2S6| 2,43S 9.729| 19.459 || 079; .219 1.(S4| 1.857| 2.320i 2.475 9.S74; 19.74S .0SO] .222 1.100| 1.884] 2.354; 2.5 || 1().(); 9| 20.03S OSI .225 l. I 16 1,912| 2.3S0 2.547 | 10. 164| 20.327 | .0S2] .229 1.132 1.938 2.423 – 2.5S4 || 10.308| 20.616 | .0S4] .232 1.14S 1.967| 2.457| 2.62() 1(). 153| 20.906 .0S5] .235 1. 164| 1.994| 2.49 || 2.657 || 10.597| 21. 195 .0S6] .238 1. 180 2.02 || 2.525 2.693 10.742. 21.4S4 .OS7| .242 1.196 2.010| 2.560 2.730 || 10.SS7| 21.773 | .0SS| .245 1.212| 2.076' 2.594 2.766 || | 1.031 22,063 .0S9 .218 1.22S| 2, 104| 2.62S 2.803 || 1 l. 176| 22.352 091 | .251 1,244; 2. 131 2,662. 2.839 l l .320|| 22.64 | 092] .255 1.260 2. 159| 2.607 2.876 ll.465| 22.93() 093 .258 1.277 2. l S6 2.73 || 2.912 || | 1.6ſ)9. 23.219 094| .261 1.293. 2.213 2.765| 2.949 || 1 1.754, 23.507 005] .264 1.309| 2.241 2.799| 2.9S5 l l .89S 23.796 096] .268 1.325| 2.26S 2.833| 3.022 | 12.043| 24.0S5 09S .271 1.34 || 2.296 2.S6S 3.058 || 12. 187| 24,374 .099, .274 1.357| 2.323| 2.902| 3.095 || 12.33|| 24.663 | .100 .277 1.373| 2.35|| 2.936 3.13| | 12.476] 24.951 .101 .281 1.3S9| 2.378] 2.970. 3. 16S 12.6.20, 25.240 102 .284 1.405| 2.406| 3.005| 3.204 || 12.764| 25.52S 103 .287 1.42|| 2.433| 3.039| 3.241 || 12.90S, 25.817 105| .290 1.437| 2.461 || 3.073| 3.277 || 13,053| 26.105 .106) .294 1.453| 2.4SS| 3.107 3.314 || 13, 197| 26.391 .1(17| .297 1.469| 2.515| 3.142|| 3.350 | 13.341 26.6S2 | .108 .300 1.4S6| 2.543. 3.176| 3.387 || 13.4S5| 26.970 | .109] .303 1.502| 2.570 3.210 3.423 || 13,629| 27.25S .l 10 .307 1.518| 2.59S 3.245| 3,460 | 13.773; 27.547 I 12| .310 1.534 2.625| 3.279| 3,496 || 13.917| 27.835 113| .313 1.550| 2.653| 3.313| 3.533 14.061| 28,123 | .114 .316 1.566] 2.6SO|| 3.347| 3.569 || 14.205| 28.4ll ..] 15: .320 1.5S2 2.70S 3.3S2, 3.606 || 14.349| 28.699 | 16 .323 1.593] 2.736 3.416| 3.643 || 14.493. 28.986 I 17| .326 1.615 2.763| 3.450 3.679 || 14.637| 29.274 .l 19| .329 1.63|| 2.79 || 3.4S5 3.716 || 14.781| 29.562 | .120 .333 1.647| 2.81S 3.519. 3.752 || 14.925, 29.850 | . 121 | .336 1.663. 2.S.[6] 3.553| 3.789 || 15.069| 30.137 | . 122] .339 1.679| 2.S73| 3.5SS 3.825 | 15.212. 30.425 | . 123 .342 1.695 2.001 || 3.622| 3 S62 | 15.356; 30.712 124 .346 1.71 || 2.92S 3.656| 3.89S | 15.500 31.000 | .126] .349 1.72S 2.956| 3.69 || 3.935 | 15.643| 31.287 | .127| .352 1.744| 2.9S3| 3.725, 3.972 | 15.787| 31.574 | .128} .355 1.760| 3.01 || 3.759| 4.00S 15.93|| 31.8Gl 129| .358 1.776|| 3.039| 3.794| 4.045 16.074| 32.149 | .130 .362 1.792. 3.065 3.S2S 4.0S1 || 16.218] 32,436 13| | .365 1.S09| 3.09.4| 3.862 4.1 18 || 16.361| 32.723 133 .368 1.825| 3.121 || 3.897| 4.155 | 16.505 33.01.0 13.4 .371 1.S4 || 3.149| 3.93|| 4.191 | 16.648] 33.296 135 .375 1.857| 3.177| 3.965| 4.228 || 16.792. 33.583 136 .378 1.873| 3.204 4.000. 4.265 | 16.935| 33.870 | . 137| .381 1.S.00| 3.232 4.034| 4.301 || 17.078] 34.157 .138 .384 1.906| 3.259| 4.069| 4.338|| 17.222| 34.443 .140; .388 1,922. 3.287| 4.103 4.374 17.365| 34.730 14 iſ .391 TABLE II. 141 LONG CHOR US. T A B L E I l. LONG CHORDS. § ey. Pººr 2 Stations. | 3 Stations. || 4 Stations. 5 Station5. 6 Stati, ns. § 16 200,000 290.909 399.90S 499.996 599.993 20 199.999 .997 .902 ..!}S.] .970 30 .908 .94)2 .9Sl .962 .933 40 .907 .9S6 .9(36 .9:32 ..SS2 5() .905 .979 .947 .S94 .815 | 0 199.992 299 Q7() 399,924 499.848 599.733 | 0 .990 .9.59 .896 .793 .637 2ſ) .9S6 .9.15 ,805 729 .526 30 .9S.3 ,932 ,829 .657 .4ſ! I 40 .979 .9 15 .7S9 .57 200 50 .974 .89.S .744 .4SS . 1U5 2 0 199.97 290.S78 300.695 499.391 50S.934 10 .964 .857 .643 .285 .750 20 .959 .834 .5S6 . I 71 .550 30 .9.52 .S1() .524 •(3.49 .336 40 .946 7S3 .459 498.9 || 8 . . (M5 50 .939 756 .339 .77S 597.862 3 0 199.931 299,726 399.315 49S.6.30 597.6()4 | () .924 .695 .237 .474 .331 20 .9 || 5 .662 . 154 .3ſ 10 - .043 30 .907 ,627 .06S . 136 596.740 40 .898 .591 39S.977 497.955 .423 50 .8SS .553 .765 .091 4 0 199.878 209,513 39S.7S2 497.566 595.744 10 .86S .47 | .679 .36!} 3S3 20 .857 .42S .57 I 145 .007 30 .846 .353 .459 4:36.921 594.617 40 .8. .337 .343 .689 ,212 50 .822 .2S9 .223 .4.19 503,792 6 0 199.S10 209.239 39S.099 406 2(H) 593.35S | () .797 ..] S7 397.07 () 495,944 592.9l 19 20 .783 . 134 837 .67S 4-16 3) ,770 .079 .7ſ iſ) 4().5 591.96S 40 .756 .()23 5.j9 . 123 .47 50 .741 20S.964 .4 || 3 494. Sº 59ſ).970 6 0 199.726 298.904 397.264 43-1. 534 500,449 10 .7 | () .843 | | {} ..??? 5S9.913 20 ,695 .779 336,932 493 Q1 2 .364 3ſ) .67 S .7 l 4 . . . iſ] .5SS 5SS.S(X) 4() .662 .64S ,623 257 .221 50 .6-14 ,579 453 49%.917 587,62S 7 0 199627 29S.5ſ).9 396.278 492.56S 5S7, 21 ! {} .6{}9 438 (199 .2] 2 5SG.4(){} Yi I .59 | .364 395,916 491,847 5S5.765 3() .572 .2S9 .729 .474 . l l 5 4!) •553 ,212 .538 .093 5S4,451 5{} .633 . 134 342 490.704 5S3.773 | S () .513 29S.054 395. 42 490.306 58.3 (IS! 142 TA Bi E III. — TABLE IV. T A B L E I I I. ... CORRECTION FOR THE EARTH'S CURVATURE AND FOR REFRACTION. § 105. | D. d. D. d. D. d. D. d 300 .002 |S00 .066 3300 .223 || 4800 472 400 .003 1900 .074 3400 .237 4900 .492 500 .005 2000 .0S2 3500 .251 5000 .512 600 .007 2100 .09ſ) 3600 .266 || 5100 .533 700 .010 2200 ,099 37(10 .2S1 5200 .# 800 ,013 2300 .108 3S()0 .296 || 1 mile .57 l 9ſ)0 .0 7 2400 ..] IS 39(KO .312 || 2 “ 2.285 1000 .020 2500 .128 4000 .328 || 3 “ 5.142 1 100 .025 2600 .139 4 100 .345 || 4 “ 9.142 J200 .030 2700 . 149 4200 .362 || 5 “ 14.2S4 1300 .035 2S)0 . 161 4300 .379 || 6 “ 20.568 1400 .040 2900 . 172 4400 .307 || 7 “ 27.996 1500 ()46 3000 . 184 4500 .4 || 5 || 8 “ 36.566 1600 ,052 3100 . 197 4600 .434 || 9 44 46.279 I 700 .059 3200 .2 ſ () 47()() .453 || || 0 --ee- 156 TABLE x. TRIGoNOMETRICAL ARD l 2 13 l4 15 16 l7 18 9 20 2 l 22 Solution of Oblique Triangles (fig. 58). R} Fig. 58. ZN 2. Cl Z ZA N A b - C Given. Sought. Formulae. a 8 n. B A, B, a b b = sin. A A, a, b L8 sin. B — º ". (4. — b) tan. - a, b, c A — Btan. (A — B) = º ºra H B) Ifs= (a +- b + c), sin. A = NE = e), l. A = A lº (º Tº) 4 , G=O)G=e Cl, b, G A 1 cos. 3 A AJE. , tan. A = s (s — (2) sin. A = *=#º=º-ºº=º. a2 sin. B sin C A, B, C, a area area = T2,sin. AT A, b, c area area = 3 b c sin. 21. - --- a, b, c area s=3 (a + b + c), area= v/s (s-a) (s—b) (s—c). General Trigonometrical Formulae. sinº A —H cos.” A = 1. sin. (A =E B) = sin. A cos. L3 =E sin R cos. A. cos. (A =E B) = cos. A cos. B =E sin. A sin. B. t sin. 2 A = 2 sin. A cos. A. - cos. 2 A = cos.” A — sinº A = 1 — 2 sin A = 2 eosº A-I. sinº A = —3 cos. 2 A. cos.” A = +- 3 cos. 2 A. sin. A +- sin. B = 2 sin. 3 (A + D) cos. 3 (A — E). sin. A — sin. B = 2 cos. 3 (A + B) sin. (A — B). cos. A —- cos. B = 2 cos, 3 (A +- B) cos. (A - B). cos. D — cos. A = 2 sin. (A + B) sin. 3 (A — B) sinº A — sinº B = cosº B — cos.” A = sin. (A +-B) sin. (A—Bº. cosº A — sinº D = cos. (A + B) cos, (A — B). * MiscellANEous ForMULE. sin. A 32 tan. A = cos Á cos. A 38|cot. A =#| 34;tan. (A + B) = 5tan. A + tan. sin. A — sin. B Sought. Area of Circle Ellipse Parabola 5 4 4G 47| Regular Polygon Surface of Sphere 49| Zone 5 0 Sphericall’olygon Solidity of Prism or Cylinder Pyramid or Cone Frustum of Pyr- } amid or Come : : tan. A i.tan B i E. tan. A tan. B B = sin (A + B) cos. A cos. B sin. (A + B) 36!cot. A + cot. B = + sin. A sin. B 37 sin. A + sin. B -*. tan. * (A + B) ‘sia. A - sin. B T tan. ; (A – B) sin. A + sin. B -* * * * : *. l. ( A 38 cos. A + cos. B T tan. ; (A + B). sin. A + sin, B --- - ! ſ ºf — 39. H.H = cot. (4 B). sin. A — sin. B 40 cos. A + cos. B T tan. ; (A – B). --- l 4] cos. B — cos. A T cot. § (A1 + D) sin. A 42;tan. ; A = 1 + cos. A - sin. A 43}cot. § 21 = 1 +...os. A Miscellaneous Formula. Given. Radius = r Semi-axes = a and b Chord = c, height = h } Side = a, number of sides = n Radius Radius . Base = b, height = h Base = b, height = h Bases = b and bi , height = h E V = r, height == Radius of sphere=r sum of angles = S number of sides = n } h ! } Formulae. It r*. 71 a b. 3 ch.” 1S00 2 - & # a” n cot. -- 4 ºr r". 2 m r h. S—(n–2) iSö9 ſtrºx–Isº- b h. § b h. # h (b+ bi + Vb b.) * The area of a circular segment on railroad curves, where the chord is very long in proportion to the height, may bo found with great accuracy by the above formula 158 TABLE x. MISOELLANEOUs ForMUL.A. Soucht. Given. Formulae. Solidity of 54|| Sphere Radius = ?” # ºr ré. Radii of bases = r K tº Q e ry" 55 Spherical Segment } and r1, height = h # Th ("+ r.” + 47%). | 56. Prolate Spheroid sº | # Trab?. 57 Oblate Spheroid |*.* | # tra”5. - Radius of base = r l 58| Paraboloid } - 9 |} ºr 7.3/. £) height = h | |* 7T 7**/), Tr = 3.14159 26535 89793 23S46 26433 83280. Log. t = 0.49714 98.726 94.133 85435 12682 88.291. TJnited States Standard Gallon = 231 cub. in. = 0.133681 cub. ft. {{ { { {{ Bushel = 2150.42 “ = 1.244456 {{ British Imperial Gallon = 2.77.27384 “ = 0.160459 t{ Length of Seconds Pendulum, at sea-level, at Equator, 39.0152 in. {{ {{ {{ * { {{ {{ “N. York, 39.1017 “ 4 ( {{ {{ {{ { { (4. ( { London, 39. 1393 { { Weight of a Cubic Foot of Pure Water, according to Rankine: At 39.1° Fahrenheit, 62.425 lbs. ; at 62°, 62.355 lbs. Figure of the Earth, according to Clarke, C. S. Report, 1877: Equatorial radius, 6 378 206.4 metres = 20 926 061.8 feet. Polar radius, 6 356 583.8 “ – 20 S55 120.8 “ Degrees in arc equal to radius 57.295'78 Minutes “ “ “ “ “ 3437.74677 Seconds “ “ “ “ “ 206264.80625 To change common logarithms into hyperbolic multiply by .434 294 4S ; the logarithm of which is 9.637 7843. 3.3 3:5 27 & Sin. a = a – — - J.C. in a F * ~ 2.3 + 2.3.1.5 - 2.1.5.0.7 * * 32 3.4 3:6 - Cos. 2 = 1 — - + + &c. 2 " 2.34 T 2.3.4.5.6 sin. 2: + sin.” + 3 sin.ºz. + 3.5 sin."z = Slil. -- * = sin. " + -āj-F-51H + -21.6% a = tan. a. — tan.* + k tan.*z – 3 tan."2 + &c. Let a = length of a flat circular arc, c = its chord, R = radius, D = deflection angle. g3 cº 24A. T 24R T + &c. Then approximately a - c = T A F3 E, E XI. SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AN ID RECIPROCALS OF NUMBERS FROM I TO AQ34's 160 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No Squares. Cubes. Square Roots. | Cube Roots. | Reciprocals. i I | 1.0000000 1.0000000 || 1.000000000 2 4 8 1.4142136 1.25992 || 0 .5()(){}{}{}()()() 3 9 2? 1.7320308 1.4.4.22496 .333333333 4 | 6 6 | 2.00()(){}(0 1.5874() || .25(){}{}(){}{}{} 5 25 | 23 2.23606S0 1.7(1997.59 .20()(){}(\{}()0 6 36 2| 6 2.449-1S)7 I.S.I 7 1206 , 166666667 7 49 343 2.0-157513 1.91 2.93|2 . . 42S57 143 8 6-1 512 2.S2S-1271 2.0000000 ..] 25ſ)|)0{}{}{j 9 8| 729 3.0000000 2.0S00S37 . I l l l l l l l I 10 I ()0 1000 3 1622777 2.) 5443-17 .100000ſ)()() | | | 2 | 1331 3.31 (562-18 2.2239S01 .(10090909 | 12 | 44 72S 3.464.1016 2.2S942S6 ,0S3333333 13 | 69 2197 3,60555 [3 2.35133.17 ,076923()77 14 106 2744 3.74 16574 2.4 101,422 .07 14:2S57] j 5 225 3375 3.8729S33 2.4662| 21 .066666667 | 6 256 4096 4.0000000 2.519 S421 .0625()()()00 17 2S9 4913 4. 123 l ().56 2.57 12S] 6 .05SS23.529 18 324 5S32 4.2426-1()7 2.62()74 4 .055555556 19 361 6859 4.33SSJS) 2.66S4016 .052631579 2ſ) 4()() 8000 || 4.4721360 2.7144177 | .050000000 21 44 l 926| 4.5S25757 2.75S92-13 ..(?-176 19048 22 4S-1 1064S 4.6994 || 5S 2.S()2(1303 .(145-1545-15 93 529 | 2 | 67 4.795S315 2.S433670 .04347S26 24 576 13S24 4.SøS9795 2.SS-1499 | .04 | 666667 25 625 15625 5.0000(){}0 2.92.40.177 .(){{}{}{}{}{}{}0 26 676 17576 5.09:}{}l 95 2.9624)6() ,033-16 | 538 27 720 196S3 5, 1961524 3.0(;(10000 .037 (137 (137 28 7S 1 21052 5.2.0 ſ 5(126 3.0365SSO .03:57 14:2S6 29 841 24339 5.3851648 3.07.2316S .0344S2759 30 900 27000 5.477'2256 3.107:23:25 .033333333 31 98 | 297.91 5.56776-14 3. 14 13.806 .03.225S065 32 1024 3276S 5.6.56S542 3.174S(21 .03125(){}{}0 33 I () S9 35037 5.74-15626 3.2075.313 .030303(130 34 | | 56 393ſ)4 5.83()9519 3.23961 IS .0294 | 1765 35 | 225 2S75 5.9 16()79S 3.27 10663 .02S57 14:29 36 | 296 46656 6.0000000 3.3() 19272 .027777778 37 1369 50653 6.0S 27625 3,33222) S. .027().27()27 3S 1.4.14 54S72 6. 16-14 |40 3.361975.4 .0263 57SQ 39 1521 593.19 6.24-190S0 3.39.121 14 .0256-11026 40 I 600 61000 6..}345553 3.4 100519 .025000000 4 I 1681 6S921 6.4(1312-42 3.44S2172 .024.3002-14 42 | 76.1 74(MSS 6.48()74ſ)7 3,476().266 .023S(Y9524 43 | S49 79.507 6.557,1385 3,50330Sl .02.3255S 14 44 I {}.36 85184 6.6332496 3.5303-4S3 .02:27:27.273 45 2025 9| 125 6.70S2(139 3.556S933 .022222222 46 2116 97.336 6.7S233U)() 3.5S30.479 .0217.3%) l 30 47 22(19 103S23 6.85365.46 3.6()SS261 .02 12766(M) 48 2:304 | 10592 6,92S2032 3.63:12'ſ Il .02(JS33333 49 2101 117649' 7.0000000 3.659.3057 .02040S163 5() 25ſ)0 125000 7,0710678 3,6S-40314 .020000000 5i 20|| | 1.3263; 7, 14 F-42S-1 3.7()S429S .0196()7$43 52 2704 i4()608 7.2| | |{126 3.7325 | | 1 .010230769 53 2Sſ)0 |ASS77 7.2SOI ()99 3.7562S58 0| SS67925 54 29 || 6 15746-4 7.34S4692 3,7797631 01 S5 l8519 55 3025 166375 7.4 161985 3.8()29.525 .01 SlSł SlS 56 3] 36 175616 7,4833| 48 3.82.5S0.24 .017857 143 57 3249 185103 7.549S.344 3.84S5() || .0|7543St.0 58 3.364 1951 12 7,615773| 3.S7()S766 .01724 |379 59 3481 205379 7,6811457 3.8929965 .016049.153 60 3600 216000 7,745.9667 3,914SG76 ,016666667 6| 3721 226.981 7.8102497 3,9364972 ,016303443 62 3S44 23S32S 7.S740079 3,957S915 .0161290.32 cube Roots, AND RECIPRocALs. 161 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 63 3069 2500-47 7.937.2539 3,9790571 .015S73016 64 4096 262| 44 8.0000000 4.0000000 .015625000 65 4225 2, 4625 8.0622577 4.02(17256 .0153S4615 66 4356 2S7496 8. 12403S4 4.04 12401 .015 151515 67 44S) 300703 8.1853.528 4.06154SU .014925373 68 4624 3.14432 8,2462) 13 4.0S 16551 .0.14705SS2 69 476i 32S509 8.3066239 4.1015661 .014492754 7 4900 343000 8.3666003 4. 1212S53 .0l 42S571.4 7 I 504 | 3.5791 I 8.4261.498 4.140S 178 .0140S4507 72 51 S4 373248 8.4S52S1.4 4. 1601676 .013SSSSSS) 73 5329 3S0017 8.54-100.37 4. 1793,390 .01369SG30 74 5476 405224 8.6023253 4. 19S33 .01 35.13514 75 5625 421S75 8.6602540 4.2.17 1633 .01.3333333 76 57.76 43.3976 8.7 177979 4.235S236 .013] 57S95 77 5929 456533 8.7749644 4.2543210 .01.29S7013 78 G(RS4 474552 S.S.317609 4.27265S6 .012S2(1513 79 624 | 49.3039 S.SSS1944 4.290S.404 .01.265S228 SO 6400 512000 8.944.2719 4.30SS605 .012500000 81 6561 531441 9.0000000 4.32674S7 .012345679 82 6724 551368 9.0553S5l 4.3444S15 .012| 95 12.2 83 6SS9 57 17 S7 9. I 104336 4,3620707 .01.204S193 84 7036 59.2704 9. 1651514 4.3795.191 .0 [ ] 904762 85 7.225 6|| 4 || 25 9.2195445 4.3.96S296 .01 || 7647.06 86 7396 636056 9.27.361 S5 4.41-40049 .0l 1627,907 7 7569 65S503 9.327.37.91 4.4310476 .01 1194.253 88 77-14 6SI q.72 9.3S()S3 15 4.447.9602 .0.1 ! 36.36.36 S0 7921 7()4969 9.4339S1 | 4.46-1745.1 .01 12.35955 9() S|{ſ0 7.20(K)() 9.4SGS3:30 4.4S140.47 .0 l l l l l l l 1 9 | S2Sl 73357 | 9.5.303)2() 4.497,941.4 .0; USS301 | 92 S464 77 S6SS 9.59 (,630 4.51.4357.4 .0} {}S69565 93 Sö49 8(). 1357 9.6-13650S 4.53065-19 .()] (17526SS 94 SS36 S3()3S4 9.693,3597 4.546S359 .01 (Hö3S29S 95 9025 857.375 9.7.467943 4.5629026 .0105.26316 96 92 || 6 SS4736 9.797.9590 4.57SS570 .01 (1416667 97 9.4|19 91.2673 9.84SS57S 4.5947()09 .01 (130927S QS 98)4 94 l 192 9.8994949 4.6l 04363 | .0102040S2 99 9S01 97.0299 9,949S744 4.6260650 .010101010 100 10000 1000000 10.0000000 4.6–115SSS .010000000 |(}l 102ſ]] I (330.301 } 0.049S756 4.657 (1095 .009900990 102 1(k4(k+ 106 ||20S 10.099.5049 4.67232S7 .000S(13922 I ()3 | {}{}U9 IU92727 1( ), 14SSS) 16 4.6S754S2 .(jū97 US73S I () { 10S 16 | 124S64 i(). 19St|390 4.7()26694 .0]96153S5 I ().5 1 1025 1 157625 10.2469508 4.7 l 769-10 ,009523S10 l()6 1 ſ 236 I 191016 10,2956.301 4.7326235 .0094.33962 107 | | 4:19 12250.43 10.3-140S04 4.7474594 .(){}9345734 |(}S l 1664 12597 12 10,3023048 4.7622032 .009:259259 109 | ISSI 1295020 10.4403065 4.776S562 009174312 | || 0 12100 1331,000 10.4SSOSS5 4.7914 199 .003000009 | | 1 12321 136763] 10.5356538 4.805S955. .009009009 1 12 12544 140-192S 10.5S30052 4.S2)2S45 .()(ISQ2S571 | 13 12769 1442S37 10.6.301 45S 4.S345SSl .00SS/355S 114 12906 14S1544 10.677 U.7S3 4. SHSS076 .00S77 1930 | | 5 132.25 I 52(\S75 10,723S()53 4.S6294-42 .00S695652 | | 6 13456 15{}()S06 10.7703296 4.S76999() .0(S620690 l 17 136SQ 1601613 1 G.Sl 6653S 4.SS)[]97.32 .0(\S3470US) | |S 13924 1643()32 10.Sö 27 S05 4.904 S6Sl .00S47.4576 l 19 1416.1 16S5159 10.90S7 121 4.91S6S47 ,00S403.361 120 14:100 172S000 10.954-1512 4.9324242 .00S333333 12! 14641 1771561 11.0000000 4.9460S74 .00S26-1463 122 14SS4 1815S48 11.0453610 4.9596757 .00S1967:21 123 15129 1S60S67 l 1,0905.365 4.97.31898 .00SI 300Sl 124 15376 1906624 11, 13552S7 4.9S66310 .00S064516 162 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. | Reciprocals. 125 15625 1953125 | I 1.1803399 5.0000000 | .008000000 126 15S76 2000376 11.22497.22 5 0132979 .007.93650S 127 I6 129 204S3S3 || 1 1.2694277 5,0265257 | .007S74016 128 163S4 2097 152 11.31370S5 5.0396S-42 .0078] 2500 129 1664 | 21466S9 || 11.3578.167 5.05277.43 | .00775}938 130 16900 2(97000 || | 1.4017543 5,0657970 .00769230S 13| 17 | 61 224SO91 || 1 | .4455231 5.07S753] . .0076335SS 132 17424 229996S | 1 1.4S91253 5,0916434 .007575758 133 176S9 235.2637 11.532:56:26 5. 104.46S7 .00751S797 134 17956 2406 104 || | 1.5758369 5. 1 172209 | .007.4626S7 135 18225 2.460375 || 1 1,618.9500 5. 1299278 || 007407407 136 18496 2515456 || 1 1.66 1903S 5. 1425632 | .007.3529.4| 137 18769 2571353 || 1 1.7046999 5, 1551367 .007:299270 133 19044 262S072 11.7473401 5, 1676403 || 007246377 139 19321 2685619 || | 1.789S.261 5. 1801015 | .007 194245 140 19600 27.44000 || 11.832.1596 5, 1924.941 .007 112S57 141 19SS1 2S0.3221 11.8743,421 5.204S279 || 00709:2199 142 20164 2S632SS | | 1.9l 6375.3 5,217 1034 .007 0-12254 143 2[]449 292.4207 l l .9582607 5.2293.215 .006993()07 144 20736 29S59S4 || 12.0000000 5.2414S2S | .0069.4444 | 145 21025 304S625 12.04.15946 5.2535S7 .006S96552 146 21316 3| 12136 || 12.0S30460 5.26:56.374 .0068.4931.5 147 2I 609 3176523 12. 1243557 5,2776321 | .006S02721 148 21904 32.11792 || 12.1655.251 5.2S957.25 .0067.56757 149 22201 3307949 || 12.2065556 5.301.4592 | .0067.11409 150 22:500 33.75000 || 12.2474487 5.313292S ,006666607 151 22SOI 3.142951 || 12.2SS2057 5.3250740 .006622517 152 23104 35 | | SOS 12.32SS2SO 5.336S033 .00657S947 153 23.409 3531577 | 12.369.3169 5.34S4S12 .0065.359.4S 154 23716 36.52264 || 12.4096736 || 5.36010S4 .00649.3506 155 21025 3723S75 | 12.449S996 5,3716S54 .006:45 1613 156 24336 3796.416 12.4800960 5.3S32126 .006410256 157 24649 3S69893 || 12.5299641 5.39.46907 | .006360427 158 2496.1 39-14312 || 12.569S051 5.4061202 | .()()63291 14 159 25281 40] 9679 || 12.6095202 5,41750.15 .0062S930S 160 25600 4096000 || 12.6491106 5.42SS352 .006250000 16] 25921 4.1732S1 || 12.6SS5775 5.4,401218 .0062) ſ 180 162 26244 425152S 12,7279221 5.45136].S | .006 172S40 163 26.569 4330747 | 12.7671453 5.4625556 .006134969 164 26S96 44.10944 || 12.S0624S5 5,4737037 .006097561 165 27.225 4492.125 | 12.8452326 5.4S4S066 | .006060606 166 27.556 45.74296 || 12.8$400S7 5,495S647 | .006ſ)24096 167 278S9 4657463 || 12.922S480 5.506S7S4 .005.98S024 16S 2S224 474 1632 12.96.14S1.4 5.51784S4 .0(159523S1 169 2S561 4S26S09 || 13.0000000 5.52877.4S .005917 160 170 2S900 40.13000 || 13.03840.48 5.53.965S3 .005SS2353 171 29241 5000211 || 13,0766968 5,550.499) ,C()5S47953 172 295S4 50SS448 || 13.1 148770 5.5612978 . .005S13953 173 29929 51777.17 | 13.1529464 5.57205-16 .0057803:47 174 30276 526S024 13. 1909060 5.5S27702 .005747 126 175 306.25 5359375 | 13.22S7566 5.5934447 | .005714286 176 30976 545.1776 || 13.2664992 5.60407S7 .0056SIS18 177 31329 5545233 13.3041347 5.6146724 . .0056497.18 178 316S4 5639752 | 13.3416641 5,625.2263 | .005617978 179 32041 57353.39 13.3790SS2 5.635740S .0055S6592 180 32400 5S32000 || 13.4164079 5.6462162 .005555556 181 32761 592974 l 13.45.36240 5.656652S .00552.4S62 182 33124 6()2S568 || 13,490.7376 5.66705 || 1 || 0(15-194505 IS3 33489 612S4S7 || 13.5277.493 5.677.41 14 || 005-1644Sl 1 S4 33S56 6229504 || 13.5646600 5.6S7734ſ) .0054347S3 185 3.4225 633.1625 | 13,601.4705 5,6980.192 | .005405405 186 3.4596 t;434S56 | 13.6331817 5.70S2675 ' ,005376344 CUBE ROOTS, AND RECIF ROCALS. 163 ſ | No. 8quares. Cubes. Square Roots. Cube Roots. Reciprocals. 187 34969 6539.203 13.6747943 5.7 1847.91 .00534.7594 1S3 35344 6644672 13.7 l 13092 5,72S65-13 .005319149 | S9 35721 6751269 13.747.7271 5.73S7936 .005291005 190 36100 6S50000 13.7S.404SS 5.74SS971 .005363H5S 191 364Sl 6967871 13.S202750 5.75S9652 .005235602 192 36S64 7077SSS 13.8564065 5.76S99S2 .(10520.8333 193 372.49 7 ISQ057 13.S924440 5.77S9966 .005181347 194 76.36 73013S4 13.92S3SS3 5,78S960.4 .005) 5:4639 195 3S(125 74.14875 13.964.2400 5.79SS900 .00512S205 196 384 16 7529536 14.0000000 5.80S7S57 .005] [204] 197 SS09 7645373 14.03566SS 5.8.1864.79 .005076.142 198 392(\4 7762392 14.07 l.2473 5.82S4767 .005050505 199 3960l 7880599 || 14.1067360 5.83S2725 005025 126 200 40000 8000000 14. 1421356 5.84S0355 ,005000000 201 4(k401 8120601 14.1774,469 5.S577660 .004975124 202 40S04 824240S 14.2126.704 5.867.4643 .004950-495 2()3 41209 8365427 14.247S06S 5.877 1307 .004926.108 204 41616 84S9664 Is...2S2S569 5.8S67653 .00490.1961 205 42025 S615125 14.317S2I 1 5.89636S5 .004S7S049 206 42436 8741816 14.3527001 5905.9406 .004854369 207 42S49 8S69743 14.3S74946 5.9154S17 ,004S30918 2(\S 43264 890S912 14.4222051 5.92.4992I ,004S07692 209 436 Sl 9129329 14.456S323 5.9344721 .0047S46S9 210 44.100 926.1000 14.4913767 5.9439220 .00.4761.905 21 I 44521 93939.31 14.5258.390 5.95334 18 .0{}4739336 212 44944 952S12S 14.56()2] 9S 5.9627320 .004716981 213 45369 956.3507 14.5945] 95 5.97.20926 .00.4694S36 214 45.796 9S003:44 14.62S73SS 5.9Sl:4240 .004672S97 215 46225 993S375 14.662S7S3 5.990.7264 .00465l 163 216 46656 10077696 14.69693S5 6.0000000 .00.4629630 217 470S9 102IS313 14.7309|99 6,0002450 .00460S295 218 47524 10360232 14.76-1S231 6.01 S.4617 .0045S7156 219 47.961 10503459 14.79S6486 6,0276502 .004566210 220 4S400 106–48000 14.S323970 6.036S107 .004.545455 221 4SS41 10793S6 I 14.S6606S7 6.0459435 .004524SS7 22.2 492S4 1094.1048 14.8996644 605504S9 .00.4504505 223 497.29 1 I(\S9567 14.933 IS45 6.0641270 .00.44S4305 224 50l 76 l 12:39424 || 14,9666.295 6.0731779 .004.4642S6 225 5U625 11390625 15.0000000 6.0S.22020 .00.4444444 226 51076 I lä43176 15.0332964 6.091 1994 | .004424779 227 51529 I 16970S3 15.0665.192 6. 1001702 0044052S6 22S 51.9S4 I 1852352 15.09966S9 6. 1091 l 47 .0043S5965 229 52441 1200SSS9 15. 1327.460 6. 1 1S0332 .004366S12 230 52900 12167000 15. I657509 6. 1269.257 .004347S.26 231 53361 1232S391 15, 1986S42 6, 1357924 .0043.29004 232 53S24 124S716S 15.2315462 6. 1446337 .0043.10.345 233 542S9 12649337 15,2643375 6.1534495 .0{}429.1845 234 54756 12812904 15.297.05S5 6, 1622401 .00-1273504 235 55225 12977S75 15.3297097 6.17 1005S .00.4255319 236 55696 1314.4256 15.362.2915 6.1797466 .0042372SS 237 56169 1331.2053 15.394S043 6.1SS462S .004219409 23S 56644 134S1272 15.42724S6 6. 1971 544 .0042016S 1 239 57.121 1365,1919 15.4596.248 6.205S218 .0041841.00 240 576.00 I3S24000 15.49.19334 6.2144650 004 166667 241 5SOS! 13997.521 15,524 1747 6, 2230S43 .004 14937S 242 5S564 141724S8 15.5563.192 6.2316797 004 13.2231 243 59049 1434S907 15.5SS4573 6.240°2515 0.04] 15226 244 59536 145267S4 15,620.499.4 6.2.4S7998 00:409S36! 245 60025 | 4706 125 15,652475S 6.2573.248 (1040SI 633 246 60516 14.SS6936 15.6S43S71 ..º.265S266 00:406504 | 247 61009 15069223 15.7162336 *č2743054 ,00-404S5S3 248 61504 15252992 15,74S()157 6.2S27613 .00-103225S 164 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes, Square Roots. | Cube Roots. Reciprocals. 2.19 62001 1543S249 || 15,779733S 6.201 1946 .00-1016.064 250 62500 15625000 || 15. SI 13SS3 6.2006053 .004000000 25 | 63(){} | 15SI 325i 15. SH297.05 6,3079).35 | . (1039S 10G-1 2.52 6.35()4 | 6′)03()()S 15, S745(J79 6.31 63,596 | .0(),306S254 2.53 64()09 16|{) 1277 | 15.90.59737 6.3247()35 | .(ſ).30.52569 251 64516 1 ($3S7()64 || 15.937.3775 6.3.3.30256 | .0()39.37 ()()S 255 6.5()25 165S1375 | 15.96S7 194 6.34 13257 | .0(1392|569 256 65.536 167772 ||6 || 16. ()()){}()(30 6.34)6()42 .()()3)()6250 2.57 660-19 | 6′)74593 16.03 || 2 || 05 6.357 S6 || .003S) { {}51 25S 66564 17 | 735 | 2 | 16.()(32.37 S4 6,3660:16S .003S75060 259 670S1 1737.3979 || 16.0931709 6.3743 || | .003S61004 200 67600 1757.6000 | 16.1245 155 6.3S25043 | . ſhſ)3S4615.4 26 | 6SI 21 177795S] 16. 155-1944 G. 39.167 (35 | .0035314 i S 202 6SG-1.4 179S472S 16, ISG-1 || | 6.30SS2’) (){138|| 67.04 263 69 69 1S 1914-17 | | 6.21727.47 0.406).jS5 | .()(3S()22Sl 264 69696 | S3997.44 16.2.18ſ)76S 6.4 506 S7 | ()().37S7S79 265 7()225 JS609625 || || 6.27SS206 6.42:31:5S3 | .0(137735S5 266 7()756 I SS21096 || 1 (3,3005()G4 6.4.312.270 . ()().3739303 267 7 12SQ 1903.4 | 63 | 16.34)|3.16 6.4:39:27.67 | . ()()37,453 IS 26S 7 IS24 1924 SS32 16.37 ()7()55 6.4.473().57 .0(1373.1343 269 72361 19-165109 || 16.4()12 195 6.455.314S .003717472 270 72900 196S3000 | 16.43 (6767 6, 46.33ſ) 1 | .003703704 7 | 7344 1 1990.25 || 16.462()776 6.47 12736 .0().36%)(H)37 272 739S4 2012364 S | 16.4)2.1225 6.4792.236 .0(13676-47 l. 73 74520 20346-1 || 7 || || 6.5.227 || 6 6.4S7 || 5 || .0(13663()()4 274 75ſ)76 2057()S24 | 16.5529.454 6,495 (353 | .00364963.5 275 75625 20796S75 16.5S3|240 6.5029.572 .003636.364 276 76] 76 21021576 J 6.6 || 3:2477 6.5 |(}S3ſ)() | .003623 ISS 277 76729 2125.3933 | 6,643.3| 70 6.51 S6S30 .0036 () 10S 278 772S4 214S1952 | 16.67333.20 6,5265|S9 .003597 122 279 7S-41 217 17039 || 16.7032931 6.5343351 | .0035S4229 2SO 7S 400 21952000 | 16.7332005 6.5421326 .00357 | 129 2Sl 7S96 | 22 [SSſ) || I 6.763().5-46 6.5-109 || 6 | .00355S7 | 9 2S2 70524 22.125768 || || 6,792S556 6,557.6722 | .0035-16009 2S3 8|}0SQ 22665|S7 | 16. S22603S 6.5654 || 4 | .003533569 284 S(\656 229()6304 16. S522005 6, 57.313S5 | .00352; 127 2S5 8|225 23| 491.25 | 16. SS 19430 6.5S0S.143 .0035()S772 2S6 81796 23.30.3656 | 16.9 || 53.15 6.5SS5323 .0034.96503 287 S2,369 236.3990.3 || 16.94 | ()7.43 6.596.2023 .0034S-1321 2SS S2944 23SS7S72 16.97 (15627 6.6().385-45 | .003472222 2S9 835:21 2-1 137569 || 17.0000000 6.61 14S90 | .0034.60208 200 S.) 100 213S0000 17,0203864 6,610.106() .00344S276 291 846SI 246 12171 17.05S722 | 6,6267 ()54 .0034.304:26 292 85264 24S970SS 17.0SSO()75 6.63.42S74 | .00342.165S 203 S5S-40 25.153757 17. 172.12S 6.64 l S522 .0034 12969 294 86,436 25 || 2 | S4 17, 146-12S2 6.6-1939QS | .003401.361 295 87()25 2567.2375 || 17, 17556 10 6,6569302 | .0033SQS31 29ſ; S76 || 6 2593 ||3:36 || 17.2ſ). 16.5().5 6,664-1437 .003378,378 207 8$209 26 [0S()73 || 17.233(3S79 6.67 19403 | .003:367(103 29S 8SS()4 26:46.3502 || 17,2626765 6.670-1200 .003355705 299 89.401 2673()S90 || 17.291 6105 6.6S6SS3| | .0033-14482 300 00000 27000000 17.32050Sl 6.69.432.05 .003333333 301 9(JGſ)| 27:2709(){ 17.3–1935 | 6 6, 7()| 7593 .003322:259 302 9| 204 275–1360S 17.378 I 472 6.7()() | 729 .003.3 || 258 303 9| S().9 27S [S] 27 || 17,406S952 6.7 || 657 (30 | .003.300,330 304 924 16 2S(19446.4 17,435.595S 6.72.3%).5(1S | . (){}32S9474 305 93(25 28372625 | 17.4642492 6.73 13155 | . ()()327 S6S9 306 936.36 286526 || 6 || 17.492S557 6.73S(364 || | |003267974 307 94.249 289.34443 17,52| 4 || 55 6.7459967 .003:257329 30S 94S64 2921 Sl 12 || 17.54992SS 6.7533 ||34 || 00:3246753 309 9:54S1 29.503629 17,57S395S 6.76(16|43 | . ()()3236246 310 96.100 2979 1000 || || 7,606S 169 6,7678995 .00.3225S06 CUBE ROOTS, AND RECIPROCALS. 165 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 31 I 967:21 300S0231 17. G35) 92.1 6.775 t 690 .003:21 5-434 312 973–14 3í).37 1328 17.66:35:217 6.7 S2:12:29 .()()32)5] 28 313 97969 3(166-1297 17.69 S()00 6.7 S96613 .(H)3 || 04SSS 314 9S$96 309.59 |44 17.7200-151 6.796SS44 .003 IS-17 | 3 315 90225 31255S75 17.74S2393 6.804(1921 .003 || 746(13 3| 6 90S36 3| 554-196 17.776.3SSS 6. S1 12S47 .003] G-1557 3| 7 || 1004S0 3| S55013 || 17. S(k1493S 6. SIS-1620 .003 ſ 54574 3.18 101 ſ 24 32157.432 17. S325545 6. S256.242 .003.144.654 3.19 1017 Gl 32461759 17. S6057 I 6. S327714 .003.134796 320 102400 3276S000 I 7.SSS543S 6.S39Q037 .003125000 321 1030-1 || 3307616.1 17,916-1729 6.8470213 .0031 15265 322 1036S4 333S6248 17.94.435S4 6. S541240 .003 || 0559() 323 I () [329 3.369S267 17.97.2200S 6. S612120 .0(3095,975 324 10-1976 34.012224 18. (1000000 6.86S2S55 .0(30S6420 3.25 10562.5 34.32S125 IS.0277.564 6. S753.443 .003076923 326 } 06:276 34645976 18.03547()| 6. SS23SSS .0030674S5 327 1 (16929 3496.57S3 1S,0S3 4 || 3 6. SS94 ISS .00305S 104 32S 1075S4 352S7552 1S. l l ()77ſ)3 6. S96-1345 .0(1304S7SO 329 10S2.11 356] 12S9 1S. 13S3571 6.9034359 .00303051.4 330 10S900 35937000 1S. 1659021 6.9 || 04232 .003030303 3.31 l(J956 | 3ö26460 I 18, 1934()54 6.9 || 73964 .003()2] 14S 3.32 1 10224 36:59.436S 1S,22(IS672 6,924.3556 .0030,120.48 333 I l{}S$9 36926()37 1S,24S2S7 6.93 1300S .G{}30(130()3 334 1 11556 3725970-4 l S.2756069 6.938.2321 .00299.40 2 335 | 12225 37.595375 1S. 3030()52 6.945 l 496 .0029S5()75 336 I 12S96 7.933ſ).56 1S.33(13028 6,9520533 .002976 190 337 1 13.569 3S272753 1S,357.5598 6.95SQ-434 .002967.359 33S l 14244 3S6 | 1.472 |S.3S477.63 6.965S 19S .0(1295S5S() 339 I 14921 3S95S219 18.41 19526 6.9726S-26 .0029.49S53 340 | 15600 30304000 1S,4390SS9 6.97.95321 .0{}294 l l 76 341 I | 62S1 3905] S2I 18.466] S53 6.9S636SI .00293-255 | 3.42 I | 6964 4ſ);}{}| GSS 1S. 4932420 6,993 |9| 6 . (1029:23977 343 I 176-49 40.35.3607 1S. 5202592 7. ()00(\000 .0029 5452 344 ll S336 407 (J75S4 1 S.5-172370 7, ()()67962 .0029(16977 345 I 19(25 4 106,3625 1S,574 1756 7. (Il 3579 | .(H)2S9S551 346 l 1971 6 41421736 IS.6ſ) 10752 7. (1203-49ſ) .(102SQ()] 73 3:47 120409 4 l 7SI 92.3 18.627,9360 7,027 (#5S .(IU2SSlS44 348 12| |(}} 42144192 }S. 65475S] 7.033S407 , ()()2S73563 349 12|S01 4250S549 1S.6S15417 7.040.5S06 .0{}2S65330 350 122500 42S75000 1S.70S2S69 7,04720S7 ,0ſ)2S57 143 351 123,201 43.243551 18.73-1994() 7. (J54(){}4 | . (RiºS 49()()3 352 123904 436 |420S I S. 76166.30 7.06ſ)6967 ,0(12S409(9 353 124609 439S6977 1S,7SS2942 7,06737.67 .0()2S32S61 354 125316 4436 1S64 1S.S. 4.SS77 7. ()74(1-44() ..(it)2S2.48.59 855 126(25 4473SS75 1S.S4|4437 7. (S(169SS .(){}2S] 6901 356 126736 451 | SO16 18. S679623 7,087.341 l .(KjøSſ)SSS9 357 | 27449 45.499.293 1S,S94.4436 7. (1930.70%) . (1()2Sūll 120 35S 12S164 45SS27 12 IS.92()SS7 7. I (105SS5 .002.793296 359 12.SSSI 4626S279 8.9472953 7, 107 1937 .0027 S55 15 36 129600 46656000 18.97.36660 7, 1 137866 .00.2777778 361 130321 47045SS1 19.0000000 7. 1203674 ,00:277 (JOS3 362 13104.4 47.437.92S 19,026.2976 7. 1269360 .()()276.2431 363 13] 760 47S32 47 19.052.55S9 7. 13349.25 .002754S21 364 132496 4S22S5.14 19,07S7 S40 7, 13 ()()37 () .00:2747253 365 133225 4Sö 27 125 19, 1()497.32 7. [465695 . (1027.307.26 360 133956 49(127S96 19. 131 265 7, 1530901 .00:273-2240 367 13-16SS) 49.430S63 19, 1572441 7. 15959.SS .002724796 36S 1354.24 49S36032 19, 1833.261 7. 1 GG()\};7 .00271 7391 369 136161 502.13409 19.20937.27 7. 1725S09 .0027 10027 70 136000 50653000 19.2353S41 7, 1790544 .0027trº703 71 1376-41 51()64S11 19,261 3603 7, IS55162 ,(\{1269.541S 37.2 13S3S4 5147SS4S 19,2S730.15 7, 1919663 ..(?()26SS172 166 TABLE X1. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. | Cube Roots. Reciprocals. 373 || 130129 518951 17 | 19.3132079 || 7. 19S4050 .0026S0965 37.4 || 139876 52313624 || 19.3390796 || 7.204S322 .002673797 375 | 1.40625 52734375 | 19.364.9167 || 7,21 12479 .002666667 376 141376 53157376 | 19.3907 194 || 7.2176522 .002659574 377 14.2129 53.5S2633 19.4164S78 || 7.224()450 | .002652520 37S I 42SS4 540.10152 | 19.4422221 7,230-126S .00264.5503 379 143641 54:439939 || 19.4679223 || 7.2367.972 .002638522 3S0 144400 54S72000 || 19.4935SS7 || 7.2431565 .00263.1579 3Sl 14516.1 55.3063.41 || 19.51922.13 || 7.2495045 .002624672 3S2 145924 5574296S | 19.544S203 || 7.255S415 | .002617801 3S3 || 1466S9 56 [S] SS7 | 19.570385S 7.2621675 | .0026.10966 3S4 | 1.47456 56623) ()4 || 19.5959.179 || 7.26S4S24 .002604 |67 3$5 | 1.48225 570666.25 | 19.6214169 || 7.2747S64 .002597.403 3S6 || 148996 57512456 | 19.646SS27 | 7.2S10794 | .002590674 3S7 149769 57960603 || 19.6723156 || 7.2S73617 | .0025S3979 3.SS 150544 53.41 1072 | 19.6977'156 || 7.2936330 | .002577320 3S9 || 151321 5SS63S69 || 19.7230S29 || 7.299S936 | .002570694 390 | 152100 593]9000 || 19.74S4177 7.3061436 .00256.4103 39 | 152SS1 5977617 | | 19.7737199 || 7.3123S2S .002557545 392 || 153664 602362SS | 19.7989SQ9 || 7.31 S6 l l 4 || .00255] 020 393 || || 54449 6069S457 | 19.8242276 || 7.3248.295 | .002544529 394 || 1552.36 6] 1629S4 || 10.8494.332 || 7.3310369 | .002538071 395 || 156025 61629S75 | 19.S7.46069 || 7.33723.30 .00253.1646 396 || 156816 62099.136 | 19.SS974S7 || 7.3434.205 .002525253 397 || 157609 62.57.0773 || 19.924S5SS | 7.3495066 | .002518892 39S 15S404 630.44792 19.9499.373 7.35.57624 .0025.12563 399 || 15920] 63521 199 || 19.9749S44 || 7.3619.178 | .002506266 400 | 160000 64000000 20.0000000 7.36S0630 | .002500000 401 160S01 644S1201 || 20.0249S44 || 7.374 1979 | .002493.766 402 | 161604 6|964SOS 20,0499.377 || 7.3S03227 . .0024S7562 403 || 162409 65450S27 20.074S599 || 7,3864.373 .0024S1390 404 163216 65939264 || 20.0997512 7.39.254 18 .0024752.48 405 | 164025 6643()125 | 20. 12461 IS 7.39S6363 .0024691.38 406 164S.26 66923416 || 20, 1494417 || 7.4047206 | .002463054 407 || 1656-19 674 19143 20, 17424.10 || 7.4107.950 .002457002 40S 166464 679.173| 2 | 20, 1990009 || 7.416S595 | .002450980 409 || 167281 6S-117929 || 20.22374S4 || 7.4229142 | .002444988 410 | 16S 100 6S921000 20.24S4567 || 7,42895S9 .002439024 4 ll 16S921 69426531 20.2731349 || 7.43-19938 . .002433090 4] 2 | 169744 6993-1528 || 20.2977S3] 7.44 101S9 | .002-127 IS4 4 || 3 || 170569 70444997 || 20.3224014 || 7.4470342 . .00242.1308 414 || 17 lº)6 709,57944 || 20.3469S90 || 7.4530399 || .002415459 4 || 5 || || 72225 7| 473.375 20,3715 ISS 7.4590.359 | .002409639 4 || 6 173056 7 1091296 || 20,39607Sl 7.4650223 .002403S46 417 173SS9 725 1713 || 20.4.205779 || 7.47099.91 .00239SOS2 4 IS 174724 73034632 20.4450483 || 7.4769664 .0023923-14 419 175561 73560059 20,4694S95 7.4S29242 .002386635 420 || 176400 740SS000 | 20.49390.15 7.4SSS724 . .0023S0952 421 17724] 7461S461 | 20.51S2S45 7,494SI 13 .002375297 422 || 1780S4 75.1514.4S 20.54263S6 || 7.5007.406 | .00236966S 423 || 178929 756S6967 20.566963S 7,5066607 | .002364066 424 179776 76225024 20.59|2603 || 7,5125715 .00235S491 42.5 180625 767656.25 20.61552Sl 7.5184730 | .00235'2941 426 | 181476 77.30S776 | 20.639767.4 || 7,524.3652 .0023.474 IS 427 | 182329 77S5'4S3 || 20.6639783 || 7.5.3024S2 .00234 1920 428 183184 7840.2752. 20.6SS1609 || 7.5361.221 | .002336449 429 | 18404 | 789535S9 || 20.7123152 || 7.54 19867 .00233.1002 43 184900 70507000 | 20.7364414 || 7.547S423 .002325581 431 185761 80062991 20.7605395 || 7.5536SSS .002320186 432 | 186624 8062156S 20.78:16097 || 7.5595263 .00231.4815 433 187489 SllS2737 20.80S6520 || 7.5653548 .002309469 434 1SS356 81746504 | 20.8326667 || 7.571 1743 .002304147 CUBE ROOTS, AND RECIPROCALS. 167 Reciprocals, | Squares. Cubes. Square Roots. | Cube Roots. 1892.25 82312S75 | 20.8566536 7.5769S49 .002298S51 190096 82SS1 S56 | 20.8806130 7.5S27S65 .00229357S 190969 S345.3453 20,9045450 7.5SS5793 .0022S$330 1918.44 S4]27672 20.92S4495 7.59.436.33 .0022S3|05 192721 S46045.19 20.9523268 7.60013S5 .002277904 193600 S5IS4000 || 20.9761770 7.605.9049 .0022.72727 1944S1 85766 121 || 21.0000000 7.61 16626 .002267574 195364 S6350SSS 21.0237960 7.6l 74 l 16 | .002262443 1962.49 86938307 21,0475652 7.6231519 .002257,336 197136 S752S3S4 21.07 1307.5 7.62SSS37 .002252,252 1980:25 SS 12 || 1:25 21.0950231 7.6346067 .0022.47 191 1989 || 6 SS7 16536 || 21. 1 | S7 121 7.6403213 .0022.42 152 1998(19 893.14623 21. 1423745 7.6.460272 .002237 136 200704 S99 15392 21. 1660) ()5 7.65.17247 .002232143 201601 90518849 || 21.1896201 7.657413S .002227171 202500 91 125000 || 21.2ſ 32034 7.6630943 .002222222 203-101 9| 73.3851 21.2367606 7,66S7665 .002217295 204304 923.4540S 21.2602916 7.6744303 .0022 123S9 205209 92.95.9677 21.2S37967 7.6S00S57 .002207506 200 l 16 9357.6664 21.307275S 7.6S57328 .002202643 207025 94 196375 21.3307290 7.69137 17 .002197 S02 207936 94S1 SSI 6 || 21.354 1565 7.697.0023 .0021929S2 SS-40 954,43993 || 21.37755S3 7.7()26246 .002] SSIS4 2097.64 9607 1912 21.40093:46 7.70S23SS .002183406 2106Sl 96.702579 || 21.424.2S53 7.713S44S .002, 78649 2] 1600 97.336000 || 21.4476106 7.7194426 .002173.913 212521 797.21SI 21,4709106 7.7250325 .0021691.97 213444 9S6] 1 12S 21.494 l S53 7.73061.41 .002164502 214369 99.252S47 21.517434S 7.736|S77 .002] 59S27 2[ 5296 99S97344 21.5406592 ,74 || 7532 .002155 72 216225 10054,4625 21.563S5S7 7,7473] ().9 .002150538 217 156 I0) 19:1696 || 21.5S7033l 7.752S606 | .002] 45923 21 SOS9 101S-47563 21.610 [S2S 7.75S4(F23 .002I41328 2190:24 102503232 21.633.3077 7.763.936] .002136752 21996.1 103.161709 || 21.656407S 7.769.4620 | .0021321.96 220000 103S23000 || 21,6794S34 7.7749SO1 .002127660 221841 104.487 l l l ; 21.7025344 7.7S04904 || .002123142 22:27S4 1051540.48 21.7255610 7.7S59928 .002l 18644 223729 105S23SI 7 || 21.74S5632 7.7914S75 .002] 14165 224676 1064964.24 21.771 54] 1 7.79697.45 0021097 05 2256.25 107.17 1875 21.7944947 7.S02453S .002105.263 226576 107S5017 21. S17 4242 7.80792.54 .002100S40 227529 10S53].333 || 21.8403297 7.8133S92 .002096436 22S4S4 1092.1 5352 21 S632 || 1 7.81 SS456 .002092050 229-1-11 10990.2239 || 21,8S606S6 7.8242942 .0020S76S3 230400 | 10592000 || 21.90S9023 7.8297.353 .0020S3333 23| 36|| 1 | 12S4641 21.9317 122 7.83516SS .002079002 23:23:24 1 11980.16S 21,95449S4 7.8405949 .0020746S9 233,2S9 I 1267S5S7 || 21,9772610 7,84601 34 .0020703.93 234.256 | 13379904 22.0000000 7.8514244 .0020661 16 23.5225 1140S4125. 22.02.27 155 7.856S2S] .00206] S56 236 t 96 I 1479| 256 22,0454077 7.8622242 .002057613 237] 69 1 1550.1303 22.06S0765 7.867613() .0020533SS 23S144 1 1621.4272 22,0907220 7.8729944 .002049 ISO 2391.21 1 16930169 22, 1 133444 7. S783GS4 .002044990 240.100 I ITG49000 22, 1359.436 7,8S37352 | .002040SI6 24l (Sl 1 IS37()77 22, 15S51.9S 7.8S90946 .002036660 242(164 ll 90954SS 22. 1S) 0730 7. S944-46S .002032520 2430.49 | 19S231.57 22.2036033 7, S907.917 .0ſ).202S398 244036 | 2055.37S4 22.2261 10S 7,905 1294 .0{r^{12429] 245(125 1212S7375 22,2485955 7.91 04599 || .0020202ſ)2 246() || 6 1220.23936 22.27 10575 7.91578.32 .00.3016129 168 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, -tº- No. Squares. Cubes. Square Roots. | Cube Roots. | Reciprocals. 407 247009 1227634.73 22.2034.96S 7.9.210994 .002012072 498 24S004 1235()5992 22.3|59 || 36 7,926. (MS5 .0020ſ)S()32 499 249001 12425.1499 || 22.33S3079 7.9317 104 || .00200400S 500 250000 125000000 || 22.360670S 7,937.0053 .002000000 501 25|001 125751501 22.3S3().293 7.9422031 .00I '99600S 502 252004 12650600S 22.4053565 7.94.75.739 .00190.2032 503 25.3009 127263527 22.4276615 7.9.52S477 .0019.SS072 504 25-10 || 6 12S024.064 22.4499443 7.95S] 144 .00 19S4 127 505 - 255025 12S7S7625 22,4722051 7.963.37.43 .001.9S() [98 5()6 256036 129554216 22.494-143S 7.96SG271 .001976.2S5 507 257049 130323S43 22.5 | 66605 7.973S73] .00 19723S7 50S 25S064 | 3 |0965 12 22.53SS553 7.979 | 122 .00196.S5ſ).] 509 2590SI 13|S72229 || 22.56 102S3 7.9S43444 .00 1964637 510 260.100 13265 1000 22.5S3ſ 796 7.9S95607 .001.96ſ)7S4 5l l 26 || 21 1334.32S31 22.605.309|| 7.99.17SS3 .00 19560-47 512 262,144 1342| 7728 22.6274 l 70 8.0000000 .001953.125 513 263.169 1350ſ)5697 22, G495033 8,0052049 .001949318 5l 4 264 196 135796744 22.671 56Sl 8.01()4032 .0019-15525 515 26.5225 136590S75 22,6936 || 4 8.0|55946 .00194 17 18 5 || 6 266256 1373SS006 || 22.7.156334 8.02()7794 .001937.9S4 517 2672S9 13SISS4 || 3 || 2:2.7376340 8.0250574 .0019342.36 518 26S324 13S99|S32 22.7596,134 8,031 12S7 .001930.502 519 26936! 1397.9S359 22.7S 15715 8.0362935 .001926732 520 270.400 14060S000 || 22.S0350S5 8.04 (4515 . .001923077 521 27] 44 | 14142076 | 22.8254244 8.0466030 .0019 19386 522 2724S4 1422.3664S 22.8473193 8.0517479 .0(1915709 523 273529 143(133667 22.869 1933 8.056SS62 .0019 12()46 524 274576 143877S24 22.89 10463 8.06.201SO .00190S397 525 27.5625 144703125 22.912S7S5 8,067 |432 .001904762 526 276676 1455,31576 22.93.16S99 8,0722620 .00100 l l 41 527 2777.29 146363| S3 22.9564S06 8.0773743 ,00|S97533 528 27S7S4 I 47 197952 22.97S2506 8.0S24S00 .001 S93939 529 279S41 14S035SS9 23.0000000 8.0S75794 .001800.359 530 290,000 14.SS77000 || 23,02172S9 8,0926723 .001SS6792 531 2S1961 l 407.21291 23,043437.2 8.09775S9 .001 SS3239 532 2S3024 I 5056S7GS 23.065 1252 8. 102S390 .001 S79699 533 28.40S9 15||| 9 |37 23.0S6792S 8. 107.912S .001 S76 | 73 534 2S5 56 1522733()4 23. 1 ()S4400 8, 1 129S03 .00 1872659 535 2S6225 15.3| 3(375 23. 1300670 8, 1804 || 4 .00|S60 l 59 536 287296 15399(1656 23. 151673S 8, 123(1962 ..(ſ) (S65672 537 2SS369 154854 || 53 23, 1732605 8, 12S1447 .(){}| SG2197 538 2S9.444 155720S72 23, 19.1 S270 8, 1331 S70 .()0|S5S736 539 290521 156590S19 || 23.2163735 8, 13S2230 | .00IS552SS 540 291600 15746.4000 23.2379001 8. 14.32529 .001851852 54 | 2926Sl 15S340421 23.2594()67 8, 14S2765 .00184S429 542 293764 I 592200SS 23.2S()S935 S. 1532939 .001 S.450lS 543 29.4S49 160103007 || 23.3023604 8, 15S3051 .001 S4 1621 5.44 205936 1609S91 S4 23.323S076 8. 1633102 .001S3S235 5.45 297025 16|S7S625 23,3452351 8, 16S3()92 | .00; S34S62 546 29SI 16 16277 1336 23.3666.429 8. 1733020 ,00IS315()2 547 299200 163667323 23.33S03 I l 8, 17S2S88 .00 182S154 548 300304 164566592 || 23.400399S 8. IS32695 .001824SIS 549 301401 165460149 || 23,4307.490 8, 1882-141 .001S21494 550 302500 I66375000 || 23.45.2078S 8, 1932.127 ,00lSIS182 551 30.360/ 1672S-1 || 5 | 23.47.33S92 S. 19S 1753 .001 S14SS2 552 30-1704 168|9660S 23.4946S()2 8,203 || 3 |9 .001 Sl I 594 553 305S09 169 12377 23.5 [59520 8,2080S25 .001SOS3].8 554 3069 || 6 17003 || 464 || 23, 537.2046 8,2130271 .001S05054 555 30S025 170953S75 23.55S43S() 8.21796.57 .00lSOIS02 556 3(19136 171S796||6 || 23.5796522 8.22289S5 .00179S561 557 3| 02:49 172S()S603 23.6(){}S-174 S.227S254 .001795332 55S 3| || 364 17374] 1 12 || 23.62.20236 8,2327463 .001792|| 15 CUBE ROOTS, AND RECIPROCALS. 169 Squares. Cubes. Square Roots. Cube Roots. | Reciprocals. 3.12481 174676S79 || 23.6431SOS 8.2376614 .001783909 3t 3600 175G | 6000 23.66-1319|| 8.24257.06 .0017S57 [4 31,4721 17655S4Sl 23.6S543s(; 8.24.7474() .0017S2531 315S-14 1775()-132S 23.7(K65392 8.25237 H5 .0ſ, 1779.359 316069 I 7S-1535-47 23.7276210 8.257.2633 .001776|| 99 3|S006 179-106 || 44 23.74SGS42 8.2621492 .001773(150 3102.25 ISU362. 25 23.76972S6 8.267 ().294 .001769.912 32().356 ISI 32.1496 23.79ſ 75-15 8.27 19039 .0(17667S4 32|4S9 1822S4263 || 23.S1 || 76|S 8.27677.26 .00176366S 322024 1S,525()432 || 23. S327506 S.28 16355 ,00|760563 323761 1S42200 JJ 23.S537209 8.2S64928 .001757469 32 1900 IS5] 93000 || 23.874.672S 8.2913.444 ,00|7543S6 326() { | 1SG | 694 l 1 23. S956()63 8.296.1903 .00|75|| 3 || 3 3.27 | S-1 1 S7 1492-18 23.9 1652.15 8.301ſ1304 .0!) . 74S252 32S329 | SSI 325||7 23.937.4 S4 8.305S651 .00174520.1 320476 lS9 || 19224 23.95S2971 8.3 l (1694 l .00|7.42160 3.3ſ 625 I90|09375 23,9791576 8.3| 55||75 .0{}l 739 130 33 1776 19 || 02976 24.0000000 8.3203353 .00|736 || 1 33:29:29 192100033 24,020S243 8.325 1375 .001733 02 33-1()S4 1931 (10552 24.(H 163(16 8.32995-12 .00 | 73()|(}{ 33524 i 194 104530 24.06241SS S.3347 553 .001727 1 || 6 336400 1951 12000 24.0S3 ISQI 8.3395500 .001724 l 38 3.37561 1961 229.4| 21. I ().394 | 6 8,34434 0 .00|721 170 33S724 197l 37.36S 24, 1246762 8.349 || 256 .0017 | S213 3.39SS9 19S 1552S7 24. 1453929 8.3539()47 .00|7 || 5266 34 l ():56 199| 7 (3704 24. 1660919 8,35S67S4 .0017 2329 3.422:25 2002ſ) G25 24. 1S677.32 8.363-1466 .0017094()2 34.3396 201230056 24.2074369 S.36S.2095 .00l ſ()61S5 344569 202262003 24.22SOS 29 8,372966S . (101703578 345744 203297,472 24.24S7 || || 3 8.3777 ISS .00|7006SO 346921 204336469 24.269,3222 8.382-1653 .001697.793 34S100 205379000 24.2S09 156 S.3S72065 .001 604915 3492Sl 2064.2507 l 21.310-1916 8.39.19423 .00 | 6′)2()47 350.164 20747-16SS 24.33 l ()āUl 8.3966729 .001 GSūl S9 35l 6-19 2ſ)S527S57 24.35l 5013 8.40] 39S 1 .00 6S634 I 352S36 2005S-45S4 24, 372 || 52 S.406 || S() .0() (6S350.2 3.5-1(125 2|{}644S75 24, 3026.218 8.4 (YS326 .0(\ | 6S0672 3.352 | 6 21 17()S736 24,413 l l 12 8.4 1554 19 .00 | 677S52 356-1(19 21277617.3 24.43.35S34 8.4.202460 .00 l (37.50.42 3.576()4 2138.47 19:2 24.45403S5 8.42-1944S .001672241 35SS01 21-1921799 24.4744765 S.42963S3 .001069-149 360000 2I 6000000 24.401S974 8, 1343.267 .001 666667 36 | 201 21 70S l SOI 24.5l 53013 8.439000S .00 l 663S04 3624 ()–: 2|S| 672(\S 2.1.5350SS3 8.4436S77 .00 | 66 || 30 36.3609 219256227 24,556(15S3 8.44S3605 .00 165S375 364S16 22034 SS64 24,576-1 || 15 S,45302Sl .001 655629 366025 || 2:21:1451.25 24.5967.478 || 8.45769(16- . .001652S33 367236 2225-45016 24.617 (1673 8,4623.479 .00l 6501.65 36S449 22.364S5.13 24.637 3700 8, 467 (H001 .001647,446 339664 || 2:247557 | 2 || 24,657 (560 | 8.47 1647 i .001 6.14.737 370SS1 225S66529 24,6770.254 8.4702S92 .00164.2036 37:2100 2260Sl()()() 24.60S 17St 8.4S0926 l .00 || 630344 37.332 | 22S()00 t 31 2-1.7 | SH 142 S.4S55579 .00 | 636(561 3745.14 2:29:22); 2S 24.73S6339 8,490 ISAS .00 16330S7 375,760 23(1346,397 24,75SS30S S. 494 S()65 .0{}| 63| 321 376996 231475544 24.77.00:234 8.499.4233 .00 l 62S664 37S225 23260S375 24,700 1935 8.50.1035() .001626016 379456 23374-1S96 24. Sl 9,3473 8.50S64 l 7 .00 l (; 93377 3S06SQ 23.1SS5 || 3 24,S394S47 8.5 ! 3.2435 .0() 620746 3S 1924 236()20032 24, S596()5S 8.517S4(13 .0016 1SI 23 3S3 | 6 | 237 | 70659 24, S797 l ()6 8.522.4321 .0016 | 550.9 3S4-100 23S32S()()0 24.SS)97992 S. 52701S9 ,0016 123(#3 170 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. |Square Roots. Cube Roots. | Reciprocals. 621 3S5641 2394S3061 24.910S716 || 8.5316009 | .001610306 622 || 3S6SS4 240641S4S 24.9309278 8.536.1780 . .0016077 17 623 3SS129 24 ISU4367 24.9599679 || 8.540750l .001605.136 624 3S0376 || 24297.0624 24,9799920 | 8.5.453173 | .001602564 625 | 39.0625 244140625 || 25.0000000 | 8,549S797 | .001600000 626 39.1876 245314376 25.0199920 | 8.5544372 .001597.444 627 | 303129 || 24649 ISS3 || 25.03996S1 || 8.55S9899 || .00159-1896 628 3943S4 || 247673152 || 25.05992S2 8.5635377 | .001592357 629 || 3956.41 24SS581S9 || 25.079S724 || 8.5680S07 | .0015S9S25 630 || 396900 250047000 25.099S00S. 8.5726.189 .0015S7302 631 39S (61 || 25.1239591 || 25. I 197134 || 8.577 1523 | .0015847S6 632 309424 || 252435.96S 25, 1396.102 || 8.5816S09 | .0015S2278 633 4006S9 || 253636137 25, 1594913 8.5S62047 | .00157977 634 401956 || 254S40104 || 25, 1793566 8.590723S .0015772S7 635 | 403225 256047S75 25, 1992063 || 8.59523S0 | .001574S03 636 | 404-196 || 257259456 25.219ſ)4()4 || 8,5997.476 | .001572327 637 405769 25S474S53 25.23SS5S9 || 8,6042525 | .001560S59 63S | 407044 || 25969.4072 || 25.25S6619 || 8.60S7526 .001567398 639 40S321 || 260917119 || 25.2784493 | 8.6132480 | .001564945 640 | 409600 || 262I44000 || 25.29S2213 || 8.6177388 | .001562.500 641 || 4 |0SS1 || 263374721 || 25.3179778 || 8.6222248 .001560(162 642 || 412164 2646092SS 25.3377.189 || 8.6267063 | .001557.632 643 4 13449 265S47707 25.3574447 8.63| 1830 .001555210 644 || 4 || 4736 || 2670S99S4 || 25.377 IS51 | 8.6356551 | .001552795 645 || 416025 26S336125 || 25,396S502 || 8,6401226 .0015503SS 616 || 4 |73 || 6 || 2695S6136 || 25.4165301 || 8,6445S55 .0015479SS 647 || 41S609 || 270S40023 || 25.4361947 | 8.6490437 | .001545595 648 || 4 |9904 || 272007792 || 25.455S441 || 8,6534974 | .001543210 649 || 421.201 || 273359449 || 25.4754784 8.6579465 | .001540S32 65 422500 274625000 || 25.4950976 || 8.662391 || | .00153S462 65 423801 || 275S94451 || 25.5147016 || 8.6668310 .0015.3609S 652 || 425 (04 || 277 16780S 25.5342907 || 8.6712665 | .001533742 653 || 426409 || 27S445077 || 25.5538647 || 8,6756974 .0015.31394 654 || 4277.16 || 2797.26264 || 25.5734237 8.6S01237 | .001529052 655 || 4290.25 || 2SIOI 1375 25.5929678 || 8.6S45456 .001526718 656 || 4:30336 2S2300416 || 25.6124969 || 8,6889630 | .001524390 657 431649 || 2S3503393 || 25.6320] 12 || 8.6933759 | .001522070 658 432964 || 2S4S90312 || 25.6515107 || 8,697.7843 | .001519757 659 || 4342S1 || 2S6191179 || 25.6709953 || 8.7021882 .001517451 660 || 435600 2S7496000 || 25.6904652 | 8.7065S77 .001515152 661 436921 2SSS047S1 || 25.7099.203 || 8,71098.27 .001512S59 662 : 43S244 2901 17528 || 25.7293607 || 8.7153734 . .001510574 663 || 439569 || 201434.247 || 25.7487864 8.7 197590 .00150S296 664 || 440S96 || 292754014 || 25.7681975 | 8.724.1414 | .001506024 665 || 442225 | 29.4079625 || 25,7875939 8.72S5.187 | .0015037.59 666 || 443556 205408296 || 25.8069758 || 8.732S918 .00150.1502 667 || 444S39 296740963 || 25.826.3431 || 8.7372604 || .001499250 668 || 446.224 | 20S077632 25.8456960 8.74 lb.946 .001497.006 669 447561 2994 18309 25.8650343 8.7459S46 .001.494.76S 670 44SQ00 || 3:00763000 || 25.884.3582 8.750340l .001402537 671 || 450241 3021 | 17 | 1 || 25.9036677 || 8.7546913 | .001.4903.13 672 4515S4 303-164448 || 25,9229628 || 8.7590383 . .0014SS095 673 || 452029 || 304S21217 || 25.942.2435 | 8.7633S09 | .001485SS4 674 454276 306 IS2024 25.9615100 || 8.7677192 .0014S3680 675 455625 || 307546S75 || 25.9807621 | 8.7720532 .001481481 676 || 456976 30S915776 26.0000000 || 8.7763S30 .001.479290 677 45S320 3UO2SS733 26.0192237 || 8,78070S4 .001.477 105 678 || 45.96S4 || 3 | 1665752 || 26.0384331 8,7850296 .001.474926 879 || 46104 || || 313046S39 26.0576.284 || 8.7893466 .001.472754 680 462400 || 314432000 || 26.076S096 || 8.7936593 .001.470588 68l 463761 || 315S21241 26.0959767 8.7979679 .00146S429 682 46.5124 || 317214568 l 26.1 151297 || 8.8022721 | .001466.276 CUBE ROOTS, AND RECIPROCALS. 171 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 683 466489 31861 1987 26, 13426S7 8.80657 .001464129 6S4 467856 320013504 26, 1533937 8.S10S6SI .00146 1988 685 4692.25 321419 125 20. 17250.47 8. St 5 [598 .001.459854 6S6 470596 322S2SS56 26. 1916017 8.S194474 .001 457726 687 47 1969 3242,42703 26.2106S4S 8.823.7307 .00 1455604 6SS 473344 32566067.2 26.2297541 8.82S0099 .001453-4S8 6S9 474721 3270S2769 26.24SS095 8.8322S50 .001451379 690 476100 32S509000 26.267S51 1 8.8365559 .001449275 69i 4774S1 3299.39371 26.2S6S7S9 8.84()S227 .001 44717S 692 47SS64 331373SSS 26.305S929 8.8450S54 .001 4450S7 693 4S0249 3.328.12557 26.324S932 8. S493-440 .001 443001 694 4S1636 3342553S4 26.343S797 8.8535985 .0014.40922 695 4S3025 335702375 26.362S527 8.857 S4S9 .00143SS49 696 4S44 16 337 1535.36 26.3S181 9 8,8620952 .0014367S2 697 4S5S09 33S60SS73 26.4007576 8.S663375 .00 l 434.720 69S 4S720.4 340()(3S392 26.4.196S96 8.S705757 .001 432665 699 4SS601 341532099 26,4386ſ)Sl 8.S74SO99 .00 143(1615 700 490000 34.3000000 26,457.51.31 8, S790.400 .001 42S571 701 491401 3.14-172101 26.47.64046 8. SS32661 .001426534 702 492S()4 34594840S 26.4952S26 8. SS74SS2 .001 q24.501 7()3 49-12ſ).9 3:47,12S927 26.514 1472 8.8917063 .00 [42:2475 7()4 4956] 6 34S913664 26,53299S3 S.S959204 .001420-455 705 4970.25 350402625 26,551 S36 I 8,900 1304 .001418440 706 49S436 351 S95S16 26.57(6605 8,9043366 ,00I 3 16431 707 490S49 353393243 26.5S0-1716 8.90853S7 .001414427 70S 50 1264 35-4S9-1912 26.60S2694 8.9127369 ,001412429 709 5026SI 356400829 || 26.6270539 8.91.6931 1 .001.4 l'O437 710 504 100 3579] 1000 26,645S252 8,921 1214 ,00140S451 7| | 505521 359.425.431 26.66458.33 8,925.307S .001406-470 712 506944 360944 12S 26.6S332S] 8,929.4902 .001 q0-1494 713 50S369 362467097 26,702059.S 8.93.36687 .001 q02525 714 509796 36399.4344 20.72ſ)7 7 S4 8.937S433 ,001 400560 715 51 1225 36552.5S75 26.7394S39 8,9420140 .00139860] 716 5 12656 367061696 26.75S] 763 8,946 1SO9 .00139664S 717 5140S9 36S60IS13 26.776S557 8.950343S .001394700 71S 515524 3701 46232 26,795.5220 S.95450.29 00139275S 719 51696.1 37169-1959 26. S14.175. 8.95S6581 ,001390S21 720 518,400 37324S000 26.S32S] 57 8.962S095 .0(1138SSS9 721 51.9S41 374S()5361 2i}.S51-1432 8,966.957 .0013S6963 722 52 [2S4 376.367048 26, S700577 8.97 l l ()07 ,0ſ) 13:S5042 723 52.2729 37.793.3067 26. SSS6593 S.975.2406 ,00l 3S3] 26 724 524 176 37950.3424 26,90724S1 8.97.93766 .0{}l 3S1215 725 5256.25 3SI ()7S125 26.925S240 S.9S350SS) .001379.310 726 52.7076 3S2657l 7 26,944.3872 S.9876.373 .0013774 10 727 52S529 3S4240583 26.96.29375 S.99 7620 .001 3755I6 72S 5299.94 3S5S2S352 26.98|4751 8,995SS29 ,0ſ) 1373626 729 531441 3874.204S9 || 27.0000000 9.0000000 | .001371742 730 532900 3S001 7000 27.01S5122 9,004 II.34 .001.369S63 731 53.4.361 3906.17S91 27.03701 17 9.00S22:29 .001.3679SQ 732 53.5S24 3922.231 68 27,05549S5 9.01.232SS ,00|306120 733 5372S9 393S32S37 27.0739727 9.0164309 .001.364.256 734 53S756 395.446904 27,0924.344 9,0205293 .00|36239S 735 540225 39.7065.375 27, 110SS34 9,0246239 .001 360544 736 541696 39S6SS256 27, 12931.99 9.02S7 149 .00) 35S$96 737 5-13169 4003|| 5553 27.1477439 9,032S021 .001356S52 73S 54,4644 401947272 27, 1661554 9.036SS57 .001 355014 739 546 121 40.35S3419 27, 1845544 9.0.409655 .001353180 740 5,476.00 405224000 27,2029.410 9,045041'ſ .001351351 74.1 5490Sl 406S69021 27,221.3152 9,049 | 1.42 .001349528 742 550564 40S5|S4SS 27,239.6769 9,053.1831 .001347709 743 55.2049 4 10172407 27.25S0.263 9,05724S2 ,001345S95 744 553536 41 1S307S4 27.2763634 9.0613098 | .0013440S6 172 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. | Cube Roots. | Reciprocals. 745 5550.25 41349,3625 27.2046881 9.065.3677 .0013422S2 746 5565.16 4 || 5160936 27.31300ſ)6 9,0694.220 .00 i 34(k483 747 55S009 416S32723 27.3313007 9,073.1726 .00 || 333688 748 559.504 4 IS50$992 27.3495SS7 9.0775 | 97 .00!336S98 749 561001 4201S9749 27.367S644 9.0S 15631 .001.335113 750 562500 421S75000 27.3SG1279 9.0S56030 .001.333333 751 56.4001 423:56:4751 27.40437.92 9. (S06392 .00|33 || 55S 752 565504 42525900S 27.4226 [S4 9.09367 || 9 .00 1329.787 753 567009 426957777 27.440S455 9.0977010 .001 328021 754 56S516 42S66 1064 27.459(1604 9. 1017265 .00 1326260 755 570025 43036SS75 27.477.2633 9. 1057.4S5 .00 1324503 756 57 [536 4320S 1216 27.4954542 9. 1097 (569 .00 1322751 757 57.3049 43379S093 27.5 ! 36330 9. 1 137SIS .00 l 32 1004 75S 574564 43551951.2 27.531799S 9, I 1779.31 .001319.261 7.59 576OSI 4372.45479 27.5499546 9, 121 S010 .001317523 760 57.7600 43S976000 27.56SO975 9. 125S0.53 .0013157S9 761 579 (21 4407 l l ()SI 27.5S622S4 9. 120S06 | .00 31 4060 762 5S0644 4424,5072S 27.60.13475 9. 133S0.34 .001312336 763 5S2I 69 4-1-1 19-1947 27.62.245-16 9. 137797 I .00131 0616 764 583696 44594.3744 27.6-105499 9. [4 || 7874 .001 30S901 765 585,225 447697 125 27.65SG3:34 9. 1457.742 .0013(17 | 90 766 5S6756 449.455(196 27.67670.50 9. 1497.576 .001.3054S3 767 5SS2S9 4512 17663 27.69476-18 9. 1537.375 .00 3037Sl 76S 5S9824 4529S4S32 27.7 | 2S129 9. 1577 139 .00 || 3020S3 769 591.361 454756609 27.73OS402 9. 1616S69 .00l 300390 770 592900 4565.33000 27,74SS739 9. 1656565 .00129S701 771 59444 1 4583140 | 1 27.766SSGS 9. 1696.225 .0012970) 7 72 5959S4 46009964S 27.7S4SSSO 9, 1735S52 .00 (205337 773 597.529 46 18899.17 27.802S775 9, 1775.445 .00] 293661 774 5990.76 4636S4S24 27.820S555 9, 1815()()3 .00 120 1990 775 600625 4654S4375 27.83.SS218 9. 1854527 .001%29()323 776 60217.6 4672SS576 27.8567766 9, 18940.18 .00] 2SS660 777 603729 469ſ)97433 27,8747 IQ7 9, 1933.474 .00I 2S700l 778 6052S4 4709 10952 27,8926514 9. 1972S97 .0012S53-47 779 606S41 4727291.39 27,91057 15 9, 20122S6 .00] 2S3697 780 60S400 47455.2000 27,92S4S0] 9.20516-11 .001282051 7Sl 609961 476,37954 | 27.946.3772 9, 2090962 .001%2S()410 782 61 1524 47S2|| 76S 27.9342629 9,2130250 .0!), 27S772 783 6|| 30S9 4SO(\{S637 27.9S21372 9, 2I 69505 .001:277 139 784 614656 48.1890304 28.0000000 9.22()S726 .0() 12755 10 785 616225 4S37.36625 28.01785.15 9.2247914 .00] 273SS5 786 6||7796 4S55S7656 28.0356915 9, 22S706S .001:272265 787 619369 4S7443-103 28.0535203 9,2326 189 .001:270648 788 6209.14 489303S72 28,0713377 9,2365277 .00] 269036 789 622521 491 169069 2S.0891438 9.240.4333 .001.267.427 790 624 100 493039000 2S. 10693S6 9,2443.355 ,001.265S23 79 | 62.56S 1 4949 13671 28. 1247222 9.2482344 .00 126-1223 792 627264 49679.30SS 28. 1424.946 9.252|300 .00 1262626 793 62SS49 40S677257 2S. 602557 9.256()224 .00] 261034 794 6.304:36 500566 | S4 28. 17SO()56 9.2590 | 1.4 .001 259,446 79.5 63.2025 5.02450S75 28, 1957.444 9.2637973 .00 1257 S62 796 6336|| 6 50.135S336 28,2134720 9.267679S .001.256281 797 635209 50626.1573 28.23 ( 1884 9.27 15592 .00|254705 79S 636 S04 508169592 2S.24.SS93S 9.275.4352 .00I 25.3133 799 63S401 5100S2399 2S.2665SSI 9.27930S1 .001251564 S00 640000 512000000 2S.2S42712 9.2S3] 777 .001:250000 80| 64 1601 51392.2401 28.3019.134 9.287()440 .00] 248.439 802 64.3.204 515S4960S 28.3 1960.45 9, 2009()72 .001%246SS3 803 6-14809 5177S1627 28.3372546 9.2047671 ,00I 2.45330 804 646 116 5.197 18:464 28.354S938 9.29S6239 .0012-13781 805 648025 5216601.25 2S,3725.219 9,302.4775 .001:242236 806 649636 52.3606616 28.390 1391 9.3063278 .0012.40695 CUBE ROOTS, AND RECIPROCALS. 173 Reciprocals. No. Squares. Cubes. Square Roots. Cube Roots. 807 651249 525557943 28.4077.454 9.3|0|750 .001.239157 808 6:52S64 5275 || 4 || 2 || 28.4253.40S 9.3| 40.190 | .00I 237624 809 654481 529,475129 2S.4420253 9.317S599 .0012.36094 810 656100 531441000 || 2S.460-1989 9.3216075 .001234568 8|| 657721 5334 l 1731 2S,47S06.17 9,325.5320 .001.233046 812 659,344 53.53S7328 28,4956 |37 9.3293634 .0(11231527 813 660969 537.367.797 2S.5131549 9.33,31916 .00] 230012 814 662,596 539353144 || 2S.5.306S52 9.337.0167 .00I 2.28501 815 66.1225 541343375 28.54S204S 9.340S3S6 .001 226994 8| 6 66.3S56 543.338-196 2S,5657 [37 9.3446575 .001.225.490 817 674 S9 5.4533S.513 || 28.5S321 19 9.3484731 .00 12.23990 SIS 669| 24 547343432 || 28,6006993 9.3522S57 .001222.494 S19 67.0761 549353.259 || 2S 6181760 9.3560952 | .001.221001 820 672400 55] 3GS000 || 2S, 63.56 [2] 9.3599016 | .0012.1951.2 821 674ſ)4 | 5533S766 | 2S. 6530976 9.36370-19 .0012 IS(127 S22 67.56S-1 5554 12248 || 2S,670.5-124 9,3675051 .001 21 65.45 S23 077329 55744 i767 || 2S.6S79766 9.37 13022 .001 215067 S24 67S976 559.176224 2S. 70.54002 9,3750963 .00 1213592 S25 6S()(32.5 561515625 | 2S. 722S132 9.378SS7 .001 21212: S.26 6S2.276 50.3.559976 2S,740.2157 9.3S2(3752 .00l 21(Yô54 827 6S3929 56.5600253 2S,7576077 9.3S64600 .001209190 S2S 6S55S-1 5676635.52 2S,7749SQl 9,3902.] [9 .00I 2077.29 829 GS7241 56.97227S9 2S,792.3601 9.394()200 .001206273 830 6SS000 57l 7S7000 || 2S.S097,206 9.3977964 .0{}l 204S19 Sº I 69(1561 573S56 [0] 2S.S.270706 9.40l 5691 .001 203369 8.32 6922.24 57593036S 2S.S.444 102 9.40533S7 .00I 201923 8.33 693SSS) 57S0)}537 2S.S6 | 7394 9.409 || 054 .001 2004S0 834 69.5556 5S0003704 2S.S7905S2 9.4 12S69ſ) ,00l 199041 S35 697.225 5S2IS2S75 | 2S.S963666 9.4|| 66.297 .00 | 197605 836 60SS96 5S-1277,056 2S.9136646 9.420.3S7 ,00l 1961.72 $37 700.569 5Sö.3762.53 2S. 9300523 9,424 420 .00l 1947.43 83S 702244 5SS-SO472 2S,94S2.297 9,427 S936 .001 193317 839 703921 590.589719 || 2S.9654967 9.4310-123 .001 191S95 840 70,5600 50270 1000 2S.9S27.535 9,4353SSO .001 190476 S4 | 7072Sl 59 |S233:21 20.000000ſ) 9.430 1307 ,001 || S9061 842 70S964 596947.638 || 29.017236.3 9.442S704 .001 IS764S 843 7|(}649 500077 107 29.03446.23 9,4466072 .00 l l S6240 S44 7 23.36 6() 121 1584 29.05167SI 9,450.34 l () .001 IS-1S34 845 71.4025 | 60335||125 | 29,06SSS37 9,45407 19 | .00 l l S34.32 S46 7 157 || 6 6)5-195736 29.0SGſ)79 | 9,457.7999 .001 || S3033 847 7] 7409 64)7645-123 20, 10326-14 9,4615249 .00 l l S0638 S4S 7| 9 |(}{ 500SQQ192 29, 1204396 9,4652470 .001 1792.45 849 720SUI 61 1960049 29. 13760-16 9.46S9661 .001 177S56 S50 722500 614 125000 || 29, 1547,595 9.4726S24 .00117647.1 851 724.201 6 6295()51 20. 1710ſ);3 9,47639:57 .00l 1750SS 852 7259ſ)4 6|S.47020S 29, lS90390 9.4S01061 .001 173709 S53 727.609 620650477 29,2061637 9.4S3S136 ,001 172333 85. 729.316 622S35S64 20.22327 S4 9.4S751S2 ,00l 170960 855 73 l ().25 625026375 29.2103S3) 9,4912200 .00l 1695.91 856 73:27.36 627222016 || 29.257.4777 9.4949 ISS .001 || 6′S224 857 734449 620-1227.93 29.27-156.23 9.49S6] 17 .00 | | 66S61 85S 736 | 64 63 (62S7 12 29, 2016:370 9.5023078 .001 || 65501 859 737SS1 63,3339779 29.30S701S 9.5059980 | .001164144 860 739600 6360.56000 || 29.3257566 9.5096S54 .00I (627.91 SGl 74 1321 63S2773SI 29.342SO 15 9,5133699 || .001 16144() S62 743() {4 6:10503028 29.359S365 9.5170515 . .00 | 160093 863 7.44769 6.42735647 29,376S616 9.5207303 .00 l l 5S749 $64 746-196 64497.25:14 29.393S769 9,5244063 .00l 157407 865 74S225 647.214625 29.410SS23 9.52S0794 .00l 156069 866 7499.56 64946 1S96 || 29.427S779 9,5317,497 .001 154734 867 7516S9 651714363 29.444S637 9,5354172 . .00l 153403 86S 7534.24 65397.2032 29.461S397 9,5390Sl8 ' .001152074 174 TABLE XI. SQUARES, CUBEs, square Roots, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. | 869 755161 656.234909 29.4788059 9.5427437 .001 150748 870 756900 65S503000 || 29.4057624 9.5464027 . .001149.425 871 75Sö4 | 660776311 29.5127091 9.5500:589 | .001 || 4S106 72 76ſ)3S4 66.3054S48 || 29.5296461 9.5537.123 .001 [467S9 873 762129 £6533S$17 | 29.54657 9.5573630 | .00) 145-175 87. 7(53S76 6676.27624 29,5634910 9.561010S | .0011441.65 875 7656.25 669921875 29.5S03989 9.56.46559 | .001 || 42S57 87 767376 672221376 29.597297.2 9.56S2982 .001 || 41553 S77 769 129 674526133 || 29,6141858 9.57 19377 .001140251 878 770SS4 676S36152 29.63.10648 9.5755745 .001 138952 879 77.2641 67915.1439 || 29.6479342 9.57920S5 | .001 137656 880 77.4400 681472000 || 29,6647939 9.5S28397 .001136364 8SI 77616] 6S3797841 || 29.68164.42 9.5S646S2 .00) 135074 8$2 77792.4 6S612S968 ; 29.69S4S4S 9.5900.939 || .00I 133787 S$3 7796S9 68S4653S7 29.7.153159 9.5937169 .001132503 884 || 781.456 690S07 104 || 29.7321.375 9.5973373 .001131222 885 7S3225 693 l 54.125 | 29.74894.96 9.600.9548 .001 129944 886 78-1996 695506456 29,7657521 9.6045696 | .001 12S668 7 786769 697864 103 || 29.7S25-452 9.60S1817 | .001 127396 888 7885-44 700227072 || 29,79032S9 9.6l 1791 l .001 I26126 8S9 790321 702595,369 || 29.8161030 9.6153977 | .001 124S59 890 792100 704969000 || 29,832S678 9.6190017 | .001 123596 891 793SSI 707347971 || 29,849.6231 9.6226030 | .001 122334 892 795664 7097.322SS 29.8663690 9.626.2016 | .001 121076 893 797449 7 1212 1957 29.8S31056 9.6297975 | .001 | 19821 894 790236 7I 45 169S4 || 29.S39S32S 9.6333907 .001 | | S568 895 S01025 7169,17375 || 29,9165506 9.6369S12 .001117318 896 802S16 7 19323136 29,9332591 9.6405690 .001 | 16071 897 804.609 72.1734273 29,9499.583 9,644 542 | .001 I 14S27 898 806.404 724 (50792 || 29.9666481 9.6477367 .001 II 35S6 899 80S201 726572699 || 29,98332S7 9.6513166 .001 | 12347 900 810000 729000000 || 30.0000000 9.654893S | .001 II l l l l 901 S! ISOl 731432701 || 30.01666.20 9.65846S4 .00I 100S78 902 813604 733S70SOS 30,0333148 9.6620403 .00110S647 903 815409 736314327 || 30,04905S4 9.6656096 .001 107420 904 817216 73S763264 30.0665928 9.669.1762 .001 106195 905 819025 74 1217625 30.0832] 79 9.6727403 | .001 |(}1972 906 820S36 7436.77416 30.0998339 9.67630|7 | .001 || 03753 907 8226-19 746 (42643 30. 1 164-107 9.679S604 || .001 102536 908 824464 74S6133 12 30, 13303S3 9.683-1166 .001 101322 909 826281 7510S9429 || 30, 1496.269 9.6S69701 | .001 100110 010 82S100 75.357 1000 || 30, 1662063 9,69052] 1 | .00109SQ01 9 || 82992.1 75605S03 || || 30, 1827765 9.69.40694 | .001097695 912 831744 75S55052S 30, 1993377 9.697.6151 .001096491 913 8.33569 76104S497 30.215SS99 9.701 1583 | .001095290 914 S35396 763551944 || 30.2324329 9,7046989 . .001003092 9 || 5 S372.25 766060S75 || 30.24S9669 9.70S2369 | .00) 092S06 9| 6 8390.56 76S575296 || 30.2654919 9.71 17723 .001091703 9 |7 840SS9 77 10952| 3 || 30.2820079 9.7.153051 .001090513 9|S 84.2724 773620632 || 30.29S514S 9,7188354 | .0010S9325 919 844561 776151559 || 30.315012S 9.7223631 | .0010SS139 920 846.400 77S6SS000 30.3315018 9,725SSS3 | .0010S6: 57 921 84S241 7S 1229.961 30.3479S18 9,729.4109 .0010857.7 922 8500S4 783777448 || 30.3644529 9,7329300 .00108.4599 923 85.1929 786330.467 || 30.3S0915I 9,73644S4 .0010S3424 924 85.3776 7888S9024 || 30.397.3683 9.7399634 .001082251 925 855625 791.453125 || 30.413S127 9,743475S .001 (S10Sl 926 S57476 79.1022776 30.4302481 9.746,9857 .001079914 927 8593.20 7965979S3 30,44667.47 9.7504930 .00107S749 92S 86] 1 S4 7991.78752 30,4630924 9.7539979 .001077586 929 S63041 80.1765089 30,47950.13 9.7575002 | .001076426 93ſ) 86-1900 804357000 || 30.4959014 9.7610001 | .001075.269 & Fºr CUBE ROOTS, AND RECIPROCALS. 17 5 No. Squares. Cubes. Square Roots. Cube Roots, Reciprocals. 931 S66761 80695449 I 30.5122926 9,7644974 .0010741 14 932 86S624 8095,5756S 30.52S6750 9.7679922 .001072961 933 S70489 812166237 || 30,54504.87 9.7714S45 | .0010718II 934 872356 814780504 || 30.5614136 9.7749743 | .001070664 935 874225 817400375 || 30.5777697 9.77S46 || 6 | .001069519 936 87.6096 820ſ)25S56 30,594 || 171 9.7819.466 .001068376 937 877969 822656953 30,610:4557 9.78542SS .001067236 938 8798.44 825293672 30.6267S57 || 9.7S890S7 .00I 06609S 939 881721 827.936019 i 30.6431069 9.7.923S61 .001064963 940 SS3600 8305S4000 || 30,6594.194 9.795S6II .001063830 94 | 885,481 833237621 30,6757233 9.799.3336 .001062699 9. 12 SS7364 835S96SSS | 30.69201S5 9.802SO36 | .00105 || 57.1 943 8S9249 83S561807 30,70S305} 9.S0627 | 1 .00I 06(k445 944 891 136 84 12323S4 || 30.7245S30 9.SO97362 | .001059322 945 89.3025 84390S625 30,740S523 9. S13 1989 | .00105S201 946 89.4916 846590536 30,757 | 130 9.8 166591 .0010570S2 947 896S09 8492.78123 || 30,773365] 9 S201 169 .00105.5966 948 89S704 85.1971392 || 30,78960S6 9.8235723 | .00105.4852 949 900601 854670.349 30.805S436 9.827.0252 | .001053741 950 902500 857.375000 30,8220700 || 9.8304757 .00105.2632 951 90.4401 860085351 30,8382879 9.S33923S .001051525 952 906304 86.280.1408 || 30,8544972 9.837.3695 .001050120 953 90S209 865523177 || 30.87069Sl 9,840S127 | .001 01931S 954 9101 16 86S250664 30.8S6S904 9.S4.42536 | .0010:48.218 955 9 12025 8709S3875 30.9030743 9,8476920 .001047120 956 913936 873722816 30.9192497 9.851 12SO .001 046025 957 915849 876467.493 30,9354166 9,8545617 | .0010-14932 958 9.17764 879217912 30.9515751 9.85799.29 .001043S4 I 959 91.96S1 88.1974079 || 30.967725] 9.85.14218 .00104.2753 960 92I 600 8S4736000 || 30.9S3S66S 9.S6484S3 .001.04 1667 961 23.521 8875036Sl 31,0000000 9.S6S2724 . .001 (){[]5S3 962 925.444 89027712S 31.0161248 9.S716941 .001039,501 96.3 927,369 89.3056347 || 31,0322,413 9,875l 135 | .00|03S422 96-4 929.296 895S41344 || 31.04S3494 9.8785.305 | .001 0373.14 965 93.1225 89S632125 || 31.0644491 9.SS19451 .00) 036269 966 933) 56 90142S696 || 31,0805.405 9.SS53574 .001 0351.97 967 9350SQ 904231()63 || 31.0966:236 9.8887673 | .001034126 96.3 937024 907039232 31. I 1269S4 9.892.1749 .001 033).5S 969 93S961 909853209 || 31, 12S764S 9.8955S01 | .00103.1992 97. 940900 91267.3000 || 31, 144S230 9.89S9S30 .00103092S 97 | 942841 915.4986 II || 31, 160S729 9.9023S35 | .001020S66 72 944784 91S33004S | 31. 17691.45 9,9057817 | .00102SS07 973 946729 921167317 || 31, 1929.479 9,909 1776 .0010277.49 97.4 94S676 9240.10424 || 3 | .208973] 9.9125712 .001026694 975 950625 926S59375 31.2249000 9.91596.24 .00102564}. 97 95.2576 9297 l 4176 || 31,24099.S7 9,91935.13 .001024590 977 95.4529 9325748.33 || 3 | .256.9992 9,9227.379 .001023541 978 9564S4 935441352 || 31.2729915 9,926.12.22 .001 02:2495 979 9584:41 93S3137.39 || 31.2SS9757 9.92950.42 .001021450 9S0 960400 941 192000 31,3049517 9,932SS39 .00102040S 981 962.361 944076141 || 31,320.9.195 9.936.2613 .001 01936S 9S2 96432.4 94.696616S | 31.336S792 9.93.96363 .00101833ſ) 983 966.2S9 9498620S7 || 31.352S30S 9.94.300.92 | .001017294 9S4 96S256. 952763904 31.36S7743 9,946.3797 ,0(), () l 626() 985 97.02.25 955671625 31.3S47097 9,9497.479 .00) () 15:22s 9S6 97.2196 95S5S5256 || 31.4006.369 9,953 I lºS .001014 | 99 9S7 974 169 96.1504S03 || 31,4165561 9,956-177 .001013 || 7 | 9.SS 97.6144 954,430272 31.4324673 9,959S3S9 .0010121.46 989 97812! 96736,1669 || 31,4483704 9.9631.9SI .00101 | 122 990 980.100 97.0299000 || 31.4642854 9.9665549 .001010101 991 9S2081 97.3242271 || 31.4S01525 9,96990.95 .001009082 992 984064 9761914SS 31,496.0315 .001008065 9,9732619 176 TABLE XI. SQUARES, CUBES, &C, No. 8quares. Cubes. Square Roots. Cube Roots. Reciprocals, 993 9S6049 979146657 31.51 19025 9.9766120 .001007049 994 988036 9S2107784 31,5277655 9.9799599 .00 l (106()36 995 990025 9S5074875 31.5436206 9.9S33055 .0010050.25 996 99.2016 9.SS047936 31.5594677 9.9S664SS .001 004(116 997 99.4009 99.1026973 31,575306S 9,9899900 .00 [{j03009 90S 996004 99.401 1992 31.59113S0 9.99.332S9 .001002004 999 99S001 997.002999 31.6063613 9.9966656 .001001001 1000 1000000 1000000000 31,6227766 10,0000000 .001 000000 1001 1002001 100300300! 31.63S5S40 10,ſ)033,322 .00099900] 0 1002 1004004 10060.1200S 31.6543S36 10.006(3622 .0009980ſ)4() 1003 1006009 100.9027027 31.670 1752 10.0099899 .00(1997U(190 1004 100SO16 10 12048064 31.6S59590 10.0) 33|55 .0009960159 1005 101.0025 1015075,125 31.7017349 10,01663S9 .0009.950249 1006 101.2036 101810SX 16 31,7175030 10.010960| .000094()358 1007 1014049 1021 147343 31.7332633 10,02327.91 .000093ſ)4S7 100S 1.016064 1024 192512 3I.74901 57 10,026595S .000992(J635 1009 101SOSI 102/2437.29 31.7647603 10.0299.104 .00099.10S03 1010 1020.100 1030301000 31.7804972 10.033222S .0009900990 101 || 102.2121 1033364331 31,7962262 I 0.0365.330 .00098.91 197 10] 2 1024144 1036,433728 31.8l 19474 10,039S410 .0009.SSI 123 1013 1026169 1039509197 31.8276.609 } 0.043 l 469 .0009S7166S 1014 10281.96 104259.0744 31, S433666 10.0464506 | .0000S61933 1015 1030225 1045678.375 31.8590646 10.0497.521 .00098522] 7 1016 1032256 I04S772096 31.87.475-49 10,0530514 .00098.42520 1017 10342S9 1051871913 31.890.4374 10.05634S5 .0009S32S-12 101S 1036324 1054977S32 31.906] 123 10,0596435 .0009S23] S3 1019 || 1038361 105S089S59 31.9217794 10.0629364 .0009S13543 1020 1040400 I06120S000 31 93743SS 10,0662271 .0009SO3022 102I }042441 106.1332261 3 (.953ſ)006 10,0695) 56 .000979-1319 1022 104.44S4 1067462648 31.96S7347 10.072SO20 .00097 S4736 1023 1046529 1070599,167 31.9S 13712 10,0760S63 .0009.77517 | 1024 104S57 10737.41S24 32.000(1000 10,07936S4 .00(197656.25 |(}25 1050625 107GS90625 32.() I 562.[2 l().0S264S4 .00097.3609S 1026 105.2676 I ()S0045576 32.03 ſ 23:18 10,0859262 .00097.465S9 I ()27 1054729 IOS32066S3 32. (H6S407 10,0S92019 .00097.3709S 1028 10567S4 10S6373952 32.06243) 1 10,0924.755 .00(19727.626 1029 105SS4I 10S9547389 32.07S029S 10.0957.469 .00097 1817.3 J030 1060000 1092727000 32.0036131 10,0990] 63 .000070S738 1031 1062961 100591.2791 32. I (191SS7 I (). 1022S35 ,000.969932! 1032 1065024 1099 || 0476S 32. 124756S 10. 1 ()554S7 .000.96S9922 1033 10670S9 1 102.302037 32. I 403 | 73 I (). 10.SSI 7 .00096S05-12 1034 1069156 1 1055073()4 32, 155S704 10. I 207:26 .000967 | | SO 1035 I07 1225 I 10S7 17875 32. 17|4|59 10. l l 53314 .000966l S36 1036 1073296 11 i 1034656 32. 1S605.30 10. 1 1858S2 .000.96525 || 0 1037 1075369 1 | 151576.53 32.2024844 I (). 12|S42S .0009643202 103S 1077.444 Il 183S6S72 32.21 S007.4 10. 1250953 .00096.339|| 1 1039 1079521 1121622319 32.2335:229 10. 1283457 .00096,24639 1040 1081600 1124864000 32.2490310 10. 1315941 .00006153S5 1041 10836SI I 12SI 11921 32.2645.316 10, 1348,403 .00006061.48 1042 1085764 11313660SS 32.280024S 10. 13S0S45 .0(109596929 I (k43 }(|S7849 I 1346.26507 32,2955 105 10, 14 13.266 ()0005S772S 1014 10S9936 I 1378.93| S4 32.3 l (1988S 10, 1445667 .000957S544 ] ()45 1002025 | 1.4 l l (361.25 32,326459S I (). 147S()47 .0000560378 1046 1094 || 6 I 14-1-1453.36 32.34 19233 10, 15|{}106 .00(1956()229 I () 17 10962(19 | | ||7730S23 32.357.3794 10, 15.127-1-1 .()()()055 (){}S 1()4S I ()0S3{}4 1 151()22592 32,372S2Sl 10, 1575(162 .0(){}}}}} .4276] : i º é 33 .437462 #. j; ſº | #; ; jº, % * 4 32156 || ". ‘. . . . .06 36962 | "...,' .56303S 25 33 | .4 * --> 77, 40 999.S38 * .43696 76.63 55S440 || 2: 31 436:00 76.5S 355S$4 ,06 .44 1560 75, S3 º 24 ;| #| || | #| | | # § j . . * | *; ; j . . #;| ≤ | #; # ..., | .43%;" | };}} | .º. g ,45507 .53 || “...º. & ; : ºf j| | | }. § | ##|zi * * : 9,9998 16 8.43.42 | 72.06 S2S 19 8 463665 ** .06 - & Ame .531S 4G - º 7.2.00 909S13 468! 72 71.35 5275-46 18 # | }}; i3 j| ºff ; 70.66 º! 17 3 || 473263 º .999809 ºf 76693 º .523307 ; .47615s ...}} .9998.05 || 0: †: ; 5.1910S 16 t •º - 99S01 e •º: 69,31 e 950 15 44 .4S0693 69,24 .9995 .06 485050 *: 51495 - * Tº Al Jº * Q9797 6S.65 510S30 | 1.4 43 || 484.4; §§ 9337 # isºſ | }; is: 13 lf; 4SS963 67.93 .999.94 .07 49.3250 ~ : S .506750 A. * 93()40 ... . .333799 || 07 º; 3.33 | tº 13 #| #|##|##| || ºff #|##|| #9 5010S0 | }. º: * Isº º t;| || | | *; g543 |*; # | jºb | #; º; ; 51 .9US$4 64.89 - §§763 e 51300S Šišš ºn pººr 33 || 512S3. § .333.39 o? 51696.1 | ..." .4S3039 || 7 * - 65 *} * 63, S2 *y 6 33 .513726 § .999. 97 | jº) .479210 || 6 as p- *.x s I # * 63.26 f. {} #| # ºf j | dº | j 3. .475414 § .524343 36; .299757 | y, §§§ $3.73 || 3:13; 4 -**_ * 53 sº 62, 18 7920 ! 3 § 523193 || 3 || || 939.53 || 3% 532(YSO .467920 ** *. *† = sy 61.65 4) Y 2 ; j| is j| | | {:}; .464221 ;| #| || 5 | ##| | | }; § | 3; h ; j| Šoš jº; .07 | jº * .4569 16 60 54281 - * th Ta M -* D. 1". Cotang. D. 1". | Tang. - ine. D. l'. Sine. M. Cog 880 0 1 2 198 TABLE XIII. LOGAR1THMIC SINES, 2O 1776 M. Sine. D. 1". Cosine. D. 1". I Tang. D. 1". Cotang. M 0 || 8.542S 19 9.999735 8.5430S4 1.45996 || 60 i || 3:2 . . . ºf | | | ºf #| 3; jº 2 .540995 | . .9997.26 ºš £3% º #3 3 .553539 59.06 ( * J. .0S tº ºº 59. 14 * ** Tº = 5S 5S .9997 22 OS .553S 17 5S 66 .4461S3 57 4 5; ; ºf # 5:3:6 | #. ,442664 56 § ºf ; jºi: | # j| ##| 4:3: 55 § #3; ; gº || 3 | jºi| ##| 3: ||. jºi| #}}| ſº | }| ſº | ##| 3:23 53 #| #| ##| #| | | #| ##| #|; 10 | 8.577566 | ..., | 9.9996S9 || 3 || 8.577877 º, 1422 50 ii is sº | #. j| 3 || "...sº | #| “... ;|| i: | #3 | ##| |º # sisti | #}| 4:455 | is }} | {...}} | {{..} | ...; ºš I ºff ; 3iºn; : # º! | #3 .9999.9 || 3 || 59.1031 || 3.3% .438949 || 45 !; .333333 || 3:... . .99955; . . . .54383 . . .405717 || 45 i; ºf: #| ºff ; ſº | #| Tº |3: i: | dº | #| º | | | Sº #% ºš |3: #| #| #| #| | | #| #| : 19 .606623 gig. .999645 || 3 | .606978 || || | .393022 || Al 20 | 8.609734 a 9.990640 8.6 100.94 |A | 1.3S0906 | 40 2 ºš | }}|† | | | Eij | }}; 'ºï ; 2: ºg | }.}}| ſº | | | Eið | }. ºš |3: 3 gº | | | ºf | | | ºf ; ºf 3. 2 ºiº || | | j| | | ºff; | }}| 3:3 ||3: 3. sº | | | jià || || || 2: #| 3:4:: 3. 25 ſº | 3 | j | | | ºff ; ºf 3. 2. ºil | #; jºš | }| sº | ##| ##|3: : ºš #}} | . . . . ; ; ; ; 29 .636776 is...}} | .999592 || 3 || 637.184 || 3:... . .362816 || 31 30 S.6396SO 9.9995S6 8.640003 | . . | 1.359907 || 30 à º ºft| "...si || || |º #| “...is § § ºš #; ºf j . º. #| 3:41.7 ||3: § ºf ##| ºf ºf ##| 3:3: 2. 31 || 65||1}} | {..} | .33358. j . .353. 37.3% jº, 36 § ºil | 3:... ºš | 3 || 3:33 43,3} | .3456; 25 § gº #; ºś| }}| sº | ##| 3:51 |3: 3. gº ##| |º : ºš #; º 33 § sº | ##| ji | {}| ſº | #% ºil ||3: § #| ::: | #35 | | | #33 | ##| #|3. 40 | 8.667689 | ..., | 9.999529 || "..., | 8.66S160 | | | | 1.331840 2 3i | Gº | #% ºf | | | ºff §§ ºft ; # j| ##| º || || || 3:3 | #; ::: is 33 ºi #; ºf | | | º ##| 3: i. # gº #; ºf | | | j| ##| 33ſ. is 4; sº | ##| | j| |}| sisti | #; ºf i. 46 $366; 33.4} | .33:393 || 3 || $4172 || 3:. . .333 || 1: 47 || 3:32.3 || 3:...is ºff ºší | 333 || 3:33.6 | 13 4; j| #; ºisi || || | | | | ##| 3 gig i. 49 .691438 || 33.3% .999475 || 3 | .691963 || 3:# .308037 || 1 50 8.6939.9S 9.999469 S.694529 1.305471 iſ jã ##| ºft| || || “...is ##| "...ºf g 53 | .39% 31.5% ºf if I Gº || 3:03 || 3:03.3 || 8 53 Zºlº || 3:... ºãº || || || 73.33 || 31% ºft| || 7 # 7.1099 || 3:...; ºłł | # 7.1% *i; } .293.4 || 6 53 || 706377 | #3; ºłż || || 7 ||38|| 3: : | 3:2:30 5 §§ | 709019 || 363; ºł3| | || || Zſºlº || 3 || 3 || 3:3: 4 57 || 7 || 0 || 35% º?: .712083 A. .2S7917 | 3 5S .713952 | X. gjiš | | | | ||7 40.85 r. * - 40 51 * * * * * *. l I l .714534 40 62 .285.466 2 59 .7163S3 . . . .9994 l l .716972 | X. 2S3028 || 1 º 40,29 - 1 | e 40.40 || “...º.º. 60 | .7 ISS00 .99940.4 ,719396 .2S0604 || 0 M. Cosine. D. 1'', Sine. D. 111. I Cotang. D. 1" Tang M. GENTs, AND COTARGENTS. 1760 s 1. cosin ES, TAN D, 11. Cotang. M - 3O f / Tang. * * * * ----- - D. 1 60 - D. 17. Cosine. w 1.2S0604 ºt M. | Sine. - 8. 40.17 | *; #3 *- sº 9.999404 Il %; 33.35 | # 58 0 8.718800 | 40.06 .999338 iſ .724.204 || 3:57: .2734 12 57 } | #| 333 j :# | 7355Ss | #. 27.1041 56 } | .7:33; §63 .999334 . l I .723959 || 3:31 36$633 55 3 .7239.2 ji .993.373 ii .73/317 | 35.10 366337 54 4 7:337 3.1% .99.9371 . . . . .33333 33.39 || 33 ºf 53 #| #| #| #| || %; 335 | #| || 7 | 7353.4 || 3: 57 .939350 13 140525 āş; * 7S 50 8 .737687 | 3:33 .999343 i3 . g) 1.2370s 49 9 .739969 || 33.3 3.999336 12 || 8. ; §§ .2547 ; 4S re- g *...*.*. g .745. f * & 525. • ; $0 S.7+???? | 37.96 .939323 13 747.479 37.65 § 47 | | }; 3.73 || 3: # | ºff #; 24sſii 46 !? | 7:3392 | ##: .99933 13 ºšiºsº | #: 235773 || 45 13 .7.1903 | #3; #; i3 | }; 3.1Q || 2:3547 || 43 # ſº | ## .993.301 || 13 756453 33.93 || 3:32 || 43 !; 733333 36.33 .9992Q4 13 .733603 || 35.73 333i:35 | 42 } | . 3.j j : | dº | #: 236935 41 #| #| 333i | jº # | #3 || 3: . . 754 || 40 }} | .7%;" || 3: ; 999272 12 || " 1.23.4754 19 | .702337 || 3:31 , . oc: 8.7632.46 36.18 2325-3 || 39 }} | 8.7;|| || 36.06 .999257 }} | #3 33.83 || 33373 37 3| | 7383.3 3.j jº i3 #: § | 333i: | 36 : 78.23 3. .9992.42 !} | ##$6 33.43 || 2:00; 35 23 .770979 35.53 .333333 i3 #| 333i | #| || ; : ; §§ j : ºšiū § 219773 || 33 25 .773.223 35. 18 .999.220 . 13 7Sö323 34.97 2. 76-0 || 32 37 77943; 㺠9992Q3 i3 .73440s | #;" | . 514 || 30 7S 524 67 7 • & © - 2|3514 28 . : | 34. .999.19 .13 S6 - | 1. 29 29 || 7S3605 || 3: 8.786486 || 31.47 .211446 2. tº gº tº ºr *; 13 |*:::::: 34-31 || 3); 23 30 8.78567 § 34-31 || “..jjiš. 13 | ?gnéjà 34:13 | 207333 27 31 º; 34-3 | již • 3 || ?º 3393 || 305.09 26 32 .78978 §§ ºiš | }; #| || 333 | . . ; 33 Ž;| 3: sº ##| 3 | # 33.83 2012;3 || 24 # | 733; 33.70 | jiā) . 13 79373 || 3: 53 195237 || 23 § Zºśl §§ 3.3142 | tº is ſº | : ig335 23 ; . § ºf: | }; § | # iš213 21 37 º, 33.23 999] 26 3 | Sºtºs 33.07 |, ... 20 § 80; . . . ºils | # sº Q ..., | 1.19325s 9 39 803876 || 3:33 () §§§ 3392 | "...; 19 S05S52 --, 9,999 || || # sº | }} lSQ317 | | S 40 *:::::: 337; jiē; . 14 $103.3 || 33.63 . IS7359 || 17 4 | º 33.63 jºgg -H | slºgai 33.43 is; it it; 42 º 32.49 -ºš6 ił siąś ; . iši i. 43 .81 § 32.34 jnr. . 14 sió529 33.9 | isſ;35 | iſ #| #;|3}} | ...; # sistſ. #} | . ; 3 || 45 º; #!? | jºgi .# | Sºnáši 3i ji ižº iş || # º 34-31 | jj;3 -kº | Sºgs § | ##| || & ºf #3; º!! ii sº | #| | * 4S .821343 31.63 99.90.36 14 tº •e 1.173S07 10 49 823210 || 3: * 8.833.93 || 31.50 17200S 9 25.130 9,999.027 . 14 .S.27992 3.36 || 1:01:36 $ 5ſ) 8.825 º l 31.36 .999) [9 . 14 S2987.4 31.23 iósg33 7 3| | 8270 3!? | join tº sºlºis 3i og ić63s? | 6 53 || $2.834 31.QS .999002 3 || 3336||3 36.96 ičí529 5 53 § . 30. 95 .99$993 .l-l s3547 i 30's? & l §§§ 4 ; º' | § #| | | # 3.70 | # § *):) º .9 .99SQ7 . 15 S39| 63 30.57 işonſ; § $º 35. j is ºft| 3. à | #| || 3. $3.39 || 3. sº iş Şış §§ .# 0 § ºff § .99S950 16 || $1614 * , I Dº § ºf . .99S941 • Q 1". I Tang. M. 60 | .84.35S5 Si D. 1ſt. I Cotang. D. 1". -* *-am= M. | Cosine. D. 1 Rºtº § 60 9:30 200 T A BLE X 40 LE XIII. LOGAR1TIIMIC SINES, 1753 M -* -- Sine. D. 1". Cosine. D. 1ſt Tang D 1ſt — 0 | 8.843.35 30.05 || 9.99$941 º . . Cotang. M. I .845 .05 - § t; 8.8446 # jº, gº j | j. 3020 | *; 60 #| #| 333d jº }} | ºff ; .####| || 3 } | #| 2.35 | j # j| #; #| # 5 .852525 29.55 .º. i. S5] S46 29.S3 ... 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S * .30 195 $ , 69S 2 | | .204029 }}}} ºi: | # ºf }; .697.393 || 39 33 || 3:463 | }. # .99.1397 33 || “...}} | 16.j || 3:3732 38 # 3; ; ; ; ; ;|3}} | .35 | .303914 | #; £380s; 3. 24 §91.3 ió. # ..991346 35 | .301587 iñS6 .695.433 36 : | };}}}} | {}}} | {:}}}| || 3 || 3052]; #3; £347.2 35 # 3.1% iðjó | #3; 33 ºsº | }. ºil 3, 34 3. ºš jºš $3.9 || 3: . .306.13 #s gºts || 33 # 3; ºš | 3:3: | 3 || 397163 | #3, 632832 32 O{ •. * * g as gº • C [...]” ºn * 29 299034 #3; 991218 || 3 || 307s.6 | #; 692184 31 § 9%; 10.34 || 9:33 43 93.4% 107, 0.391537 30 § ºś ič | #167 || 3 || 3:109 | #3 tº 22 33 .300:35 | i. § º!! #1 33 .309734 I 8%4 .690246 28 33 .301514 io 3) | }}}}} ... . .310309 § .6S980I 27 # 33132 | #| º # 3iº || | b% .6SS95S 26 § ºš #3; ºf . . .3116S5 ; § .6SS315 25 § º ič, $3; 3 || 3:3:27 | #6; $376.3 24 3. º ##| | ºff. ; | 3:3: | }}; ºšić | 33 3. 31.3 #3; ºf j: | 3:350s ##| || 3:633 ||3: 39 305207 | #35 | .999960 | 3 | .314247 # .6S5753 21 40 | 9.305S19 9.99ſ)934 9.314SS5 0.6S5] 15 20 41 .306.430 }}}} .990.90S # .315523 #: .6S4477 I9 # ºil | #6 ºš2 | # 3īāſā }} .6S3S4 l 18 # 3 ſº | };}} | º . . .315:35 | # & in # 3º # ºś | #| 3.74% #; sº | 16 ; ºff # ºsº | # 3isſº; # ºsſº is # ºf ##| º # 3īščº | }}; Estáš tº {{| 3 || 9 || |||}| ſº | # | 3:330 | }}; ºn i3 ; 3|||}| # ºf # 3ijëi | }}; sº i. * | 3112sº | #| 900697 | f | 3% #| | #ds ii 50 9.31 IS93 9.99067.1 9,321222 0.67S77S 10 5l .312495 §§ .990645 º: 35iS; 1948 || 3:5145 || "g § 3; ; ºś | # 323ſº | }}; gºi $ § #3; # ºf # | 333i: | # ºf # | #3. 'º'; ºš | #| 3:33 | }; .676267 || 6 § | ##| | j ºš | # | 333; ##| | § 3; §§ ºil | # 3313.3 $º ºf § | #| #| ºš | | | 3: | {}}| sº | 3 § | ##| || 3 || $º || || || 32:33i ič | .37378.9 2 § | #73; §§ º! ..., | 326553 ‘. . .673147 | 1 60 | .317S79 g .990404 9 || 3:47; 10.36 jºš | 6 M. Cosine. D, 1ſt. Sine. D. 1". Cotang. D. 1ſt. Tang. | M. IOIO 780 208 TABLE X ill LO3ARITHMIC SINES, 12O I 67C M. Sine D. 1". Cosine. D. 1" Tang. D. 1". Cotang. 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Codino. D. 1. Sino. D. 1". Cotaug. | D. 1". Thng. | M. 69G coSINES, TANGENTS, AND COTANGENTS. 21C M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". 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Cº- 19 || 560531 | #3; 969223 || 3 || 59.1308 || 3: | 408692 || 41 30 9.5%; 539 9; 3 sº | 9.5%lºs, 6.21 || 04:3 || || 3| | #1; ; ; ; ;12; § 59204 5 .407946 || 39 ; jºiáº; ; ; ºš # #233; # | 3:4 |3s ; ºf ; ºf # 5:35 | }} | 4:3" | 3: 2-1 | .5621.46 §§ .96S976 $3 .533171 20 406S29 36 : jºić # ºš | 3 || 53.4% | }} | 4:3 ||3: 25 sº | #: gºss. # jºij | } .4060S6 34 : | #13 || 3: ºš: | # 5:33; § 45.5 |3: § | #| #| |º #| #ſº | }} | 4:3 ||3: & cº-ºp-gº- - Sºx) - tº Cy ºf a º -1 Chº? 29 563755 g; 958728 || 3 || 595027 | #3 | 404973 || 31 30 9.5%; 5.34 9.98S9.3 sº 9.593398 ... 0.404602 || 30 3; #39; § ºš | 3 | "º #% .33333 ||3: ; º!... . . . .º & .52613S § 17 | 403S63 2s § ºff; | #: ºš | } | ºffs #; 4:3 ||3: # #535 | }; § | 3 | jºšš | | | 4313 ||3: ; #5 #: ºš j . ºf # | 4:3 ||3: 35 | #5 | }. º || 3 | ºffié | # | 4:3 ||3: 3. #| : º | # ºš | }}; ºf 23 § | #| | | ºff; | # sº | }}; 4 lºg | 3: * º 5.30 .96S22S .84 .59S722 6. 14 º º 9.5672 r: 9.96S 178 9.509091 .40ſ,309 iſ ; ; ; isºs | | | "º | }}; ºil is 4; º! | # isºs | # º #; ſº is #| 3: . . . sº | # sº | }}; ºf iſ # º j . º. ; gº #3 || 3:35 | is # ºg j ºf j . º # | 3:0; 15 43 #3 | # ºšić | # ºg | # ºf 3 # ºš | }. ºsº | #| ſigº #| || 3:33; 3 ; j| #; j| #| Gº || || || 3: i. 19 .570120 | . . .967725 si | .602395 it, .397605 || || * | *ś 5.25 9.2.É. si 9.603.6l 6 to 03.339 10 3. 57ſº . . .ºzi | # & | }}} | .3965.3 a 33 57.1% . . .9375. ... ºłºś | }} | 3:50 Š § ſº | }. º? | 3: ºš §§ .396142 || 7 # jºš # sº | 3: | 6′33 || || || 3:57: | 6 § jº, § ºi $: gºsº | }} | #3 § #: #: º # sº | }} | 3: § | jºš #: ºśń ºf Gº || || || 3:53 || 3 § § ; ºš j| sº | }. ºš 3 § ºff; | j . ºf j . ººg | | | 3:3: 6) 57.3575 °. ºtö6 S5 ºil, 6.06 || 3:590 || 0 M. f Gairyo. D 1ſt Sino. D. 1". I Cotang. D. 1". Tang. | M. ! I 12 Aºi Kriº 218 TABLE XIII. LOGARITIIMIC SINES, 222 1570. M. Sine. D. 1''. Cosine. D. 1". I Tang. D. 1". Cotang. | M. 9 93.33% 5.21 || 0.37.1% 85 9.5%:110 | 606 || 0.333330 gº | | .573S 52ſ .967 | 15 º .30873 || 3: .393227 | 59 3 || 3:4200 #3. 9.70% | 3 || 60713. j || 333333 || 33 3 | #33 º 95.013 || 3: | 60.500 | }. .332500 | #. 4 | .574S24 §§ .96696) †: .607863 §§ .3921.37 || 56 § .373}} | {i} | .96919 § .608225 | 3. § .39.1775 55 § ºft. i3 | ºffº § .gºš | | | | 331412 || 3: 7 .57575S #. iš .96:03 ... . .303350 Šºš .391050 || 53 § ºft| ## ºš ... 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T. tº e tº A • º ºf A "Tow - r y fºrm * * * * . . . . . . . . . . ; . . . . Jö; ), 4. {}Ult . O. . ºf J2, l ; iſ jºiº || || || "º gº #: ºš is 43 sº | | | ºffſº | #| 3:3: # #3 is 43 || 5:5:3 || || || 3:33i j || 3:3 # | #3; iſ 44 537; ; ºſsº | 3 | ºff ; 37; is 4; 53.3% | }} | ºffs25 || 3 | ºffi # | 3: is 35 | ###| | | ºffiz; j . ºf ; 37. iſ 4. ºš | | | ºff ; ºš ; ; ;3| | |3 § sº | | | ºff º .62×523 | #; 37%.7 | {} ... [...] § |...] § |...] § . . . {DU ºff. :) 0.964560 9,624. 5 A, 37; ãi jigſ | }} | .jj § .6246S3 ; ,375.317 | 9 53 | #33 j . ºff; † | #25.35 | }; 3:3: § 5} | .5.82 j || 3:4400 º .625383 | # | 374612 || 7 54 ºš j | {{{347 º .625.41 | #; 37.9 || 6 § ºš. j | 934394 | 3 || 636093 §§ .373907 || 5 56 || 5 ||}} | ...; 964240 º .#26415 | # 373; 4 3. ºf 3.3% | {{{..., | 3 .625797 | # | .37393 || 3 § #2; # | ºffſi: | }| ſº | #; 3; 3 § ºãº j . ºftſº .89 || 6′3750i §§§ .3734% 60 | .59 [S78 - jčíð36 |, .89 || 627852 - .372148 || 0 M. | Cosine. D. 1'ſ Sine. D. 1'ſ. I Cotang. D. 1". Tang. 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Cº.). º 0 | 9.623229 }{} *. .97 §03; §§ 335297 | 12 ăl ſº 4.55 9.33783 9.66536 § 331965 Il 52 .533.74 3. .3578% .97 | *; 6 5.52 03:34 k § | #| }; § j †. ; #3; 1. ší || 333i: | 4.54 957687 .6330?? | }. jī ; ; #| 3:3 j # j| : ºf I | 8 e ~ • * * *_º - 6:40 ; #| 3 || | # # ºf ; ; ; 7 • U 3-1||DW.J. • *-** - .667 5.51 .333.309 6 ; #| 3: jº ... I fººl *) 33297 5S 5: º 3745. j | #733 5.5l 332979 5 § º #; #; j | ºffſ; §§ | #3 go . 3. º §37; .9S .G6S()|3 5.50 .33:318 3 - .625948 .52 557376 .9S .668.343 5.50 .33:1987 2 i; M. : Cosine , ºf Of 23 f ºś 5.50 331;7 || || | . | D. 111. Sine. D, 1". Cotan .331327 | 0 1 149 & g.T. D. In." Tang |M|. 65C COSINES. TA 25o , TANGENTS, AN * , AND COTANGENTS. M. Si 0 SłDe. I) l H. Cosine. D R!! Tung 1 5 4 3 9.6% - , D, 1ſt. # º; 4.51 9. ; 98 || 9.663573 Cotang. 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D. 1" * ºić6 5.21 .293.146 tº a tº Sine. D. 1ſt .29 . 1". Cotang. D. 1" . 1". Tang. sº ess, , TANGENTS , AND COTANGE NTS. 223 279 M. Sl Sine. D. 1". Cosine D 152d O r - . 1ſt Ta - 9.657047 ng. D. 1". Cota * .657235 # º 1.07 || 9.707166 otang. M. .6575.42 . 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D. in Tong, M. [2] D 11 582 228 TABLE XIII. LOGARITIiblic SINES, 32O 14.7% Ai. | Slne. D. 1". Cosime. D. 1". I Tang D. 1". Cotang. | M. 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I $23919 || || 3 || $449 || 3 || , 18S590 T 4 57 | 7355.25 | #3; $23:37 . . . .31 lºſ | 3 || ..] SS3| 3 || 3 53 | 735749 ..., | .923.5% | | | || $1.3% 3 ºf ..ISS036 2 £9 | 73.914 || 3 | .923673 | 1.3% $1334|| || 3 || .187759 | 1 60 | .736) ()9 & .923591 w .8l 2517 - . 187483 || 0 M. Cosine. D. 1". Sine. D. 1ſt, I Cotang | D. 1" Tang M. | L !220 57 & .i. l S. 29 83 2 M. Sine. D l' - l'ſ (Josipe. D 1" 14 º 0 9.736|09 lit. Tang. D. 1'ſ # º, 3% 9 323:591 . 1". Cotang. M 3 .73&493 3.24 j | }.3% 9.S1251.7 -- 4 .73092 § 3; 9:137 ...? | Słżºł 3.61 0.187433 60 r. ºš6 || 3:33 º; | 1.37 $13.0% | 3:3: . 1S/200 || 59 § | 73 tº 3:33 jºš 1.3% .S133-17 4.6l išj | Š § | #3; 33 jāti | }.3% sº | 491 . 1866.53 57 § .737.4% 2. jš | 1.37 .S13S99 4,60 . 1S6377 56 9 .73768! § 2; .ºllº }.} | .84176 4.00 |SGIO 1 || 5.5 .737S35 § 22 #. | .37 º i 4452 : . G| | # 54 10 9.73S ..., | .92235i ...; $14723 , ()() 5548 || 5 |*:::::: 32 |*; ; Siâni | }; lº: § É ºšiº || 3:33 jºš | 1.38 9.S. 52SO 3.g., .184996 || 5 3 | 73S03. 3.3 || 3:303 i.; $1.35% 460 |0. Sº 5. | | #20 || 33 gº | 1.33 j || 4.99 isiſ; ſº .739013 3.2 | ºš | 1.33 .8 || 6 |07 4.59 .išić is º; 33} 333; 1.33 Şişş3 || 3:39 iš | }. ºš | 3:?! jº | 1.33 sigg; 4.5% jśīš | 6 .73%);90 3.2] jºso | 1.33 .S1($9.33 4.59 . IS33.12 45 .7397S3 3, 20 jºin; 1.33 .S 172 10 4.59 . I S.3067 : 9.7.3997.5 3.2] 322023 33 s 74S4 4.59 - lsº | | 43 º; 329 |*; }} | sizij | }; #| || ºn | 3:30 jś | 1.3% 9. SIS(135 º, IS2241 || 4 | 7 ºn | 3.3% jº | 1.33 Siși, 4.38 0. IS 1965 | 40 .730.43 3.19 ºl i..., | Sº 3: . . [S]ºſ | 39 7.0931 || 3 |9 jiš); lº sij || 3:38 išijiš 3s 74 iſ 35 | 3 |} jiši 1.33 .8.1913.5 4.5S isi;0 | 37 74išić 3.1% jiji | 1.32 .8.194 10 4.5S . IS(1865 36 '74 isºs | 3 |9 jºi; 1.33 Siº || 3:3 | Sū590 $5 74.1395 || 3 ||3 jºtº, 1.33 |siº § . Sºlº 3. gºij | * j i ij | # 35 | }. ; .742(JSO 3.18 9,92] 107 I.39 .S2(15(JS 4. #; 32 .742.27 1 3.13 || $21ſº 1.39 9.830.83 ; : .179492 || 31 733.63 || 3 ||3 ºj | 1.3% sºn; 4.57 0.179217 30 733632 || 3 |7 º; | 1.4% | sº | 4:37 . 17S943 29 .742.43 3 : 7 º; | 1.4% ,8216(16 4.57 ..] 7S66S 28 .74%: § {} | .520ſ; i.j Sºlºsſ) 3.3% 73. 27 7.43333 3, 17 ºf 1.49 ºiáſ || 4:37 . . # 3. #| || 3 iſ jº 1.35 | Sº ...; .177,46 || 3: .743602 3, 6 º, 1.4% .82.27(13 4, 57 . I 7757 l 3. 9,743702 §§ 920352 i in $º 3.3% .1773. 33 § | 3 || | *; i.j | S23251 4.57 177033 22 7.j; ; 3. It jūsī | 1.49 9.823524 3.6 .176749 21 .744.361 Šiš º i.j $33,33 4.56 0.17%;" | ? .7445.50 § {, , .º 1.40 $3 (nº 4.5% 1762()2 0 .74-1739 3. l 5 .9 || 093 | 1.4 l .8.2.13:15 4.56 . 7592S 19 .744933 3, 15 .919.S.46 1.4 l .8.2.1619 4.56 . 175655 l 8 .745.17 §§ .919.62 1.4 l sº 4.5% . 1753.81 17 745,06 || 3 || júñº | 1.4| sºng; 4.3% iſsidy ſº #| 3 || | jº }} | Sº | }. #| || | 9.7456S3 3. ] { jj | {{! .S25.13 4.56 .17361 | 13 .745.1 3. lºt 3.0 , 9124 1.4 l ,8259S6 4.55 .174287 | 12 .74.060 3. I { | 910339 14t 9.82ſ23 4.55 . 174014 || 1 || .7462.48 § {, , §§ 1.4 I .826.532 4 55 0.737. ºù | }; #. iii º 3 : 17:43 I0 .74%24 § 3 ; 9 ſº 1.4 l .827 07S 3. .73% 9 #; #13 || ||. {j} | Sºi #; . º; ; .246999 3.13 º! ; | 1.42 .827624 4.55 ižº 7 73; is; 3, 13 giš0 | 1.4% j} | 4:35 1:23:5 6 #| #13 | }; # sº | . . . . § 73;563 || 3. 12 #; łł. º: 4.54 ..if sºft : ~ .9|S57 33 || Sºl; 4.5 . I ? 55 Cosine. D. In slue 4 º łł º ; D, 1ſt. Cotan . l ; 1(\l 3 0 &. D. lº T sº- ºr ang. Mi. 5 6% 230 TABLE XIII. LOGARITHMIC SINES. 34 O 1 4 3C | M. Sine D. 1". Cosine. D. 1". Tang. D. 1". | Cotang. | M. i 0 || 9.747.562 9.9 |Sj74 9.82S9S7 - 0, 17|013 || 6() I ºff #3 | "...iº || || || "...sº | }; iijó | }. 3 74793; 3.}} | .318104 || || 3 || $39532 || 3 || | .1743 | #3 3 || 2:31:3 || 3 || || 3:35 | j || $29:05 | }} | .179185 5. * | 733319 || 3 || || 9,13233 || || 3 || $300. 3.3 . 169923 56 | | 73.9. #| || Alsº | }}} | .83319 || || || 1ſº 55 § | 738& 3.}} | .918002 || || 3 || 3:06:41 3. . 1693.79 54 Z Zºº | 3 || || 9,179.6 || || 3:. . .830sº †: .1501ſº 53 Złºś #| || 317 ºn | }.}} | .83|165 | }; 1653. 52 9 || 749243 jö .917805 || || 3 || 831437 || 3 || | .16S563 5. 10 9.742.29 || 3:10 9.917.719 | 1.43 || 9.83709 || 4.53 0.133391 || 30 | | 73.3% | 3 | | .217.534 || || 3: . .8333 || 3: . I 6S010 || 49 !? | Zººl ##| || 91.75ls | {}} | .833453 | . . . .17747 || 3: }} | .7:33, ##| || 9,1745? | }.3 || $2.25 | . . . .167475 || 4. H | 73.72 #} | .9173.5 | | | | .833.96 || 3:. . .15.204 || 45 # Zāº; ; ; .31739) || ||33 || 3:30; 3. . 16693-2 || 4: #3 Zºº | 3 || || 91721 || || 3 || 3:30 #3; . 1666(3 || || 4-1 2 º j . .317||s i ; $338|| || 3 || | {:}} | {3 | | 73}} | #}} | .317|32 || | | | .8333S2 | . . . .156|| || 3% 19 .751099 || 3 || || 916916 || | | | .834154 || 3: . .165S46 || 4 20 9.7512S4 9.916S50 9.S3].425 co 0.165575 40 2 ºi:53 ##| gº | | | | sº | }; iń 33 || 7:16:4 || 3 || || 21ſº | | | || $31,57 #. ..] 65033 || 38 33 73.333 || 3 || || 316600 | | | | | S㺠3: . 1617 62 || 37 24 º || 3 || || 31%ld | }.}} | .83:303 º . 16449 || || 36 25 | 73:203 || 3 || || 91.27 1.33 || $3.750 | 3. § . 16.12.20 || 35 35 | 7:393 || 3 || || |&#| | }.}} | .833}| #. ..] §39-19 || 34 27 75,2576 §§ .º.º. iº || $33333 3. ;: . . 63678 || 33 3. Zºë0 || 3 || || 91916. i.; .833333 3 § .163407 || 32 29 | .752944 • *- .91 GOSl ‘. . .836S64 . . 163136 31 3.06 1.45 3.51 30 9.733.33 || 3.06 9.95934 | 1.45 9.83.33 || 4 51 || 0 |º 3. 3| | 733312 || 3 || || 31590? | | | | Sºlà | } }} | {{#3 39 32 | 73.93 || 3 || || 3:33.9 || | | | ºff; | º ..] 62.325 28 33 || 7.3% | 3 || || 3:57.33 || | | | Sºſº | | } , j62ſ).j4 || 27 31 | 73}} | . . .91564. ; : .83:316 | }} ... l 6 7 S-4 26 35 | 734016 | }}; 915550 j; ºf ; ; ..] 6 || 5 || 3 || 25 36 | .75-1229 §§ .915472 i. # .83.37 || 3 || . . 6 1243 || 24 37 || 754112 || 3 § 915385 i. 3. 33903. ; ; . 16(1973 || 23 3. ...; §§ { };}} | {:}; ſº | };} | {º} | . . º 3.05 º 1.46 º 4.50 wº º 4 .75-1960 .9 || 5 || .839S * . 16()) {} 3i ºf § gº || || || "...is | };} | .jſ, ſº 43 | 73.33% §§ ºl{Q}} | {..}} | {{{373 #; . 1596.22 || ||8 43 | 733; 3.}} | .213869 | }}; $40.3 || 3 || | | |3}} | . 44 Zāº ຠ.314.73 || || 3 || $10917 | 3 || || 3:(; ; ; 4; º? | 3 || || 3:45.3 || || || $41.87 }: ..] SS| 3 || || 5 35 | ºffſ; §§ .345; #3; $44.7 || 3.3, ºš || || 47 | 73% | . . .31431Q | }. 8473. 3.35 | {:} | {3 4; ºl; §§ | 3:3: | | | | 8 || || | | | | |##| |} tº. - *-*.. , gº º & ºt, - fºr fºr ºf 49 º 3.03 º 1.46 º 4.49 wº . . 50 9.7567S * .91 42 | - 9.842535 ..] 57.465 à || "...ſº | }; iii. # "º | }}} | .ij | §3 | 737. § ºſº | }}} | {{{..., | 3 || | # § 53 Zºº; ; ; 33.3 | }.}} | {{{3}{3 || 335 | #3. . # | 737:07 || 3 || | |3}} | { } || $4363 | }}} | .333 || 6 § | 7378. §º ºšº | }.}} $43,3} | {{j . 156] 1 S 5 § º §º $13.3 || || || $4; lº! 3.43 . I 55S°49 || 4 57 | 73.30 || 3 || $13530 || || # $420 | } { . 1555S() 3 § 7.8% | 3 || || 3:3.4, || § 1959 || 333 .155311 2 § Zāºl | 3 || || $13.3 || || 3% § is ºf: 60 | .75S59 | g .91.3365 - .845227 - .15.4773 ſ () M. cosine. I dº. sino. D 1.T co-ang TD 17. Tang. M. -T- -- - - ----- - ------------- =l 124 C. 55°. COSI NES. TA 35& , TANGENTS - , AND COTANGENTS. §ſ. Sine O To 9.75 Cosine. . 1'ſ. Tang D. 1ſt 144 .75S5 s tº gº - wº |º] | 391 |*ś Cotang. | M. 2 .75S952 3.00 jišč | 1.47 9.845227 0. I 5477 º- 3 | 733i: | 39 #; i is ; 4.48 || “...i. 7.3 | tº 4 .7:59,312 3.00 .9 i 30:39 1.48 .845.64 4.48 #; 59 * | }; à | }; # sº | #; . #; . § | ##3 || 3 || # iş | # 435 | }; . } | {j}| 3 || | # }} | Sºo || 3: {; . § | }; ; ; ; # sº | }; #;" | } | }}}} | . . ; # #is #: .# 10 - 3.35 | .912566 i.35 | 847326 3.3% .33.3% 9,760390 # sióti | # . 152624 iſ º 3:33 9.912477 9.847 3.4% .152356 | | #; Žº #; 148 9.837.913 0.15 #| #| 3 | }; # "sissi | #. #; #| #| | | #| #: # | 3:3: | # #| #;| ≤ | }; # sisº | }. .# ié | #iº || 3:33 #; iſ jº # #; i; º ºš jº i ſã | #: 3.3% | #: § #3; ºf #; # sº | # | . £48 #| #; | 3 | }; # sº | # | .# 20 | 9.7 #; giióñ | }; #| || 333 | };}} 9.7C2] 77 Q 1.49 .S50325 4.46 . 149943 3i | # || 3:37 9.91 l 5S4 9.S505 3.; .149675 # | }; ºf #; 1.40 9.850593 0.14 3. Żº | #; # Săsăi | #: i 9.407 24 ºš | 3:33 gii.313 | 1.4% .851 129 3.35 | {{2|39 #| 3: | 3: | }; #. sjiāº; #: # ; #; gº #; # sºiº || 3: .# * | }; gº ; #| || $º # #; 23 ºn | 3:36 glūjāś | {{!! .852.99 3.3; .43369 25 º? | 3:33 : iń j 3.35 | 13750, 30 9.7 #; gióñó | }; j| 435 | # 9,763.954 iš, .853001 3.35 | {{???? 3i ºf | 3:33 9,9106S6 9 Sj. 3.33 .146999 32 76430S 2.95 jiūjó | 1.3% 9.853268 - || 0 || 467 #| 3 | | }; }} | sº | #: .# § | 3 || #; # sº | # i 6465 § | 3 | | }; 1.5 §§§ # . Hºlº º; | 3:3: gigº, . S5433G 4.45 .# #| 3: | }; # sº | # #; § #| 3: | }; # S㺠# 45397 § | #| 3: | # # sº | #: .#130 40 || 9.765 #; ºśń | }; #| 44; #; 9.7657.20 i.;; .855671 4.45 . 144596 iſ 7:30, 2.3% 9,909782 * * 4.44 . 144329 43 º || 3:33 jojööt | 1.51 9.S5593S 0.144 #| #| < || || }} | sº | # .144062 # º 3.93 309310 | 1.51 .836+71 3.3; .1337; #| #; gº ; }} | {j} | # H3:33 ić ºf 3.93 §§§ | 1.33 Şıoğ 4.3% . 143263 # | }. Žºg | }; # sº | }; .#3; #| 3:. . . . . ; #: sº | }; .#3:30 § | #;| & | }; # sº | # . 142463 § 97.77. • ** * i.; S5S338 3.35 | {{{33} § Zºº 2.91 9.33873 9.S5S 3.43 .141664 52 ºf 3.9% jūšší | 1.5% .85$602 O ; #;| 3 || | }; # sº | #: | ". 14139S ; #| gi j # sº | #: .14! 132 55 | 73315 || 2:31 jošū; .33 .S59400 3.3; Hºº 56 iès; #. º; iş j 3.33 .1439 ; ; gº j #; sº | #: .140334 §§ ºssi || 3:0 j i sã j 3.33 | . Hº ;| #| #| #| }; # | 333 | }; ; : ; 335 | #: }; sº | #: 33.3% M.T. #| iş j 3.33 || 3:0 d. Cosine. D. 1" jºi 4.43 .139005 ---- Sine. D. Itſ. C . I 3S739 otang. D. 1ſt. T ang. TABLE X111. ~ y ſº * . S LOGARITIIM IC S. NES, Cotang. H Tang. D. 11. Cosino, D. 1/l. 0.438.33 S$ne. g 9.S6] 261 3.43 || “...ij}; 9,90793 | 1.53 -Sºl 327 | 3.3. . 13S20S 9.7%.3 || 2:00 .997S35 | i. Sºlº #3; . 1379-42 78.3393 || 3.5 .907774 i. .8%2C3 3.3: . . .37677 .7%366 j .907032 i. ..Sº 3.33 . 1374 ll .769.4% 2. 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S. iº $51.437 | ** | . --- .S2537 I 2.34 #1673 * * 1ſt Tang. | M. .8255 I 1 * D. 1". I Cotung. D. 1". - ine. D. 1". Sine. & - Cosine, - *A §3 238 LOGAR1THMIC SINES, TABLE XIII. 430 137C M. Sine. D. 1". Cosine. D. 1ſt Tang. D. 1'ſ. Cotang. MI 0 || 9,8255] 1 , 9.87 1073 9.954:437 0.045563 || 60 i $2365i | 3:# sº | 1.99 i ji 4.33 || "...tº, j 2 $25%i #3; sº | }.}} | jº, 4:3 | dº | #3 3 || 33333i | 3: ºš | }.99 jº, 433 | dº | #7 4 || $260, i | 333 sºft | 1.99 j | #33 tº #5 5 32&ti | 3:33 || sº | 1.30 jºš 433 | dº | . § | 3:335i | 333 || sº | 1.99 || jčí || 4:3 | tº gº 7 || $2646 3. § sº | : º #: 0.437s; 53 § $3:31 || 3: . .87916 || || || || 3:3469 || 3:33 .943531 || 33 9 826770 || 3: . .870047 | | | | 936723 || 3: | 013277 || 51 10 9.3283.9 2.32 9.8%33 | 1.91 9.956377 4.23 0.043023 59 | 1 | .827049 || 3:3 | .869SIS .95723] ‘. . .042769 || 49 j} | .337.j || 333 jº; 1.9 | jºš 433 | tº || 3: 13 || $3.3% | 333 j | 1.3, ºš | 333 | dº | ? i4 || $37.67 || 3:33 jº, 1.9 jj; 433 | dº | 36 i; ºf #; sº | }} | ...sº | #3; ºf: |3: so ºr 4 ºf ** {A^* tº) iſ . g * Q: - tº {}| #| 333 j|iº | jº 33 || || || iš sº | 3:3} | jº, $3 | jºš | 4:3 | tº ſº iš | #3; 33} | .j | };} | ...; 433 ºš iſ sº 2.31 g 1.92 || “” 4.23 º 30 || 9.8333¢. 2.31 9.85s, Sã | 1.92 || 9.9593.6 4 23 0.04.0484 || 40 21 .82S439 ..." .86S670 . . . .9597.69 ... . .040231 || 39 22 .82S578 # .86S555 }; .960023 #: .039977 38 SN * < w ... ººms ºf ( )? & #| #| #| #| #| #| #| #|: #| #| 33 j ig | #| 3: #|: 3. sº | #; sº | }; jºij | }; ºš |3: 3. $234% 335 | Sºść2 | }..., | 93.543 || 333 || 93455 32 29 || 829545 || 3:3 867747 i.; .961799 || 3:33 || 03S201 || 31 30 9.8296S3 * 9.867631 9,962052 t 0.037948 30 3i | "...sºſ || 2:30 || "...sº (; ; 1.93 || "...gººg 4.23 ºf 35 33 || $2.jpg | 3:39 ºg | 1.33 jščo 43 | dº | 3: 33 || $30037 § sº #; $62sić § ošíš7 27 # | {:}} | 33, $37,137 || || 3 || 3:30. #3; .93%3 ||3: 3; §373 || 333 || 83.931 i. ::::::: 4.23 º: ; 36 .830509 º' .866935 | * * .96.357 e g 2.29 º 1.94 ºš | 4.23 * 3. ºšić | 333 ºl; i.; ºš | 3: ſºlº ||3: 39 .83092 || 3:2; .866586 i.; .961335 | 333 || 035665 21 40 9.83|05S 9.866470 9.9645SS o 0.0354.12 20 4 || | .831 195 ;: .866353 }; .961342 #: .933.3 13 #| #| #| #| #| #| #| #|# | }. 33% 335 | {{{ſº} | .35 º #33 ſº | }; i. $ = º! : e ſº ſº. * f #| ###| 333 j| is j| 333 | #; # º 2,2S ... 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To || 9,5337s; | Nº || 9.86±27 9,969656 | . oo 0.030344 60 * 2.26 || “... 1.96 tº gº 3 °º & 4.22 º fº | | j . º. º! iſ #| 3: #| || 3 j || 2:35 | #3 | 1.97 jºig #33 | dºj | #7 3 | ##| 2:35 | # | 1.37 jū; 43 | dºi| 56 # sº | 333 j| 1.97 | jºij || 3: | dºjoſs || 55 & j || 3:23 j | 1.37 jºiº, #33 || 02:35 | #4 } | jº, 2.25 | ji | 1.37 jºiáº) || 4:3; ºf 53 § $34s65 § $631.83 # jjićsº #: ozsäiš 52 9 || 83.1999 || 3: . .863964 || | | | 971935 | 333 || 028065 51 |*; 224 |*; 19s |*; 422 |%; # # sº | 2:2; sº | 1.33 i jºš 4.3% | 0273); 45 3 j| 2.34 j | 1.93 jºš | 333 tº 47 iſ j 2.34 jºi | 1.33 jºi | 3: | dºg 46 i; j7 || 2: j;3 || || 3 | jºšiši | 4:3: | dºg; 45 ić jºt 2.34 j | 1.93 gº 3.3 jºš | 44 tº jº, 2.34 jī; 1.93 ºn | #33 | dº | 43 is sº | #; sº | }; ºš #; ºš |3% 19 || 836313 || 3: S61877 || || 3 | .974466 || 3:35 | 023534 || 4t º, jº, 2.33 | #3 | 1.93 i jºš | 333 | dº | 3: 33 || $36s; 2.23 $300 | 1.92 | jºj || 4:3 || 03:32 || 37 3 || 3:013 || 2:33 ºšū | 1.99 || 37.5%; 4% | 02:35 | 36 º; ºlig 2.33 || $16 | 1.33 | jºš #3 | dºioſ; 35 23 || $º 222 jiài | 1.99 jºš i 333 || 03:63 ||3: 37 | #313 || 2:22 j | 1.99 || 376391 || 4:3 | dºg 33 2. ºãº # sº | }} | ºf ; ; ; ; ; ; 29 || 837679 | . . . .8608s2 | . . . .976997 || 333 023003 || 31 § jºš | ??? | j || 3 ||Q ºš #3 ºil ||3: § jii 2.22 | j || 2 ||Q jºšinj | 3: | dºi 27 § $314 | ??! I sº | 3:00 | jºšć 3% tº 26 § º || 2:21 | jogº || 3:00 jºši; 3.33 | dºlº, 25 § $10 || 2:31 j || 2:00 ºššš 3.33 tº ºf § j} | 2:31 | jºi 2.0l jºi 3.3 | dºg | 33 § sº. # sº | #}} | ...; #; ºf . 39 830007 || 3:3: 8591so | 3 || | .979527 || 3:3: . .020473 21 - * * 7978 tº o * | *; 221 |*; 20 |*; 423 | "...; ; 4; jani || 3:39 jiù | 2:01 | još 3-3 | dºi1 | is 4; sº | ??? | j| 2:01 | j || 4:3 | dištěž 17 à | jºš | 3:30 | jºy | 2:01 ºf 4.3 biºg | 16 4; jj 3:39 || 3:5756 || 2:03 || Siúi 3.3% otégé is # sº | }} | . . ; ºši; # is; i. * * * * * * - tº Sº I & * * * * sº I tº ſº * * Q.4 f. #| #| #| #| 3: | #| # #| || 49 | 840823 || 3 || || 85S272 || 3: . .9S2056 || 3:3: .017944 11 50 9:0:39 2.19 9.58||5|| || 2.02 || 9,933}| 4.21 0.07%" | 1Q !: - º, Sſ). 2C * Qº) ſº) - *d ºr #| #| #| #| #| #| #| #| || #| #| 3 || | #; gº | #|| 3 | #; . § Šimić | 3 || | S㺠2.93 jà || 4:3; jiē; ; §§ #iº || 3 |8 || sº | 3:03 || ºë 3.3, oilº, #7 | #t373 || 3 || || 3:00 || 2:03 jº, 43, jiāº 3 §§ 54ing | }}} | jºš 2.93 | j || 4:3; oij || || § | #d | } § ºš §§ ºšiša # jišić i 60 .841771 |_.856934 || “. .9S4S37 | * .015103 || 0 | M. Coalue. D. 1''. Sine. D. l'. I Cotang. | D. 1". Tang. M. 1332 3. €º 240 TABLE XIII. LOGARITIl M1C SINES, &c. 4 +O M. | Sine. D. 1", Cosine. D. 1ſt Tang. D. 1". Cotang. 0 9.S.1 1771 9.856934 9.9S4S37 r ! | #; # j} | #| "...sº | #3; º ; #| 3 is ; gº j| 33i | }; } | #| 3 is j gº j| 43i | }; * | ##| 3 || | #; Žº j| 4:3: | }; 6 | #2; 3, 17 § 39; j | }} .0 || 3:S99 } | j || 3 || | #| 3: | #| 33i | }; § sº | 3 |7 | jº, 3.9% j 3.3t 9:33.93 } | #| 3 || | #| Zſt: ; ; 42 | }; .8:1307 9.8557] I : 9.9S7365 0.01263.5 | | | .84.3206 3. }} | .sº 33 | "...; 43. º; | 2 | .843.336 | ..." § | 393 | jši 421 | #3 i5 sº | 3 || | j 393 | jº, 43. .Q12129 iſ #3; 3.1% | j | 393 | jºg | 4:34 ſ!!. i; #; | 3 || | j || 3:03 | j 43. .0l 1624 ić | #3; 3 || || 3:; 393 | j 431 #: i.7 j | 3 || | j || 3:03 ji:1 || 4-2 | . !!! 13 is Siſiº || 3 || | #7 3 tº $334 || 3 ºf 91.6% 2, 16 - 5-1 27 2.06 .9S93S7 4. .01 (1613 19 | 844243 || 3 || | .851603 || 3 || | .989810 3. # .01.0360 20 9,844372 | | 9.S541S0 | ..., | 9.989S93 • 0.010 * | *ś| 3:15 |*; 206 |*; 42 | ". ſº % jí || 3 |5 | #: 393 | j 42. º; 23 .844760 # jin, 3% jj; ; 4.3. º; 3. #; # ºść #: 350503 # dogog, 5 .8450(S . . . .853S62 | . 99.1156 .. on'ss. 26 jiຠ3.1% tº 2.06 | . 3 ºf QºS.1% #| #; giš j| 3 | #| || 3 || || || #| # }} | ºff #% ºil | }} º; 9 .845533 | " §§§ | 3.97 | jºić; 43 | tº 2. 1 - e * .007833 30 || 9345632 2 º 9.853242 : 9.99.2420 º loº § #| 3 || | # #} | ºg | #3 | "º #| #| #| || #| #| #| º º { .846 || 75 . . * ** 45 • *- "º 4.2 | - Fº § | #6. 2.14 # 2.07 # 33; º? 35 | #3; ##| 3:2:35 | # º; 43; # § | #| 3:3 | # § j| 3 | dºsſ 39 j 3.13 || 3:47 ºš I º! || 3 || | |0ſº old. S16 || 3:3 | .852122 || 3:35 | .993604 || 3 || | .005306 40 || 9.846944 * - • * * Ai $º 2.13 o § 2.08 *::::::: 4.21 |0. º 42 | #7135 | 2, 3 || 3:1747 || 2:03 | jºš | 4:3" | . | 43 .847327 § | : j6% 2.93 jū; 4:34 #; #| |&##| #3 ºligº #| || 3: | # ºf: § $473.3 || 3:3 || $3:373 # jššío 4:3; oºgo 43 || $47.02 || 3:3 || 85.24% §§ jół63 || 4:3; on 537 47 | 847.3% | 3 || || $31,121 || 3. jºij || 4:3 | tº: § ºff; | #; sº 3. § jó963 || 4:3 | dºn; 49 | 843091 || 3 || || 850870 || 3: . .997:22. § .002779 50 S m F: - - - - - *** - |*ś| 212 |"; 209 |*; 42 |"; 52 | .84S472 # # .850.493 # § jº 43| | ºngº 53 | .8.18599 || “. sºs || 3 | jšší || 4:3 | doi:gº § ####| 3 || || 3: | 3 || | j || 4:3; objāj: #5 || $3; | 3 | | | joić | 3 |Q | jº, 431 | § 56 j || 3 || | Sºn | 3 || | jšº || 4:3; onio #7 | još | 3 || | j || 3 |Q | jº: | 4:34 | 1 º 3.H | 843; 3 is $33:2 ; .000758 § | #| ##| 3: | }} | ºff # ſº § | 3:3339 || 3 || || 8:36:1 || 3 || |...}}}. 431 ontº;3 60 | .849485 * .S.40485 | * 0.00000 || 4.21 || 000000 M. | Cosine. D. 1". Sine. D. 1". Cotang. D 1ſt. Tang. 4. 5 O TABLE XIV. N A T U R A L S I N E S A N D C O S IN E S. 242 NATURAL SINES AND COSINES. TA BLE XIV. 2|.006:40 3|. (){}960 4|.01280 5|.0) 309 5|.01600 02 13 33 4-O Sine. .00000 .00029 ,000.58 ,000S7 .00|| 6 .001.45 .00 | 75 .00204 .0()233 .0]262 .0029 | .0032) .00349 .00.378 .0{}407 .00436 .00.465 .00495 .00.524 .00553 .005S2 .000 l 1 Cosin, One. One. One, One. One. One. One. One. One. One. One. .99999 .99999 .999%).9 .99999 .99999 .99999 .99999 .99999 .99993 .9999 S .9999.S .99998 .9999 S 99.99$ .99997 .99997 .99997 .99997 .99996 .99996 ,00902 .99996 .0093 ||.99996 .9999.5 .9999.5 .99995 .99995 .9999.4 5|.99994 .9999.4 .99%)03 3|.99993 .99993 .01251 | .99992 .99992 .9999 | .9999 | .9999 | .90990 .99990 .99%)S9 .99989 .99999 .99988 .909.SS .99987 .999.87 .990S6 .99986 .99985 00669 .0069S .00727 .00756 .007S3 .00S14 .00S.14 .00S73 .000S9 .0.133S .01367 .01396 .01-42.5 .014.54 ,014S3 ,01513 .015.42 .0157] .01629 .01658 .01687 .01716 .01745|.99985 Cosin. Sine. .99966 ||. .99966 . . .99965 . .99963 . .99963 . .99962 |. .99957 . . .99952 . . .99950 |. .99949 . .99948 ||.0: .99947 |.05 jjiši. .99944 . . .99943 : . Sine. .03.490 .03519 .0354S). .03577|. .03606 |.9% .03635|. . (3661 |. .0369.3|. .03723|.9% .03752|. ,03781 |. .03S10|. 03S39|. .03S6S] .9992: .03S97 |. ,03926|. .03955|.9% .039S4 |. .04013|. .04042|.9% ,0407 ||. .04 100]. .04 129}. .04 || 59 . .04 ISS|. .042|| 7 |. .04246|. .04275|,999(19 4 || 90907 .99904 } | .99901 5|.99$96 4 .99$94 .99.893 3|.99SQ2 2|.99S90 .99SS9 .90SSS .99.SS6 .99SS5 ,99SS3 .99.SS2 .99SSI 4|.99$79 .99$78 .99876 .99$75 .99$73 .99S72 .998.70 .99S69 .99S67 .99S66 .99S64 .04769 .0479S .04S27 .05I 46 .05175 .05205 .05234|.99S63 Cosin. Sine. 33.99966 || .99905 ||. ºjö3 || .909ſ)0 . .90S9S I. .99$97 . - .0630S |.9% .06337 |.9 .06366 |. .06395 ||. .06424 |. .06453|. .064S2 |.9 .06.5 l l .9 .0654() |.9 .06569 |.9 .0659.S. ,06627 . .06656|. .066S5|. .067.14|. .06743|. .06773|. .06S02|. .06831 |. Cosin. .99S63 3|.99S61 2}.99S60 .99S5S .99S57 .99S55 .99854 37 | .99852 .06S60 .06SS9 .0691S .06947 .06976 Cosin. .99S51 39S49 | 890 Sine. Cosin. .01745|. .01774 .99984 .01803 .99984 .0IS32|.99983 .01S62|.99983 .01S9||.999S2 .01920 | .99982 .0}949 .9998 | .01978 .99950 .02007 .99980 .02036|.9997.9 .02065 .99979 .0.2094 | .9997S .02 [23] .99977 .02 152|.99977 .02I S1 | .99976 .022 || | | .99976 .0224ſ)|.99%)75 .02.269 |.9997.4 .0229S .9997.4 ,02327 .9997.3 ,02356|.9997.2 .023S5}.9997.2 .02414|.9997 | .02443.99970 .024.72: .99969 .02501 .99969 .02530|.9996S .02560. .99967 .02580 .02618 .0264.7 ,02676 .0270.5 .02734 .02763 .02792 .02S2] .02S50 .02S79 .0290S ,0293S .02967 .02996 ,03025 .03054 .030S3 .03 || || 2 .03141 . ()3| 70 .03 || 99 .0322S ,03257 03:2S6 .03316 O3345 .03374 ,03403 .034.32 .0346 | .03.190 .99964 .9996] .99960 .99959 .99959 .9995S .99956 .99955 .99954 .99953 .99952 .99951 .99946 .99942 .9994 l .99940 .99939 Cosin. Sine. SS D 87O Sine. Cosin. .06976|.997.56 .07.005|.997.54 .07034|.997.52 .07063).99750 .07092|.09748 .07 121 |.99746 .07 150|.99744 .07179 .99.742 .997:40 .9973S .997.36 5|.99734 24 |,997.31 3|.997.29 .997:27 .997:25 .997:23 .997:21 .99719 .99716 5|.99714 5|.99712 4|.997 10 .99708 2|.997.05 .99703 .99701 .99699 .996.96 .996.94 90692 .99889 .996S7 .996S5 .996S3 .996S0 .9967S .9967 G .99673 ,9967] .9966S .99666 .99664 .9966] .99659 99657 .99654 .99652 .99649 .99647 .99644 ,99642 .99639 .99637 ,99635 ,99632 .99630 .99627 ,99625 .99622 ,07875 .079()4 .07.933 ,07962 ,0799] .0S020 , ()S{}{9 .OS()78 .0S 107 ,0S136 .0S 165 .0S194 .0S223 .0S252 .0S281 .0S310 .0S330 ,0S36S .0S397 .0S426 ,08455 .0S4S4 .0851.3 .08542 .0S57] .0S600 ,0S629 .0S658 .0S6S7 .0S716,996.19 Cosin. Sine. 85O -— TABLE 243 XIV. 2: A TU RAL, SINES AND COSINTES. lM ; . | : 50 6C 89 90 Sine. .08716 ,0S745 .0S77. .0SS03 .0SS31 .0SS60 ,0SSS9 .0S9 IS .0S9.47 .0S976 .09:005 .09034 2}.09063 ,09092 .0912I .09150 .091 79 .0020S .09237 .09.266 .09.295 .09324 .09353 ,093 S2 .094] I .09.440 ,09469 .0949S .09527 ,09556 .095S5 .096 (4 .09642 .09671 || .09700 .0972)|. .0975S]. .097S7|. .09S16|. .09S45|. .09S7 .00003}. .09932|. .09961 |. .09990|. ... 10019|. $!. 1004S}. 7|. I0077|, . 10106 |. . 101.35|. . 10164|. . 10192|. 2|. 10221 |. . 10250), . 10279|.g . 1030S), . 10337|. . 10366|. ... 10395), , 10421 . . 10453|. Cosin, Cosin. Sine. Cosin. Sine. Sine. .996 19 .996.17 .996.14 .996 12 .99609 .99607 .99604 ,99602 .99599 .99596 .99594 .9959 | .99.5SS .995S6 ,995S3 .995S0 .99578 .995.75 .99.572 .99570 .99567 .9956.4 .99562 .99559 ,995.56 .99553 .9955] .99.54S .995.45 .99.542 .99540 .99.537 .995.34 2 . . . 21 S7 . 10453 . 104.82 . I (951 | . 10540 . 10569 . I 0.597 ... 10626 . 10655 . 106S4 . I 1320 . I 1349 . 1 137S . I 1407 . I 1436 ... l 1465 . I 1494 , l l 523 . I 1552 . I | 5SC) . I 1609 . l l 63S . 1 1667 . 1 1696 . l 1725 . l l 754 . I 1783 . 1 [S] 2 . 1 1840 . I 1869 . 1 ISSS . I 1927 . . 1956 . I 1985 . 12100 . 12129 . 1215S Cosin. .99452 .9944) .994-f6 .99443 ,99440 .99.437 ||. .99.434 |. .99.431 . .9942S |. . 10713]. . 10742|. . 10771 |. ... 10SU0|. ..] OS29 . . 10S5S|. . 10SS7 |. ... 10916|. . 10945}. . 10973.9% . 11002|.9 . 1103 ||.9 . I 1060|. . l IOS9|.9% . I l l l S}. ..] I 147|. ... l l l 76|. . 1 1205|. ..] 1234|. . I 1263|, . I 1291 .9 .99.357 ,993.54 .9935i .99.347 .99344 .993.4] .99.337 .99.334 .993.31 .99327 .99324 .99320 .99.317 .993 |4 .993 l () .99307 .99303 .99.300 .99.297 .99.293 .99.290 .992S6 .992S3 .99.279 . 12014]. . 12043]. . 12071 |. ,99265 .99.262 .9925S .99.255 Sine. , 12|S7|. , 122 || 6 |. . 12.245}. . 1227-1 |. . 12302|. . 1290SI. . 12937|. . 12966|. . 12995|. . 13024|. . 13053|. , 130SI |. . 131 10|. . 13139|.0% . 1316S . . 13197|, . 13226|.9% . 13254 . . 132S3|.9 . 13312|. . 1334 ||. ..] 3370}. . 13399 |. , 13427 .9 . 13156}. . 134S5|. . 13514|. , 13543|. . 13572. . 13600|. . I 3629 |. . 1365S]. . 136S7 |. . 13716|. . 13744]. . 13773|. . 13S02|. . 13831 . , 13S60| ,990.31 ,990.27 Sine. . 13SS9 . 13917 Cosin. . 13917 . 13946 . 13975 . 14004 . 14033 . 1-4061 . I.409ſ) . I 11 19 . 1414S . 14 177 . 14205 . 14234 . 14263 . 14292 . 14320 . 14349 . 1437S . 14407 . 14436 ..] 4:464 . 14-193 . 14522 . 14551 . 145S0 . 1460S . 14637 . 14666 . 14695 . I-1723 . 14752 . I q.78] . 14S1() . 14S3S . 14S67 . 14S36 . 14925 2 . . ] 1954 . 149S2 ..] 50.1 ! . 15040 . 15069 . 15097 . 15126 , 15155 . I 51S4 . 15212}. . 15241 |. . 1527()|, , 15299|. . 15327 |. . 15356|. .15385|. . 15414 |. . i5442|. . 15471 |.9 15500|. Cosin. .99ſ)27 .99.023 .990.19 .90015 ,990 l l .99006 .99.002 .9S99S .9S994 .9S990 .9S9S6 .9S9S2 .98978 .9S973 .9S969 .9S965 .9S96] .9S957 .9S953 .9S94S .9S944 .9S940 .9S936 .9S93] .9S927 .9S923 .9S919 .9S914 3Sgiól. .9S906 ||. .9S902 .9SS97 . .9SS93 . .9SSS9 |. .9SSS4 Sine. . 15643 . 15672 . 15701 . 15730 . 1575S . 15787 . 15816 . 15S45 .15S73 . 15902 . 1593] ... I 5959 . I 59SS . 16017 . 160-16 . 16074 . 16103 . 16132 ..] GH 60 ..] 61 S9 ... }621 S . 16246|. . 16275|. . 16304|. . 16333}. . 1636]. I. . 16390}. : 6 : : : .9SSSO || , 16 .9SS76 , .9SS7 | .9SS67 . .9SS63 . .9SS5S . .9SS54 .9SS49 . .9SS-15 .9SS4 I . 15529 |.9S . 15557 |. ..] 55S6|. . 156] 5 |. . 15643|, Cosin. Cosin. .9S769 .9S764 .9S760 .9S755 .9S751 .9S746 .9S741 .98737 .9S732 .98728 .9S723 .9S718 .9S714 .9S709 .9S704 .9S700 .9S695 .9S69() .9S6S6 .9S6SI .98676 556.33 5|ºsé25 533|.9S624 .9S551 .9S546 .9S541 .9S536 .9S531 .9S526 .9S521 4|.9S516 .9S5] I .9S506 .9S501 .9S496 .9S49 | .9S4S6 .9S4Sl Sine. S33 829 SOS 15 14 i l § 244 TABLE XIV NATURAL SINES AND COSIN ES. 103 1 10 129 132 14C M. Sine. Cosin. Sine. Cosin. I Sine. Cosin, Sine. Cosin. Sine. Cosin.M. 0|, 17365|.9S4S1 |. 190SL |.9S163 .20791 |.978 || 5 || .22 195|.97437 || 2 | 102.97.03() 6() | |. 1739.3|.9S476 |. 191')9].9S157 |.20S20|.97S(19 .22.523|.9743() .2ſ22ſ)|.07(23| 59 2|. 17422|.9S47 | | . 1913S1.9S 152 |.20S4S}.978()3 .22552|.97.42.1 | .2424)|.97()15] 5S 3|. 17451 |.9S466 |. 19167 .9S146 .20S77|.97.797 .225Sſ)|.974 || 7 | .24277|.970ſ S. 57 4|. 17479|.9846 . 1919.5|.9S140 | .20905|.977.91 .2260S .974 | | | .2.1305 |.9700 || 56 5|. 1750S |.9S455 . . 19224|.9S135 .20933.977S4 .22637 .974() { | .24.333|.96994| 55 6|. 17537|.9S450 | . 1925.2|.9S 129 .20962.97.77S .22665|.97.39S .2.1362|.969S7| 54 7|. 17565.9S445 |. 192S1 |.9S124 .20990|.97.772 .22603 |.97.39|| .24390|,969SO 53 8|. 1759.4|.9S44() |. 19309|.9S IS I.21019|.97766 | .227:22|.973S4 .214 |S|.96073| 52 9|. 17623|.9S435 | . 19338|.9S1 12 .21047|.97760 | .22750|.97.37S .24446|,96966 51 10|. 17651 |.9S430 | . 19366|.9S107 | .21076|.97754 .2277S].97.37 || .24474|.9695); 5() l 1 |. 176S0|.9S425 | . 19395|.9S101 i.21 104 |.977-1S .22S07 |.0736.5 ! .245(3|.95952] 49 12|, 1770S|.9S420 || 19423|.9S006 |.21 132|.977.4.2 . .22S35|.97.35S | .2.1.53 ||.96945 q8 13|. 17737|.9S4 || 4 || 19 (52.98090 | .21 | 6 ||.97735 H .22S63,9735 | | .245.59 .96937, 47 14|. 17766|.9S409 || 1948 ||.9SOS-1 | .21 lS0|.97729 | .22S92| 97.315 .245S7|.96930|46 15|. 17794 |.9S4 H | . 19509|.9S079 .212 IS].97723 .22020|.973.3S .24615|.969.23| 45 16|. 17823|.98.309 i. 1953S).9S073 .21216|.977 || 7 | .2294S i. 9733| | .21644.969 |G| 44 17|. 17852|.9S394 | 19566|.9SQ67 .21275|.977 | | | .22977 .97.325 | .24672|.969(10| 43 18|. 17SS0|.9S3S9 19595|.9S416 || .21303.977 U.5 .23005|.97.318 . .217()() (96902| 42 19|. 17909|.9S3S3 | . 19623|.9SQ56 .2133| |.976)S 23(133}.973| | | .247.2S].96S94| 4 | 20|. 17937|.98.378 || 19652|.9S0.50 .21360|.976)2 | .23062.973) | | .21756|.96SS7 4() 2||. 17966|.9S373 || 196SO |.9SO44 .21338|.976SG | .2319() .97.29S | .247S4|.96SSO 39 22|, 1799.5}.9S36S 19709|.9S039 || .214 || 7 |.976Sſ) .23| |S|.9729 .24S13|.96S73| 3S 23|. ISO23|.9S362 |. 19737|.9SO33 .21415 |.97673 .23146|.972S4 | .2484 ||.96SG6||37 24 |. 18052|.9S357 |. 19766|.9S027 . .21474|.97667 | .231755.97.27S | .24S69|.96S5S 36 25l. (S08 ||.9S352 |. 1979.4|.9S02 | | .21502|,9766 | | .232)3].97.27 | | .24807|.96S5 || 35 26|. IS109|.9S347 | . 19S23|.9S016 || .21530|.97655 | .2323| |.97.264 | .24025.96S44; 34 27|. 18138|.98.34 ||. 19S5 ||.9SOI () .21559|.9764S .23260 .97.257 | .24954 |.96S37| 33 28|. 1S166|.9S336||. 19SS0|.9S004 ||.215S7 .97642 | .232SS|.0725 | | .249S2|.96S29| 32 29|. 18195|.9S33| |. 1990S] .97998 || .21616|.97636 .233| 6 |.97244 .25010|.96S22| 31 30|. 18224|.9S325 | . 19937 .97992 | .21644|.9763() .233.45|.97237 | .2503S).96S15|30 31 |. 18252|.9S320 | . 19965}.979S7 . .21672.97.623 .23373}.97.230 | .25066.96S07| 29 32|. 1S281 |.9S315 . 1999 ||.979Sl .21701 |,97617 | .2340] | 97223 .25094 |.96S00| 2S 33}. 18309|.9S310 || 20022|.97.975 .21729}.976 l l .23429 |.972| 7 | .251.22.067.93| 27 34 |. 18338|.9S304 || 2005 ||.97960 | .21758.97604 ||.23-15S .972.10 .251 51 |.967S6| 26 35). 18367|.9S299 || 20079|.97963 .21786.9750S I.234S6|.97.203 | .251.70 |.9677S 25 36]. 183951.9S294 | .2%) IOS .9795S .21814.97592 | .2351.4|.97 196 .25207 |.96771 24 37|. IS-124|.9S2SS | 20136|.97952 . .21843 .975S5 .235.42|.971S9 .25235|.9676-1 23 38|. 18452|.9S2S3 | .201 (;5|.97.046 .21871 |.97.579 .2357 ||.971S2 | .25263.96756. 22 39|. 1848||,9S277 .2ſ)|Q3|.97.94ſ) | .21S99).97.573 . .23509|.97.176 / .25201 |.96749| 21 40|. 1S509|.9S272 .20222|.979.34 .21928.97566 | .23627 |.97 169 .25320|.967.42| 20 4 ||. 1853S).9S267 .20250,9792S .21956|,97560 .23656 .97l G2 .2534S |.9673-1| 19 42|, 18567|.9S.261 .20279| .97922 .21985|.97553 .236S4|.97 l;5 | .25376|.96727| 18 43]. 18:595|.9S256 || 20307|.97916 . .22013|.975-17 | .23712|,9714S | .25-104.96710| 17 44|. 1S624|.9S250 | .20336|.9791() .2204 ||.97.54 | | .2374()|.97 || 4 || | .25432|.96712| 16 45|. 1S652}.9S245 || 2036-1|.97905 || .22070.97534 .23709|.97134 || .25460|.96705] 15 46]. 1S6S ||.9S210 20303|.97S99 || .2209S .9752S .23707 .97 127 | .254SS}.966.97| 14 47|. IS710}.9S234 .2042||.978.93| .22126|.97.521 |.23S25|.97 120 .255.16|.96690 13 48]. 18738|.9S229 .20450|.97SS7 | .22155|.975 ( 5 I .23S53.97 l 13 | .255.15|,966S2] 12 49ſ. 18767 .9S223 .2047S |.97SSI .22183|.9750S 23SS2}.97 106 | .25573|.96675; l I 50|. 18795|,982. S I .20507|.97875 . .22212|.97502 | .239|(}|.97 100 | .2560||.96667| 1() 5||. ISS24|.9S212 .20535|.97S69 |.22240|.97496 || .23938 .97093 | .25629|,96660. 9 52|, lSS52}, QS2(17 | .20563|.97S63 .2226S).97.489 .23966|.970S6 .25657|,96653. 8 53|. 18881 .9S201 | .20592|.97S57 .22297 .974S3 .23995}.97.079 | .256S5|.96645| 7 54|. 18910|.9S196 .20620|,97851 | .22325|.97476 .240.23|.97072 | .25713].966.3S 6 55| |SQ38|.9S 190 .20649|.97S45 .22353|.97470 .2405 ||.97005 | .2574 |,06630 5 56}. 18967|.9SIS5 .20677|.97S30 .223S2|.97463 .24079|.97()5S .25769|.966:23| 4 57|. 18905.9SI 70 .20706|.97S33 .224 (0.97.457 .2410S).97.05 | | .2570S),96615| 3 58|, 19024 |.93174 .2073 ||.97827 .22438|.97.450 .21 136].9704-1 | .25826|.9660S 2 59|. 19052|.9816S .20763|.97S21 | .22467|.97444 .24164 |.97(137 .25854 .96600|| 1 60|. 10081 |.9S163 .2079 ||.97815 .22495.97437 .24102|.97.030 .25SS2|-96.593 () M. Cosin. Sine. || Cosin. Sine. Cosin. Sine. I Cosir. Sine. Cosin. Sine. M. 7 :) J 780 77 o 7 () > 750 - -º-º-º-º-º-, ----- X I V. NATURAL SINES AND COSINES. ". i i l 160 17O 182 190 Sine. .25SS2 .25%) iſ) .2593S .25\}66 .26|{}7 .26 35 .26163 .2619 | .26.219 .26247 .262/5 ,26303 .2633l ,26359 ,263S7 .264 15 .26-143 .2647 i .26500 .2652S .26.556 .265S4 .266 [2 .26640 .2665S .266.96 ,26724 .26752 .26780 .26SOS .26S36 .26S64 .26S92 .26920 .26%) AS .26976 }|.27(10.] .27032 2} .2706ſ) .27()SS .27 | | 6 5|.271 44 ,27172 .27200 .2722S ,272.56 ,272S4 .27312 2|.2734() .2736S .27396 .27-124 .27.152 ,27.1Sſ) .2750S .27536 .2756-1 . . Cosin. .96547 . .965-40 | . .96.532 .27 ,9652; 3657l 2. .96509 .96502 ||. .9649-1 964s6 || .96479 . .96471 .9646.3 . .96.456 .2S .964-1S .2S .964.40 . .964.33 ||. .964.25 .2. .964 17 . .964 I() { ...; .964()2 . .96394 963$6| ,96379 ,9637 | .96363 . .96355 . .96347 H. .96340 . .963.32 . .963:24 .963| 6 ||. .9630S . .9630 | .96.293 . .962S5 . ,96277 .2 .96:269 . .9626|| 56.253 || .962.46 . . ,9623S ,9623() ,962:22 .96214 .96206 .961.9S .96 || 90 .96 || 82 .961.74 .96 | 66 ,961 5S ,96 50 .961-12 ,96 || 3-1 .96126 25|.95S16 ||. 2|,95S07 , .2SS47 .2SS75 .2S903 .2S93] .2S959 .2S9S7 .200) 5 .29()42 .29070 .2900S .291.26 .2915-1 .29 US2 .2920ſ) .29237 Cosin. .961.26 2.961 is .961 10 .9602 55|.96013 .96005 ..Q5997 ,959.SS) .959Sl 5}.95972 .95964 .95956 .95948 .95940 ||. .95931 2|.95923 . .959 || 5 || . .95907 . $1.95S9S . .95S90 ||. ** .95SS2 .3 ,95S74 . .95S65 . S5}.95S57 . 3| .95S49 . .95S4 I .95S32 , .95S24 Sine. Cosin. Sine. .29237 .20265 .29.293 ,2932! .2934S .29376 .294.04 .29432 .29460 .294.87 .295] 5 7 .29543 .2957 I .20599 .29626 .2965.4 .296S2 .29710 .29737 .95790 .3ſ ,957.91 .957S2 ||. 4} .95774 . 2|.95766 . .95757 .95.749 ,957.10 .95732 ,95724 . . .957 15 . .95707 . ,95698 ||. 3569) * ,956Sl .95673 .95664 . .95656 . .956.17 .95639 .956.30 .30902 .956.30 ,95622 .956.13 95605 .95596 ,955SS .95579 .955.71 .95562 .95554 .955-15 .95536 .9552S .95519 .955] 1 .9550.2 .95493 .954S5 .95476 .95.467 .95459 .95450 .9544 l .95:433 95.424 .954 15 .95.407 .9539S 5|.953S0 3}.953S0 .9537.2 .95363 .953.54 .95.345 .95337 . .9532S , .95319 5|.95310 2|.953()1 .95203 ,952S4 .95275 .95266 .95.257 .95.248 .95:210 .95231 .95.222 .952.13 .95204 ,95195 .95 l l 5 . .95106 || . .30902 .300:29 .30957 .309S5 .31012 .3104() .3|06S .31095 .3| 123 .31 151 .3] 178 .31206 .31233 .31.261 .31289 .31316 ,313-14 .31372 .31.399 ,31427 .31.454 .314S2 .31510 .31537 .31565 .31593 .3]620 .31648 ,3]675 .31703 .31730 .3175S .317S6 ,31 SI 3 .31S96 .31923 .31951 .31979 .32006 .32034 .3206 l .320SQ .32! 16 .32144 Cosin. Sine. 9.5106 3255 .95097 jºss' .32 .95079 .32 .95070 . .95061 .95052 .3: .95043 | . .950.33 || 3: .95024 . .950l 5 | . .95000 . ,32914|. ,32942|. .32969|. ,32997|. .33024 |. .33051 |. ,33079|. .33105}. .33 ||34 |. ,331.61 |. .94SQ7 . ,94SSS . .94S7S , .94.869 . .94S60 . .94.997 .949SS ,94979 .94970 .94.96 | .94952 .94.943 ,94933 .94924 .94.915 ,94906 ,9485.1 ºisſºl. .94.832 . .94S23 . .94S14 ,9-1805 .94795 : , .947S6 .94777 |. .9476S . .9475S . .94749 .9474() .94.730 .9472] . . .947 || 2 | . .94702 .946.93 .3217 ||, .32199 |. .32227 |, 32254 |. .322S2]. .95|S6 . . .95177 . .9516S . .95159 . ,94561 .9.4552 .33764 . ,337.92 .33S19 .33S46 .3387.4 .33901 .33929 .33956 .339S3 .340 || || .3403S ,34065 .34093 .34 120 .34 (47 .34 175 .34202 Sine. Cosin. Sine, Cosin, Sine, Cosin- Sine. Cosin. Cosin. .94.552 .945.42 2.94533 9|.94523 .94514 .94504 .94.495 .94 l l S .94] OS .9409S ,940SS .9407S .9406S .9405S .94049 .94039 ,940.29 .940.19 .9.1009 .93.999 .939S9 .93079 .93969 Sine. 740 ºr 3o 739 71O 700 57 55 54 53 52 50 TA B L E NATURAL, SIN ES AN O COSI N ES. X I V. M. 10|.. 209 2 10 2 32 23O 24 O Cosin 2|. 93969 .93959 .93949 .939.30 ,93929 ,93919 56|.93909 3|.93S99 .938S9 .93S79 5|.93S69 73|.93359 .93S49 .93S39 .93S29 2|.93S 19 .93S09 56 .937.99 .93789 .937.79 S|.93769 .93759 3|.937.4S .9373S .9372S .9371S .9370S {}|.9369S .936SS 3|.93677. .93667 .93657 5|.93647 2|.93637 .93626 .936 || 6 4 |,93606 .93596 .93585 293575 .93565 ,93555 ,935-14 .93534 2|.93524 .93514 .93503 .93493 .934S3 .93472 5|.93462 .93452 .93441 .9343| .93420 ,934.1() .93400 .933S9 .9335S Sine. 690 $2| 33379 | .93.368 . Sine. 35S,37 ,35S64 ,35S9 | .359 US .359.45 .35973 .36000 .36027 .360.54 .360Sl .36108 .36:135 .36.162 .361.90 .362t 7 .36244 .3627 | .36298 .36325 .363.52 .36379 .36106 .36434 .36461 .364 SS .365.15 .36542 .36569 .36596 .36623 .36650 .36677 .367.04 .3573| .3675S ,36785 .36812 .36 S39 .36S67 .36S9 | .36921 .36948 .36975 .37002 .37029 .37056 ,37083 .371 UC) .37 [37 .37 (6 | .3719 | .3721S .37245 ,37272 Cosin jºš ,9334S .93337 .93327 .933| 6 .93.306 . .93295 |.. ,932S5 .37 .93274 . .93264 .93253 . ,93243 |, .93.232 . .93222 .. .932. I .9320] Sine. Cosin, .37.46 l ,374SS .93190 .37 ,931 S0 . .931 69 |. .931.59 .93|4S .931:37 .931:27 . .93] 16 |. .93|{}6 . .93095 . .930S4 .93074 . .93ſ)63 . . .930.52 . ,93042 . .3S295|. .3S322]. ,38349). .92999 || .3837 .929.8S . .9297S . .92967 .: .92956 . . ,93031 .93020 .93ſ)! () ,929.45 .92935 .92.92.1 .929 || 3 ||,3S ,92902 .92S92 .3S .92S81 .92S70 .92S59 33sſ) || .92S3S .92S27 .: .92816 ||. .92.805 i. .927.94 32:54 i. .92773 . .92762 .92751 .927.40 .92729 . .9271S Sine. .927 [S .926S6 .926.31 3| 32630 || .926.09 .92587 3333i .923 || 0 ||. 3|,922.99 ||. .922S7 . .92276 |. .92265 .922.54 .92243 .922.3 | .92220 .92209 #| gigs ,921 S6 .921.75 5|.92164 2|.92 (52 .920.50 Sine. 33707 || .92697 . .92675 . 5|.92664 2|.92653 . . . ,92642 |. .92598 : , .92576 . S|.92565 5|.92554 .92543 . .92532 . 3|.92.52 | | . 3|.925.10 . ,924.99 ||. Cosin. .92050 ,92039 .92028 53|.9.2016 .399.02 .3992S ,399.55 .4016S .401.95 .4022 | .402-1S .40275 .40301 .40328 .40355 40381 .40408 .40434]. .40461 |. .404881. .40514 |. ,40541 |, .40567 |. .4039.4|. .40621 ..10647 .40674 Cosin. T | .92005 .9.1994 | . .919.S2 . ,9197] .9 1959 . . .9.1948 . .9 1936 jiši. .919 || 4 ||. .9.1902 .91891 ,91S79 .91S6S S|.9.1856 5|.91845 .91833 ||. S|.9|S22 . ,918) () . .9 [799 .917S7 . .01775 ,9} 764 . S|.9) 752 . 5|.91741 |. ,91729 , .917 IS .91706. ,91694 ,916S3 .9167 I .9 1660 .9 1648 5|.9 1636 2|,9| 625 .9 || 6 || 3 5|.9l 6(11 .91 590 . .91578 . . .91566 . .91 555 . . .91543 |. ,91531 .91519 .9 |50S .91496 .914S4 .91472 662 Sine. J0673 .407(M) .407.27 .40753 .41892 .4 1919 .41945 .4 U972 .4 |99S ,42024 .42051 .42077|. 5 : .42104|. .42130ſ. ,42156|. ,42 IS3 |. .42209|. .42235|. .42262|. Cosin. Cosin. .9 || 355 .91343| { .91.33:1 .91319 .91307 .91295 3|.912S3 .91272 }|,91260 .91248 .9 1236 .91224 2|.912H2 .9 1200 .91 ISS .91176 .9l 164 5|.9l 152 .9l 140 .91 12S .91 16 .9l 104 7|.91092 .910SO ,9106S .91056 3|.91044 .9.1032 .91020 .9100S .90.996 .909S4 .90972 .90960 .9094S .90936 .9ſ)924 5|,909 || | .90S90 .90SS7 |.90S75 .90S63 ,90S51 .9ſ)S3%) .90S20 .90S1.4 .9ſ]SO2 .9079() .9077S .90766 .90753 .30741 .90729 | 1 Xl V. NATURAL SINES AND COSINES, M I. i 259 26O 270 283 290 Sine. ... Sine. Cosin. .42262|. .422SS|. .42315|. .42341|. ,42367|. .42394 . .42.420/. 42446|. .42473|. .4.2499|. .42525 .90,507 .42552|. - .42578|. .42604 |. ,4263| .42657 .426S3 .42709 .42736 .427.62 .427SS .42S15 .42S4 I .42S67 .42S94 .42920 .42.946 .42972 .42999 .43025 .43051 .43077 .904.46 .90433 ,90421 .9040S .90396 ,90371 .9035S 2 . .43942 5 .44020 S | .44203}.S. .903S3 || 4. 90346; .43S37 .43S63 .43S$9 .43916 89S79 .S9 S67 .S9S54 .S9S4] .S9S2S .4396S .43994 .44046 .44072 .4409S .44 | 24 .44151 .441.77|. .89777 .89764 .S9739 .4-1229 |. .44255].S. .44281 |. .44307|. .S9493 2| SQ 167 .8945.4 ,8944 | S|.S93S0 .S9350 S}.S932 | .449S4|.893 l I .45(; 10 .45ſ):36 .45062 ,450SS .451 14 ,45140 .45166 .45.192 .4521S .45243 .45'269 ,45295 .4532! .453.47 .45373 .45.399 Cosin. .892S5. .S9272 ,80259 .S9245 ,89232 .892.19 .S9206 ..SS: Q1 . Sine. (333 .S9SI6 . .89S03 I. .S9790 . .S97.52 .4% .897.26 . $35,13]. .S9506 . .894S0 . .8042S . .S91 || 5 ||. .S) 10:2 . . . .S9376 . .S9363 . . .S9337 . , S929S , ,89193 . .89 ISO . .89.167 I, .S9153 , , S9140 . .S.9127 . .89] 14 , ..SS006 . .S7979 |. .S7965 |. Sine. |Cosin. .4S4S1 |,87462 .4S506|. .4S532}. .48557 |. .4S5S3|. .4S60S}.S7.39|| .4S634 .4S659|.87363 .4S6S-4|. .487 10|. .487.35|. .4876] .4S7S6 .4SS| 1 .4SS37 .4SS62 .4SSSS .4S913 .4S93S .4S964 .489S9 .S727S .S7235 .8722 | .872O7 .S7 193 .87] 7S 4|.87164 .S7 150 .87 136 .S7] 2 I ,491.16||.S.71 ()7 .4914 l |.S709.3 .49 66 .87079 .49192|.870.64 ,492.17 | S7050 .492.42|.S7.036 .4926S].S7021 .S7007 .S6993 .86978 .86964 ,86949 .S6935 .S6921 .86906 .86$92 .S6S7S .S6S63 .86S49 .86S34 .86S2ſ) .S6S05 .S679) S6777 .S6762 .4974S | S673S .49773}.S6733 ,49798|, 49.495 .4952 l .495.46 .4957 l , 49596 ,496.22 .49647 , 4967.2 .40697. ,49723 ,49S90 |.S .499.24 .86646 , 49.950 |.S663.2 ,4997.5 S65 | 7 .50000; S6603 Cosin. Sine. 600 ,87264 || 4 .S7250. < 2|,43104 5|.431 S2 S}.43261 9|.432S7 .90.334 ,9032] .90309 .90284 .90271 .90259 .43.130.90221 43156 .901 S3 .90|7| .43209 .43235 .433 13 .43310 .43366 ,43392 .434 IS .43445 .43471 .43497 .43523 .43549 .43575 .43602 .43628 .43654 ,436S0 .437(16 ,43733 ,437.50 .437S5 ,438|| .43S37 Cosin. 0 +O .90IOS ,900S2 .9007) .900.57 .900-15 .900.32 .900. 9 .90007 .8999.4 .899Sl .S$).96S .89956 .8994.3 .89930 .S$918 ,890ſ).5 , S989? .90296 . .90246 , .90233 . .9020s .90.196 |. ,901 5S . .90146 , .90.133 |. .90120 4. .9()005 . Cosin. .89101 .89087 .89074 .S9061 26 SS674 52|, SS661 .SS634 ..SS553 ..SS530 ..SS526 ..SS5 2 ..SS499 , SS4S5 , SS472 ..SS-15S .465S7 ,466 || 3 .46639 .46664 .46690 ,46716 .46742 .8S431 ..SS417 .8S404 ..SS390 :SS377 .8S363 ..SS349 ..SS336 ..SS322 ..SS30S .SS295 Cosin. Sine. G20 2 : .47255 2 . .47332 ..SS6SS , .8S6-17 | .47 , SS62ſ) . 5 : SS607 .47 ..SS593 .47 ..SS5Sſ) 47 33|, SS566 .47 ..SS445. Sine. .46947 .46973 .46999 .47024 .47050 .470.76 .47 101 .47 127 .47 153 .471 7S .4720.4 .47229 Cosin. .8S 295 .8S2S] .8S.267 .88254 .8S24() .8S226 .8S213 ..SS199 .8SlS5 .8SI 72 ..SS 5S .8S144 ..SS130 .8Sł 17 .8SIG3 .8SOS9 ..SS075 .88062 .8SO4S .8S034 ..SS020 .472S] .47306 .S7993 .87951 .S7937 .S7923 5|.S7909 .87S96 .87882 .S7S6S .87S54 5|.S77S.1 .S777ſ) .S7743 .S7729 .877 l 5 .S770 .S76S7 .4S009|.S7673 .4S124 |.S7659 .48150|.S7645 .4S175|.S7631 .4S20t|.87617 4S226|.S7603 .4S252}.S75SQ .4S277|.S7575 .4S303}.S756] .48328|.S75.46 .4$35.4|.S753.2 .4S37)|.875 IS .4S405|.87.504 .4S430|,8749ſ) .4S456|.S7.476 .4S4S1 | .87462 ,47997 .4S02.2 .4S0-1S .4S073 Cosin. Sins. .S784() , .87S.26 |. 4}.878] 2 . 0|.S779S . }|.S7756 |. 6 1 O * -- - --— — ...— ... -- 248 TA BL E XIV. NATURAL Sl NES AND CO SIN ES, -. 3U > 31 o 3233 33O 34 D M. Sino. Cosin. Sine. Cosin. Sine. Co sin. Sine. Cosin. Sirie. Cosin M. 0.5nio SG603.558577.525928 Stó 56.S3S6755T, S2965 l.50J25.86588 51529.85702.530l 7,8-1789.54. SS.83S5l .559-3 .S.2SS759 2".50)50.86.573.5l 554.85687.5304l .8-1774 .545 3.83S35 .5596S.82S7 58 3.50076.86559.5l 579.85(72.53066. S-75) .54537.S3S 9.55992.S2S55 57 4 5U0 .86544.5604.85657.53)9l .8-7-3 .5456l .S3S04 .56) 6.S.2S3956 5.50126.86530.51628.85642.53l 15.S. 172S .545s G.S.37S8 56040. S2S22 55 6.501 5l .865l 5.51653.85627.53l 40.S. 17 2.546l 0.S.3772 56)64 .S2S)654 7.5Ul76.8650l .51678.856l 2.53l 34.S 697.50:35. S3756 .56)SS.S279U 53 8.5020 .86.186.5l7(3.S5597.531S9.846Sl .54659.S.37 10.56l 12.S277.352 0.50227.8647l .5728.S55S2 53214 . S4666.546S3.S.3724 .56l 36.827575 l 0.50252.S6457 .51753.S.5567.5323S.S4650.547(S.S.37(S.56l G0.S274 5) 50277 . S6442.5 778.8555 53263.S.4635.54732.83692.56l S4 . S2724.49 2.5U.312.86427.5 S03.85536.532SS.846l 9.54756.83676.562()S.S27(S 48 lo.5 327 .86 3.5 S2S.8552 . 533l 2.S 6}4 .547Sl .S.366) .56232.82002 l7 4.5 352.86398.5852 .S55 6.53337.S 158S.54 S().5 .S36 5.56256.S2G756 15.50377.863S4.5. S77.851) .5336 .84573.54S20. S3629.562S0.8265945 l6.5).103.86369 .59)2. S5476.533S6.S.4557.54S54. S36 3.563 5.8261344 l 7.5).42S.S6354.5927.S546 .534 l. S4542.5 S78.S3597.56320.8262643 S.5(1453.8634().5 ).52 .S.54 6.53435. S4.526.5-1902. S35S .56353.S2; () 42 9.5 7Sj.86325.5l 977.S 5:3.536).S.15l .5 1927.S3565 .56377.S.2503 4 l 2().5}5(3.863l () .52 (2.85 6.534 S- .84-95.54)5 .S3549.564() .825774) 2.5(52S.8325.52(26.854 () .53509.8 s() .54975.83533.56-125.8256 39 22.5).553.862Sl .52)5l S53S5.5353 . S4 164 .54999.S33 7.5644) .825.143S 2:3.5).578.86.266 .52)76,8537().5355S, S- S.55}24.8350 .56-73. S252S 37 24.506 3.8625 .520.85355.535S3, S4.433.5504.S.S3-1 S5.56-97.825 36 25.5062S.86237.521 6.S534 .536)7.844 7.55072.83-6} .5652 .S249535 26.5t)65-4.86222.52 5l .85325.53632.S-4(2.55097.S3453.565-15.8247S34 27.5679.86207 52 75.853l 0.533,50 .843S6.5512 . S3.137.56569.821G233 28.507).4 .SG 92 522(0.852)4.536Si .84370.55l 45.8342 .56593.8246.32 29.50729.85 l7S 52.225.85279.53705. S-355.5569.83-(J5.566 l7.8242031 3).5075-4 .S663 .52250. S5264.53730.81339.55 94.833S).5664 .824 1330 3l .50779.S6 S 52275.85249.5375. .S 1324.5521S.S3373.56665.82.306 29 32.50S0 .S6 33.522)).S523. .537.79.S.43}S 5522.83356.5f3S9.823SO 2S 33.50S2).86 9.5232- .S52 S , 53S() .84292.55266.8334() .567l 3.S2363 27 3 1.5) 35.1 . SG 0 523.19 .S.5203.53S2S.8-1277 .5529l .S3324 .56736.82.317 26 35.5)S79, SG()S) .52374.85 SS.53853.8426l .553l 5.S330S .56760, S233025 36.500) .8G()74 5230).85 73.53877. S-245 .55339.832)2.567S4.823 4 24 37.5)929.8605) 52.423.8557 .5392.81230.55.363.83276.56SOS,S2297 23 38.5095 .S6U45.524. S.S.5 12.53026.8-2l 4.553SS.8326) .56S.32.S.228 22 39.50979. S6030.5247.3.85 27.5305l .S. Js .554 12.S.321-1.56S.56.8226-1 2 l 4U.500 .8605 .524)S.85l 2.53)75. S- S2 55-135.S3228.56SS0.822 S 20 4 .5l (20. S600m .52522.S.5006 .54l 0.S 67.55460.83212.5600 l.S223l 19 42.5105-1 .S50»S5 .52547.850S .54() 2.4.8. 15 .554 S- .83l 95.5602S.S22 4 l S 4:3.5079.8507().52572,856; .5-) 1.8 35 .5550).83 70,56):52.82l 0S l7 4.4.5 (4 . S5956.52597 . S5 5 5 (173, S42 .555.3.3. S3 63.5t973. S2 Sl l6 45.5l 129.S594 5262 .85035 .54)07.S4 () .55557.S3l 17.57000.82165 l5 46.5 54.85026.526-6.8502( . .5-22. S-10SS.555Sl .S3l 3l .5702.1 .82 4S 14 47.5 l79.859 5.237 .85()(5 5 46.84072.556(15,83 l5.57047.8232 13 48.5l 204.85S93.52606. SQRQ ,5 l7 .S-1057.55630.83(3S.5707l .S.2 5 12 49.5l 220.85SS .5272) .S-1974 54 95.8404 .5565 .83082.57095. S2093 ll 50.5125-1 .S.5S66.527.45 . S-4059,5-220.84(25.5567S.S3066.57l l9.82082 l0 5l .5l 279.85S5.5277() .S. 1).13.542 .S-(X9.55702.8305().57l 43.82065, 9 52.51304.85S36.527Q4.81)2S .5426) .S.3994.55726.8303 .57 l67.82(4S 8 53.5l 329.85S2 .523 9.840 3.54203.8307S.55750.83) l7.57l 9l .S2032 7 54.5l 354 .85S06.52S S4S97 .543 -7.S3962.55775.8300 57.215.820 5 6 55.5 (37) .S5792.52SG9.84SS2.543.42.S.3916 .55700.820S5.5723S.8999 5 56.51401.S.5777 .52S93. S4SG6.5-306 ,S303() .55S23.82069 57262,81982 4 57.5l 120.85762.52) S.S.4S5 .543) .S.39l 5.55S47,82053.57%286.81965 3 5S.5l 454, S5747.52043.84S36,54- 5.83S99.55S7l .82).33.573l ().8919 2 59.5 479.85732.52967.8 S2O .54 40.83SS3.55S95.S.202) 5733-1 .832 l 60. 515)4.S.577.52)92.84SU5 54.64 ,83S67.55919.82)01 5735S.895 0 M. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Slne. Cosin. Sine. M. 59O 58o f57 o 5 G b 5.5 o ºr TA BLE NATURA L SIN F.S AND COSINES. 249 XIV. 15 16 35C 36O 37C 390 Sine. .5735S .573Sl .57405 .57429 .57453}. .57477|. ,5750||. .575.24 |, .5754S|.S .57572|. .57596 |. ,57619). .576-43 |. .57667 |. .57691 . fº-y ºr .577 15 . .5773S|. .57762|, .577S6}. .57Si ()|. .57S33|. .57857 |. .57SS| |. .57904 |. .5792S|. .57952|. .57976|. .57999|. .5S0.23|. .5S)-17 |. .5S070|. .5S094 . .581 i 81. .58||4|| |. .58|| 65|. ,581 SQ}. .5S212}. .5S236|. .58260|. .5S731 .5S755 .5S770 . . Cosin. Cosin. .8|915 .81 Sø9 .8ISS2 .8.1865 sii/3 .81157 .81 140 .8l. 123 .8] I ()6 .810S9 3|,81072 .81055 .81 03S .8l ()2] .8100.4 .S09S7 .Sü970 ,8(1953 ..SO936 .S0919 .80902 Sine. 54o Sine. .5S779 .5SS{12 .5SS26 .5SS49 .5SS73 .5SS06 .5S920 .5S9-13 .5S967 .5S990 .590 4 .59037 .59061 .590S4 .59 [OS .591.31 .5915.4 .59] 7S .59:20 I .592:25 .59248 .59272 .59295 .593 l 8 .59342 .59365 .593S0 ,594 12 Cosin. ..SO902 .80SS5 ,80S67 .8(S50 ..SOS33 .80S 16 .S0799 .80765 .8074S , .80730 . .807 || 3 ||. .80696 . ..SO679 |. .80662 : . .S0644 .80627 .806) () . .80593 |, .80576 .8055S . .S0541 .80524 ,80507 ||. .804S9 |. .80472 . . .80455 . .80–438 . . ,59436|. .59.159|. .594S2}. .59506. .595.29 |. .59552|. .50576|. .59599 |. ,596.2.2]. ,59646|. .59600 . .5989.3|. .597 | f |. 5973) : . .59763|. .597SG|. .59SO9 |. .59S32|. ,59856|. .598.79 |. .59902 |. ,599.26|, .599-19}. . (599.72|. ,59095 |. .60()19 |.7% .60(){2|. .6(1065|.7995 .600SQ |.79 ,601 12|. ,60135|.79. ,6015S]. .601S2|.7 -y- Sine. Cosin. ,001 S2 .60205}. .6022S|. .60251 |. ,60274 |. .6{}29S]. .60321 |.7 .S07S2 . .60S53|. .60S76|. .60S99). ,60922|. .609-15 |. .6096S}. .6099 ||. 2 . .61(1) 5|. .6l 03S). .6 || 06 ||.7 .6|(}S4]. .6l 107|, .6 || 130|. .6 || 53|. .6l 176 |. .61 l 99 |. .612221. ,61245|. ,61268].7: ,61291 |.7 ,613S3 |. .61406|. .61429 |. .61451 . .79S64 .6l 474 .7SS7 ,61497 , 61520 .6|543 .61566 Costn. rºy J r * , y NS: }{D .7SS37 .7SS 19 .7SS01 Sine. .62479|.7 ,62502|. ,62524|.7 ,62547 . .625.70 |.7 .62592|. .62615|.77 .6.263S .77 .62660}, 626S3|, .62706|,77 .6272S].77 ,6275||. ,62774 |. .62796|. .62S19 |.77 .62S42}. .62S64 . .62S87|. .6.2909|. .62032 Cosin. .777 l 5 Sine. Cosln. 63d 539 *---ee-amº-------------------- 5 ISP Sine. Cosin.) M. .62932 .62955 .62977 .63000 .63022 .63045 .6306S .6.3090 ,631 [3 ,63|35 .6315S .631.80 .63203 .63225 .63248 .6327 I .63293 .777 |5| 60 .77696. 59 .77678; 58 .77660, B7 .77641 56 .77623 55 .77605| 54 .775S6; 53 .77568||52 .77.550, 51 .7753 || 50 .77513| 49 .7749.4|| 48 .77476|| 47 .7745S 46 .77439| 45 .7742|| 44 .7740.2] 43 .773S4| 42 .77366|| 4 | .77347| 40 .77320|| 39 .773)0 38 .77292, 37 .77.273 36 .77255] 35 TA BI, E N AT l; R A L SIN ES AND CU)SINES. XIV. M | 402 4 R J 29 43O 440 Sine. .64279 .64301 .64323 .64346 6436S .64390 .64412 .64435 .64457 .64479 .64501 .64524 .64546 .64568 .64590 .64612 .6.4635 .64657 .646.79 .64701 .64723 .6.1746 .6476S .64790 .64967 .649S9 .6501 1 .650.33 .65055 .65077 .65100 .65122 .65144 .65166 ,651SS .65210 .65232 .65254 .65276 .6529S .65320 .653.42 .65364 |. Cosin. .76(504 .765S6 .76567 .7654S .765.30 .7651 I .76492 .76473 .76455 .76436 .764 |7 .7639S .763SO .7636] .76342 .76323 .76304 .76286 .76267 .7624S .70229 .76.210 .76.192 .76||73 .76 |5. .76 13.3 .76 l l 6 .76097 .76078 3,760.59 .76041 .76022 .76003 .759S4 .75965 .75946 .75927 .7590S .75SS9 .75S70 .75S5] .75S32 .75S 13 .75794 .75775 ,75756 .7573S ,75719 .75700 .653S6|.75 .65408|. .65430|. .65452|, 4|.65474 .65496 .65518 .65540 .65562 .655S4 .65606 . . Cosin. .755S5 . .75566 |. .75547 . .7552S . .75509 . .75490 . 499 Sine. .65606 .656:28 .656:50 .65672 .65694 .657 | 6 .6573S .65759 .6578] .65S03 .65S25 .65S47 .65S69 .65S9 | .65913 .65935 .65956 .65978 .60000|. , 66022 ,6604.4 ,66066 .66088 .66109 .66 |3| .6615.3 ,661.75 ,661.97 .6621S .66240 .66:262 .662S4 .66.306 .6G327 .66349 .6637 I .66393 .664 l 4 .66.436 .6645S .6G4Sſ) ,6650i .66523 . 66545 . 66566 ,665SS .66610 .66632 .666.53 .66675 .66697 .66718 Cosin. Cosin. .75.47 | .75-452 .75.43.3 ,754 14 ,75395 .75375 .75356 .75337 .75318 .75299 .752SO .75261 .75241 .75222 .75203 .75184 .75165 .751 46 75.126 .75 || 07 .750SS .75060 .75050 .75030 .750|| .74992 .74973 ,74953 .74934 .74915 .74S96 .74S76 .74S57 .74S3S .74S18 .74799 .7.478() .74760 .74741 .7.1722 .74703 .7.46S3 .7,466.1 ,74644 .74625 .74806 .745SG .74567 . .745-48 ||. .74528 .67 .74509 ||. .744S9 | . .74470 . .74451 .74431 5|.74.412 . .74392 ||. S}.74373 . 70|.74353 . .74334 .74314 Sine. 482 Cosin. Sine. 4. * Sine. .6691.3 .66935 .66956 .6697.8 .66999 .67021 ,670.43 .67064 .670S6 .67 107 .67129 ,6715] .67172 .67 194 .67215 ,67237 ,6725S .672SO .67,323 .67344 .67366 .673S7 .674.09 .674.30 .67452 .67,173 .67495 .67516 .6753S .67559 .675SO .67602 .67623 .67645 .67666 .676SS .677().) ,67730 .67752 .67773 .67795 .67S 16 .67S37 .67859 .678S0 .673ſ) ||. .743.14 .74295 .7x1276 .74256 .74237 ,74217 .74 ISS .74 || 78 .7.4 I 59 .74 139 .74 I2() .74 100 .740S0 .74061 .74041 ,74022 ,74002 .739S3 3963 .73944 .7392.4 .7300.4 .73SS5 .73S65 .73S46 .73S26 .73S06 ,737S7 •737.67 .73747 .7372S .7370S .736SS .73669 .73649 .73629 .73610 .73590 .73570 .73551 .73531 .7:35 ll .73191 ,73472 ,734.52 .734.32 .734 13 3|.73393 .73373 5|.73353 ,73333 .733 14 .732.94 .73274 ,73254 .73234 .732) 5 .73195 .73175 .73155 ,73135 .6S200 .6s221 .68242 .6S264 .6S2S5 .6S306 .6S327 .6S349 .6S370 .6S391 ,6S4 12 ,6S-134 .6S455 .6S476 .6S497 .6S518 ,6S530 .6S56] .6S5S2 .6S6(j.3 .6S62 | .6S6 15 .6S666 .6S6SS ,6S709 .6S730 .6S75l. .6S772 .6S793 .6SSH 4 .6SS35 .6SS57 .6SS78 .68899 .6S920 .6S941 .6S962 .6S983 .6900.4 .69025 , 69ſ) 16 , 69067 .690SS .69109 ,691.30 .6915] .69172 .691.93 .69214 .69235 .69256}. ,692.77|. .69298 |. .69319|. .693.40|. .69361 i. .693S2|. .69403|. .69424 |. .694.45|. ,69-166 . . Sine, ſ Cosin. Cosin. ,731.35 .731 || 6 .73096 .73076 ,73056 .730.36 .730 [6 .72996 .72976 .72957 .72937 .729.17 .72S07 .72S77 .72357 .72S37 .72S] 7 .72797 .7:2777 .72757 .72737 .727 17 ,72607 .72677 .72657 .72637 .726.17 .72597 .72577 ,72557 .72537 .725.17 .72497 .7:2477 .72457 .72437 .724 [7 ,72397 .72377 .72357 ,72337 .72:317 .72:297 .72277 .72257 .72236 .722) 6 .721.96 .72176 .72156 .71934 Sine. 4, 62 Sine. ,694.66 ,694$7 .6950S .69.529 .69549 .6957ſ .69591 .696 || 2 .696.33 .6965.4 ,696.75 69696 .69717 .697.37 .6975S .697.79 .69S00 , 69S2] , 69S42 ,69S62 ,63SS3 , 69904 .69925 .6994.6 .69066 .699S7 .7000S .7.0029 ,70049 .70070 ,700.91 .701 12 .70 || 32 .70) 53 .701.74 .70] 95 .70215 .70236 .70257 ,70277 ,7020S .70319 .70339 ,70360 .703S1 .70401 .70422 ,704.13 .70463 .704S4 .70505 .70525 .70546 .70567 .705S7 .7060S ,7062S ,70649 .70670 .70690 .707 | 1 Cosin. 455 Cosin. .7 1934 .71914 .7 ISQ4 .7 IS73 .71S53 .7 [S33 .71S 13 .7 1792 .7 l 772 .71752 .71732 .7 | 7 || || .71691 .7 | 67 | .71650 .7] 630 .7 || 510 .7 1590 .7I569 .7 1549 .71529 .7 150S ,7148S .7146S .7] 147 .7 1427 .71407 .71386 .71366 .71345 .71325 .7|305 .712S4 .71264 .7 1243 .71223 ,71203 .7] [S2 .7] 162 .7 || 14 t .7 || 21 .71 100 .7.1080 .7.1059 .71039 .7 1019 .7099S .70978 .70957 .7.0937 .70916 .70S96 .70S75 .70S55 .70S34 .70S13 .70793 .70772 .70752 .7()73] .707] I Sine. |M. M. 60 59 58 57 56 55 54; 53 52 5| 5ſ) 49 48 47 46 45 44 43 42 4] 40 39 3S 37 T A B L E X V NATURAL TANGENTS AND COTANGENTS, 12 252 TABLE XV. NATURAL TANGENTS AND COTANG ENTS. OO 1O 29 3O M. Tang. | Cotang. | Tang. Cotang. Tang. TCotang. Tang. TCotang. |M. 0 .00000 |Infinite. .01746 57.2900 || 03:32 || 2S6363 || 05241 TT9 (Si) || 60 1 | .00029 || 3437.75 .01775 || 56.3506 (352 2S,3994 | .05270 || 1s. 97.55 59 2 .00058 || 17 18.87 | .01SG4 55.4415 || 03550 2S. 1664 | .05299 || |S.S7 || 1 || 5S 3 .00087 || 1 145.92 || 0 || $33 5-1.5C13 || 03579 27.937.2 | .0532S | 18.767s 57 4 .001 16 839.436 | .01S62 53.70S6 | .03600 27.7 || 7 | .05357 || 1 S.6656 || 56 5 .00143 | 687.549 .01 S91 || 52.8S2} | .0363S 27,4899 || 053S7 lS. 5645 55 6 .00175 572.957 | .01920 52.0S07 || 0.3657 27.27 15 | .Q5416 | 18.4645 54 7 | .00204 || 491. 106 | .01949 || 51.3032 .03696 27.0566 .05445 | 18.3655 53 S .00233 || 429.7 | S .01978 || 50.5-185 ()3725 26. S450 || 05474 | 18.2677 52 9 .00262 | 3S1.97 | | .02007 || 49.8157 | .03754 26.6367 | (5503 || 1s. 170S | 51 10 .00291 || 343.774 | .02036 || 49, 1039 || 037S3 26.4316 | .05533 || |S.0750 | 50 | | | .00320 || 312.521 | .02066 4S.4121 || 03S12 || 26.2296 || ,05562 17.QS(12 || 49 12 .00349 2S6.47S .02095 || 47,7395 .03S42 26.0307 || 05501 || 17.SS63 || 48 13 | .0037S 26-1.44 l .02124 || 47.0853 || 03S7 || 25.S34S (15620 17.7934 47 14 | .00407 || 2:15.552 | .02153 || 46.44SQ .03900 25.64 [S .05640 17.70 || 5 || 46 15 .00436 229. 182 | .02182 || 45.S294 | .03029 || 25.4517 | .0567S | 17.6106 || 45 16 .C0465 214.85S .022] 1 || 45.2261 .0305S 25.2644 .0570S 17.5205 || 44 17 | .00495 || 202.219 .022.40 44.63$6 .030S7 25.07% S .05737 17.4.314 || 43 18 .00524 190.9S4 | .02260 44.0661 | .040 16 24.S97S .05766 17.3432 || 42 19 .(\{).553 || 1 S0.932 .022)S 43.50SL | .0.1016 || 24.7 [S5 .05705 || 17.255S 41 20 | .005S2 | 17 l.SS5 ,0232S | 42.964 | | .0-107.5 || 24.54 S .05S24 17. 1603 || 40 21 | .006 || 1 | 163,700 | . 02357 || 42.4335 | (14 I(\{ | 24.3675 .05S54 17.0S37 39 22 .00640 || || 56.25%) || 023S6 || 41.91 5S .O.ſ 133 24, 1957 | .05SS3 | 16.9990 || 3S 23 .00669 || 149.465 || 024 (5 41.4 106 || 01162 || 24,0263 | .05912 | 16.9, 50 | 37 24 .00698 || 1:43.237 .02+44 40.9174 (1419 || || 23. S393 | .0594 | | | 6.8319 || 36 25 | .007.27 | 137.507 | .02.473 || 40.435S .0122ſ) || 23.6945 | .0597 () | 16.7-196 || 35 26 .00756 || 132.219 (12502 || 39.96:55 | .0425() || 23.532| | .05999 || 16.66S1 || 34 27 .007 S5 127.321 ,0253 | | 39.50:59 ()4279 || 23.37 IS ,06029 16.5S74 || 33 28 ' .00815 122.774 .02500 || 39.056S | .01:30S 23.2137 || 0605S | 16.5075 32 29 .00S-14 || | 18.540 .02.5S0 || 3S,6177 .04337 23,0577 | (6(S7 | 16.42S3 || 2 | 30 | .00S73 || || 14.5S9 .02610 || 38. ISS5 || 0-1366 22.903S | .06 | 16 | 16.3499 || 30 3) .00002 || || 10.S.)2 ,026.1S 37.76S6 || 04305 22.7510 || 06145 | 16.2722 | "..) 32 .0093I 107.426 ,02677 || 37.3579 || 0-1424 22.6]2() (16175 | 16. 1952 28 33 .00360 | 104. 17 | | .0.27(16 || 36,9560 .04.454 2,43-1 || | . (620-4 || || G. l l 90 27 34 .009S9 l{}l. 107 | .027.35 | 36.5627 | ()44S3 22,3(ISI .06.233 16,ſ}435 26 35 | .01018 9S.2179 .02764 || 36, 1776 .0-1512 22, 164() .06262 15.96S7 || 25 36 .01(k+7 || 95.4S95 .02793 || 35.S()(16 || 0454 || || 22.02| 7 | .06.291 15,SQ45 24 37 .01.076 || 92.90S5 | .02S22 || 35.43| 3 || 0457() 21.SS13 . .(1632| | | 5.S2] 1 23 38 .01 105 90.4633 . .02S51 35.0695 || 0-1500 21.7426 (16350 15.74S3 22 30 .0l 135 | SS. 1436 .02SS1 || 31.7l 51 || 04628 21.6(56 || 06370 15,6762 21 40 | .0; 164 85.93)S | .02010 || 34.367S .0465S 21,47()4 || 0640S 15.6048 20 41 .01 | 03 || 83.8-|35 | ,02939 34.0273 (116S7 21.3.369 | . (6-137 || 15.5340 | 19 42 .01222 || 81.847() | .0206S 33.6935 | .04716 21.2040 | .06467 || || 5.463S 1S 43 .01251 || 70.9-134 || 02007 || 33.3662 || 04745 21,0747 | .06496 || 15.3943 17 44 .01280 || 78. 1263 (13026 || 33.0452 || 0:47, 4 || 20.9.16ſ) .06525 | 15.3254 16 45 .01309 || 76.3000 | .03055 || 32,7303 | .04S(3 20. SISS ] ,06554 | 15 2571 15 46 || 0133S 74.7292 .030S4 32.4213 .04S33 20.6032 ,065S-1 || 15, 1893 | 1.4 47 .0l 367 | 73. 139() | .03 || 4 || 32. [ ISI .(4S62 20,5691 .06613 15, 1222 || 13 48 || 0 || 306 || 7 1.6151 .03143 || 31.S2(15 .04SQI 21.4465 .06ſ, 42 15. (.557 | 12 49 .01425 || 70. 1533 .03 || 72 31.52S4 || 04020 20.3253 .0007 || || 14.9SQS | 1 | 50 || 01455 | 68.750ſ .0320|| || 31.24 || 6 || 04)4) || 20,2056 || 067(10 || 14.92.44 10 51 .014S4 67,4010 | .03230 30.9599 .0497S 20,0S72 (167.30 14.8596 || 9 52 .015 [3 | 66. 1 ().55 .03250 30.6S33 .050ſ)7 | 19.97 (12 || 06750 14.7954 || 8 53 || 0 || 542 6-1. S5S0 || 032SS 30.4 || 6 || ()5ſ)37 | 19.S546 .06? 14.73| 7 || 7 54 . (1157 63,6567 | . ()33 || 7 || 30. I 446 ()5()(36 | 19.74ſ)3 || 06Sl 7 || 14.66S5 6 55 || 0 || 600 || 62.4992 | .03346 || 29.8S23 (15(105 || 19.6273 .06S47 || 14,6050 5 56 | .0] G20 61.3S29 || 03376 29.6245 .()5] 24 | 19.5156 (16S76 14:543S 4 57 .016.5S 60.3(15S (3-105 || 29.37 || || 05153 || 10.405 || || 069(15 14.4S23 || 3 5S .(JI GS7 59.9650 || 0.3434 20. [220 | .0.5 ! S2 | 19 2059 (16:13.4 || 14.42t 2 || 2 59 || 0 || 7 || 6 || 5S.2612 || 03:46.3 || 2S.S77 || (15212 | 1Q. IS79 (16963 || 14.3607 || 1 6() .01746 || 57.2%)[] . .().3492 || 2S,6363 || 0.324 | | 19.ſls [ ] . .06993 || || 4,3007 || 0 M. Cotang. Tang. Cotang. | Tang. Cotang. Tang. (Cotang. | Tang. |M. 893 882 87 O 860 TABLE xv. NATURAL TANGENTS AND COTANGENTS. 253 4-O 5C GO go M. º, Cotang Tang. | Cotang. | Tang. Cotang. Tang. | Cotang. |M. ūſº | H3; Fºſsi ſã TI330|| Tº jū35|Tºš| SH35|G|. i ºffſ; iºti isºs | ii.3313| iółó | 34sisi .iº || 3.12isi |53 3| giggſ | Hº! ºsſºſ | || 3:19) 19:53) #3314|| 3:33 šiū;6||3: 3| ºn | Hix; sº iii.; tā ( Šºš jzā āś ; 4|| Kiiſ, iłºś ºšić ii.2% iºs ºf tº § |56 : § ºiš tº ºš5 ii.247 iſºſ || 3:30; .342; $º 35|| g| ºriš i3.jā; ºš ii.2 is ióš 3.33724 .2455 §§2 is # § § §§ § #; jºiº || 3:13; i3; Šºš | #3 27 | 3.837s QSQS išić iñ735 | 333399 .i.2515 7.965S 53 #| #| #| #| #| #| };|..};| ####|# 10|.972S5 13.7267 .99942 || ||.95% iš. ; .# # # 11 .07314 || 13.6719 .03071 | | 1.0237 19834 9.23910 || 1:303 ###| || i} ºf iáši; dāini iſ is: Isº ºić| 3:3| º is #| #|##| #| #;| #| j :#| #|}} if ºğ i3.j: jiā iñāſ; ſº jää jº: ; Śiś #| ºft| iá; ; ºš| #| 3: 13; }; ; #6 .07461 13.4039 || 09:218 10, S4S3 | . 10981 9. 106. ..., | . . 17 | .97490 13.3515 . .09.247 | 1Q.S139 #| #}}|{:}|{#|: is ºf i3.35% ºf ióżº jià | ºffs; 12Si3 7.56623 |4% | ſº | }}}}| ſº | }}}#}} {{!!}} };}}|...}; Žiš |3: # # #; § #. j| j| iáš ####|}} * | }. iſgi ºš5 ióðis; iiiş §§ 12539 || 7.752; § #; §§ º; #| # ; #; § § j| i3 is ſº ióðſis iii;| Sºsº | jºš ºilij ||3: 24 .07635 | 12.9962} .09353 | 1Q.57S9 ##| ###| #|##|; ; #: #; ºš| #| ###| Šišiš) iſ; ; }; ; . Uf f : $gsi | 033ii ióšîăg| ii.2% §§§§§ i3017 | 736 tº 3. ; ; ; ; ; ; ; ; ; ; ;|3: ‘Y fºr ** r * … I •xxº • . I 1305 || S.S.4551 | . 13076 || 7.64732 3 28 .07S12 || 12.8014 || 09570 || 10.4491 | . 1 13: sº 7.6.47.32 33 $20 ſy” Sº º) ºr fºr s t - .1 1335 | 8.S2252 | . 13106 || 7.63005 29 .07S41 12,7536 | . (19630 || 10.4172 * 31% 7.63005 ||3: à ºð iáº; º;| #3; .# § .# # % 31 .07S99 || 12,6591 .0965S 10.353S º, sºl º ; : #|º §|}}}}}|...}}|{{j};|; • U 4 JJ 5666 ºf ióżgſ; iiiş Šºši iş54 | ? §§ #| ºš i3.jſ jºić º! #iſ $6% 3.254 || 7,544S7 || 27 sº * ... . . " - . I 151 1 || 8.6S701 | . 132S4 || 7,52S 35 .0SQ17 | 12.4742 .09776 || 10.2294 | . ] I 54 *; 7.52SO6 26 ºn fa. :::::::: . - . l l 54 I | 8.664S2 | . I33i 3 || 7.5i 13: * 36 .0S046 | 12.42SS | .09SQ5 10. 19SS | . I 157 $2|-|33;3| 7.5/132 25 º; ºf $3, $3; ‘. . ." ..] I 570 S.64275 . 13343 || 7,49.465 37 .0S075 12.3S3S .09S34 10, 16S3 . I le §375] .3313| 7,434; 24 3S ,0Sl()4 12.339|| ºš64 ióiši . I 1600 8.620IS 13372 7.47806 || 23 *Rū ſº Sº • ..] 1629 8.59893 | . 13402 || 7.46154 22 39 .(S134 12.2946 || 09SQ3 || 10, 10SQ ..] 165 rº- 102 || 7.4615.4 22 - 5,2,3 . . Sº I * ..] 1659 || 8.577 lS . 13432 7.445() 40 .QS163 | 12.2505 .09923 iº * ºn fºr ºf ºf 4 . 4500 21 # º; {3}...] §§ ičiš #; § .# # ; 3) ºf iáis; j| ióðīš| ###| #j| # #3; # ºft| #| |j} | j| ###| # 13.521 7,396 | 6 || || 8 Sosº | | S A-3, I 9.9S931 . . I 1777 ; 8.4912S . 13550 7.37 44 .0S2S0 | 12.0772 | . 1 (0.40 9.96007 || | IS * 50 7.37999 || 17 A* I ºf litt 4 t tº a SÜ * * QF) ºy Q ſº ºxº #| tº ºft| j| ºff #; § #; #: | 46 .08339 11.9923|.1Q009 9.90211 . . . IS65 8.42795 .13639 † y 47 ºs | || 3:04 idiºs 3.3733; iis; 8.4(iii; i: .33190 | 1.4 43 ºf liºsi joiás || 3.543s; ii; §§§ .# 7.31600 || 13 § ºft|ii; iii., §§ {ij}| sº #;|3}}|{} #| || |};|}}}|...};|..}}}. §| ##|###|}} isis; iiis;| figã jº, işină Šºš #| #3; 3| $3; | }}}}}}|...}{3.5 3.733,7|,42042 § †. £3. 310|| 3 § {j}|###|};|##|{#|sº #| #| } #| ||...}|##| 3: ; ; ; ; ; ; 7 ; º; #; º: ; .# j :#| #; : sºſ) [ . UCYUt]… I I I - 3\c 3. .62.205 || | | . Sº I " ...: l ; ; ;2, 57 | (Sºft| | | 1.5.16||.10422 ;|}}}| 3: ;| #| | 5S (S630 || 11.50;2| iſjö2 9.56%i işig Šišić 13:35 | 7.1671 3 59 |.0S720 II.46S5 ióisí | 9.53166 jºij | S. §| };} ..];| ? (30 (JS7.4%) | | 1,43() : º: ..]:49 || $163. SH-1424 || 7 ||3}{2| | - I | . 10510 || 9,51436 . I2278 || 8. 14435 .14054 || 7. I 1537 || 0 M. Cotang. Tang. Cotaug. Tang. Cotang. Tang. Cotar T - 856 84O 4. §. 1g, Tang. |M. - S39 82O 254 TABLE XV. NATURAL TANGENTS AND COTANGENTs. So 99 100 I TO M. Tang. | Cotang. Tang. Cotang. Tang. Cotang. | Tang. T Cotang. M. 0 | 1.4034 || 7. 11537 | . 15838 6.31375 .17633 || 5.67128 .1943S | 5.14455 |60 l | . 14084 || 7. 1003S | 15S68 6.30189 ..]?663 || 5.66.165 | . 19468 5.1365S 59 2 | . 141 13 || 7.08546 | . 15898 || 6.29007 | . 17693 || 5,65205 . 19498 || 5.12S62 || 5S 3 | . 14143 || 7.07059 |. 15928 || 6.27829 | .17723 || 5.64248 | . J.9529 || 5, 12069 || 57 4 . 14173 || 7.0557 1595S 6.26655 . 17753 5.63295 | . 19559 || 5.11279 || 56 5 | . 14202 || 7.04.105 | . 1598S 6.254S6 .17783 5.62344 . 195S9 || 5.10490 55 6 | . 14232 7.02637 | . 16017 | 6.24321 | . 17813 || 5.61397 | . 19619 || 5,097.04 54 7 | . 14262 || 7.0l 174 |. 16047 | 6.23160 | . 17843 || 5.60452} . 19649 || 5.0S921 53 8 . 14291 || 6.99718 . 16077 || 6.22003 | . 17873 5.595] 1 | . 196SO || 5.0S 139 || 52 9 . 1432 l 6.9826S | . 16107 6.20S5] . .17903 || 5.5S573 . . 19710 || 5.07360 || 51 10 | . 14351 | 6.96823 | . 16137 || 6. 19703 | . 17933 5.5763S | . 19740 || 5.06584 50 | 1 | . 143S1 || 6.95385 | . 16167 6, 18559 . 17963 || 5.56706 | . 9770 5.05S09 || 49 12 | . 14410 || 6.93952 | . 16196 || 6. 17419 | . 17993 || 5.55777 | . 19SOI 5.05037 || 48 13 | . 14440 | 6.92525 | . 16226 || 6, 16283 | . ISO23 5.54S51 | . 19S31 5.04267 || 47 | 4 | . 14470 || 6.9l 104 | . 16256 || 6.15151 . ISO53 || 5.53927 | . 19S61 || 5.03499 || 46 15 . 14499 || 6.8968S | . 162S6 6.14023 . 180S3 5.53007 | . . 9891 || 5.02734 || 45 16 .14529 6.8S278 | . 16316 || 6.12S90 | . 18113 5.52090 | . 19921 || 5.01971 || 44 17 | . 14559 || 6.86874 . 16346 6. I 1779 .18143 5.5] 176 . 19952 || 5.0.1210 || 43 1S . 14588 6.85:475 | . 16376 || 6. 10664 | . IS1.73 || 5.50264 . 19982 || 5.00451 || 42 19 | . ] 1618 6,840S2 . 16405 6.09552 | . 18203 || 5.49356 .20012 || 4.99695 || 4 | 2ſ) | . 14648 || 6.82694 | . 16435 | 6.(S444 . 1S233 || 5.48451 | .20042 4.9S940 | 40 21 | . 14678 || 6.81312|| . 16465 6.(1734() . IS263 5,47548 .20073 || 4.9SlSS 39 22 | . 14707 || 6.79936 | . 16495 || 6.0624() | . 18293 5.46648 .20103 || 4.9743S 3S 23 . 14737 6.78564 , 16525 6.05]43 . 1S323 || 5.45751 .20133 4.96690 37 24 .14767 || 6.77199 | . 16555 | 6.04ſ)5] | . ] S353 || 5.44857 | .20164 4.95945 || 36 25 | . 14796 || 6.75S3S | . 16585 6.02962 | . 18384 || 5.43966 | .20194 4.95201 || 35 & ) 14826 6.744S3 | . 16615 || 6.01S7S | .184 (4 5.43077 | .20224 || 4.94460 | 34 27 | . 14S56 || 6.73|33 || , 16645 || 6.00797 . IS444 || 5.42.192 | .20254 || 4.93721 || 33 28 .14886 || 6.71789 . . 16674 || 5.997:20 | .1847.4 5.4 1309 .20285 || 4.929S4 || 32 29 | . 14915 || 6.70450 | . 16704 || 5.9S646 | . 18504 || 5.40429 | .20315 4,92249 || 31 30 . 14945 || 6.69116 | .16734 5.97576 | .18534 || 5.39552 .20345 || 4.91516 || 30 31 | . 14975 | 6,677S7 | . 16764 5.96510 | . IS564 || 5.3S677 | .20376 || 4.90785 29 32 | . 15005 || 6.66.463 .16794 5.95448 . 18594 || 5.37805 | .20406 || 4.90056 || 28 33 .15034 || 6.65144 .16S24 || 5.94390 . 1S624 || 5.36936 .20436 4.S)330 27 34 | .15064 6.63S3| | . 16S54 || 5.93335 | . 1S654 5,360.70 | .20466 || 4.8S605 || 26 35 .15094 6.62523 .16884 5.922S3 | . 1S6S4 || 5,35206 .20497 || 4,878S2 25 36 | . 15124 || 6.61219 16914 || 5.91236 | .18714 || 5.34345 .20527 || 4.87162 24 37 . 15153 || 6.599.21 | .16944 5.90191 | .18745 5.33487 | .20557 || 4.86444 23 3S . 15183 6.5S627 | . 16974 || 5.89.151 . 18775 5.3263] .2058S | ?.85727 22 39 . 15213 || 6.57339| . 17004 || 5.881 14 . 1SS05 || 5.31778 .20618 4.85013 || 21 40 | . 15243 || 6.56055 . 17033 || 5,870SO | . ISS35 | 5.3092S | .2064S 4.84300 20 4 || | . 15272 6,54777 . 17063 || 5.86051 | . ] SS65 5,30080 .20679 || 4.83590 19 42 | . 15302 || 6.53503 | . 17093 5.85024 .18895 || 5.29235 | .20709 || 4.828S2 18 43 . 15332 || 6.52234 . I 7123 5.84001 | . IS925 || 5.28393 | .20739 || 4,82175 || 17 44 .15362 6.50970 | . 17153 5.829S2 . IS955 ; 5.27553 | .20770 4.81471 | 16 45 | . 15391 || 6.497.10 , 17.183 || 5.81966 | . 180S6 || 5.26715 .20S00 4,80769 15 46 . 15121 | 6.48456 | . 17213 || 5.80953 . . 19016 || 5.25SS0 .20830 || 4.8006S 14 47 | . 15451 | 6.47206 | . I 7243 || 5.79944 | . 19046 5,25048 .20861 || 4.79370 13 48 . 15481 6.45061 | . 17273 || 5,7893S .19076 5.2421S .20891 4.7S673 12 49 . 15511 || 6.44720 | . 17303 || 5.77936 | . 19106 || 5.23391 . .20921 4.77978 || 1 || 50 | . 15540 6.434S4 | . 17333 || 5.76937 | .19136 5.22566 .20952 4.772S6 10 51 | . 15570 6.4.2253 | . 17363 5.75941 | . 19166 || 5.21744 .209S2 4.76595 || 9 52 .15600 | 6.41026 .17393 || 5.74949 .19197 5.20925 | .21013 || 4.75906 || 8 53 | . 15630 || 6.39S04 | .17423 5.73960 | . 19227 | 5.20107 | .21043 4.75219 || 7 54 . 15660 | 6.38587 , 17453 || 5.72974 . 19257 5, 19293 .21073 || 4.74534 || 6 55 | . 15689 || 6.37374 .174S3 || 5.71902 . . 19287 || 5.18480 . .21 104 || 4.73851 5 66 | . 15719 6.36.165 .17513 || 5.7 1013 | . 19317 5, 17671 .21 134 4,7317() 4 57 | . 15749 || 6.34961 . 17543 5.70037 | . 19347 || 5. 16863 .2] 164 4.724.90 || 3 5S | . 15779 || 6.33761 , 17573 5.69(164 | . 19378 || 5, 1605S .2] 195 || 4.7 1813 || 2 50 | . 15809 || 6.32566 . 17603 || 5.6S094 | . 1940S 5.15256 .21225 || 4.7 ||37 || 1 60 | . 15838 || 6.31375 . 17633 || 5,6712S . 1943S 5. 14455 .21256 4.7.0463 _0 M. Cotang. Tang. Cotang. | Tang. Cotang. I Tang. Cotnog. Tang. |M. | 81 O SO3 7 Qo Tºo TABLE XV. NATURAL TANGENTS AND COTANGE, NTS. 255 I2O 130 14 O 15O M. Tang. TCotang. Tang. TCotang. Tang. TCotang. Tang. TCotang. M. 0 21256 || 4.7.0463 .230S7 || 4.33I4S .24933 || 4.01.078 .26795 || 3.73205 || 60 | I .212S6 4.6979| | .231 17 || 4.32573 . .24964 || 4.005S2 | .26S26 3.72771 j 59 2 | .21316 || 4.69 121 | .23148 || 4.3.2001 | .24995 || 4.000S6 .26S57 3.72338 58 3 | .2|347 || 4.6S452 | .23179 || 4.31430 | .25026 || 3.99592 .26SSS | 3.7 1907 || 57 4 .21377 || 4,67786 .23209 || 4.30S60 | .25056 || 3.990.99 || .26920 3.71476 56 5 .2140S 4.67121 | .23240 4.30291 .250S7 || 3.9S607 .26951 || 3.7 1046 55 6 | .2143S 4.66458 | .23271 || 4.29724 . .251 18 || 3.981 I? .269S2 || 3.70616 54 7 | .21469 || 4.65797 .23301 || 4.29159 | .25149 || 3.97627 | .27013 || 3.701SS 53 8 .21499 || 4.65138 .23332 || 4.28595 .251 S0 || 3.97139 .27044 || 3.69761 52 9 | .21529 || 4.644SO | .23363 4.2S032 | .252] 1 || 3,96651 .27076 3.69335 || 51 | 0 | .21560 4.63S25 | .23393 || 4.2747] . .25242 | 3.96.165 | .27 107 || 3.6S909 50 | | | .21590 || 4 63171 .23424 || 4.269] 1 | .25273 || 3.956S0 | .27) 38 || 3.6S4S5 || 49 12 | .21621 4.62518 .23455 4.26352 .25304 || 3.95196 | .27169 || 3.6S(61 || 48 | 3 | .21651 || 4.61868 .234S5 || 4.25795 | .25335 | 3.947 [3 | .27201 || 3.67638 47 I4 .216S2 || 4.61219 | .235.16 || 4.25239 .25366 3,94232 .27232 3.672| 7 || 46 15 .217 12 || 4.60572 | .23547 || 4.24685 .25397 || 3.9375] | .27263 || 3.66796 || 45 I6 .21743 || 4.59927 .2357S 4.24132 .2542S 3.93.271 | .27294 || 3.66376 || 44 17 | .21773 || 4.59283 .2360S 4.235S0 .25459 || 3.9.2793 | .27326 3.65957 || 43 18 .21 S04 || 4.5S641 | .23639 4.23030 .25490 3.92316 .27357 3.65538 42 19 .21834 || 4.5S001 .23670 || 4.22481 .25521 3.91839 .273SS 3.65121 : 41 20 | .21864 4.57363 .23700 4.21933 . .25552 3.91364 .27419 } 3.64705 || 40 2l .21895 || 4.56726 .2373.1 4.21387 .255S3 || 3.90S90 .27451 || 3.64289 || 39 22 .21925 || 4.56091 .23762 || 4.20S42 .25614 || 3.904 || 7 | .274S2 || 3.63S74 3S 23 .21956 4.55458 .23793 4.2029S | .25645 || 3. Sq945 | .27513 3.63461 37 24 .21986 || 4.54826 .23S23 4. 19756 .25676 || 3.SS);74 .27545 || 3.63048 || 36 25 .22017 4.54.196 .23S54 || 4, 19215 .25707 || 3.89.004 || .27576 || 3.62636 || 35 26 .22047 4.53568 .23SS5 4, 18675 .25738 3.SS536 .27607 || 3.62224 || 34 27 .22078 4.52941 .23916 || 4. 18137 | .25769 || 3.SSOGS .27638 3.6.1814 || 33 28 .2210S 4.52316 || .239.43 4. 17600 . .25S00 || 3.S7601 | .276.70 || 3.61405 || 32 29 .22139 4.51693 .23977 || 4, 17064 .25S31 3.87136 .27701 || 3.60996 || 31 30| .22169 || 4.5107 I .2400S 4, 16530 .25862 || 3.SG671 .27732 || 3,605SS 30 | 3 | .22200 || 4.5045i .24039 || 4. I5997 | .25S93 || 3.8620S .277 3.601 S1 || 29 32 .222.31 || 4.49S32 .24009 || 4. 15465 .25924 || 3.SS745 .27795 || 3.59775 28 33 .22261 4.492.15 .241C0 || 4. 14934 || .25955 3.852S4 .27S.26 || 3.59370 27 3-1 || 22:292 || 4.4S600 | .24131 4. 14405 | .259S6 || 3.S4S24 .2785S | 3.5S966 26 35 22322 4.479S6 | .21162 || 4. 13877 .26017 | 3.S43 27SS9 || 3.5S562 || 25 36 22353 || 4.47374 .24193 4.13350 | .26048 || 3.83906 | .27921 || 3.5Si60 24 37 223S3 4.46764 | .24223 || 4.12S25 | .26079 || 3.83449 | .27952 || 3.5775S 23 3S 22414 || 4.46155 .24254 || 4, 12301 | .26110 || 3.82992 | .279S3 || 3.57.357 22 39 .22444 || 4.4554S .242S5 4. l l 77S .2614 3.S2537 .2SO15 || 3.56957 21 40 .22475 4.4494.2 . .24316 || 4 || 1256 | .26172 3.820S3 | .2SO46 || 3.56557 20 41 .22505 || 4.4.433S .21347 4, 10736 | .26203 || 3.81630 | .2S077 || 3.56.159 19 42 .22536 || 4.43735 | .24377 4.10216 | .26235 3.S. 177 | .2S109 || 3.55761 IS 43 | .22567 || 4.43134 .24408 || 4.09699 || .26266 || 3: S0726 2S140 55364 17 44 .22597 || 4.42534 .24439 || 4.09182 .26297 3.80276 | .2S172 3.5496S 16 45 .2262S 4.4.1936 | .244.70 || 4.0S666 .26328 || 3.79S27 | .2S2O3 3.54573 || 15 46 | .22658 || 4.4.1340 | .24501 || 4.0Si62 | .26359 || 3.7937s .2S234 3.54.179 || 14 47 | .22689 || 4.40745 .24532 || 4.07639 || 26390 3.7S93] . .2S266 || 3.537S5 || 13 48 .22719 4.40152 | .24562 || 4,07127 | .2642, 3.7S4S5 .2S297 || 3.53393 || 12 49 .22750 4.39560 .24593 || 4.06616 | .26452 3.7SO40 .2S329 3.53001 || 1 | 50 | .22781 || 4.38969 | .24624 4.06107 | .264S3 || 3.77595 | .2S360 3.52609 || 10 51 .228 ll 4.3S3S1 | .24655 || 4.05599 .26515 3.77.152 | .2S391 || 3.52219 || 9 52 .22S42 4.37793 .246S6 4.05092 | .26546 || 3.76709 | .2S423 || 3.51829 || S 53 .22872 || 4.37207 | .247 | 7 || 4 045S6 .26577 3.7626S .2S454 || 3.51441 7 54 .22903 || 4.36623 .24747 4,040SI .2660S | 3.75S2S .2S4S6 || 3.51053 || 6 55 | .22934 4.36040 | .2477S 4,03578 .26639 || 3.753SS 2S517 | 3.50.666 5 56 | .22964 || 4.35459 .24S09 || 4,03076 | .26670 3,74050 | .2S5.19 || 3.50.279 || 4 57 .22995 || 4.34879 | .24S40 || 4.62574 | .267.01 || 3,745.[2] .2S5S0 || 3,40S94 || 3 58 .23026 || 4.31300|| .24S; I 4,02074 .26733 3.73075 .2S612 || 3,49509 || 2 59 | .23056 4.33723 .24902 || 4.01576 .26764 3.73640 2s633 3.39.125 | 1 60|_23US7 || 4.33I4S) .24933 || 4.01078 .26795 || 3.73265 .28675 3.4S74i | 6 M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang.T.M. 775 76O 750 74O 256 NATURAL TANGENTS AND cotANGENTs. TA BLE XV. -- 10 IGO 17O 190 Tang. .2S675 .2S706 .2S7. .2S769 .2SS00 .2SS32 .2SS64 .2S895 .2S927 .2S958 .29621 .29653 .29685 .297 6 .2974S .297 SO .298] I .29S43 .29875 .29006 .2903S .29970 .3000l. .300.33 .30065 .300.97 .301.23 .3ſ) 16ſ) .30.192 .30224 .30255 .302S7 .30319 .30351 .303S2 .3()573 Cotang. Tang. Cotang. 3.4S741 3.4S359 3.47977 3.47596 3.472| 6 3.46S37 3.4645S 3.400SO 3.45703 2.45327 3.44951 3.44576 3.44202 3.43S29 3.43.456 3.43084 3.427.13 3.42343 3.4.1973 3.347.32 3.34377 3.34023 3.33670 3.33317 3.32065 3.32614 3.32264 3.310|| 3.31565 3,31216 3.30S6S 3.30521 3.3017.4 3.20S29 3.29.4S3 3.29.139 3.2S795 3.2S452 3,2S109 3.27()S5 .30573 .3(1605 .30637 .30669 .30700 .30732 .30764 .30796 .3082S .30S60 .30S9 | .3(1923 .30955 .309S7 .3|019 .31051 .310S3 .31 115 .31 147 .3 l l 78 .31210 .31242 .31274 .3|306 .31338 .31370 .31402 .31434 .31466 .31498 .31530 .31562 .31594 .31626 .3165S .31690 .31722 .31754 .31786 .31818 .323.96 .32428 .32 j60 .32402 3.27()S5 -- - - 3.26745 3.26406 3.26067 3.25729 3.25302 3.250.55 3.247 19 3.243S3 3.24049 3.237 13 3.233S1 3.23048 3.227 15 3.223S4 3.22053 3.21722 3.21392 3.21063 3.20734 3.29.1% 3.20079 3, 19752 3. 19-126 3.191 ()0 3. lS775 3.18451 3, 18127 3. I 7804 3.174S1 3.17159 3.16S3S 3. 16517 3. 16197 3, 15S77 3, 1555S 3. 152-10 3. 14922 3. 14605 3. 142SS 3, 13972 3. 13656 3.1334|| 3.13ſ)27 3, 12713 3, 12400 3.12087 3, I 1775 3. l 1464 3. l l l 53 3. ] ()S.42 3. 1053.2 3. 10223 3.09914 3. ()Sſ)73 3.0776S 6573 Tang. | Cotang. .34.133 || 2.90.421 .34,465 2.90|47 .34-198 || 2.SSS73 .335.30 2.SS600 .34563 2.SS327 .34596 || 2.89055 .34628 || 2.83.783 .34661 || 2.8S51 | .34693 2.8S240 .34726 2.87.970 .34758 || 2.87700 .347.91 || 2.S7430 .34S24 || 2.87 | 6 | .3-1856 2.86S92 .34SS9 || 2.86624 .34922 || 2.86356 .34954 2.S60S9 .349S7 2.85S22 .352|6 || 2.S3965 ,3524S 2.83702 .352S1 || 2.83439 .35314 || 2.83176 .35346 2.829 4 .35379 2.82653 .354.12 || 2.82391 .35445 || 2.821.30 .35477 || 2.S 1870 .35510 || 2.81610 .35543 || 2,81350 .35576 || 2.81 (.91 .3560S 2.80S33 .3564 || || 2.S()574 .35674 || 2.S03 || 6 .35707 || 2.SO050 .35740 2.79S02 .35772 2.79545 .35S0.5 2,792S9 .35S38 2.79033 .35S7 || || 2.7S778 .35904 || 2.78523 .35937 2.78.269 .35960 2,78014 .36()02 || 2,7776 l .36()35 2,77507 .3606S 2.77.254 .36101 2,77002 .36||34 || 2,76750 .36 | 67 || 2.7649S .36199 2.76.247 .36232 2.75996 .36'265 2,75746 .3020S 2.75.406 .36:33 l 2.75246 .36364 || 2.74097 .36397 2.74748 M. Cotang. Tang. Cotang. Tang. Cotang. | Tang. Cotºng. Tang." 739 720 7 10 700 TABLE XV. NATURAL TANG ENTS AND COTANGENTS. 202 210 223 23O M. Tang. Cotang. Tang. Cotung. | Tang. Cotang. | Tang. Cotang. |M Öſ. 36.307 || 2.7474S .3S3s 6 ||26-1509 || 4) iſ? || 2:47.509 | .42147 2.355S5 | 60 l .364:30 || 2.74-194) | .3842) | 2.6()2S3 | .40436 || 2.47302 | .424 S2 2.35395 59 2 .36463 || 2.7425 | | .3S453 || 2.6ſ 1057 | .4(1-170 2.42 095 | .425 | 6 || 2.35205 || 5S 3 | .30496 || 2.74 104 || ,3SHS7 2.5%)S3 | | .405(14 || 2.4(isss | .42.53 | | 2.35015 57 4 | .36520 2.737.56 | .3S520 | 2.596(16 | .4053S 2.46(3S2 .425S5 || 2.34S25 || 56 5 .36562 | 2.735(10 | .3S553 || 2.593S1 | .40572 2.46476 | .42619 || 2.34636 55 6 .305.05 2.73.263 | .3S5S7 || 2.59.156 | .40606 || 2.46270 | .42651 || 2.34447 || 54 7 | .36628 || 2.73| | | 7 | .38620 | 2.5S032 .40640 || 2.46065 | .426SS | 2.3425S 53 8 . .3666 | | 2.7277 l .3$654 || 2.5S70S | .40674 || 2.45S60 | .42722 || 2.34069 || 52 9 | .36694 || 2.72526 | .3SGS7 2.5S4S4 .40707 || 2.45655 .427.57 2.33SSI 51 10 | .36727 | 2.722S1 | .3S72| || 2.5S26 | | .4074 | | 2.45451 .427.91 || 2 33693 50 | | | .36760 | 2.72036 .3S754 || 2.5S03S | .40775 | 2.45246 | .42S26 || 2 33505 || 49 | 2 | .36793 2.7 1792 | .3S7S7 || 2.57S15 .40S()9 || 2.45043 | .42860 | 2.333| 7 || 4S i 3 | .36S-26 2.7 154S | .3SS21 2.57593 | .40S43 || 2.44S39 .42S94 2.33130 || 47 14 | .36S50 | 2.7 30.5 ! .3SS54 || 2.5737 | | .4(JS77 || 2.44636 .42929 2.32943 || 46 15 .36S92 2.7 1062 | .3SSSS | 2.57 150 . .4091 | | 2.44.133 . .42063 2.32756 || 45 16 .35925 || 2.70S19 .3S$2| || 2.5692S .400.15 2.44230 .4299S 2.32570 44 | 7 | .383.jS 2.7(1577 | .3S955 2.56.707 | .40979 || 2.44027 . .43||32 || 2.323S3 43 lS | .363S | 2,7033.5 .3$9SS 2.56.4S7 || 4 || 0 || 3 || 2.43S25 | .43(167 || 2.32197 || 42 19 | .37 (24 || 2.7(1094 | .30ſ)22 2.56266 .4 ()47 2.43623 | .431 () | | 2.32012 || 4 | 20 .37 (137 2.69853 | .390.55 2.56(H6 .4 IQS1 || 2.434.22 .43136 2.31826 40 21 | .3703() 2696 || 2 | .390S9 || 2.55S27 || 4 || 1 || 5 || 2,4322() .43| 70 2,3164 l 39 22 .37 23 || 2.6937 | .39 22 || 2.5560S .4 t 49 || 2,43010 | .432(15 2.31456 || 3S 23 .37 15: ; 2.6913| | .391.56 || 2.553S3 || 4 | | S3 2.42S19 .432.39 2.3127 I 37 24 .37 100 2.6SS92 .3919ſ) || 2.55 70 || 4 |2| 7 || 2,42618 | .43274 2.3 OS6 || 36 25 | .37223 || 26S653 .33223 2.54952 || 4 || 2.5l 2.424 is . .433/19 2.30902 || 35 26 .37256 2,6S-4 14 .30257 2,5-1734 . .4 12S5 2,422 S .13343 || 2.307 IS | 34 27 | .372S9 || 2.6S175 .39:200 || 2,54516 || 4 1319 || 2.42) 19 | .4337S | 2.30534 || 33 28 .37322 2.67:37 . .39324 || 2.54.299 .41353 2.4 IS19 .43412 || 2.30351 || 32 29 | .37355 2.677)\} | .30357 || 2.540S2 || 4 || 387 | 2.4 1620 | .434.47 || 2.30167 || 31 30 .373SS 2.67462, .39391 || 2.53S65 | .41421 2.41421 i .434S1 || 2.299S4 || 30 31 | .37422 2.67.225 33:25 || 2.536.1S | .41455 2.4 1223 .43516 || 2.29SO1 29 32 .37455 | 2.66989 .3945S 2.53432 || 4 |490 || 2.41 (25 | .43550 | 2.29610 |28 33 .37ts | 2.66752 .3J43. 2.532.17 || 41524 || 2.40S27 | .435S5 | 2.294.37 27 34 .37521 | 2.66516 .39526 2.53(H)|| 4155S | 2.40620 | .43620 | 2.29254 |26 35 | 37554 2,662S1 | .39559 || 2 527S6 || 4 1592 || 2.40432 | .43654 || 2.29073 || 25 36 | .375SS 2.66(146 .39593 || 2.5257 .41626 2,40235 | .436S9 2.2SS91 || 24 37 .37621 | 2.35S] I | .30626 2.52357 | .41660 | 2.40(3S | .43724 || 2.2S710 |23 38 .37654 || 2.65576 | .39600 | 2,52142 .41694 | 2.39S41 .4375S 2.2S52S 22 39 .37687 || 2.653.42 | .39694 || 2.51929 || 4 || 72S 2.396.45 .43793 || 2.2S34S 21 40 | .37720 2,651 (10 | .397.27 | 2.5 ! : 15 || 417(;3 || 2.304-10 || 43S2s 2.2S167 20 4 l .37754 2.64S75 .39761 2.515(12 .41797 2.39253 | .43S62 || 2.27987 19 42 .377S7 2.64642 .397.95 2.512S) || 4 |S31 2.3905S .43S97 || 2.27S06 is 43 .37 S20 | 2,644 () .39S29 2.51076 || 4 |S65 2.3SS63 .43932 || 2.27626 17 14 .37353 || 2.64177 .39SG2 2,50S64 || 4:S09 2.3SGGS .43966 2.27.147 | 16 45 .37SS7 2.63945 .39896 || 2.5(1652 || 4 || 933 2.3S473 | .44601 || 2.27267 15 46 | .37320|| 2.637 14 .39930 2.504.10 || 4 105s 2.3S279 || 44036 || 2,970SS | 1.4 47 | 37953 || 2.634S3 | .39963 2.5:229 .42002 || 2.3SOS4 .4407 2.26909 || 13 18 .37986 2.63252 | .39997 || 2 500[S .42036 || 2.37SOI .44105 || 2.26730 12 49 .3S020 2.63)21 | .40(131 || 2.49S07 || 4207ſ) 2.37697 .44140 2.26552 || 1 | 50 | .3S).53 2.6279| | .40065 2.49597 | .42105 || 2.3:50.1 ! .44 175 2.2837.4 10 51 | .3S0S6 2.6256 | | .4009S 2.493SG | .42139 2.373] 1 | .4420 2.26.196 || 9 52 .3312) 2.62332 | .401.32 2.49177 || 42173 2.37 US .442-14 || 2.260 18 || S 53 .3S153 2.62103 | .40166 2.4S967 . .42207 || 2.36:125 | .442;g 225S40 | 7 54 .3SIS6 2.6 (S74 | .40200 2.4S75S .42242 2.35733| 44.314 2.25663 || 6 35 | .3522) 2.61646 .40234 2.4S549 | .42276 | 2.3654 .4.1346 2.254S6 || 5 56 .33333 2.614 IS | .40207 2.4S340 .42310 | 2.36340 343; 2.25309 || 4 57 | .3S2S6 || 2 til 190 .4113ht | 2.4S132 | .42315 2.3615s | 44; is 2.25133 || 3 §§ .3S320 || 2 (; 1963 .40335 | 2.1792 || 42379 || 2.3596; .444;3 || 2:21956 5 39 .33333 2.60736 .4(369 || 2.47: 16 || 424.13 | 2.35776 || 4 tiss || 2:247s) || 1 | 60|_333SG | 2.60309|| 40.103 || 2.47509 | .42147 || 2:355S5 || 33323 || 2:2;604 || 0 M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. |M. 60C 682 679 669 |tº lIN Il ‘8qugoO oº:9 oº:9 of 9 oº::0 “Subj, ‘Suu,00 3ubj, ‘Āuthoo '3ubſ, ‘Susqoo!' IOS6S I 98668. I 69006. I 80306. I Z8806. I Özi:06. I A.0906. I IP/06. I 9.ES06' I 3, IOI6' I ZFI I6'ſ 2S&I6'I 8 IFI6' I #99 [6' I 069 I6' I 9&SI6' I 396 I6' I S6036"I 98.336' I 0&I96. I I92.96.' I '30.9300 IZI89;" 8.98%G' 9 [8&Q' 62&G’ Žižg' 90&&.3" S9) ºg' I£IZg' #60&Q' Z9029" 0&0&g" £8619" 956 [g' 606 IQ' &SIG" QESIG" S6/ I G' I92 IG" #32 I.G.' SSQIG" I99 IG" f I9 IG" AZSIG" O#919.' 803 IG" Z9PIg' 03FIg' 869. IG" 998.I.G.' 619. IG" £83, IQ' 952, Ig" 60&IG" £/ I IG" 98 I Ig’ 660IG" 990 IG" 930IG" 6S609" 99.603° |Tºua, 19296' I &Of 96 I pj996. I GS996' I A&S96. I 69696. I l l l 16"I 893,26. I G68/6' I 889/6. I 18926' I £3,826. I 996/6. I 0I ISG.' I 89&S6' I 969S6' I OpgS6' I $8986. I SČSS6' I 3.1686. I 9] [66. I 193,66. I 905-66. I 09966' I Q6966. I lf:S66. I 9S666. 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I6SF’ ISSSF’ GłSS5' 60SS5' 8/ZS:7' 09.090'2, 3.81.90% 888.90% GSp30'3, A8990 & ()64.90% ʲ'3, f6090'3, A.f390 & 00F90'3, 899.90% 90/90'3, 09S90 & Fl()/.0% Z9 IA.0% IZQZ0'3, 92P/0% 089403 SGH-60'Z, Hºggé0'2, I IS60’?, 69660'3 93, IOl'Z, is?,0I '8, ZFF0I '8, 00901 & SG20I '8, 9160I '8, GZ0I I & ££& II 2. 368] I & 2.99 || I'z, I [A | I & IZSI I & 0802, I’2, 06 IZl 2, 0988, I '8, IIQ2, I’8, I/92, I’8, £8. SøI’º, £668, I & Hºg Ig I-3, 9189 I & Ajº 3 6899 I & IOS8 2, £969 I & G2. If I'?, SSZFI & IG F# I & £Z/Sp' A8:/S7' I()2Sp," ç99Si." 6&98?” £63S5' Z998?' 12,987' 9SFSF’ A484/7.' I#82P." GOEZF' 2ZZp' j-93.25° 66 IAp" 99 IAF' S&I ºp' 2.60ZF' 990.15" 1302.5" GS695' 0969;" flé9?" 6&S95” 95895' SOS9?' 3.1/95' A $.19?" Žſ 495" 99.995' I9995' IGF; I'Z, pl 9p 1 % All? I & Of6? I'3 f{) IQ I & S939 I & 38 fºg I Z. 9699 I & O9/91 & Ø69 I & 0609 Iº, 99.391 & (J&F91 & 98.99 I & IG/9] "3, Z [69 I & 8:S02. I'8, 6FZZl 2, 9IfZ I & ŽSg. I '8; 6FZAl "3 9 IGA. I & fºs()8] 3, IGöSI & 6] FSH & ASGSI 3 99/SI "3 £3,6S I & Č606 I'3 I936 I & 08F6 I?, 6696 I & 6926 I & S866 I'6, SOIO2, 3 S/2.02, 3, 6FF0%, ’8, 61903 & ()6/03'3, 1960&’, &8 II?,’ & #08. I?,’ & GZFI2, 3, 1F918, 8, 6 ISI3'3 Ż6618, 8, #9 IZZ, & Z883.3% 0.19%. 3, £S9??, ?, AgSOE 08082, & #029.3% SZ893 & £GQ92, & A3/83, 3, Ž068.3% A.0fºº, ØØhº,'º', S&H #2, 3, #09fZz, I999F," 96.999' 0999p" 80995." £1995' SSQ95' 3.0997" Z9Pºp" Zgłºg}" Z699p" 2.99g 5’ ZZ8GP" Ž6ZGF’ ZQZG;" Z23,95' ZSIGſ," 3GIGF’ A.I IGF’ ŽS09 p" Zł0g?" ?, I09?" ZZ6F5° Zł6##" Z06tº' 2, 185?" AgSpf' 30Shp' £925?" £925?" A69Bºb" 2.99FF' AZ95.5° £695.5° SçGłºp" £ZGFB" ‘Subj, 3u8300 ‘8UBJ, ‘5uenoO ‘5ual o,22% O'Q2; oº:: of 2, f *S.I. NSI) NWJLOO (1 NW SJLNgiº) NWL TV 1ſt LV NI "A X SI"IgVJ. TABLE XV, 25.9 NATURAL. TANG ENTS AND COTA N G ENTS. | 28O 292 300 3 LO M | Tang. | Cotang. | Tang. | Cotang. Tang. | Cotang. Tang. | Cotang. TO |.5317| | ISS073 || 554.3TTS0405 T57735 |T| 73205 || 600S6 |T.6642S I .5320S 1.87941 .55469 | 1. SO2S1 | .57774 ; 1.730S9 .601.26 | 1.66318 2 .53246 I.S7809 | .55507 || 1.801 5S .57813 | 1.72973 .60165 | 1.66209 3 .532S3 | 1.87677 .55545 | 1.80034 .57851 | 1.72S57 .60205 || 1.66099 4 | .53320 | 1.87546 | .555S3 | 1.7901 || | .57S90 | 1.72741 .60245 | 1.65990 5 | .53358 | 1.87415 | .55621 | 1.7978S | .57929 | 1.72625 | .602S4 | 1.658SI 6 | .53395 | 1.872S3 .55659 | 1.79665 [ .57968 1.72509 || 6(1324 | 1.65772 7 | .53432 | 1.87 152 .55697 | 1.79542 | .5S007 | 1.72393 | .60364 1.65663 8 .53470 | 1.87021 .55736 1.794 | 9 || .5S046 | 1,7227S .60403 | 1.65554 9 .53507 | 1.86S9 | | .55774 | 1.79296 | .5SOS5 | 1.72163 | .60.143 | 1.65445 10 | .53545 | 1.86760 .558.12 || 1.79 |74 | .5S] 24 | 1.72047 | .604S3 | 1.65337 11 | .535S2 | 1.86630 | .55850 | 1.7905 || | .5S 162 | 1.7 1932 .60522 | 1.65228 12 .53620 | 1.86499 || .558SS | 1.78929 .5S201 | 1.7l SI 7 | .60562 | 1.65120 13 | .53657 | 1.86369 .55926 | 1.7SS07 .5S240 || ſ.7 1702 .60602 | 1.650] I 14 | .53694 | 1,86239 .55964 1.7S6S5 .5S279 | 1.7 [5SS | .60642 | 1,64903 15 .53732 | 1.86 [09 .56003 || I.78563 | .5S31S | 1.71473 | .606S1 | 1.64795 16 .53769 || 1,85979 .5604 || || 1.7S441 .5S357 | 1.7135S 60721 | 1.646S7 17 .53S07 | 1.85850 | .56079 i.7S319 .5S396 | 1.7 [244 .6076] | 1.64579 18 .53S44 | 1.85720 | .56l 17 | 1.7S19S | .5S435 | 1.7 | 129 .60S01 | 1.64471 19 .53882 | 1.8559) .56156 I,78077 .5S474 1.7 1015 .60S41 | 1.64363 20 .53920 | 1.85-162 .56.194 | 1.77955 | .5S513 | 1.70901 | .60SS1 | 1.64256 21 .53957 | 1.85333 .56.232 | 1.77S34 .58552 | 1.707S7 | . 6092] | 1.64148 22 | .53995 || 1,85204 || .56270 | 1.77713 | .5S591 | 1,70673 .60960 | 1.64041 23 .54032 | 1.85075 | .56309 | 1.77592 .5S631 1,70560 | .61000 | 1.63934 24 .54070 | 1.84946 .56347 | 1.77471 .5S670 | 1.70446 .61040 | 1.63S26 25 | .54107 | 1.84S18 .56385 | 1.7735l .58709 | 1.70332 .61GSO | 1.63719 26 .54145 | 1.846S9 .56-124 | 1.7723) | .5S74S | 1,70219 .61 l?0 | 1.63612 27 .541S3 | 1.84561 .56.462 | 1.771 10 | .5S7S7 701 06 | .61 60 | 1.63505 28 .54220 | 1.84433 .56501 | 1.76900 .5SS26 1,60092 .61200 | 1.63398 29 .54258 | 1.84305 | .56539 || 1.76S69 || .5SS65 | 1.69S79 .61240 | 1.63292 30| .54296 | 1.84177 .56577 | 1.76749 .5S905 | 1,69766 .612SO | 1.63.185 31 | .54333 | 1.84049 .56616 || 1.76629 | .5S944 1,69653 .61320 | 1.63079 32 | .54371 i.S3922 .56654 1.76510 | .5SQ83 | 1.695.41 .61360 | 1.62972 33 | .54409 | 1.83794 | .56693 | 1.76390 .500:22 | 1.6942S .61400 | 1.62S66 34 .54446 | 1.83667 | .56731 | 1.76271 .59061 | 1,69316 | .61440 1,62760 35 | .54484 || 1,83540 .56769 | 1.7615] | .59101 | 1.69203 | .614SO | 1.62654 36 .54522 | 1.83413 | .56SOS | [.76032 .59.140 | 1.6909] . .61520 1.6254S 37 .54560 | 1.832S6 | .56S46 | 1.75913 | .59179 | 1.6S979 .61561 | 1.6.2442 38 | .54597 | 1,83159 | .56SS5 | 1.75794 .59.218 1.6SS66 | .61601 | 1.62336 39 .54635 | 1.83033 | .56923 | 1.75675 | .5925S | 1.6S754 | .61641 | 1.62230 40 | .54673 || 1.82906 .56962 | 1.75556 .50297 | 1.6S643 | .616S1 || || 62125 4l .5471 l | 1.827S0 | .57000 | 1.75437 .59336 1,6S53] . .61721 1,62019 42 .54748 1.82654 .57039 || 1.753.19 | .59376 | 1.6S419 .61761 1,61914 43 .54786 | 1.8252S .57078 | 1.75200 || 59.415 | 1.6S30S .61S01 | 1.6l SOS 44 | .54S24 1.82402 .57ll & 1.750S2 .59454 | 1.6S196 | .61842 1,61703 45 | .54862 | 1.82.276 .57155 | 1,74964 ,594.94 | 1,6SOS5 | .61SS2 | 1.6159S 46 .54900 | 1.82150 .57 193 1,74S46 || 59533 1,67974 || .61922 1,61493 47 | .54938 | 1.82025 .57232 | 1.74728 .59573 1,67S63 | .61962 | 1.6l3SS 48 .54975 | 1.8.1899 j .5727 | | 1.74610 ,5961.2 1,67752 .62003 | 1.612S3 49 | .55013 | 1.8.1774 .57309 | 1.74492 | .59651 | 1.67641 .62043 | 1.6l 179 50 .55051 | 1.81649 .57348 | 1.74375 .5969] | 1.67530 .620S3 | 1.61074 51 .55089 1.81524 .573S6 | 1.74257 || 597.30 | 1.67419 .62124 1,60970 52 .55127 | 1.81399 || .57425 | 1.741.40 597.70 | 1.67.309 || 62164 | 1.60S65 53 | .55165 | 1.81274 .57464 | 1.74022 | .59809 | 1.67198 || 62204 || 1.60761 54 .55203 | 1.81 150 .57503 | 1.73905 | .59849 | 1.670SS | 62.245 | 1.60657 55 .55241 1.8.1025 | .57541 1,737SS .59SSS | 1.65978 || 622S5 1,60553 56 .55279 | 1.80901 | .575S0 | 1.73671 || 5992S | 1.66S67 | .623:25 | 1.60449 57 | .55317 | 1.807.77|| .57619 | 1.73555 .59967 1,66757 || 62366 | 1,603:15 58 .55355 | 1.80653| .57657 | 1.7343S 60007 | 1,66647 .62+06 | 1.60241 59 .55393 | 1.80529 | .57696 | 1.73321 .600.46 | 1,6653S | 62446 | 1.60.137 60 .55431 | 1.80405 | .57735 | 1.73205 .600S6 | 1.6642S | .62187 | 1.60033 M. Cotang. Tang. Cotang. I Tang. Cotang. | Tang. Cotang, Tang | 6I13 (50.2 503 589 14 M.: 200 COTANGENTS. TABLE XV. NATURAL, TANGENT'S AND 32O 330 340 350 Tang. .6.2487 .62527 .6256S ,62608 .62649 .626S9 .627.30 .62770 .62S] ] .62S52 .62892 .62.933 .62973 .630.14 .6.3055 .63095 .631.36 .631 77 .63217 .63258 .63299 .63340 .633S0 .63421 .63462 .63503 .63544 .635S4 .63625 .63666 .63707 Tang. Tang. Cotang. Tang. .64941 .649S2 .65024 .65065 .65106 .6514S .651 S9 .6523] .6:5272 .65314 .65355 .65397 .65438 .654S0 .65521 .65563 .66189 .66:230 .66272 .663 [4 .66356 ,66398 .66140 .664S2 .66524 .6G566 .6660S .66650 .66692 .66734 .66776 .66S18 .66S60 .66902 .66944 .669S6 .67028 .67071 .67 || 13 .67.155 .67]97 .67239 .672S2 .67324 .67366 .674.09 .67451 .67451 .67493 .67536 .67578 .67620 .67663 .67705 .6774S .67790 .67S32 .67S75 .69761 .69S04 .69S47 .69S91 .69934 .69977 .7(0.21 Cotang. Cotang. 1.4S256 1.4S163 1,48070 I.47977 I.47SS5 1.47792 1.47699 1.47.607 1.47514 1.47422 1.47330 1.472.38 1.47 146 1.47053 1.46982 1.46S70 I.4677. 1.466S6 1.46595 1,46503 l l l 1 l 1.454.10 1.45320 1.45229 1,45139 1.45049 1.4495S l 1.441.49 1,44060 1.43970 1,43SSl 1.43792 1.43703 1,43614 1.43525 1.43436 1.433.47 1,4325S 1.43169 1.430S0 1.42992 1.42903 1.42S15 .70021 .70064 .70.107 .70|5|| .70.194 .7()23S .70281 .70325 .7036S .704 12 .7(1455 70499 .70542 .705SG .70629 .70673 .70717 .70760 .70S04 .70S.48 .70S9 | .70935 .70979 7] 023 .7 1066 .713:29 .71373 .7 1417 .7146|| .7.1505 ,71549 .7 1593 .71637 .716S 1 7 1725 7 1769 7 IS13 .71S57 .7 1901 .71946 .7 1990 .72034 .7207S .72122 .72167 .722 || 1 .72255 72299 .72344 .723SS .724.32 .72477 .7252 I .72565 .726 || 0 .72654 Tang. Cotang. Cotang. Cotang. Cotang. 1.60033 1.59930 I.59S26 1.59723 1.596.20 1.59517 1.59414 1.593] 1 1.5920S 1.59105 1.59002 1.5S900 1.5S797 1.5S695 1.5S593 1.5S490 1.5S3SS 1.5S2S6 1.5S184 1.5S0S3 1,579S1 1.57879 1.5777S 1.57676 1.57575 1.57474 1.57372 1.5727 1 1.57 170 1.57069 1.56969 1.56S6S 1.56767 1.56667 1.56566 1.56-166 1.56366 1.56265 1.56165 1.56(365 1.55966 1.55S66 i.54685 1.53556 1.539S6 1.53SSS 1.5379 | 1.536.93 1.53595 1.53,497 1.53400 1.533(12 1.53205 1.53107 1.53010 1,529.13 1.52S 16 1.52719 1.52622 1.52525 1.52429 J.52332 1.52235 1,521 39 1.52043 1.5.1946 1.51850 1.5175 1.5165S 1.51562 1.5.1466 1.51370 1,51275 1.5l 179 1.5.1084 1.509S8 1.50S93 1.50797 1,50702 1,50607 1,505 i2 1.50417 1.50322 1.50228 I 50.133 1,50038 1.49944 1 49S49 I 49755 1.49661 1.49566 1.49.472 1.49378 1.4929.4 1.49190 1.49097 1.49003 1.4S909 1.48S16 1.487.22 1.48629 1.48536 1,484.42 1.4S349 1.482.56 Tang. Tang, 1.42S] 5 1.42726 1.41409 1.4 1322 1.41235 1.4 | 1.48 1.4 1061 1.40974 1.40SS7 1.40S00 1,407 14 1.40627 1.405-40 1.4045 1.40367 1.40281 1.401.95 1.40|09 1.45}{}22 1,39936 1.39850 1.39764 1,306.79 1.39593 1.395()7 1.30421 1.39336 1,392:50 1.391.65 1.39079 1.3S994 1.3S909 1,37891 1.37S()7 1.37722 1,3763S Tang. 560 550 540 Ml TABLE xv. NATURAL TANGENTS AND cotANGENTS. 201 369 37o 38O 390 : M. Tang. Cotang. Tang. Cotang. | Tang. || Cotang. | Tang. | Cotaag. |M. TO Tºč54 |TI3763S 75355 | 1.32704 |.7S129 1.27994 | .S0978 | 1.23490 | 60 | | 72699 || 1.37554 .75401 | 1.32624 .7S175 | 1.27917 | .81027 | 1.233 16 59 2 || 72743 | 1.374.70 | .75447 | 1.32544 .7S222 | 1.27841 | .8.1075 | 1.23343 5S 3 .72783 | 1.373SG | .75492 | 1.32464 .7S269 | 1.27764 .8l 123 1.23270 57 4 .72S32 | 1.37302 .7553S | 1.323S4 .7S316 1.276SS | .81 17 l | 1.23.196 || 56 5 .72S77 | 1.372 IS .755S4 | 1.32304 || .7S363 | 1.276.1 ! . .84220 | 1.23123 55 6 .72921 1.37 ||34 | .75629 | 1.32224 .7S410 | 1.27535 | .81268 || 1.23050 54 7 .72966 | 1.37050 .75675 | 1.32144 .7S457 | 1.2745S .8.1316 || 1.22977 53 8 . .73010 | 1.36967 .75721 | 1.32064 .7S504 || 1.273S2|.S1364 | 1.22904 || 52 9 .73055 | 1.36SS3 | .75767 || 1 319S4 | .7S551 | 1.27306 || S14 13 | 1.22831 || 51 10 .73100 | 1.36S00 .75S12 | 1.319(H | .7S50S 1.27230 .81461 | 1.22758 50 ll .73144 | 1.367 ig | .75S5S | 1.3tS25 | .7S645 | 1.27 153 .81510 | 1.226S5 49 12 .731S9 | 1.36633 . .759ſ) 4 || 1.31745 .7S692 | 1.27.077 .8l 55S 1,22612 48 13 | .73234 | 1.36549 .75050 | 1.31666 .7S730 | 1.27001 || Si606 | 1.22539 47 14 .7327S 1.36466 .75096 || 1.315SG | .7S7S6 | .26925 | .81655 1.22467 46 15 .73323 | 1.363S3 | .76042 | 1.31507 | .7SS34 1.26S49 | .S1703 | 1.22394 || 45 16 .7336S | 1.36300 | .760SS | 1.31427 .7SSSI | 1.26774 .S1752 | 1.22321 || 44 17 .734 13 1.36217 | .76||34 || 1.3134S .7SO2S 1.2669S | SIS(10 | 1.22249 43 18 .73157 | 1.3613-1 | .76 ISO | 1.31269 .7S975 | 1.26622 | .81849 | 1.22176 || 42 19 .73502 | 1.3605 t . .76226 | 1.31 l 90 .79022 | 1.26546 .81 S98 || 1.221 (k4 41 20 .735-17 | 1.3596S | .76272 | 1.3 l l 10 | .790.70 | 1.2647 | | .S1946 | 1.22031 | 40 21 | .73592 | 1.35SS5 .76318 1.31031 .791 | 7 | 1.26305 || S1995 | 1.21959 || 39 22 || 73637 | 1.35SQ2 .76364 | 1.3(1952 . .79164 | 1,263.19 || S2(\4-1 | 1.21 SS6 || 38 23 .736St | 1.357 19 | .764 10 | 1.30S73 ,7921 2 | 1.26244 .S2092 || 1.2 S14 || 37 24 .73726 | 1.35637 || 76456 | 1.30795 .79:259 | 1.26169 .S2I4] | 1.21742 || 36 25 | .7377 | | 1,35554 | .76502 | 1.307 || 6 | .79.306 | 1.26093 .8.2190 | 1.21670 35 26 .73SI6 | 1.35472 .7654S | 1.30637 .79354 | 1.260.18 .8223S l.2159S 34 27 | .73S61 | 1.353S9 .76594 | 1.30558 || 79401 | 1.25943 .S2287 | 1.21526 || 33 28 .73906 | 1.35307 .76640 | 1.304SG | .79449 | 1.25S67 .82336 | 1.21454 32 29 .7395.1 | 1.35224 .766S6 | 1.3040l .79496 | 1.25792 .82385 | 1.213S2 31 30 .73996 | 1.35142 .76733 | 1.30323 .79544 | 1.257 17 | .82434 | 1.21310 30 31 .74041 | 1.35060 | .76779 | 1.30244 .79591 1.25642 .824S3 || I.2123S 29 32 | .740S6 | 1.3407S | .76S25 | 1.30166 | .796.39 || 1.25567 | .S2531 1.21 166 || 28 3 .74131 | 1.34S96 | .76S7 | | 1.300S7 || 7.96S6 | 1.25492 | .S2580 | 1.21094 || 27 34 || 74 176 | 1.34S14 | .76%) lS | 1.30009 || ,79734 | 1.254 l 7 | .82629 | 1.21023 26 35 | 7422t | 1.347.32 .76064 1.2993] | .797SI 1.25343 .8267S | 1.2ſ 951 || 25 36 || 74267 | 1.3465() { .77() l () | 1.20S53 | .79S29 | 1.2526S .827.27 | 1.20S79 || 24 37 || 743] 2 | 1.3456S .77.057 | 1,207.75 | .79S77 | 1.25193 ) .S2776 1.20SOS 23 38 || 71357 | 1.344S7 .77 t 03 | 1.29696 | .7992.1 | 1.251 lS | S2S25 | i.20736 22 39 || 7.1402 | 1.34405 .77 I-40 | 1.296 l'S .79972 | 1.25044 || S2S74 | 1.20665 21 40 || 74.447 | 1.34323 | .77.196 | 1.2954 l .80020 | 1.2-1969 || S2023 | }.20593 || 20 41 7:1492 | 1.34242 | .772-12 | 1.29:63 | SOU67 | 1.24S95 | .S.2972 | 1.20522 19 42 .7453S | 1.34 |60 | .772S9 || 1,293S5 .801 15 | 1.248.20 | .S3022 l.20451 | 18 43 .7.15S3 | 1.34079 || 77335 | 1.20307 | .80 l 63 | 1.247.46 .8307 | | 1.20379 17 44 .74628 1.3390S .773S2 | 1.20229 || S()2t 1 | 1.24672 | .83.120 | 1.2030S 16 45 ,74674 | 1.33916 | .7742S | 1.29152 | SO25S | 1.24597 || S3169 | 1.20237 || 15 46 .74719 I.33S35 .77475 | 1.20074 || S0306 | 1.24523 || S321 S 1.20166 || 14 47 | .74764 | 1.33754 | .77521 | 1.2S997 | .8(1354 1,24449 || 8326S | 1.20095 || 13 48 | .74S10 | 1,33673 .7756S | 1.2S919 . .80-102 | 1.24375 .S3317 | 1.20024 12 19 , 74S55 | 1.33592 | .77615 | 1.2SS42 | SO150 | 1.24301 || S3366 | . , 19953 || 1 | 50 .74000 | 1.335ll .7766 | | 1.2S764 || S(\{QS | 1.24227 | S3415 | 1. 19SS2 || 10 51 ,74946 | 1.33:130 | .7770S 1.2S6S7 || S0546 | 1.2.4 lb3 || S3465 | 1. 19Sll 9 52 .74991 | 1.33349 .77754 | 1.2S610 | S0594 | 1.24079 i .S3514 | 1.19740 || 8 53 .75037 | 1.33:26S | .77SOl | 1.2S533 .806-12 | 1.24005 || 83564 | 1. 19669 7 54 .750S2 | 1.331S7 | .7784S | 1.2S456 ..SO690 | 1.23931 || S3613 | 1. 19599 || 6 55 .7512S | 1.331 07 || 77S95 | 1.2S379 | SO73S | 1.23S5S | S3C62 | 1. 1952S 5 56 | .75173 | 1.33026 | .7794 l ; 1.2S302 || S07S6 | 1.237S4 .S3712 | 1.19457 || 4 57 | .75219 | 1.32946 .779SS | 1.2S225 | SOS34 | 1.23710 || S3761 | 1.19387 3 58 .75264 | 1.32S65 .7SO35 | 1.2Si3S | SOSS2 | 1.236.37 S3SI I | 1. 19316 || 2 59 .75310 | 1.327S5 .7SOS2 | 1.2S07 || SO93() | 1.23563 || S3S60 | 1, 19246 || 1 60 | .75355 | 1.327.04 || 7Sł29 | 1.27994 | SQ97S | 1.23490 S391 () { 1, 19175 0. | M.(?otang. | Tang. 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I 12206 |z9IFI’I g69ZS #92Sl'I 999 F8" | 81 8? | 68F90°l 90686 GSZ0I 'I f/906 || 6ZZFI' I £fg28' | fg.98 I ‘I Z09;8" | &I 69 || 19990'I ZGS86 || 6;80 I’I IZ906' 962.É I ‘I 2,6FZS' | HOFSI "I 29 FFS' II 09 || 8 |990'I | 16486 | FIFOl' I 699.06 || 999 FI' I l FFAS' | #1581 I Z0F#9" | OI 19 || 94990’ I Ğ" | SZł0I 'I 9 IQ06 || 09:F# I I 6SEZ8" | ##GSI I ZggłS' | 6 39 290 | | SS986' £HCOI I | 89 HO6 S6th I I S892S' | #198 I ‘I Z03+8" | 8 89 || 00S90 I £8996 || Z090I 'I Oliſ)6 g0gFI'l ZSZZS' | #S9SI " I | SgzłS' | Z #9 &S90’ I | S1986 Z1901 ‘I 19806' Z895.I.' I 99.228" | #g/SI " I | SOzłS' | 9 99 Ø690"I #3986" | 1820I 'I #08:06' | 669Fl'I FSLS' | #2SSI I | SGI+8" |g 99 || 18690’ I 69F86 |Z0SOI. I Igzó6' | 1925 I'I £3 IZS' | #6SSI*I 80IHS' 5 Z9 | 6tQ40' I | g Itô6 || 1980I 'I | 66 [06 | #8SFI’I ZSOAS' | #96SI I | 6GOF8" | 8 89 &II.4.0 I | 09886’ 1860I 'I 9F106 |Z06FI “Y | Ig018" | G806 I ‘I 600F8" | z 69 #4 IA0' I | 90886 96601'I £6006" | 6965 I*I OS698" | G0I6I “I 09688" | I 09 || 48.640"I &6 || 1901 l’I_{0}006 |2809 I’I_{63698 |9416L'I |OI688" | 0 ‘W 8d0300 ºup L | 3uenoo | 3a eI, 3uthoo 3UBJ, ‘āubyoo 3u81, 'IV of 5 oºij, olſ tº CO; "S.LN J 5)N WJ O J QſNW SJLNGI*) NWL TVºIſlLWN Č9% *AX GITBIVI, TABLE xv. NATUBAL TANGENTS AND cotAxgENTs. 263 4 4 O 4 lo 4 49 M. Tang. | Cotang. M. M. Tang. | Cotang. M. | M. Tang. Cotaug. M. 0 .96569 | 1.03553 60 | 20 | .97700 | 1.02355 | 40 || 40 .9S$43 | 1.01 170 i 20 1 .96625 | 1.03493 59 || 2 | | .97756 || || 02:295 || 39 || 4 || | .9SQ(\l 1.01 || 12 19 2 . .966S1 || 1,03433 58 22 | .978 || 3 || 1,02236 || 3S 42 .9S958 | 1.01.053 || |S 3 | .96738 1.03372 57 || 23 .978.70 | 1.02176 || 37 43 .90016 || 1.0(1994 || 17 4 .96794 | 1.ſ)3312 || 56 24 .97927 | 1.02 || 7 || 36|| 44 .99073 | 1,00035 | 16 5 | .96S50 | 1.03252 55 ||25 | .979S4 || 1.02057 || 35 || 45 .99 |31 | 1.0(S76 15 6 .96907 | 1.03.192 54 || 26 | .9SQ4 | | 1.01998 || 34 || 46 | .991S9 1,00818 14 7 .96963 | 1.03132 53 27 .9S098 || 1.01939 33 47 .992.47 1.00759 13 8 . .97020 | 1.03072 || 52 |2S | .9S] 55 | 1.01S79 || 32 4S | .993(k4 | 1.007 (11 || 12 9 .97.076 | 1.03012 || 51 || 29 | .9S213 | 1.01S20 || 3 || || 49 | .99302 | 1.006.42 || 1 | 10 .97 ||33 || 1.02952 50 || 30 .9S270 | 1,01761 || 30 || 50 .99.420 | 1.005S3 || 10 | 1 | .97 189 1.02S92 || 49 |31 | .9S327 | 1,01702 || 29 || 5 || | .99.478 1.00525 9 12 .97246 | 1.02S32 || 48 || 32 .9S3S4 1.01642 2S 52 .99536 | 1.00467 || 8 13 .97.302 | 1.02772 |47 ||33 | .9S441 1.015S3 27 || 53 .99594 | 1.00408 || 7 14 .97359 || 1,027 13 || 46 34 .9S499 || 1.01524 26 || 54 .99652 | 1.00350 || 6 15 .974] 6 || 1.02653 || 45 || 35 | .9S556 | 1,01465 || 25 || 55 i .997 || 0 | 1.00291 || 5 16 .974. 2 1.02593 || 44 || 36 .98613 1,01406 || 24 || 56 .99768 || 1.00233 || 4 17 .97529 || 1.02533 || 43 ||37 | .9S671 | 1.0.1347 23 57 | .99$26 1.00175 i 3 1S .97586 || 1,02474 42 3S | .98728 1.012S8 22 58 .99$S4 || 1.001 || 6 || 2 19 .97643 | 1.024 14 || 4 || || 39 || .98786 | 1.01229 || 2 | | 59 . .99942 | 1.00058 || || 20 .97700 | 1.02355 40 | 40 .9SS43 | 1.01 170 20 || 60 | 1.00(K)0 | 1.00000 || 0 M. Cotung. I Tang. |M. M. Cotung. Tang. |M. M. Cotang. | Teng. |M 4 50 * 4A 60 4 tºo 264 TABLE XVI. RISE PER MILE OF VARIO US GRADES. T A B L E X W I. RISE PER MILE OF WARIOUS GRADES. Grade Rise por Grudle IRíso per Grade Rise per Grado Riso per per Milo. ar Milo. T ilt, per Mile. Rtation. Station. Station. Station. .01 .52S .4l 21.64S .81 42.768 1.21 63.SSS .02 1.036 .42 22, 170 .82 4.3.206 1.22 64.4 | 6 .03 1.584 .43 22.7()4 ,83 43.S24 1.23 64.944 .04 2. I 12 .44 23.232 .84 44.352 1.24 65.47? .05 2.640 .45 23.760 .85 44.8SO 1.25 66.000 .06 3. 68 .46 24, 2SS .86 45.40S 1.26 66.528 .07 3,696 .47 24,810 .87 45.936 1.27 67.056 .08 4.224 .48 25.344 .88 46.46-4 1.28 67.5S4 .09 4.752 .49 25, S7. .89 46.992 1,29 6S. l l 2 . 10 5.280 .50 26.4()0 .90 47.520 1.30 6S.6-10 . I l 5,808. .5| 23.928 ,91 48.048 1.31 69. 168 . 12 6.336 .52 27.450 .92 48,576 1.32 69,696 .i.3 6. S64 .53 27.9S4 .93 49. 104 1.33 7().?:24 . 14 7.392 .54 28.51% .94 49.632 1.34 70,752 . I 5 7.9.20 .55 29.()40 .95 50, 160 1.35 7 l.2SO ... 10 8.448 .56 29.568 .9ö 50.68S 1.36 7| S(\S . 1 7 8.976 .57 30,000 .97 51.216 1.37 72,336 . 18 9.504 .5S 30,624 .98 51,744 1.38 72 S64 . 19 10,032 .59 31, 152 .99 52.272 1.39 73. 39'2 ,2) 10. 560 .60 31.6S0 1.00 52.800 1.4() 73.921) .21 11.088 .61 32.2ſ,8 | 0 | 53.328 1.4 l 74 448 .22 | | .616 .62 32.736 1.04 53.856 1.42 74,976 .23 | 2. 144 .63 33,254 l,03 54.3S-4 1.43 75.504 .24 12.672 .64 33.792 1.04 54.912 1.44 76,032 .25 13.2(X) .65 34.320 1.05 55. 4.10 1.45 76.500 .26 13.728 ,66 34,848 1.06 55.96S 1.46 77, (kº .27 | 4.250 .67 35.376 1.07 56, 496 1.47 77.6 ſt; .28 14.784 .68 35,904 1,08 57.024 1.48 78. 14 .29 15.312 .69 36.432 1.09 57.552 1.49 78.67. .3() 15,840 .70 36.960 1.10 58.050 1,50 79.200 .3| 16.368 .71 37.488 1. II 58.608 1, 5] 79 729 .32 16,898 .7 38,016 1, 12 59. 136 1,52 80.266 .33 | 7.4% .73 38.544 I, 13 59.664 1.53 80.784 .34 17.952 .74 39.072 1. 14 60.19% 1.54 81.312 .30 18,480 .7 39.600 1.15 60.720 1.55 81.840 .3% 19,008 .7 40. 128 1. 16 61 248 1.58 82.368 .37 19.538 .77 40.656 1. 17 01,776 1.57 82.886 .38 20.064 .78 4.1. 184 1.18 62.3(k4 1.58 83.4% .39 20.592 .79 41.712 1, 19 62.832 1.59 83 952 .40 $21, 120 .80 42.240 1.20 63.380 1.60 ,480. TABLE X v1. RISE PER MILE OF VARIous GRADEs. 265 gº Rise per | *...* | Rise per dº Rise per | *| Rise per e.g. Mile. per Milo. pe Mile. per Alile. Station. Station. Station. Station. 1.61 85.00S 1.81 95.5GS 2. 1 () | | |0.SS0 4.1() 216.480 1.62 85.536 1.S2 96,096 || 2.20 1 16. 160 || 4.2() 221.760 1.63 86.064 1.S3 96.624 || 2.30 | 121.440 4.30 227.040 1.64 86.592 1.84 97. I 52 2.40 | 126.720 || 4.40 || 2:32.320 1.65 | 87. 120 1.85 97.6S0 || 2.50 | 132,000 || 4.50 237.600 1.66 87.64S 1.86 9S.208 2.60 | 137.280 || 4.60 242.SSO 1.67 SS, ) 76 1,87 9S. 736 2.70 || 142.560 || 4.70 || 248. 160 1.63 SS,7()4 1.88 99.264 2.80 147.840 || 4.80 || 253.440 1.69 | 89.232 1.S9 99,792 || 2.90 || 153. 120 || 4.90 || 2:58.720 1.70 || S9.760 1.90 || 100.320 i 3.00 || 158,400 || 0.00 264.000 1.71 90.2SS 1.91 || 100.S48 || 3. 10 | 163.680 || 5 || 0 || 269,280 1.72 90.S 16 1.92 || 101.376 || 3.20 | 16S.960 5.20 274.560 1.7 91.344 1.93 || 101,904 || 3.30 || 174,240 5.30 279.840 1,74 || 91.872 1.94 || 102.432 || 3.40 || 179,520 5.40 285, 120 1.75 92.400 1.95 || 102.960 || 3 50 184.800 5.50 290.4(10 J.76 92.92S 1.96 || 103.4S8 || 3.60 | 190.0S0 5.60 295.6SO 1.77 93.456 1.97 || 1 ()+.016 || 3.70 || 195.360 5.70 || 300,960 1.78 93.9S4 1.98 || |(}4.544 || 3.80 200,640 || 5.80 || 306.240 1.7 94.512 1.99 || 105.07; 3.90 205.920 5,90 || 311.520 | 1.80 || 95.040 2.00 105,000 || 4.00 21 l.200 || 6.00 || 316.800 266 TABLE XVII. TANGENTS AND SHORTEST DISTANOEs FROM TABLE XVII. TANGENTS AND STIORTEST DISTANCES I'ROM INTERSEC, TION POINT OF A ONE-DEGREE CURWE. For chords of 100 feet the radius of a one-degree curve is 5729.65 feet. To find its tangent for any intersection angle I, we have, by § 4, T = R tan. I, and to find the shortest distance from the in tersection point to the curve, we have, by § 71, b = T tan. # I. By these formulae this table is computed. To find T and b for a curve of any other degree (chords 100 feet), divide the values given in the table for the proper intersection angle by the number of degrees, whole or fractional, of the curve. Thus, to find T and b for a 3° 20' curve we should divide the proper tabu- lar values by 33. This process supposes the radii of curves to be inversely proportional to their degrees. This is not strictly true, as may be seen by referring to Table I. The resulting errors, however, are in general too small to be of practical importance in the ordinary use of such a table. For a 10° curve and an intersection angle of 90°, the value of T obtained from the table is too small by only 7, and that of b too small by only .3, For curves of smaller degrees and for smalier intersection angles the errors diminish rapidly. INTERSECTION POINT OF A. ONE DEGREE CURWE. 267 T b. I T b. I T. Ö 50.0 .22 || 6° 300.3 || 7.86 11° 551.7 26.50 54.2 .26 B' | 304.5 8.08 5’ 555.9 26.90 58.3 .30 10 || 305.6 8.31 10 || 560.1 27.31 62.5 .34 15 313.8 8.53 15 564.3 || 27.72 66.7 . 39 20 || 3:17.0 | 8.76 20 568.5 28.14 70.8 .44 25 33i: | 8.3% 25 || 572.7 28.55 75.0 .49 30 || 335.4 3.33 30 576.9 28.97 '79.2 .55 35 | 339.5 9.47 35 581.2 29.40 83.3 .61 | 40 || 333.7 || 3:... 40 5$5.4 29.82 87.5 .67 45 337.9 || 3-3 || 45 583.6 30.25 91.7 .73 50 || 342.1 | 10.30 50 593.8 || 30.69 95.8 .80, 55 || 3:46.3 10.45 55 598.0 || 31.12 100.0 .87 7 350.4 || 10.71 || 12 602.2 31.56 104.2 .95 || ". 354.6 | 10.3% 5 606. 4 || 32.00 108.3 I.02 10 || 358.8 11.2 10 || 610.6 32.45 112.5 1.10 15 363.0 | 11.49 15 614.9 || 32.90 116.7 | 1.19 20 || 367.2 | 11.75 20 619, 1 ; 33.35 120.9 1.27 25 | 371.4 12.02 25 623.3 : 33.80 125.0 | 1.36 30 375.5 12.39 .30 627.5 34.26 129.2 | 1.46 35 | 379.7 | 12.53 35 | 631.7 34.72 133.4 1.55 40 | 383.9 12.85 40 635.9 || 35.18 137.5 1.65 45 3SS.1 13.13 45 640.2 35.65 141.7 | 1.75 50 392.3 13.41 50 644.4 36.13 145.9 | 1.86 55 | 396.5 13.70 55 648.6 || 36.59 150' 0 | 1.96 || 8 400.7 13.99 i 13 652.S 37.07 154.2 2.07 5 | 404.8 14.28 5 : 657.0 37.55 15S. 4 || 2.19 10 | 409.0 14.58 10 | 661.3 38.03 162.5 2.31 15 || 413.2 14.SS 15 665.5 38.52 166.7 2.42 20 417.4 15.18 20 669.7 39,01 170.9 2.55 25 421.6 15.49 25 673.9 39.50 175. I 2, 67 30 425.8 15.80 30 67S. i 39.99 179.2 2.80 35 | 430.0 16.11 35 | 682.4 40.49 1S3.4 2.93 40 || 434.2 | 16.43 40 686.6 40.99 1ST .6 3,07 45 || 438.4 16.74 45 69t).S 41.50 19] .7 3.21 50 || 442.5 17.07 50 695.1 42.00 195.9 3.35 55 446.7 17.39 55 699.3 42.51 200:1 3.49 || 9 450.9 || 17.72 703.5 4: 204.3 || 3.64 5 # ##|* s #; #; 208.4 3.79 10 || 459.3 ; 18.3S 10 || 712.0 44.07 212.6 3.94 15 463.5 18.72 15 || 716.2 44.59 216. S 4.10 20 || 467.7 19.06 20 | 720.4 45.13 220.9 4, 26 25 471.9 19, 40 25 | 724.7 45.65 225.1 4.42 30 76. 1 19,75 30 728.9 46.18 229:3 || 4, 59 35 || 480.3 20.10 35 | 733.1 46,71 233.5 4.75 40 || 484.5 20.45 40 | 737.4 7.25 237.6 4.93 45 || 488.7 | 20.80 45 || 741.6 47 - S0 241.8 || 5.10 50 492.9 21, 16 50 | 745.8 48.34 246.0 5.28 55 || 497.1 21, 52 55 || 750. 1 # 48.89 250.2 5.46 10 501.3 21.89 is '754. * - sº 254.3 || 5.64 5 #; #: |* a j . . . 25S.5 5. S3 10 || 509,7 22,62 10 762.8 50.55 262.7 6.02 15 513.9 23.00 15 767.0 51, 12 266.9 6.21 20 || 518.1 23.37 20 ºf .3 51.68 271.0 6.41 25 || 522.3 23.75 25 | 775.5 52.25 275.2 6. (51 30 526.5 24, 14 30 779.8 52.S2 279.4 6. S1 35 | 530.7 24.52 35 | 784.0 53.39 283.6 7.01 40 534.9 24.91 40 788.3 53,97 287.7 7.2.2 45 || 539. 1 || 25.30 35 | 733.5 54.55 291.9 7.43 50 || 543.3 25.7 50 796.8 55,13 296.1 '7.65 55 547.5 26.10 55 801.0 5 .7: 268 TABLE XVII. TANGENTS AND SHORTEST DISTANCES FROM * I. T. b. I. T. b. I. T. b. 16° 805. 56.31 21° 1061.9 97.58 26° 1322.8 150.7 5' | 809 56.90 5/| 1066.2 || 98.36 5' | 1327.2 151.7 10 | S13 57.50 10 || 1070,6 || 99.15 10 || 1331.6 152.7 15 818 58. 10 15 1074.9 99.95 15 $1336.0 | 153.7 20 822 58.70 20 | 107.9.2 | 100.7 20 1340.4 | 154.7 59.91 30 | 10S7.8 || 102.3 30 || 1349.2 156.7 60.53 35 | 1092.1 || 103.2 35 | 1353.6 157.7 61.14 40 1096.4 104.0 40 1358.0 | 158.7 61." 45 1100. S | 104.8 45 1362.4 || 159.7 - 62.38 50 | 1105.1 105.6 50 1366.8 160.8 55 | 852.0 | 63.01 55 | 1109.4 106.4 55 | 1371.2 | 161.8 17 856, 3 63.63 22 11:3.7 | 107.2 27 1375.6 162.8 5 | 860.6 || 64,27 5 1118.1 108.1 5 1380.0 163.8 10 || 864.8 64.90 10 | 1122.4 1(18.9 10 || 1384.4 164.9 15 869. 1 65.54 15 1126.7 109.7 15 || 1388.8 165.9 20 | 873.3 | 66.18 20 | 1131.0 | 110.6 20 | 1393.2 167.0 2 5 7 0 .3 3. §§ 59, 31 25 || || 083.5 | 101.5 25 | 1344.8 155.7 0 3 5 S 20 1010.3 | SS. 39 25 1270.2 | 139.1 30 1535.3 | 202. 5 | 1014.6 | 89.14 5 1274.6 140.1 5 | 1539.7 203. 10 || 1018.9 | 89.89 10 | 1279.0 | 1.41.0 10 | 1544.2 | 204. 15 1022.2 | 90.64 15 1283.4 142.0 15 1548.7 205. 20 i 1027.5 91.40 20 1287.7 142.9 20 || 1553. 1 || 206.8 25 | 1031.8 92.16 25 | 1202. 143.9 25 1557.6 207.9 30 1086.1 || 92.92 30 1296.5 144.9 30 1562.1 209.1 35 | 1040.4 || 93.69 35 | 1300.9 || 145.8 35 | 15('6.6 210.3 40 || 1044.7 94.46 40 | 1305.3 || 1:46.8 40 | 1571.0 211.5 45 | 1049.0 | 95.24 45 || 1309.6 | 1.47.8 45 || 1575.5 212.7 50 | 1053.3 || 96.01 50 1314.0 148.7 50 | 1580.0 213.9 55 | 1057.6 || 96.79 55 | 1318.4 149.7 55 | 1584.5 | 215.1 25 | 877.6 | 66.82 25 | 1135.4 || 111.4 25 | 1397.6 | 168.0 30 | 881.9 67.47 30 || 1139.7 112.3 30 1402.0 | 169.0 35 | 886, 1 68. 12 35 | 1144.0 | 113.1 35 | 1406.5 170.1 40 || S90.4 6S.77 40 | 1148.4 || 113.9 40 || 1410.9 171.2 45 | 894.7 | (;9.43 45 1152.7 | 114.8 45 1415.3 || 172.2 50 | 898.9 70.09 50 | 1157.0 | 115.7 50 1419.7 | 173.3 55 | 903.2 || 70.75 55 | 1161.4 || 116.5 55 1424.1 | 174.3 18 907.5 71.42 23 1165.7 || 117.4 || 28 1428.6 || 175.4 5 911.8 || 72.09 5 || 1170.1 118.2 5 1433.0 || 176.5 10 | 916.0 | 72.76 10 || 1174.4 119.1 1() || 1437.4 || 177.6 15 920.3 | 73.44 15 1178.7 | 120.0 15 1441.8 || 178.6 20 | 924.6 || 74. 12 20 || 1183.1 | 120.9 20 1446.3 179.7 25 || 928.9 74.80 25 | 11S7.4 121.7 25 || 1450.7 180.8 30 | 933. 1 || 75.49 30 || 1191.8 122.6 30 || 1455. i | 181.9 35 | 937.4 76.18 35 | 1196.1 | 123.5 35 1459.6 | 183.0 40 || 941.7 76.87 40 | 1200.5 | 124.4 40 || 1464.0 | 184.1 45 || 946.0 | 77.57 45 1204.8 || 1:25.3 45 1468.5 185.2 50 950.2 || 78.26 50 | 1209.2 | 126.2 50 || 1472.9 | 186.3 55 | 954.5 78.97 55 | 1213.5 | 127.1 55 1477.3 | 187.4 j9 95S-8 79.67 24 1217.9 | 128.0 ſ 29 14S1.8 1S8.5 5 || 963, 1 || 80.38 5 1222.2 | 128.9 5 1486.2 | 189.6 10 || 967, 4 || 81.09 10 1226.6 | 129.8 10 || 1490.7 | 190.7 15 971.7 81.81 15 1230.9 || 130.7 J5 || 1495.1 | 191.9 20 | 976.0 | 82.53 20 | 1235.3 || 131.7 20 1499. (; 193.0 25 | 980.2 83.25 25 | 1Q39.7 132.6 25 | 1504.0 194.1 30 9S4.5 83.97 30 | 1244. () || 133.5 30 | 1508.5 195.2 35 | 9SS.S | S4.70 35 | 1248.4 134.4 35 | 1512.9 | 196.4 40 || 993. 1 || 85.43 40 | 1252.8 || 135.4 40 | 1517.4 | 197.5 45 997.4 86.17 45, 1257.1 || 136.3 45 1521.9 198.7 50 | 1001.7 | 86.90 Ü0 | 1261.5 | 137.2 50 | 1526.3 | 199.8 55 | 1005.0 | 87.64 55 | 1265.9 || 138.2 55 | 1530.8 201.0 1 3 4 6 INTERSECTION POINT OF A. ONE DEGREE CURWE. 269 I T Ö. I. | T. b. I. T. b. 310 1589.0 | 216.2 || 36° 1861.7 || 294.9 || 41° 2142.2 || 387.4 5’ 1593.5 || 217.5 5’ 1866.3 296.3 5’| 2147.0 389.0 10 | 154)S.0 || 218.7 10 1870.9 297.7 10 || 2151.7 390.7 15 1602.4 219.9 15 1875. 5 || 299.1 15 2156.5 392.4 20 | 1606.9 221.1 20 1880.1 300.6 20 2161.2 394.1 25 | 1611. 4 || 2:22.3 25 | 18S4.7 302.0 25 || 2166.0 395.7 30 | 1615.9 223.5 30 18S9.4 303.5 30 2170.8 397.4 35 | 1620.4 224.7 35 | 1894.0 304.9 35 | 2175.6 || 399.1 40 1624.9 226.0 40 | 1898.6 306.4 40 || 2180.3 | 400.8 45 | 1629.4 227.2 45 || 1903.2 307.8 45 2185.1 | 402.5 50 1633.9 228.4 50 | 1907.9 || 309.3 50 2189.9 || 404.2 55 ; 1638.4 229.7 55 | 1912.5 ! 310.8 55 2.194.6 | 405.9 32 1643.0 230.9 37 1917.1 ! 312.2 42 2190.4 | 407,6 5 | 1647.5 || 232.1 5 | 1921.7 || 313.7 5 2204.2 409.4 10 | 1652.0 233.4 10 | 1926.4 315.2 10 || 2209.0 411.1 15 1656.5 234.6 15 1931.0 316.6 15 2213.8 412.8 20 | 1651.0 || 235.9 20 1935.7 318.1 20 || 2:218.6 41 4.5 25 | 1665.5 237.2 25 1940.3 || 319.6 25 22:23.3 416 3 30 1670.0 || 238.4 30 1945.0 321. 1 30 || 2228.1 41S.0 35 | 1674.6 || 239.7 35 1949.6 322.6 35 2232.9 || 419.7 40 | 1679.1 241.0 40 | 1954.3 324.1 40 || 2237.7 421.5 45 1683.6 24.2.2 45 1958.9 325.6 45 2242.5 423.2 50 | 1688.1 243.5 50 | 1963.6 327.1 50 || 2:47.3 || 425.0 55 | 1692.7 244.8 55 1938.2 32S. 6 55 || 2252.2 426.7 33 . , 1697.2 246.1 || 38 1972.9 330.1 : 43 2257.0 || 49S.5 5 1701.7 | 247.4 5 | 1977.5 331.7 5 2261.8 || 430.3 10 1706.3 248.7 10 | 1982.2 | 333.2 10 2266.6 432.0 15 1710.8 250 0 15 1986.9 || 334.7 15 2271.4 || 433.8 20 1715.3 || 251.3 20 | 1991.5 336.2 20 || 2:276.2 435.6 25 1719.9 || 252.6 25 | 1996.2 337. S 25 2281. 1 || 437.4 30 || 1724.4 253.9 30 2000.9 339.3 30 22S5.9 439.2 35 | 1729.0 255.2 35 | 2005.6 340.9 35 2290.7. 441.0 40 1733.5 256.5 40 | 2010.2 342.4 40 2295.6 442.7 45 173S. 1 || 257.8 45 | 2014.9 || 344.0 45 2300.4 444.5 5ſ) 1742.6 259.1 50 | 2019, 6 345.5 50 || 2305.2 446.4 55 i 1747.2 | 260.5 55 2024.3 347.1 55 || 23.10.1 || 448.2 34 1751.7 261.8 || 39 2029.0 348.6 || 44 2314.9 450.0 5 1756.3 263. 1 5 2033.7 350.2 5 || 2319.8 || 451.8 10 || 1760.8 || 264.5 10 2038.4 || 351.8 10 2324.6 453.6 15 17{iš.4 || 265.8 15 2043.1 || 353.4 15 23:29.5 || 455.4 20 1770.0 || 267.2 20 2047.8 354.9 20 2334.3 457.3 25 || 1774.5 26S.5 25 | 2052.5 356.5 25 2339.2 459.1 30 1779.1 269.9 30 | 2057.2 358.1 30 2344.1 || 460.9 35 1783.7 271 .2 35 | 2061.9 || 359.7 35 2348.9 || 462.8 40 1788.2 || 272.6 40 2066.6 361, 3 40 2353.8 || 464.6 45 1792.8 || 273.9 45 2071.3 || 362.9 45 235S.7 || 466.5 50 | 1797.4 275.3 50 2076.0 || 364.5 50 2363.5 46S.4 55 1802.0 | 276.7 55 2080.7 || 366.1 55 2368,4 470.2 35 1806.6 || 278.1 40 2085.4 367.7 45 2373.3 472.1 5 | 1811, 1 279.4 5 2090. 1 || 369.3 5 2378.2 || 473.9 10 | 1815.7 2S0.8 10 2094.9 || 371.0 10 2383.1 475.8 15 1S20.3 28.2.2 15 2099.6 || 372.6 15 2388.0 || 477.7 20 1824.9 2S3,6 20 2104.3 || 374.2 20 || 2392.8 479.6 25 1829.5 || 285.0 25 2109.0 375.8 25 || 2397.7 4S1.5 80 | 1834.1 286.4 30 21.13.8 || 377.5 30 2402.6 483,4 35 | 1838.7 || 2S7.8 35 | 2118.5 379,1 35 | 2407.5 4S5.3 40 | 1843.3 289.2 40 2i;23.3 || 380.8 40 24.12.4 || 487.2 45 | 1847.9 || 290.6 45 2128.0 | 882.4 45 2417.4 || 489.1 50 | 1852.5 292.0 50 2132.7 || 3S4, 1 50 | 2422.3 || 491.0 55 1857.1 || 293.4 55 2137.5 385.7 55 | 2427.2 || 492.9 270 TABLE XVII. TANGENTS AND SHORTEST DISTANCES FROM I. T. b. I. T. b. I. T. b. 46° 2432.1 || 494.8 ± 51° 2732.9 618.4 56° 3046.5 759.6 5' 2437.0 496.7 5' 2738.0 | 620.6 5’| 3051.9 || 762, 1 10 2441.9 498.7 10 || 2743.1 622.S 10 || 3057.2 || 764.6 15 2446.9 || 500.6 15 2748.3 625.0 15 3062.6 || 767.1 20 2451.8 502.5 20 2753.4 627.2 20 3067.9 7(39.7 25 2456.7 504.5 25 2758.5 | 629.5 25 3073.3 772.2 30 2461.7 || 506.4 30 2763.7 631.7 30 3078.7 774.7 7.3 35 | 2466.6 508.4 35 | 2768.8 633 9 35 | 3084.0 || 77 4() 2471.5 || 510.3 40 2773.9 636.2 40 || 3089.4 || 779.8 45 2476.5 || 512.3 45 2779.1 638.4 45 || 3094.8 782.4 50 24S1.4 514.3 50 2784.2 | 640,7 50 || 3100.2 || 784.9 55 2486.4 || 516.2 55 2789.4 642.9 55 || 3105.6 || 787.5 47 2401.3 || 518.2 52 2794.5 645.2 ſ 57 || 3110.9 790.1 20.2 5 2799.7 647.4 5 || 3116.3 || 792.7 2 : 10 || 2804.9 || 649.7 10 3121.7 795.2 1 || 15 28.10.0 652.0 || 15 3127.2 797. S 1 20 2815.2 654.3 ; 20 || 3132.6 | 800.4 * . 1 # 25 2820.4 656.5 25 313S.0 803.0 30 || 2521.1 530. 1 : 30 2825.6 || 658.8 || 30 || 3143.4 S05.6 32.1 35 | 2830.7 | 661.1 : 35 | 3148.8 808.2 1 i 40 2835.9 | 663.4 40 || 3154.2 | 810.9 2 45 2841 1 | C65.7 || 45 3159.7 813.5 2 : 50 | 2S46.3 | 668 0 : 50 | 3165. 1 || 816.1 2 || 55 2851.5 670.3 55 3170.6 | 818.7 48 2551.0 | 542.2 | f 3 2856.7 | 672.7 58 3176,0 | S21.4 5 2556.0 544.3 5 2861.9 675. () 5 || 318.1.4 824.0 10 || 2561.0 | 546.3 10 2867. 1 || 677.3 10 || 3186.9 S2(;.7 15 2566.0 548.3 15 2872.3 679.6 15 || 3192.4 || 829.3 20 2571.0 | 550.4 20 2877.5 6S2.0 20 || 3197.8 || 832.0 25 2576.0 552.4 25 2S82.8 | 684.3 25 || 3203.3 834.6 30 258! .0 554.5 30 2888.0 | 686.7 30 || 3208.8 837.3 35 || 25S6.0 556.6 35 | 2893.2 | 689.0 35 | 3214, 2 | S40.0 40 2591.1 558.6 40 2898.4 || 691.4 40 || 3219.7 | 842.7 45 2596.1 || 560.7 45 || 2903.7 693.8 45 || 3225.2 | 845.4 50 2601.1 562.8 50 2908.9 || 696.1 50 i 3.230.7 | S48.1 55 2606. 1 || 564.9 55 2914.2 || 698.5 55 || 3236.2 850.8 49 2611 .2 ſ 566.9 54 29.19.4 || 700.9 59 3241.7 | 853.5 5 2616.2 || 569 () 5 2024.7 || 703.3 5 || 3247.2 | 856.2 10 2621.2 571. 1 10 : 2929.9 || 705.7 10 || 3252.7 || 858.9 15 2626.3 573.2 15 2335.2 708.1 15 || 3258.2 861.6 20 | 2631.3 575.3 20 2940.4 710.5 20 | 3263.7 | 864.3 25 || 2636.3 577.4 25 2945.7 712.9 25 || 3269.2 867.1 3{) 2641 4 579.5 30 2951.0 715.3 30 || 3274.8 || 869.8 35 2646.5 581.7 35 2956.2 717.7 35 | 3280.3 || 872.6 40 2651.5 583.8 40 2961.5 | 720.1 40 3285.8 || 875.3 45 2656, 6 585.9 45 || 2966.8 || 722.5 45 3291.4 || 878.1 50 2661.6 5S8.0 50 2972. 1 || 725.0 50 3296.9 || SS0.8 55 2666.7 590.2 55 2977.4 727.4 55 || 33U2,5 | 8S3.6 | 50 2671.8 592.3 55 2082.7 | 729.9 60 3308.0 | 886.4 5 2676.9 594.5 5 29°8.0 | 732.3 5 || 3313.6 || 8S9.2 10 $681.9 596.6 j 0 , 2993 3 || 734.8 10 || 33.19.1 | 891.9 15 2687. () 598.8 15 2998.6 || 737.2 15 3324.7 894.'' 20 2692, 1 600 .9 20 3003, 9 739.7 20 || 3330.3 | 897.5 25 | 2097, 2 603.1 25 3000.2 742.1 25 || 3335.8 900.3 30 2702.3 605.3 30 3014.5 || 744.6 30 3341.4 903.2 35 2707.4 607.4 35 | 3019, 8 || 747. I 85 || 3347.0 906.0 40 || 2712.5 609.6 40 || 3025.2 749.6 40 || 3352. 90S.8 45 2717.6 || 611.8 45 303().5 752.1 45 3358.2 911.6 2.7 - * - INTERS EOTION POINT OF A. ONE TEGREE CURWE. 271 T. Ö. I. T. Ö. I. T. b. 3375.0 920.1 | 66° 3720.9 1102.2 : 71° 4086.9 || 1308.2 3380.6 9:23.0 5' 3726.8 || 1105,4 5' 4093.2 1311.9 3386.3 925.8 10 || 3732.7 || 1108.6 10 4099.5 1315.6 3391.9 || 928.7 15 373S.7 1111.9 15 || 4105.8 || 1319.2 3397.5 | 931.6 20 || 3744.6 1115.1 20 || 4112.1 1322.9 3403.1 || 934.5 25 || 3750.6 1118.4 25 || 4118.4 || 1326.6 3408.8 937.3 30 || 3756.5 || 1121.7 30 || 4134.8 || 1330.3 3414.4 || 940.2 35 | 3762.5 1124.9 35 4131.1 1334 . () 3.420.1 | 943.1 40 || 3768.5 1128.2 40 || 4137.4 1337.7 3.425.7 946.0 45 || 3774.4 || 1131.5 45 || 4143.8 || 1341.4 3431.4 || 948.9 50 | 3780.4 || 1134.8 50 || 4150.1 | 1845. 1 3437.1 | 951.8 55 || 3786.4 || 1138.1 55 || 4156.5 1348.8 3442.7 || 954.8 # 67 3792.4 1141.4 72 4.162.8 1352.6 3448.4 957.7 5 || 3798.4 1144.7 5 || 4169.2 1356.3 3454.1 960.6 10 || 3S04 4 || 1148.0 10 || 4.175.6 || 1360. I 3459.8 963.5 15 || 3810.4 || 1151.3 15 || 4182.0 || 1363.8 3465. 4 966.5 20 || 3816.4 1154.7 20 || 4188.4 1367.6 347 l. 1 969. 4 25 || 3S22.4 || 115S.0 25 || 4194.8 1371.4 3476.8 || 972, 4 30 || 3S28.4 || 1161.3 30 420.1.2 : 1375.2 3482.5 975.3 35 | 3834, 5 || 1164.7 35 4207.6 || 1379.0 34SS. 2 978. 3 40 || 3S40.5 | 1168.1 40 4214.0 1382.8 3494.0 9S1.3 45 || 3S46.5 1171.4 45 4220.4 || 1386.6 3499.7 984.3 50 3S52.6 || 1174.8 50 || 4236.8 1390.4 3505, 4 9S7.3 55 3858.6 1178.2 55 || 4233.3 | 1394 2 3511.1 990.2 | 68 3864.7 || 1181.6 73 4239.7 139S 0 3516.9 993.2 5 || 3870.8 || 1185.0 5 4246 2 1401.9 35:22.6 996, 2 10 || 3S76.8 11SS.4 10 || 4252.6 1405 7 3528, 4 999.3 15 3882, 9 1191.8 15 4359. 1 || 1400. 6 3534.1 1002.3 20 || 38S9.0 | 1195.2 20 4265.6 1413.5 3539.9 1005.3 25 || 3895. 1 119S. 6 25 4272, 0 || 1417.3 3545.6 10:08.3 30 || 3901.2 1202.0 30 || 4278.5 14:21.2 3551.4 || 1011.4 35 | 3907, 3 || 1205.5 35 4285.0 1425.1 3557.2 | 1014, 4 40 3913 4 1208.9 40 4291.5 1429.0 3562.9 || 1017.4 45 3919.5 1212.4 45 4298.0 | 1.432.9 356S.7 || 10:20.5 50 | 39:25.6 1215.8 50 4304.5 / 1436. S 3574.5 1023.6 55 3931.7 1219.3 5 : 4311 1 1440.7 35S0.3 1026.6 || 69 3937.9 || 1222.7 || 74 4317.6 1444.6 3586.1 1029.7 5 || 394.4.0 1226.2 5 4324. 1 || 1448.6 3591.9 || 1032, S 10 || 3950.2 1229.7 10 4330, 7 1452.5 3597.7 1035.9 15 3956.3 1233.2 15 || 4337.2 1456.5 3603.5 | 1039 () 20 :962.5 1236." 20 || 4343.8 1460.4 3609.3 || 1042.1 25 | 3968, 6 124t).2 25 4350.4 1464.4 3615.1 | 1045.2 30 || 3974, 8 || 1243.7 30 4356.9 || 146S.4 3621.0 | 1048.3 35 | 398:1, 0 | 1247.2 35 4363.5 1472.4 3626.8 || 1051.4 40 || 39Si.2 1250.8 40 4370.1 1476.4 3632.6 1054.5 45 || 3993.3 1254.3 45 4376.7 1480.4 3638.5 1957.7 50 3999, 5 1257.9 50 || 4383.3 1484.4 3644.3 || 1060.8 55 | 4005.7 | 1261.4 Ü5 || 4389.9 1488.4 3650.2 | 1063.9 || 70 4011.9 || 1265.0 75 4396.5 1492.4 3556.1 1067.1 5 4018.2 1268.5 5 4403.1 : 1496.5 3661.9 1070.3 10 | 4024.4 1272 1 10 : 4409.8 1560.5 3667.8 1073.4 15 4030.6 1275.7 15 4413.4 1504.5 §3.7 | 1976.6 20 | 4036.8 || 1279, 3 20 4423.1 : 1508.6 3.19.5 1079.7 25 | 4043.1 12S2.9 25 4429.7 1512.7 36S5.4 1082.9 30 4049.3 1286.5 30 4436.4 1516.7 3691.3 1086.1 35 | 4055.6 1290.1 35 4443.0 1520.8 3697.2 1089.3 40 | 4061.8 1293.6 40 4449.7 1524.9 3703.1 | 1092.5 45 | 4068.1 | 1297.3 45 1456.4 1529.0 709.0 1095.7 50 | 4074.4 || 1300.9 50 4463. 1 | 1533. 1 3715 0 || 1099.0 55 | 4080.6 || 1304.6 55 4469.8 1537.3 272 TABLE XVII. TANGENTS AND SHORTEST DISTANCEs. I. T. b. I. T. b. I. T. b. 76° 4476.5 1541.4 # 81° 4893.6 ISO5.3 : 86° 5343.0 2104.7 5' 4483.2 | 1545.5 5’| 4900.8 1810.0 5’| # 350.8 2110.0 10 4,89.9 | 1549.7 10 || 4908.0 | 1814.7 10 || 5358.6 2115.3 15 4496.7 | 1553.8 15 || 4915. 2 | 1819.4 15 5366.4 2120. 20 4503.4 1558 0 20 4922.5 1S24. 1 20 5374.2 2126.0 25 || 4510. 1 || 1562.1 25 4929.7 1828.9 25 53S2.1 2131.4 30 4516.9 || 1565.3 30 || 4937.0 | 1833.6 30 || 5389.9 2136.7 35 4523.7 - 1570.5 35 | 4944.2 | 1838.3 35 | 5397.8 2142, 1 40 4530.4 1574.7 40 || 4951.5 1843.1 40 || 5405.6 2147.5 45 || 453?.2 | 1578.9 45 || 4958.8 1847.9 45 5413.5 2152.9 50 || 4544.0 | 1583. 1 50 4966. 1 | 1852 6 5() 54.21.4 21:58.4 55 4550.8 15S7 3 55 4973.4 1857.4 || 55 || 5429.3 || 2163.8 77 4557.6 || 1591.6 || 82 49S0.7 | 1862.2 : 87 5437.2 2169.2 5 4564.4 | 1595.8 5 4988.0 | 1807.0 S 5443.2 2174.7 10 45; 1.2 1600.1 10 4995.4 187; S 10 || 5453. 1 2180.2 15 || 4578.0 | 1604.3 15 EC02.7 1876.7 | 15 5461.0 2185.6 20 4584.8 1608.6 20 5010.0 | 1881.5 | 20 5469.0 2191. 1 25 || 4591.7 | 1612.9 25 t017. 4 18 S6.3 ; 25 5477.0 2196.6 30 || 4598.5 | 1617. 1 30 5024.8 1891.2 : 30 5484.9 || 2:02.2 35 | 4605. 4 | 1621.4 35 5032. t 1896 1 35 | 5492.9 22i}7.7 40 || 4612.2 | 1625.7 40 503.9.5 1900.9 4(? | C500.9 || 2:213.2 45 4619. 1 | 1630.0 45 5046.9 1905.8 45 || 5509.0 2218.8 50 || 4626.0 | 1634.4 50 | {054.3 1910.7 5() 55.17.0 2224.3 55 4632.9 | 163S.7 55 || 5061.7 | 1915.6 || 55 || 55:5.0 23:29.9 78 4639 S (643 0 || 83 5069.2 1920.5 | SS 5533.1 2935.5 5 4646.7 | 1647.4 5 5076.6 1925.5 5 5541.1 22.41 1 10 || 4653.6 | 1651.7 10 5084.0 | 1930.4 | 10 || 55.49.2 2246.7 15 4660 5 | 1656.1 15 5091.5 19:5.3 15 5:57.3 2252.3 20 4667.4 1660 5 20 5099.0 | 1940.3 20 | 5565.4 2258.0 25 4674 4 | 1664.9 25 || 5106.4 || 1945.3 25 5573.5 2263.6 {0 || 4681.3 | 1669.2 30 5113.9 || 1950.3 : 30 || 55S1.6 2269.3 35 46S8 3 | 1673.6 || 35 || 5121.4 1955.2 35 | 5589.7 2275.0 40 4695.2 1678.1 40 5128.9 1960.2 40 5597.8 228).6 45 4702 2 | 1682.5 45 51:46.4 j965.3 : 45 5606.0 2286.3 50 4709. 2 | 1686.9 50 5143.9 1970.3 50 5.614.2 2292.0 55 47.16.2 | 1691.3 55 || 5151.5 1975.3 55 5622.3 2297.8 79 4723 2 | 1695.8 | 84 5159 .0 1980, 4 89 5630.5 2303.5 5 47.20.2 1700. 2 5 || 5166.6 || 1985.4 5 5638.7 2309.3 1() 4737.2 1704.7 10 || 5174.1 | 1990.5 10 5646.9 || 2315.0 15 47.j4.2 1709.2 15 5181.7 | 1995.5 15 56.5.1 2320.8 20 4751.2 | 1713.7 20 || 5189.3 2000.6 || 20 5663.4 2326.6 25 4758 3 || 1718.2 25 || 5196.8 2005.7 25 5671.6 || 2332.4 30 4765.3 1722.7 || 30 5204.4 2010.8 30 5679.9 || 23:38.2 35 4772.4 1727.2 35 | 5212.1 2016.0 || 35 5688.1 || 2344.0 40 4779.4 1751.7 40 5210.7 2021.1 40 5696.4 2349.8 45 47S6.5 1736.2 45 62.97.3 2026.2 || 45 5704.7 2355.7 50 3793.6 1740.8 50 5234.9 2031.4 || 50 5713.0 236 l .5 55 4800.7 1745.3 || 55 5242.6 2036.5 55 5721.3 || 2367.4 80 4807.7 1749.9 || 85 5250.3 2041 ." § 90 5729.7 |2373.3 5 || 4814.9 1754.4 5 || 5257.9 || 2046.9 5 573S.0 2379.2 10 4822. 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